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3898 results
A bill is not yet an implemented policy
Checks and balances create several decision points.
Open lessonLesson · Mathematics investigationA changing radius creates a changing area
How does a length rate become an area rate?
Open lessonLesson · Proof foundations: from claims to argumentsA claim needs a domain
Separate examples, universal claims and counterexamples by identifying the allowed objects.
Open lessonLesson · Social studiesA codebook makes interpretation inspectable
Qualitative analysis identifies patterns in text while retaining context.
Open lessonProject · Read like a literary detectiveA comparative literary casebook
Curate an argument about how readers judge people through incomplete information. Use at least two course texts, preserve a changed interpretation, and test your method on an unfamiliar passage.
Open projectLesson · Axiomatic Euclidean and non-Euclidean geometryA complete Euclidean coordinate model
Check incidence, distance and order in R².
Open lessonProject · Generative project; five separate transfer cases; human review pendingA controlled generative experiment
Save matched diffusion, flow and GAN snapshots, recompute samples and compare coverage, costs and failure evidence.
Open projectProject · Literary inquiry and interpretationA critical essay and annotated exhibit
Develop an interpretation of the complete Story of an Hour and compare it with The Returned Key or another work you can access and cite responsibly. Curate four exhibit captions that let a reader inspect the evidence. Use the local edition notes to distinguish textual claims from contextual facts. The original teaching positions support guided critical conversation; independent scholarship requires reading and citing an actual critical source.
Open projectLesson · Mathematics investigationA decaying equation can have a growing approximation
When does a time step invent behavior absent from the differential equation?
Open lessonLesson · Social studiesA difference is not yet a treatment effect
Psychology studies behavior and mental processes through systematic evidence.
Open lessonLesson · Differential equations: models, methods and dynamicsA differential equation constrains a changing function
Separate an equation, its solution family and an initial-value problem.
Open lessonLesson · Social studiesA feasible budget and a defensible decision
A budget expresses tradeoffs under constraints.
Open lessonExperiment · Environment and sustainabilityA full reservoir can hide a shortage
How long can stored water conceal demand greater than replenishment?
Open experimentLesson · Axiomatic Euclidean and non-Euclidean geometryA geometric model and proof portfolio
Assemble model calculations into an explicit logical comparison.
Open lessonProject · Operations project with separate transfer assessmentA model with an operating plan
Save staged release, monitoring policy comparisons and incident recovery; write an operating plan.
Open projectProject · Perception project with real-image evidence and three separate objective cases; implementation review pendingA perception system under stress
Train a convolutional classifier, freeze subgroup evaluation, inspect failures and save reports.
Open projectProject · Poetry, drama and performanceA performance score with critical commentary
Curate a poem and a dramatic scene from the course. Create a coherent text-only or performed interpretation of each, compare an alternative, and defend your choices with the words and structure of the texts.
Open projectExperiment · Environment and sustainabilityA plan must survive its resource constraints
Can a high-output land plan still work when water falls short?
Open experimentProject · Reasoning project; human review pendingA planner with uncertain observations
Save search and belief-model comparisons, write a decision report and attempt four separate transfer cases.
Open projectLesson · Social studiesA poll asks a question, not public opinion in general
Survey results depend on wording, response options, ordering and recruitment.
Open lessonProject · Rhetoric, media and public voiceA public argument in two media
Develop an argument about the supplied study-room pilot or a bounded issue you can investigate responsibly. Create a written proposal and a poster storyboard or speech script. Both must preserve the same evidence and uncertainty. Private submission is sufficient; publication and personal disclosure are optional.
Open projectProject · Classical machine learning projectA real-data model comparison
Compare a mean baseline, line and residual ensemble; freeze evaluation, report errors and attempt independent transfer cases.
Open projectLesson · Mathematics investigationA repeated rule can settle, alternate or grow
What determines whether errors around a fixed point shrink?
Open lessonProject · College compositionA researched position essay
Investigate a bounded educational or campus question and write a qualified position essay. You may complete an offline practice portfolio using the explicitly fictional seminar packet, or an independent research portfolio using real sources you inspect. Label which route you chose; packet work is not evidence of independent source discovery. Preserve your source trail, earlier thesis, draft and revision memo. No public posting is required.
Open projectProject · Stories and creative writingA revised story and adaptation
Develop a complete short story from a course seed or an original premise, revise it through a documented workshop process, and adapt a scene into another viewpoint or dramatic form.
Open projectLesson · Mathematics investigationA route, a distance and a slope are different
How do coordinates determine distance without confusing it with direction?
Open lessonAssessment · Compactness of function familiesA sharp maximum, an operator image and a grid certificate
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentProject · Trace claims and reason about public decisionsA source-backed civic evidence brief
Prepare a brief about fictional fountain repairs. Trace claims to original records, resolve incompatible comparisons, write an honest headline and compare policy benefits under a budget. Include a source table and an explicit fairness criterion.
Open projectExperiment · Physics and engineeringA stronger correction can make control worse
How do gain and delayed sensing affect a controller?
Open experimentLesson · Single-variable calculus foundationsA tangent line as a local prediction
Build a tangent model, quantify its error in a worked case and distinguish small input changes from globally linear behavior.
Open lessonProject · Discover, prototype and improveA tested path to saved work
Design and revise a small flow that helps a learner resume an investigation. Submit the user problem, original and revised screen flows, task script, outcome table and prioritized next increment. Textual screen prototypes are valid artifacts.
Open projectLesson · Point-set topology: spaces, constructions and separationA topology proof and counterexample workshop
Select invariants and constructions appropriate to a new comparison.
Open lessonLesson · Social studiesA wartime proclamation and a constitutional abolition
The Emancipation Proclamation used wartime executive authority and named the areas to which it applied.
Open lessonConcept · GlossaryAblation
A controlled comparison that removes or changes a component to investigate its contribution, including interactions and resource costs.
Open conceptLesson · Infinite cardinality through explicit constructionsAbsorb countable additions with an explicit shift
Construct and invert a bijection using a supplied sequence, repair interval endpoints, and distinguish disjoint copies from ordinary unions.
Open lessonCourse · Mathematics program unitAbstract algebra I: groups and symmetry
A proof-based group-theory unit with finite computations, structural theorems, counterexamples and complex representation theory through character completeness, semidirect products, small-order classification and geometric rotation groups. Rings, fields and Galois theory belong to later units. Submitted writing requires mathematical review; these pages alone do not establish semester-course equivalence.
Open courseCourse · Mathematics program unitAbstract algebra II: rings, fields and Galois foundations
Proof-centered lessons from ideals and factorization through finite fields, Galois correspondence, solvability by radicals, modules, Smith classification, rational canonical forms, Noetherian ideal theory, discriminant methods and quadratic integrality, Dedekind factorization, ideal-class bounds and arithmetic Galois cycle certificates. Independent proof assessment and review remain necessary; this unit does not establish universal semester-course equivalence.
Open courseLesson · Optimization: models, certificates and algorithmsAccept step s using a verifiable decrease rule
Prove backtracking termination and state global convergence conditions.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceAccepted steps, rejections and work
Execute and audit a bounded adaptive Euler step-doubling trace.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceAccount for active transport and vesicle topology
Trace pump cycles, opposing transfers and cargo compartments without mistaking endpoints for mechanisms.
Open lessonLesson · Science · Biology and living systemsAccount for atoms in aerobic respiration
Use reactant limits and atom accounting to explain a respiratory endpoint.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceAccount for biological links without inventing a synthesis mechanism
Separate peptide hydrolysis, atom conservation and actual cellular assembly.
Open lessonLesson · Science · Physics and physical systemsAccount for capacitor energy under different constraints
Compare stored energy and mechanical work with and without a connected voltage source.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceAccount for charge across an object and its reservoir
Charge accounting begins by choosing recipients and drawing the system boundary. In this packet A, B and a reservoir start neutral. Moving n electrons from A to B leaves A with +ne and gives B −ne. If the reservoir then supplies another m electrons to B, the three final charges are +ne, −(n+m)e and +me. Their total is zero even though the A+B subsystem now has negative charge. Charge has crossed that smaller boundary; it has not been created.
Open lessonLesson · Social studiesAccount for costs outside the transaction
Can profitable trades reduce the stated social total?
Open lessonLesson · Science · Physics and physical systemsAccount for current at a parallel junction
Compare branch currents while preserving a common voltage difference.
Open lessonLesson · Science · Chemistry and matterAccount for each gas in a mixture
Calculate partial pressures, mole fractions and total pressure for components sharing one vessel.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceAccount for electrical energy across a full schedule
Power is an energy-transfer rate, whereas energy depends on both rate and duration. The fictional device spends separate intervals at 24 W and 40 W, then the remainder of a 24-hour day at 2 W standby. The intervals must not overlap or exceed the day. For these constant segments, add PΔt; actual varying powers would require a finer measured schedule.
Open lessonLesson · Science · Physics and physical systemsAccount for energy during charging and discharge
Reconcile source work, stored field energy and accumulated resistor heating.
Open lessonLesson · Science · Physics and physical systemsAccount for energy when friction acts
Explain where translational energy goes during a rough-track slide.
Open lessonLesson · Science · Chemistry and matterAccount for heat in the samples, cup and surroundings
Close a signed heat balance and explain why the final temperature is not a simple average.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsAccount for hydraulic force gain without inventing energy
An increase in pressure applied to a confined ideal fluid is transmitted through it. At equal piston elevations the same gauge-pressure increase acts on both piston faces. Force equals pressure times area, so a larger output piston can balance a larger force. The transmitted quantity is pressure, not identical total force.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsAccount for lost and retained dimensions
Prove rank-nullity and use pivots without confusing row and column spaces.
Open lessonLesson · Science · University Physics I with calculusAccount for mass crossing a momentum boundary
The identity F = ma assumes a particle of fixed mass. A subsystem that loses mass needs an additional momentum-flux account. Consider an abstract one-dimensional retained body with no external force, initially at rest. An infinitesimal positive mass dμ leaves with velocity v − u relative to the inertial frame, where u is its fixed backward speed relative to the body.
Open lessonLesson · Science · Chemistry and matterAccount for neutralization before calculating pH
Use reaction equivalents and final volume instead of averaging pH values.
Open lessonLesson · Science · Earth and space scienceAccount for planetary incoming and outgoing energy
Build a global radiation ledger using the disk-to-sphere area ratio and a clear net-flux sign convention.
Open lessonLesson · Science · Physics and physical systemsAccount for pressure in an ideal-gas mixture
Track each gas in the whole shared vessel and distinguish amount fractions from mass fractions.
Open lessonLesson · Science · Physics and physical systemsAccount for pressure work in ideal flow
Use Bernoulli’s equation to explain pressure changes along an accelerating liquid stream.
Open lessonLesson · Science · Physics and physical systemsAccount for rest mass and released energy
Convert a consistent initial-minus-final rest-mass difference into Q and preserve total energy conservation.
Open lessonLesson · Science · Engineering design and scientific computingAccount for the age of a feedback reading
Relate a delayed sample to command overshoot without changing the integration interval.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsAccumulate signed velocity and total travel
Recover position while separating displacement, distance and observations.
Open lessonLesson · Calculus II: series, curves and applicationsAccumulate speed along a path
Use vector speed, displacement and parameter intervals without overlooking retracing.
Open lessonLesson · Differential equations: models, methods and dynamicsAccumulate the effect of past inputs
Derive the response kernel and keep causal time limits.
Open lessonLesson · Mathematics investigationAccumulation can cancel without erasing motion
Why can displacement be zero while distance traveled is positive?
Open lessonConcept · GlossaryAction value (Q)
Expected return associated with taking an action in a state, under specified later behavior. A learned estimate can be wrong.
Open conceptConcept · GlossaryActivation
A layer’s output after its transformation. A nonlinear activation lets a network express more than a straight line.
Open conceptConcept · GlossaryActor–critic
A learner with a policy (actor) and a return estimator (critic). A critic can supply an advantage estimate for policy updates, but inaccurate bootstrapped values can bias that signal.
Open conceptLesson · Course 1 · How does AI process language?Adapt a pretrained model
Compare full fine-tuning with a rank-two output update, inspect its arithmetic and measure new versus retained behavior.
Open lessonConcept · GlossaryAdaptation
Changing behavior for a new task using additional evidence. Parameters, temporary context, or another retained state may be involved; specify which.
Open conceptProject · AI cumulative project; human review pendingAdaptation project
Saved comparisons, replayed evidence, bounded reports and separate objective transfer assessment.
Open projectStudio · AI implementation practice; independent review pendingAdaptation project implementation studio
Run a local reference, implement a separate learner exercise and preserve reproducible evidence.
Open studioConcept · GlossaryAdapter
A transformation connecting representations expected by different components. It can be programmed or learned and must be evaluated for compatibility.
Open conceptLesson · Differential equations: models, methods and dynamicsAdd a limiting capacity and analyze stability
Connect equilibria, rate maxima and separated solutions.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesAdd arrows and find the actual resultant
Construct vector addition by components, end-to-end arrows and a parallelogram.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsAdd independent variables and change multiple coordinates
Derive supports and Jacobians before evaluating integrals.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsAdd momenta and locate the system’s motion
Linear momentum is the vector p=mv. In one dimension its sign follows the chosen positive axis, unlike kinetic energy, which depends on speed squared. Momentum units are kg m/s, equivalent to N s. A larger mass can have the same momentum as a smaller mass moving faster, but their kinetic energies need not match.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceAdd pressure phase before squaring
Two same-frequency pressure signals can reinforce or cancel locally. Let the first peak pressure amplitude be 1 Pa and the second be b Pa. Their relative phase is φ=2πΔr/λ when their source phases match and the only phase difference comes from a path-length difference Δr in a uniform medium. Local amplitudes already include any unequal spreading or attenuation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsAdd signed disturbances and distinguish beats from cancellation
In a linear wave model, disturbances at the same position and time add algebraically. Two pulses pass through one another rather than collide like rigid objects. For example, an upward displacement of 3 mm overlapping a downward displacement of 2 mm gives +1 mm at that instant. After separation, the original pulses continue under the assumed linear, nondissipative model; the overlap is not evidence they bounced.
Open lessonLesson · Science · Physics and physical systemsAdd wave displacements and interpret cancellation
Compare signed component displacements with their sum and test what cancellation does and does not imply.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsAdd work across changes of force and direction
A force can change momentum without transferring kinetic energy if it is perpendicular to motion. For a constant force over a straight displacement, work is the component along that displacement times its length: W=Fd cosθ. Resolve force and displacement in one frame before multiplying. The result is a signed scalar, not another vector arrow.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsAdjoin all roots and prove uniqueness up to base-field isomorphism
Separate a root field from a splitting field.
Open lessonConcept · GlossaryAdvantage
The difference Q − V: how much better an action is estimated to be than the policy’s state-value baseline.
Open conceptLesson · Axiomatic Euclidean and non-Euclidean geometryAffine coordinates and barycentric reasoning
Prove geometric statements without a metric.
Open lessonCourse · Social studiesAfrican American studies and community histories
Hands-on social studies course and project
Open courseLesson · Social studiesAfrican societies and the origins of diaspora
Explain African diversity and diaspora without beginning the story with enslavement.
Open lessonLesson · Social studiesAgriculture, settlement and early cities
Evaluate settlement and social organization through material evidence.
Open lessonRefresher · Math refreshersAlgebra and substitution
Practice the mathematics, then return to your lesson.
Open refresherCourse · Mathematics program unitAlgebra foundations: expressions, equations and roots
Ten lessons develop exact calculation, justified transformations and checking against original conditions. Written reasoning is self-reviewed with specific rubrics; it is not automatically graded. Further functions, systems and modeling units are still needed for full Algebra I/II coverage.
Open courseAssessment · Algebra foundations: expressions, equations and rootsAlgebra foundations: mixed unit assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitAlgebra II and precalculus: functions and structure
Twelve lessons connect representations, transformations, systems and algebraic function families. Exact calculations and written arguments supply varied practice. This unit does not complete Algebra II or AP Precalculus; trigonometric, polar and broader modeling sequences remain separate work.
Open courseCourse · Mathematics program unitAlgebra reasoning, constraints and solution evidence
A focused school-entry sequence in expression structure, algebraic operations, parameter cases, contextual equations, graph-based solutions and verified matrix methods. Written reasoning and graph construction require review; public practice is not automatic mastery.
Open courseAssessment · Algebra reasoning, constraints and solution evidenceAlgebra reasoning: mixed independent evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · MathematicsAlgebra, functions and mathematical modeling
Build and solve equations, inequalities and systems, interpret quadratic shape, and use exponential/logarithmic models within a stated domain.
Open courseConcept · GlossaryAlgorithm
A specified procedure for transforming inputs into results. Its guarantees depend on assumptions about inputs, rules, and resources.
Open conceptCourse · Mathematics program unitAlgorithm reasoning: invariants, termination and cost
Specify state transitions, discover sufficient invariants, prove total correctness and derive operation costs with integer rounding. Compare finite traces with proofs that cover every allowed input.
Open courseLesson · Social studiesAllocate a budget with marginal benefit
Which next purchase offers the most benefit per credit?
Open lessonExperiment · Applied mathematicsAllocate a constrained budget
Which combinations fit after reserving a safety margin?
Open experimentLesson · Science · General Chemistry IAllocate energy to temperature change and phase fraction separately
Solve a piecewise fixed-pressure energy account rather than applying mcΔT through melting.
Open lessonExperiment · Cybersecurity and digital literacyAllocate security maintenance
Which schedule covers more exposed accounts under the same staff budget?
Open experimentLesson · Social studiesAllotment and sovereignty: reading the Dawes Act
Compare federal allotment provisions with Cherokee institutional history and repair unsupported claims about land and government.
Open lessonLesson · Social studiesAn agreement needs a credible alternative
Negotiation compares proposals with what each party can obtain without agreement.
Open lessonProject · Application projectAn assistant that uses evidence
Save matched workflow comparisons, freeze a candidate, report evidence and attempt separate transfer cases.
Open projectProject · Research, editing and publishingAn edited research feature with source dossier
Produce a research feature for a named audience using the fictional repair-café packet or inspected real sources. Label packet practice and independent research distinctly. Include a source dossier, editorial decision trail, accessible presentation and correction plan. Downloading a private review copy completes the release activity; public posting, interviews and disclosure of personal information are not required.
Open projectExperiment · Environment and sustainabilityAn efficient replacement has an upfront cost
When does lower operating impact offset manufacturing and increased use?
Open experimentLesson · Algebra reasoning, constraints and solution evidenceAn equation describes a set of points
Test both directions of the connection between an equation and its graph.
Open lessonProject · Explain, substantiate and recommendAn evidence-backed improvement proposal
Prepare a before/after instruction, a claim-to-evidence table, a chart specification or attached chart, and a concise proposal about a fictional service. Preserve the original draft so a reviewer can inspect your revision decisions.
Open projectProject · Course 1 cumulative projectAn evidence-based AI decision
Cumulative data audit, review threshold, error costs, uncertainty and decision brief with a separate transfer case.
Open projectProject · Write with purposeAn explanatory feature a reader can use
Create a feature explaining what the temporary library hall can and cannot offer. Help a named audience make a decision, using the supplied packet faithfully and documenting your revisions.
Open projectLesson · Calculus II: series, curves and applicationsAn infinite sum is a limit of finite sums
Define convergence through partial sums and reject misleading termwise intuition.
Open lessonAssessment · Real Analysis II: uniform limits, metric spaces and local calculusAnalysis II: uniform control, spaces and local theorems
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentProject · Real Analysis I: completeness, limits and rigorous calculusAnalysis manuscript: small area error, different limits
Develop and revise one connected manuscript about two ways of measuring error between continuous functions. Construct a Cauchy sequence that loses continuity in integral distance, prove that uniform distance prevents that failure, and identify which measurements remain stable. Keep exact calculations, rejected arguments, theorem dependencies and actual feedback. This public, supported portfolio is not an unseen examination or automatic certification of proof correctness.
Open projectLesson · Complex analysis: holomorphic functions, contours and mappingsAnalytic continuation and monodromy
Continue a holomorphic germ using overlapping neighborhoods and identity uniqueness.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsAnalytic elements: build a stable finite continuation chain
Construct segment and seam margins rather than assuming one global branch.
Open lessonLesson · Science · General Chemistry IIAnalyze a selective membrane through two solvent constraints
The membrane passes solvent but blocks both nonelectrolyte solutes.
Open lessonLesson · Science · AP Environmental ScienceAnalyze demographic-transition scenarios without ranking societies
Compare rate gaps and explain why demographic transitions are conditional patterns.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsAnalyze differences before applying t
Preserve matching and define the sign of change.
Open lessonLesson · Mathematical modeling and senior investigationAnalyze interactions and conservation in continuous time
Use positivity and total balances before simulation.
Open lessonLesson · Probability protocols and expected workAnalyze restarted experiments and their actual work
Separate terminal outcome probabilities, unresolved capped runs and expected attempts, while checking that a restart really reproduces the same independent experiment.
Open lessonLesson · Social studiesAngkor: water, temples and political authority
Investigate how religious patronage and infrastructure relate to state power, without equating monuments with universal control.
Open lessonConcept · GlossaryAnomaly detection
Scoring unusualness relative to a fitted reference. An anomaly flag is a reason to investigate, not a confirmed problem; coverage, thresholds and base rates affect interpretation.
Open conceptLesson · Literary inquiry and interpretationAnswer a reading at its strongest point
Represent competing critical claims fairly and develop a response through shared textual evidence.
Open lessonLesson · Course 2 · Memory and shared working stateAnswer with evidence
Build an extractive answer with precise citations, reject unsupported claims, and make conflicts and missing facts visible.
Open lessonCourse · Social studiesAP African American Studies investigations
Hands-on social studies course and project
Open courseCourse · ScienceAP Biology — mechanisms and evidence
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceAP Chemistry preparation
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceAP Environmental Science
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · Social studiesAP European History investigations
Hands-on social studies course and project
Open courseCourse · Social studiesAP Human Geography practice
Hands-on social studies course and project
Open courseCourse · Social studiesAP Macroeconomics practice
Hands-on social studies course and project
Open courseCourse · Social studiesAP Microeconomics practice
Hands-on social studies course and project
Open courseCourse · ScienceAP Physics 1: algebra-based mechanics
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceAP Physics 2: algebra-based thermal, electromagnetic and optical systems
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceAP Physics C: Electricity and Magnetism — calculus field models
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceAP Physics C: Mechanics — calculus-based modeling
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · Social studiesAP United States History investigations
Hands-on social studies course and project
Open courseCourse · Social studiesAP World History: Modern investigations
Hands-on social studies course and project
Open courseSubject · Experiment collectionApplied mathematics
Use rates, geometry and constraints to test everyday decisions.
Open subjectLesson · Science · College Physics II — algebra-based fields and evidenceApply each polarizer to the light reaching it
Polarization describes the direction of a transverse wave field. An ideal first linear polarizer transmits half the intensity of unpolarized input. Once the light is linearly polarized, a second ideal filter transmits a fraction cos²θ, where θ is the angle between its axis and the incoming polarization. That angle must be updated after each filter rather than repeatedly measured from the original source.
Open lessonLesson · Science · AP Chemistry preparationApply molar phase energy only within the available phase inventory
Use temperature-matched vaporization data, signed heat and finite molar inventory while refusing unsupported phase-boundary extrapolation.
Open lessonLesson · Science · Physical Geology and Earth SystemsApply Snell geometry without inventing impossible rays
Body-wave paths bend when wave speed changes. For a plane isotropic interface, the ratio sin(angle from normal)/speed is conserved across a transmitted ray. A faster lower layer therefore bends the ray away from the normal, not toward it. Angles measured from the horizontal interface would use a different geometric expression.
Open lessonLesson · Optimization: models, certificates and algorithmsApproach a boundary optimum from strict feasibility
Derive the logarithmic central path and its duality-gap interpretation.
Open lessonLesson · Course 2 · Planning and reasoningApproximate a belief with particles
Propagate, weight and resample possible positions. Compare a seeded particle filter with exact inference and diagnose lost diversity.
Open lessonLesson · Mathematical statistics: derivation and model criticismApproximate a sampling law without forgetting its model
Derive empirical resampling and diagnose nonsmooth or dependent failures.
Open lessonLesson · Mathematics investigationApproximate accumulation without hiding its error
How do left rectangles, midpoint rectangles and trapezoids differ?
Open lessonLesson · Algebra reasoning, constraints and solution evidenceApproximate intersections without inventing solutions
Combine graph candidates, domains and controlled interval refinement across function families.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeApproximate measurable sets by open and compact sets
Construct Lebesgue measure from interval length.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeApproximate measurable values from below
Prove threshold criteria for measurability.
Open lessonConcept · GlossaryArc consistency
For each directed binary relationship, every remaining value has at least one compatible value in the neighboring domain. These separate supports need not combine into a global solution.
Open conceptLesson · Social studiesArchaeology: reconstruct deposition before dating behavior
An object can be older than the deposit containing it. A later disturbance can also move an object into an earlier layer. Archaeological inference therefore requires a formation history before a cultural interpretation.
Open lessonRefresher · Math refreshersArithmetic and percentages
Practice the mathematics, then return to your lesson.
Open refresherCourse · Mathematics program unitArrival processes, absorption and honest simulation
Construct a Poisson process from waiting times, prove finite-window independence and thinning, and solve absorbing chains with a termination certificate. Audit bounded simulations without discarding unresolved outcomes or mistaking repeated seeds for new evidence. These four lessons extend existing probability and directed-walk material; they do not constitute a complete stochastic-process course.
Open courseLesson · Social studiesArtifacts need context and alternative explanations
Archaeology interprets material traces through provenience, association, dating and formation processes.
Open lessonLesson · Science · Scientific inquiry and measurementAsk a question that can be tested
Separate a question, a model and a prediction.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityAsk a question that expects variation
Define units, variables, populations and operational measurements before summarizing.
Open lessonLesson · Mathematics investigationAsk how surprising the result is under a model
What does a small p-value actually say?
Open lessonLesson · Course 2 · Planning and reasoningAsk what would change under intervention
Change a causal graph, compare observed and randomized outcomes, and discover why a population effect does not determine an individual counterfactual.
Open lessonLesson · Optimization: models, certificates and algorithmsAsk whether an optimum exists
Specify objectives, feasible sets and attainment before selecting an algorithm.
Open lessonLesson · Mathematical modeling and senior investigationAsk whether different parameters can look identical
Separate structural ambiguity from noisy estimation.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceAsk whether new information can change a decision usefully
Monitoring can reduce uncertainty without changing a decision. Its decision value depends on the actions available, their consequences, the signal’s accuracy and its cost. Structured decision making separates scientific probabilities from stakeholder utility choices and identifies whether learning could improve an authorized future choice.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceAsk which nearby problem a computation solves
Relate forward error, backward error and conditioning through stable and unstable formulas.
Open lessonLesson · Science · Earth and space scienceAssemble a multi-variable front argument
Compare wind, temperature, dew point and pressure through a documented cold-front window.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsAssemble a signed three-charge energy ledger
Electric potential energy belongs to a system of interacting charges. Choosing zero when charges are infinitely far apart gives U=kq₁q₂/r for a pair. Same-sign pairs have positive energy and opposite-sign pairs negative energy. This sign is distinct from the always-nonnegative magnitude of force. It describes how much energy an external agent supplies or removes during a quasistatic assembly.
Open lessonModule · Course 2Assemble an experimental agent
Bring the pieces together in a small, measurable puzzle-solving system.
Open moduleLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsAssemble interaction energy without double counting
Distinguish system pair energy from source potential and self-energy.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesAssemble labeled components
Derive the set-of-components exponential formula.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsAssign the source field to the correct current recipient
The left long wire lies at x = 0 with current +2 A along z.
Open lessonConcept · GlossaryAssociative retrieval
Finding or weighting stored information using a query’s relationship to memory keys. Similarity alone does not verify the result.
Open conceptAssessment · Boolean specifications, certificates and circuitsAssumptions, hidden variables and circuit families
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesAtlantic trade: trace a cargo through imperial rules
Apply selected Navigation Act conditions to fictional shipments and distinguish restrictions from actual trade.
Open lessonLesson · Social studiesAtlantic voyages: incentives, claims and evidence
Compare a royal grant, a voyage report and museum evidence to test explanations of expansion.
Open lessonConcept · GlossaryAttention
A mechanism that compares queries and keys to decide how to combine information from value vectors.
Open conceptConcept · GlossaryAttention allocation
Deciding which information or computation gets priority at the system level. This is distinct from a Transformer’s internal attention calculation.
Open conceptLesson · Social studiesAttention and encoding: visible does not mean noticed
Distinguish stimulus availability, attention, conscious report and later memory when evaluating an omission.
Open lessonLesson · Social studiesAttribution: behavior under choice and constraint
Distinguish an observed action from a dispositional inference by examining choice, incentives and situational constraints.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceAudit a closed-cycle heat input before dividing for efficiency
A monatomic ideal gas executes a clockwise pressure-volume rectangle starting at 1 litre and 100 kPa. It heats at fixed volume to pressure 100a, expands at that pressure to b litres, cools at fixed volume to 100 kPa, then compresses back. A kPa litre equals one joule. This specifies quasistatic mechanical paths, not a realizable two-fixed-reservoir heat-transfer schedule.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsAudit a complete electrical-to-thermal process across changing boundaries
A correct equation can still answer the wrong system question. An ideal constant-potential source charges an initially empty capacitor through resistance, then is disconnected. The charged capacitor later discharges into a thermal arrangement that transfers only a stated fraction of energy to a gas. The gas may expand or remain at fixed volume. Treat these as separate stages with separate boundaries before adding their ledgers.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsAudit a complete thermal cycle before claiming useful work
A cyclic device returns its working gas to the starting state, so its net internal-energy change is zero. That statement does not mean every stage has zero ΔU. Compute heat and work stage by stage, then add the signed values. Forgetting one cooling or compression segment can create a fictitious device that produces work without a complete energy account.
Open lessonLesson · Social studiesAudit a dated census and representation claim
Is the population used for a national total identical to the apportionment population?
Open lessonExperiment · Data literacy and statisticsAudit a dramatic chart
Did the data change, or just the visible baseline?
Open experimentLesson · Science · Physics and physical systemsAudit a heat engine’s efficiency
Separate heat input, rejected heat, net work and power when comparing cycles.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceAudit a lineage change across reproduction, movement and detection
A population claim often passes through several stages before reaching a graph. Organisms contribute offspring, individuals move among populations, and observers detect only some of those present. A changed fraction in the final records can therefore combine biological change with an observation effect. A plausible adaptive story is not a substitute for reconstructing these stages.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsAudit a multi-stage mechanism by changing system and law
A successful synthesis begins by dividing the event into intervals and naming a system for each. A spring launches a cart, a rough segment converts mechanical energy into internal energy, a brief inelastic capture joins a second cart, and a different spring stops the joined pair. The same conserved subtotal does not automatically apply across all four intervals.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsAudit a nuclear reaction with charge, nucleons and rest energy
Nuclear binding reflects competition between short-range attractive nuclear interactions and proton Coulomb repulsion. A bound nucleus has less rest mass than its separated constituents; the mass difference corresponds to binding energy. Strong binding per nucleon can explain why some light-nucleus fusion and heavy-nucleus fission pathways release energy, but it does not mean every proposed fusion or fission is energetically allowed.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingAudit a pendulum small-angle approximation
Derive the angular restoring equation and quantify the approximation rather than asserting it.
Open lessonLesson · Social studiesAudit a real hiring-discrimination experiment
Compare randomized name signals with interview callbacks, then distinguish an absolute gap, a relative gap and an unmeasured hiring outcome.
Open lessonLesson · Science · Earth and space scienceAudit a recorded time series
Decode actual archived station values, irregular times and missing pressure.
Open lessonLesson · Social studiesAudit a score before interpreting a trait
Separate consistent measurement, valid interpretation and comparison with a reference group.
Open lessonProject · Data: from records to defensible conclusionsAudit a service dashboard and propose a fair trial
A fictional service claims it became faster because its dashboard mean fell. Build an evidence brief using a cleaned record audit, a distribution summary and a comparison design. Preserve the raw data and distinguish a changed record from a changed service.
Open projectLesson · Arrival processes, absorption and honest simulationAudit a simulation without hiding its unfinished paths
Compare model identities, sampling uncertainty and explicit resource-censored outcomes.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceAudit a supplied bandgap detector without claiming a real device
In a semiconductor, electronic states form bands rather than only isolated-atom lines. A valence-to-conduction transition leaves both an electron and an unoccupied valence state called a hole. A band gap is an energy separation, not a physical empty layer between two pieces of material. Doping changes available levels and carrier populations; it does not mean every photon is collected or that a material has one universal efficiency independent of wavelength and temperature.
Open lessonLesson · Science · Introductory MicrobiologyAudit a useful-process claim through its material boundary
Microbial metabolism supports useful food, chemical and environmental applications, but a named useful product is not a complete process-performance measure. Product identity, consumed substrate, residual feed, biomass, byproducts, quality and energy requirements have different boundaries. A high product fraction alone does not demonstrate safety, sustainability or a viable production process.
Open lessonLesson · Science · AP Environmental ScienceAudit a watershed balance without double-counting infiltration
Use watershed boundaries to distinguish storage, internal transfers and water exports.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsAudit a whole flow-power account before inferring efficiency
The final model uses a hypothetical everyday-style lifting flow to connect fluid mechanics, power and thermal measurements without prescribing any physical apparatus. Liquid density is 1000 kg/m³ and g=10. It rises 2 m from a large reservoir with negligible inlet speed to an outlet speed of 2 m/s. Both endpoint pressures are atmospheric, and a separate prescribed pipe loss is 5000 Pa.
Open lessonLesson · Science · Environmental science and sustainabilityAudit actual conductance observations
Read USGS units, dates, methods and missing days before interpreting change.
Open lessonLesson · Science · Introductory MicrobiologyAudit actual micrographs before counting or identifying objects
Two actual public-domain images accompany this lesson: a DIC image of yeast attributed to Masur, with a later rotation credited to Andrew Pertsev, and a digital-microscope field attributed to Cucumberkvp and labeled Paramecium bursaria by its creator. Exact source pages, authorship, reuse declarations, original bytes and hashes are retained. Those source labels are provenance, not a blinded identification achieved by the learner.
Open lessonLesson · Science · Physical Geology and Earth SystemsAudit actual USGS phase records before interpreting the interior
This destination uses actual public USGS phase-data records for the 6 February 2023 earthquake identified as us6000jllz, not invented seismic readings. The versioned QuakeML product supplies a preferred origin at 01:17:34.342 UTC, station picks, phase labels, angular distances and model residuals. The downloadable extract retains six P picks whose interpretation agency is us, selected near six distance targets without selecting on residual size.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceAudit amplitude and energy at a string junction
A junction can reflect part of a transverse wave even though both strings carry the same tension. The wave speed is √(T/μ), where T is tension and μ is mass per unit length. The supplied characteristic impedance is Z=T/v=√(Tμ), with units kg/s. It describes the traveling-wave transverse force-to-velocity relation; it is not an electrical resistance or an extra loss mechanism.
Open lessonLesson · Science · Introductory AstronomyAudit an actual spectrum before treating a curve as an inference
This lesson supplies an actual archived SDSS spectrum: public DR17 plate 358, MJD 51818, fiber 270, reduction RUN2D 26. The unchanged FITS source, complete 3819-row CSV, fixed selected windows, original flags and hash are downloadable. The source pipeline labels it QSO with redshift about 2.1095004 and ZWARNING 0; these are disclosed pipeline results, not a learner’s independent classification or proof of flawless data.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingAudit an impulse window before accepting endpoint agreement
Separate missing force tails, baseline accumulation and an independent momentum comparison.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceAudit an inheritance claim when surviving offspring are a selected sample
A testcross can reveal gamete combinations only through an observation pathway: gametes form, fertilization occurs, offspring survive, markers are expressed and categories are scored. A departure in the final categories need not originate during meiosis. The course-wide task is to separate an inheritance mechanism from survival and readout alternatives while retaining the original data boundary.
Open lessonAssessment · Distribution diagnostics and calibrationAudit an unfamiliar distribution-fit investigation
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceAudit an uphill transport cycle before claiming a mechanism
Moving a neutral ideal solute from lower to higher concentration requires a positive free-energy inputRTln(cdestination/csource) per mole. The logarithm uses a dimensionless concentration ratio. This supplied thermodynamic expression evaluates a state change; it does not specify the protein motions or rate of a transport cycle.
Open lessonLesson · Science · General Chemistry IAudit apparent yield against independent product purity
Use a supplied mass fraction to distinguish gross recovered material from desired product.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceAudit carbon fixation and distinguish pathway evidence
Separate carbon inputs, energy carriers and regenerated acceptors while evaluating perturbation claims.
Open lessonPath · Students, engineers and managersAudit data and debug learning
Repair an evaluation pipeline and a training run, then solve separate transfer cases.
Open pathLesson · Science · Physics and physical systemsAudit electrical power and energy
Distinguish current conservation from energy transfer and compare resistor dissipation.
Open lessonLesson · Science · General Chemistry IIAudit entropy on the full isolated thermal boundary
A component can lose entropy while the complete isolated process produces it.
Open lessonLesson · Science · AP Chemistry preparationAudit formal charge and compare resonance contributors
Use per-atom ledgers and symmetry to reject invalid or unequal structural interpretations.
Open lessonLesson · Science · AP Environmental ScienceAudit hydrogen as an energy carrier
Explain fuel-cell chemistry and upstream conversion losses.
Open lessonAssessment · Joint information and adaptive strategiesAudit joint information and an adaptive reward rule
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · Environmental Science: ecological systems and evidenceAudit measurement continuity before interpreting a trend
A time series contains physical variation and the history of how observations were made. Instrument replacements, station moves, sampling time and processing changes can introduce discontinuities. A documented artifact should be investigated rather than automatically interpreted as an environmental change or removed merely because it is inconvenient.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceAudit metabolic coupling under actual concentration conditions
Free-energy change describes the thermodynamic tendency of a specified process under stated conditions. A standard free-energy value is not automatically the value inside a cell. For the supplied reaction, actual ΔG=ΔG°+RTlnQ, where Q is a dimensionless activity ratio. Changing product/reactant conditions can change the energetic demand.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceAudit mixed evidence before revealing a worked response
An assessment should require transfer across mechanisms, not merely repeated vocabulary recognition.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceAudit nutrient efficiency denominators and missing pathways
Nitrogen-use metrics are meaningful only when their numerator and denominator are stated. Harvested nitrogen divided by fertilizer input is a partial balance ratio; it does not count all biological fixation, deposition, imported material or soil-stock contributions. A ratio above one can therefore reveal omitted inputs or stock drawdown rather than miraculous nutrient creation.
Open lessonLesson · Science · Earth and space scienceAudit recorded weather observations
Compare real station records while preserving timestamps, units, source fields and missing values.
Open lessonLesson · Science · General Chemistry IAudit sigma, pi and hybrid descriptions without literal orbital machinery
Relate localized overlap representations to bond multiplicity and rotation constraints.
Open lessonLesson · Science · University Physics II with calculusAudit source quadrature against an analytic benchmark
Approximating a continuous source by cells is a numerical method, not a new physical charge measurement. For the uniform rod −1≤s≤1 m, divide the length into N equal cells of width Δs=2/N and place each cell charge λΔs at its midpoint. Sum the signed vector kernel rather than adding magnitudes.
Open lessonLesson · Science · General Chemistry IIAudit species selectivity before inferring a solubility constant
A filtered dissolved-total assay is not necessarily a free-ion measurement.
Open lessonLesson · Science · Physical Geology and Earth SystemsAudit sulfide acid potential without pretending to predict pH
Excavating or fracturing sulfide-bearing rock can change contact with oxygen and water. Pyrite oxidation can release acidity and mobilize other constituents. The geological distribution of sulfides and neutralizing minerals matters, but a total mineral inventory alone does not specify reaction rates, drainage paths or receiving-water effects.
Open lessonLesson · Science · Introductory MicrobiologyAudit supplied dilution/count records without a culture protocol
A recorded count estimates a concentration only after identifying which fraction of the original sample was represented. The analyzed volume and the original-sample-equivalent volume are not the same when the record includes dilution. This lesson interprets a pre-existing fictional table; it does not describe preparing dilutions, plating, incubation or handling biological material.
Open lessonLesson · Science · Earth and space scienceAudit surface, atmosphere and space fluxes
Use three boundaries to distinguish internal exchanges from external energy and net heat transfer.
Open lessonLesson · Science · Introductory MicrobiologyAudit test denominators without diagnosing a person
A detection or classification test can have different error probabilities depending on the underlying state. Sensitivity is the fraction of state-positive records classified positive; specificity is the fraction of state-negative records classified negative. The fraction of positive results that actually have the state uses a different denominator and also depends on the state’s frequency in the tested population.
Open lessonAssessment · Domains, invariants and structural countingAudit the domain and certify the conclusion
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Probability protocols and expected workAudit the observation rule and expected workload
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Matching completions, families and unequal sidesAudit the quantifiers and matching assumptions
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · Introductory MicrobiologyAudit the record frame before using a resistance percentage
Resistance surveillance can inform stewardship only when the population, sampling, definitions and time frame are clear. A percentage from one selected set of records is not automatically a population prevalence or a rule for an individual’s treatment. Stewardship includes using evidence responsibly and avoiding unsupported antimicrobial use, not selecting a drug from an arbitrary classroom table.
Open lessonAssessment · Collisions, collection and waiting timesAudit the sampling mechanism and its guarantees
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Directed random walks and stationary guaranteesAudit the transition model and its conclusion
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsAudit thin-film phase shifts before predicting cancellation
Two reflected contributions can return from the front and back of a thin film. Their relative phase includes both the extra propagation through the film and any reversal upon reflection. A reflection toward a higher-index medium reverses phase by 180°; reflection toward a lower-index medium does not. Transmission alone does not add that reflection reversal. Count each interface separately before classifying the result.
Open lessonLesson · Mathematics for AI · M04Average future rewards with explicit assumptions
Compute expectation, discounted sums and support-aware importance ratios.
Open lessonLesson · Mathematics investigationAverage outcomes before they happen
Can an expected payoff be an amount you never actually receive?
Open lessonLesson · Statistics foundations: data, design and sampling variabilityAverage outcomes with probability weights
Construct a distribution and distinguish expectation from a guaranteed result.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsAverage within groups, then average across groups
Use the tower property with a well-defined conditioning law.
Open lessonLesson · Social studiesAverages can hide unequal opportunities
Stratification concerns patterned inequality in resources and opportunities.
Open lessonRefresher · Math refreshersAverages, probability, and expectation
Practice the mathematics, then return to your lesson.
Open refresherCourse · Mathematics program unitAxiomatic Euclidean and non-Euclidean geometry
A proof-based undergraduate sequence in incidence, order, congruence, continuity, Euclidean transformations and constructions, affine/projective foundations, and hyperbolic, spherical and elliptic models. Model calculations accompany written proofs; synthetic representation theorems and Gauss–Bonnet are explicitly identified dependencies, not conclusions certified by a diagram.
Open courseLesson · Axiomatic Euclidean and non-Euclidean geometryAxioms, interpretations and relative consistency
Separate geometric language from models and logical consequences.
Open lessonConcept · GlossaryBackpropagation
A method for calculating gradients through connected computations. An optimizer then uses those gradients to update weights.
Open conceptConcept · GlossaryBacktracking
Explore a tentative choice, then undo its branch-specific assignments and deductions when it fails before trying another choice.
Open conceptLesson · Point-set topology: spaces, constructions and separationBaire category and nested neighborhoods
Prove the complete-metric Baire theorem with nested closed balls.
Open lessonLesson · Science · Earth and space scienceBalance a watershed water budget
Account for monthly inputs, outputs and storage using basin-equivalent depths and volumes.
Open lessonLesson · Mathematics investigationBalance an equation, bound a decision
What changes when you divide an inequality by a negative number?
Open lessonLesson · Science · Chemistry and matterBalance atoms and identify what remains
Explain a balanced reaction using atom conservation and account for a limiting inventory.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsBalance crossed-field forces without removing the fields
An electric field exerts force qE regardless of whether the particle is instantaneously moving. A magnetic field exerts q times the velocity-cross-field direction and has a speed dependence. These forces can oppose one another, but cancellation is a condition on the complete setup, not evidence that either field is absent.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceBalance decomposition without treating nutrient immobilization as disappearance
Bacteria and fungi transform organic material and recycle elements through food webs and mineral pools. Decomposition does not simply make matter vanish. Carbon can enter microbial biomass or be respired, while nitrogen may be released to mineral pools or temporarily incorporated into new biomass.
Open lessonLesson · Science · General Chemistry IBalance electron equivalents separately from ionic charge
Combine half-reactions under a specified medium before calculating reaction yield.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsBalance exchange without merging maternal and fetal circulation
The placenta forms through interactions between fetal-derived and maternal tissues. Fetal vessels in chorionic structures exchange across a boundary with maternal blood in surrounding spaces; the circulations are not normally represented as one freely mixed blood pool. Umbilical arteries carry blood from the fetus toward the placenta, while the umbilical vein carries blood back toward the fetus, so artery/vein names still follow direction rather than oxygen content.
Open lessonLesson · Science · Physics and physical systemsBalance magnetic braking with mechanical work
Compare driver input, resistor heat and the effects of reversing field or motion.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceBalance nuclear identities and a complete mass-energy account
A nucleus has proton number Z and nucleon number A, with neutron count A−Z. Isotopes share Z but differ in neutron count. The abstract parent in this ledger has A40,Z20; selecting a mode is a bookkeeping experiment, not a claim that a particular real isotope has all those decay channels. Alpha emission removesA4,Z2. Beta minus changes a neutron into a proton while emitting an electron and antineutrino; beta plus changes a proton into a neutron with a positron and neutrino. Gamma emission changes excitation without changing A or Z.
Open lessonLesson · Science · Chemistry and matterBalance oxidation and reduction without losing charge
Diagnose coefficient proposals using separate atom, charge and electron-accounting checks.
Open lessonLesson · Science · General Chemistry IBalance solvation gains against interactions that must be disrupted
Explain a mixing enthalpy ledger without turning it into a guaranteed solubility prediction.
Open lessonLesson · Science · Physical Geology and Earth SystemsBalance volatile inventory without inventing an eruption threshold
Magma contains silicate melt, crystals and dissolved volatile species in proportions that can change during ascent. Pressure reduction can lower volatile solubility, allowing gas to form. Gas amount, expansion, escape, viscosity and conduit conditions interact; gas presence alone does not uniquely determine eruption style.
Open lessonConcept · GlossaryBaseline
A reference method used to judge whether another approach adds value. Fit a mean or majority predictor from training data and compare candidates on the same separate examples.
Open conceptLesson · Homotopy, fundamental groups and covering spacesBasepoints and induced homomorphisms
Construct the homomorphism induced by a based continuous map.
Open lessonLesson · Point-set topology: spaces, constructions and separationBases and generated topologies
Verify the basis covering and local-intersection conditions.
Open lessonConcept · GlossaryBayesian network
A directed acyclic graph and local conditional probability tables that factor a joint distribution and encode conditional-independence assumptions. The arrows alone do not establish causation.
Open conceptGuide · Getting startedBefore you start
Readiness checks and school math refreshers.
Open guideConcept · GlossaryBehavior cloning
Supervised learning that fits a policy to demonstrated action targets. Agreement on demonstrations does not establish recovery skill or successful complete attempts.
Open conceptConcept · GlossaryBehavioral compatibility
Agreement between connected components about the meaning and use of exchanged information. Matching shapes or schemas alone does not establish it.
Open conceptConcept · GlossaryBellman equation
A consistency condition relating a state value to immediate reward and discounted expected future value. Policy evaluation follows fixed actions; the optimality equation maximizes over actions.
Open conceptConcept · GlossaryBenchmark overfitting
Adapting choices to repeated benchmark feedback so the score improves without a corresponding improvement on new tasks.
Open conceptLesson · Calculus-based probability: laws, dependence and limitsBernoulli, binomial and geometric mechanisms
Specify what is counted and the stopping convention.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryBetweenness, separation and Pasch
Identify the order assumptions that diagrams usually conceal.
Open lessonConcept · GlossaryBias
A learned offset added to the weighted input. It lets the model predict a nonzero value when the input is zero.
Open conceptCourse · Mathematics program unitBinary gcd and integer certificates
Prove halving and subtraction rules, preserve integer Bezout coefficients, and distinguish a plausible answer from an independently checkable gcd certificate.
Open courseLesson · Binary gcd and integer certificatesBinary gcd: preserve the value and prove termination
Replace division by guarded halving and subtraction, tracking an external power of two and a decreasing integer potential.
Open lessonCourse · ScienceBiology and living systems
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseLesson · Boolean specifications, certificates and circuitsBoolean specifications and certificates
Translate requirements into truth conditions and choose the right evidence for satisfiability, validity, equivalence and entailment.
Open lessonCourse · Mathematics program unitBoolean specifications, certificates and circuits
Turn a finite specification into logical formulas, distinguish witnesses from universal certificates, and prove circuit correctness with an explicit work/depth model. Normal forms provide supporting depth.
Open courseConcept · GlossaryBootstrap sample
A sample drawn with replacement, so an original row can appear repeatedly or be omitted.
Open conceptLesson · Social studiesBorders do not stop every consequence
Sovereignty concerns authority within a territory; interdependence means decisions in one place affect others.
Open lessonLesson · Science · Engineering design and scientific computingBound a diagnosis by its hypotheses and instruments
Recognize model inadequacy and preserve the difference between public practice and independent diagnosis.
Open lessonLesson · Optimization: models, certificates and algorithmsBound a maximization problem from above
Derive weak duality and a finite-dimensional strong-duality proof mechanism.
Open lessonLesson · Science · Earth and space scienceBound a runoff interpretation
Separate controlled water-budget results from erosion, intervention and flood claims.
Open lessonLesson · Science · General Chemistry IIBound an apparent molar mass without assuming its identity
Let solute mass be w grams, molar mass M g/mol and solvent mass W kilograms.
Open lessonLesson · Science · General Chemistry IIBound an inferred constant and test a changed inventory
A shared calibration scale can cancel in a ratio without verifying every assumption behind that ratio.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceBound an inferred quantity without hiding model error
An inferred concentration depends on several inputs, not just the sample reading. In c=(y−b)/m, errors in signal, intercept and slope can all change the result. State whether each quoted uncertainty is a worst-case bound, a standard deviation, a confidence interval or something else; these meanings are not interchangeable.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsBound and determine a quadratic class group
Prove a planar lattice bound and exhaust the classes of integers adjoin square root of minus five.
Open lessonLesson · Science · Earth and space scienceBound climate claims with energy evidence
Separate equation checks, observational evaluation and the missing processes required for climate forecasts.
Open lessonLesson · Growth estimates and controlled approximationBound harmonic sums and explain logarithmic growth
Compare reciprocal sums with rectangles, control tails and distinguish asymptotic growth from exact thresholds.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeBound integrals of lower limits
Derive Fatou’s lemma using tail infima.
Open lessonLesson · Science · Physical Geology and Earth SystemsBound missing time without assigning it all to erosion
An unconformity records a break in the preserved sedimentary sequence. Nondeposition, erosion and removal of previously deposited material can contribute to the gap. An angular unconformity also records tilting before younger deposition; a disconformity separates sedimentary packages without necessarily changing their orientation, while a nonconformity overlies older crystalline rock.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsBound tails and state what convergence actually means
Use Markov, Chebyshev and Jensen before applying iid limit theorems.
Open lessonLesson · Generalized linear models: fitting and evidenceBounded fitting and finite existence
Bounded fitting and finite existence. Derive, calculate and qualify a fitted response model.
Open lessonLesson · Single-variable calculus foundationsBounding a function you cannot simplify
Establish a limit with upper and lower bounds, including oscillation that a sparse graph may conceal.
Open lessonLesson · Social studiesBrain evidence: activation, timing and causal claims
Match a brain-research method to its inference and avoid treating one active area as a complete explanation.
Open lessonLesson · Science · University Physics I with calculusBreak a density-gradient and origin-offset ambiguity
A reported center coordinate can combine a true mass-gradient effect with an additive origin offset. One number cannot normally identify both. Reuse the 2 m normalized linear-density family, but now treat its positive gradient α and coordinate offset δ as unknown: the reading is xcm(α) + δ.
Open lessonLesson · Poetry, drama and performanceBreak a line; change an expectation
Compare lineation choices and distinguish a line ending from a grammatical stopping point.
Open lessonConcept · GlossaryBrier score
The average squared difference between a predicted probability and a binary outcome. Lower is better; it measures probability error.
Open conceptLesson · Social studiesBrown, Baker and Shaw: different equal-protection questions
Equal protection can arise in education and representation, but the legal questions differ.
Open lessonExperiment · Money, economics and businessBudget for timing and surprises
Can a plan with ordinary monthly surplus still run out of cash?
Open experimentLesson · Science · General Chemistry IBuild a balanced equation without changing species
Construct coefficient constraints and audit an independently proposed oxygen amount .
Open lessonLesson · Differential equations: models, methods and dynamicsBuild a basis of homogeneous solutions
Use the characteristic equation and two initial conditions.
Open lessonLesson · Infinite cardinality through explicit constructionsBuild a bijection from two injections
Construct the Cantor–Schroeder–Bernstein bijection and prove its domain, cross-branch injectivity and full coverage.
Open lessonExperiment · Communication and evidence-based writingBuild a brief from evidence
Which evidence fits the decision and the available space?
Open experimentLesson · Statistics foundations: data, design and sampling variabilityBuild a coherent probability model
Use sample spaces, complements and unions without double counting.
Open lessonLesson · Science · General Chemistry IIBuild a dimensionless quotient and separate direction from speed
An equilibrium quotient belongs to a written reaction, a temperature and a declared activity convention.
Open lessonLesson · Science · University Physics II with calculusBuild a disk field from ring elements
A disk is not a collection of equally charged rings when its surface density is uniform. A ring of radius r and width dr has area 2πr dr, so dq=σ2πr dr. Omitting this radial Jacobian gives the wrong amount of charge and the wrong axial field.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceBuild a falsifiable investigation and evidence portfolio
A useful investigation begins with a mechanistic question and competing predictions.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceBuild a feasible region from constraints
Combine boundary rules, inequalities and contextual integer choices.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsBuild a field from an irreducible polynomial
Check arithmetic and prove finite field sizes.
Open lessonLesson · Finite Fourier bases, convolution and FFTBuild a finite Fourier basis from roots of unity
Derive the finite inverse transform, energy identity and real-input symmetry from exact geometric sums.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceBuild a fixed-number partition sum with exclusion
Consider two identical spin-polarized noninteracting fermions and four nondegenerate one-particle states of energies 0,delta,2delta,3delta. Spin polarization means each listed state has capacity one. At fixed N=2, allowed configurations are unordered pairs of distinct indices: (0,1),(0,2),(0,3),(1,2),(1,3),(2,3). There are six, not sixteen.
Open lessonLesson · Statistical investigations, simulation and probability decisionsBuild a forecast from observed frequencies
Turn observed frequencies into an inspectable forecasting model and identify its limits.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsBuild a null distribution from the assignment mechanism
Enumerate or simulate reassignments that the design actually allowed.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceBuild a numerical result another person can audit
Combine method choice, independent references, error budgets and implementation evidence.
Open lessonProject · Numerical methods and dynamical systemsBuild a numerical result you can audit
Prepare a reproducible numerical investigation combining an approximation method and a time-evolution model. Declare tolerance, workload and stopping rules, compare with an exact or independently checked case, and demonstrate a failure caused by an invalid setup or coarse step. Use at least two mechanisms; a smooth-looking plot is not validation.
Open projectLesson · Science · Earth and space scienceBuild a one-layer greenhouse energy balance
Separate solar transmission, thermal absorption and two-direction atmospheric emission.
Open lessonLesson · Write with purposeBuild a paragraph that goes somewhere
Reorder and revise sentences so that each advances a paragraph’s controlling idea.
Open lessonLesson · Science · AP Environmental ScienceBuild a period total fertility rate from age-specific rates
Calculate a simplified TFR and distinguish it from births and population growth.
Open lessonLesson · Calculus II: series, curves and applicationsBuild a physical integral from local force
Specify units, displacement, density and depth before accumulating work or hydrostatic force.
Open lessonLesson · Mathematics investigationBuild a point from directions
Can two selected vectors reach every point in the plane?
Open lessonExperiment · Programming and software engineeringBuild a price calculation tool
Does changing calculation order preserve the contract?
Open experimentLesson · Measure, Lebesgue integration and the probability bridgeBuild a product measure from rectangles
Prove measurability of sections by a monotone-class argument.
Open lessonLesson · Social studiesBuild a question and a chronology
Turn a broad topic into a question that dated evidence could answer.
Open lessonLesson · Mathematical modeling and senior investigationBuild a question from claims and their assumptions
Critique sources without confusing summary with novelty.
Open lessonProject · Evaluate resource choices over timeBuild a resource decision brief with a stress case
Choose an energy, water, transport or land-use decision. Compare alternatives on an explicit service and time basis, account for resources or impact, and test an adverse scenario. Use captures from at least two experiments. All supplied rates and indices are synthetic; a real recommendation needs appropriate local evidence.
Open projectLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsBuild a resultant field from source components
A field describes how a small probe would interact with fixed sources at each location. Its definition is E=F/q for a sufficiently small test charge; a positive probe feels force along E and a negative probe opposite E. The field belongs to the source configuration, not to the chosen sign of the probe. Making the probe large enough to rearrange the sources breaks that test-charge approximation.
Open lessonLesson · College compositionBuild a sequence of reasons
Organize an essay by argumentative purpose and make relationships between paragraphs explicit.
Open lessonLesson · Triangle-free coloring certificatesBuild a shadow graph and check its coloring
Construct the Mycielski shadow graph, count its edges and degrees, and prove triangle-freeness by checking every possible triangle type.
Open lessonLesson · Science · University Physics II with calculusBuild a signed source-to-observer field kernel
A point-source field is a vector-valued function of the observer position. Keep the source coordinate separate from the observation coordinate: the displacement P−rs points from source to observer, and its magnitude enters the inverse-square law. Combining the unit vector with the distance gives Kq(P−rs)/|P−rs|³. This representation makes sign and geometry auditable rather than hiding them in an arrow sketch.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsBuild a space using linear combinations
Test subspace closure and describe spans, including polynomial examples.
Open lessonLesson · Science · AP Chemistry preparationBuild a structural explanation of acidity
Match an acid comparison to its structural mechanism, construct oxidation bookkeeping, and recognize leveling and confounded comparisons.
Open lessonLesson · Course 2 · Learning without losing old skillsBuild a trustworthy practice loop
Generate examples, filter them with a verifier, train a candidate, and protect earlier skills before promotion.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsBuild accumulation from small contributions
Use partitions, sample points and units to distinguish a finite estimate from a limiting integral.
Open lessonLesson · School trigonometry: proofs, inverse branches and periodic decisionsBuild addition formulas from rotated coordinates
Derive rather than assume sine/cosine formulas, then justify tangent division.
Open lessonLesson · Course 2 · Assemble an experimental agentBuild an architecture that earns its complexity
Train a policy/value network, compare search and memory, freeze a rule-transfer test, and plan a disciplined research program.
Open lessonPath · Students, software engineers and managersBuild an assistant that uses evidence
Define an assistant contract, retrieve and cite evidence, enforce document boundaries and compare complete workflows.
Open pathLesson · Science · AP Chemistry preparationBuild an elementary-step sum and test its rate prediction
Construct event counts, track intermediates and regenerated participants, then separate net-equation agreement from controlled rate-law evidence.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceBuild an energetics argument from paired oxygen evidence
Correct matched light/dark records, normalize independent preparations and distinguish net exchange from conditional gross production.
Open lessonLesson · Science · AP Chemistry preparationBuild an equilibrium particle diagram with an honest scale
Construct monomer and dimer counts, conserve constituent units, and convert weighted symbols and stated volume into the equilibrium quotient.
Open lessonLesson · Science · Earth and space scienceBuild an evidence-limited cosmic argument
Separate historical observations, modern distance evidence and synthetic mechanisms.
Open lessonLesson · Social studiesBuild an exhibit that can be revised
Communicate a historical interpretation while preserving evidence, uncertainty and revision.
Open lessonLesson · Science · General Chemistry IBuild an identification argument across independent chemical models
Separate observation, fitted parameter, model prediction and an identification claim across chemical scales.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesBuild an inclusion–exclusion sieve
Derive derangement counts by forbidden fixed points.
Open lessonLesson · Science · Biology and living systemsBuild an inheritance prediction from gametes
Combine parental allele probabilities before applying a dominance assumption.
Open lessonLesson · Science · General Chemistry IBuild an ionic array ledger without double-counting pairs
Distinguish nearest-pair attraction from a surrounding charge array.
Open lessonLesson · Science · General Chemistry IBuild an orbital occupancy ledger with charge and spin constraints
Connect nuclear identity, electron count, orbital capacity and Hund filling within a declared small-atom domain.
Open lessonLesson · Abstract algebra I: groups and symmetryBuild and audit a complete group-theoretic argument
Combine quotient, action and counting methods without circular reasoning.
Open lessonLesson · Social studiesBuild and challenge a qualitative codebook
Code historical fragments, explain ambiguous decisions and test how a revised definition changes an interpretation.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsBuild and classify solutions of Pell equations
Use norm multiplication to generate Pell solutions.
Open lessonLesson · Social studiesBuild and criticize a fixed basket
Can two households experience different inflation?
Open lessonProject · Algebra, functions and mathematical modelingBuild and defend a model-based comparison
Compare two fictional plans using linear costs or constraints, then add a nonlinear growth or quadratic relationship. Solve relevant boundaries, check them by substitution, and explain the domain where the model is useful.
Open projectProject · Course 1 milestoneBuild and evaluate a classifier
Train, validate, test, and export a model card.
Open projectProject · Course 2 milestoneBuild and evaluate a puzzle agent
Train a policy/value network and compare exact search.
Open projectLesson · Arrival processes, absorption and honest simulationBuild arrivals from independent waiting times
Connect a waiting-time construction, arrival epochs and interval counts before simulating.
Open lessonLesson · Science · AP Chemistry preparationBuild balanced coefficients—and test whether they are unique
Construct four coefficients, diagnose independent atom budgets and find distinct balanced product-mixture proposals.
Open lessonLesson · Partial differential equations and Fourier methodsBuild boundary-adapted orthogonal modes
Derive Green’s identity and the regular spectral framework.
Open lessonLesson · Linear Algebra II: operators and spectral structureBuild chains where eigenvectors are insufficient
Construct nilpotent chains, recover block sizes from kernel growth, and state Jordan uniqueness precisely.
Open lessonLesson · Partial differential equations and Fourier methodsBuild forced solutions from point responses
Track Green-function signs and boundary terms.
Open lessonLesson · Abstract algebra I: groups and symmetryBuild groups from automorphism actions
Construct and verify external and internal semidirect products.
Open lessonLesson · Partial differential equations and Fourier methodsBuild harmonic boundary responses from modes
Solve rectangle and disk boundary problems.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceBuild interpolation one node at a time
Derive divided differences, nested evaluation and incremental interpolation.
Open lessonLesson · Science · Chemistry and matterBuild ionic formulas from charge and structure
Use a signed charge ledger to derive simplest ionic ratios and distinguish formula units from molecules.
Open lessonLesson · Mathematical statistics: derivation and model criticismBuild one test for a one-sided family
Use monotone ratios and stochastic ordering to control a composite null.
Open lessonLesson · Science · AP Chemistry preparationBuild phase cartoons while conserving intact molecules
Place molecules, check count and spatial organization, then identify what a still diagram cannot establish.
Open lessonLesson · Precalculus: constructing and validating polynomial modelsBuild polynomials from meaningful constraints
Construct cubic and quartic models while keeping dimensions, domains and assumptions visible.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsBuild pressure from equilibrium and a stated reference
Consider a thin horizontal fluid slab at rest. The upward force from pressure below balances downward pressure force above plus slab weight. For constant density, Pbottom−Ptop=ρgΔh, where Δh is positive downward separation. This force balance explains why pressure increases with depth in a static fluid without requiring fluid to be flowing.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsBuild rotational inertia from five planar particles
Rotational inertia describes how strongly a rigid mass arrangement resists angular acceleration about a specified axis. For a point particle the contribution is mr², with r the perpendicular distance to that axis. For several particles, add the contributions: I=Σmr². Its unit is kg m². Total mass alone does not determine it.
Open lessonLesson · Calculus synthesis: regions, motion and approximationBuild section areas and split changing boundaries
Translate a region into section dimensions and choose boundary order separately on each interval.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsBuild sine and cosine graphs from rotation
Map unit-circle coordinates to graphs and transform their periods and heights.
Open lessonLesson · Abstract algebra I: groups and symmetryBuild solvable groups from abelian pieces
Prove derived-series and extension criteria and the A₅ obstruction.
Open lessonLesson · Science · AP Chemistry preparationBuild strong-acid/base inventories and test the water approximation
Repair a hydroxide coefficient and final-volume denominator, then distinguish conserved excess from equilibrium ion concentration.
Open lessonLesson · Course 2 · Testing whether improvement is realBuild the baseline before the extension
Replay versioned data, fitting rules, seeds and metrics before changing the model.
Open lessonLesson · Abstract algebra I: groups and symmetryBuild the full S₃ character table and decompose a new action
Connect matrices, irreducible characters and actual invariant subspaces.
Open lessonLesson · Homotopy, fundamental groups and covering spacesBuild the fundamental group
Prove concatenation descends to endpoint-homotopy classes.
Open lessonLesson · Science · Physics and physical systemsBuild the harmonic series from fixed-end modes
Count loops and nodes, relate allowed wavelengths to string length and compare harmonic frequencies.
Open lessonExperiment · Applied mathematicsBuild the largest enclosure
Which rectangle gives the most space for a fixed fence?
Open experimentLesson · Integral calculus: accumulation, methods and applicationsBuild volume from cross-sectional area
Derive disk and washer integrands from distances to a named rotation axis.
Open lessonLesson · Social studiesBureaucracy: a comment process is not a referendum
Delegation gives an agency responsibility to implement a statutory goal, often through technical rules and discretionary judgments. Participation can expose assumptions and alternatives, but popularity alone does not expand statutory authority.
Open lessonLesson · Science · Physical Geology and Earth SystemsCalculate a clock age without counting inherited daughter as decay
A radioactive clock compares surviving parent with daughter produced since a relevant closure event. For a one-to-one parent→stable-daughter model, radiogenic daughter D* equals measured daughter D minus independently known initial daughter D 0. Initial parent is then present parent P plus D* if the system stayed closed.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceCalculate allele frequencies and test equilibrium claims
For a diploid two-allele locus, allele frequency p equals (2AA+Aa)/(2N), and q=1−p.
Open lessonLesson · Calculus through supplied graphs and measured tablesCalculate and estimate derivatives from tables
Use input correspondence in chain and inverse rules and distinguish sampled secant slopes from exact derivatives.
Open lessonLesson · Statistical investigations, simulation and probability decisionsCalculate and interpret Pearson correlation
Use a reproducible technology workflow for nontrivial Pearson correlation and explain its limits.
Open lessonLesson · Inference workflows: calculate, check and concludeCalculate categorical evidence and identify its study design
Build expected counts and interpret chi-square tests as association or distribution comparisons according to sampling.
Open lessonLesson · Science · Earth and space scienceCalculate Darcy flux and volumetric flow
Connect head drop and conductivity to flow while keeping flux distinct from pore-water velocity.
Open lessonLesson · Science · Physics and physical systemsCalculate gas work along different paths
Compare signed pressure-volume areas for identical endpoint states.
Open lessonLesson · Science · Earth and space scienceCalculate moment ratios from logarithmic magnitude
Derive a seismic-moment ratio and keep approximate energy scaling separate.
Open lessonLesson · Distribution calculations: events, tails and cutoffsCalculate normal regions and verify inverse cutoffs
Keep the event, tail direction, units and standard deviation visible while moving between probabilities and thresholds.
Open lessonLesson · Science · Physics and physical systemsCalculate photon energy without confusing it with power
Convert wavelength to frequency and photon energy, then calculate photon rate from beam power.
Open lessonLesson · Science · Physics and physical systemsCalculate pressure force from both sides
Use pressure differences rather than one-sided absolute pressure to find a panel’s net load.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsCalculate quadratic ideals as integer lattices
Connect multiplication matrices, ideal membership and finite quotient indices.
Open lessonLesson · Social studiesCalculate spending rounds and challenge their assumptions
When does an initial purchase generate repeated spending?
Open lessonLesson · Science · Earth and space scienceCalculate suspended-sediment transport through time
Combine flow and representative concentration in each interval before summing transported mass.
Open lessonLesson · Science · Earth and space scienceCalculate the small rest-mass difference
Use isotope-specific atomic masses to distinguish processed fuel from converted rest mass.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesCalculate with a real and an imaginary coordinate
Add, subtract and multiply complex numbers by distributivity and i²=−1.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsCalculate with classes while respecting cancellation
Prove congruence arithmetic and distinguish units from zero divisors.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCalculate with residue classes without invalid cancellation
Prove modular operations and characterize multiplicative inverses.
Open lessonAssessment · Single-variable calculus foundationsCalculus foundations: limits and local change assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Calculus II: series, curves and applicationsCalculus II: convergence, approximation and geometric models
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitCalculus II: series, curves and applications
Eighteen lessons extending differential and integral calculus through convergence, Taylor approximation, parametric and polar curves, length, surface area and physical integrals. This sequence does not by itself establish full BC examination readiness or semester equivalence; written work needs rubric-based review.
Open courseCourse · Mathematics program unitCalculus synthesis: regions, motion and approximation
Six focused investigations complete specific integration, spatial, planar-motion, polar and Taylor-data actions. Public worked practice and written rubrics support learning; they do not establish independent AP examination readiness or general calculator proficiency.
Open courseCourse · Mathematics program unitCalculus through supplied graphs and measured tables
Interpret discontinuities, derivative information, signed accumulation and unequal-time measurements. Exact graph models and uncertain measured data are kept distinct. Public written practice requires review; this unit does not establish AP examination readiness or graphing-calculator proficiency.
Open courseCourse · Mathematics program unitCalculus-based probability: laws, dependence and limits
Lessons develop discrete and continuous laws, transformations, multivariate probability, expectation and limiting arguments. Original exercises include numerical checks and written proofs. This sequence is not a claim of complete independently assessed probability-course coverage; measure-theoretic foundations, stochastic processes and advanced convergence remain further work.
Open courseCourse · MathematicsCalculus: local rates and accumulated change
Connect limits, derivatives and integrals, use the chain rule and partial derivatives, and justify an optimization claim with its domain and constraints.
Open courseLesson · Science · AP Chemistry preparationCalibrate a calorimeter before inferring reaction enthalpy
Infer apparatus capacity from independent pulses, subtract a matched blank and test which boundary assumptions support molar reaction heat.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCalibrate cell dimensions from a shared image scale
An image is evidence only when its scale and provenance are known. A microscope objective label describes an optical setup, but a displayed picture may later be cropped or resized. A scale bar that has undergone the same uniform resizing as the object preserves a ratio that can still be used to recover physical length.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsCalibrate position before interpreting a speed estimate
A motion record begins with a measurement rule, not just a column of numbers. An instrument can have an origin offset, a scale error or both. In the supplied model, indicated position r=sx+o, where x is true position in metres, s is a dimensionless scale and o is a fixed offset in metres. Subtracting two indicated positions cancels o, but the difference remains s times the true displacement.
Open lessonConcept · GlossaryCalibration
Agreement between predicted probabilities and observed frequencies over comparable cases. A calibrated probability is not a guarantee for an individual case.
Open conceptProject · Linear Algebra I: systems, spaces and linear mapsCalibration dossier: noisy differences around a network
Build and revise one calibration dossier for a network of relative-reading sensors. Recover what can be identified, reconcile an inconsistent loop using declared nonuniform weights, verify the solution by genuinely different methods, and defend what the result does not establish. Keep the original measurements unchanged while separating fitting, later inspection and a changed-network transfer. This is a public supported portfolio, not automatic mastery, an unseen examination or a certification that any physical sensors are accurate.
Open projectLesson · Course 2 · Connecting tools and specialistsCall a tool, then check the evidence
Propose structured calls, inspect independent runtime checks, and recover from simulated failures.
Open lessonLesson · Social studiesCampaign finance: classify the transaction before applying a rule
Contributions, independent spending and coordinated communications are different categories. A statement that one category is not subject to contribution limits does not mean all financing is unlimited or undisclosed. Classification depends on facts, not the label chosen by a sponsor.
Open lessonLesson · Read like a literary detectiveCan you trust this narrator?
Compare a narrator’s claims with reported actions and distinguish limited knowledge from possible self-justification.
Open lessonLesson · Social studiesCanal communities: power, markets and local costs
Compare industrial promotion and canal petitions, diagram different effects of infrastructure, and distinguish a proposed remedy from a completed one.
Open lessonLesson · Science · University Physics I with calculusCancel internal impulses only inside the correct system
For a fixed set of particles, total momentum is M times center-of-mass velocity. Internal forces cancel in equal-and-opposite pairs when both recipients are included. This statement does not mean that every component has constant momentum or that internal energy cannot change.
Open lessonLesson · Science · General Chemistry ICancel spectators from an equation without deleting them from solution
Reconcile net ionic change with a complete residual-ion charge account.
Open lessonLesson · Calculus II: series, curves and applicationsCancellation, absolute convergence and error
Check decreasing magnitudes and bound the remainder without ignoring absolute convergence.
Open lessonLesson · Infinite cardinality through explicit constructionsCanonical digits and injections from a square
Resolve positional ambiguity, prove canonical expansions exist uniquely, and encode two real numbers without mistaking an injection for a bijection.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsCarry capacitor state across a topology change
The capacitor begins uncharged and charges through 2 kohm from a 6 V source.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceCarry measurement uncertainty through a net environmental budget
A small net balance can be the difference of two much larger fluxes. The reported means alone do not show whether the sign of the net is well resolved. Keep uncertainty attached to the actual measured terms and distinguish standard uncertainty, a distributional interval, a hard bound and an unknown structural error.
Open lessonLesson · Science · General Chemistry ICarry named entities through mass and mole conversions
Treat the specified entity as part of an amount-of-substance calculation, not an optional label.
Open lessonLesson · Science · University Physics I with calculusCarry position and velocity across a contact-loss event
A support begins at rest with upward coordinate ys=−k t³/6, so its acceleration is−kt. A2 kg point initially rests on it. If contact holds, the point must share this acceleration. Newton's law gives N−mg=−mkt, or N=m(g−kt). A contact surface can push upward but cannot pull the point down.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsCarry timing bounds into an inferred spring constant
A period measurement must count complete cycles using the same phase and direction at both ends. Timing from the positive endpoint to the negative endpoint gives half a cycle, not a full one. Repeating a recognizable crossing can help, but a mistaken cycle count produces a systematic error that averaging does not fix.
Open lessonConcept · GlossaryCatastrophic forgetting
Substantial loss of older capabilities after parameter updates for new tasks.
Open conceptLesson · Course 1 · What has the network learned?Catch a model taking a shortcut
Change the background, keep the shape, and investigate what a perfect score was hiding.
Open lessonLesson · Distribution diagnostics and calibrationCategorical fit and model revision
Compute specified and fitted Pearson discrepancies.
Open lessonLesson · Social studiesCategories shape what an observer notices
Language organizes distinctions, but vocabulary alone does not mechanically determine thought.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsCauchy estimates, Liouville and the fundamental theorem of algebra
Bound derivatives from circle bounds using Cauchy’s formula.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusCauchy sequences and completeness
Control pairs of tail terms and locate exactly where completeness is used.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsCauchy’s integral formula and derivatives
Derive the interior value formula by subtracting the singular part.
Open lessonConcept · GlossaryCausal intervention
In a specified causal model, replacing a variable’s ordinary assignment mechanism with a controlled setting while leaving other mechanisms unchanged. This differs from observing its naturally occurring value.
Open conceptConcept · GlossaryCausal mask
An information boundary that allows each decoder position to attend only to itself and earlier positions.
Open conceptLesson · Homotopy, fundamental groups and covering spacesCell attachments and surface groups
Derive the normal-closure relation imposed by attaching a disk.
Open lessonLesson · Statistical PCA: coordinates, reconstruction and scalingCenter data and construct principal-component scores
Compute the actual covariance, directions and coordinates of a small dataset.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCertify a maximum flow by a cut
Prove flow value is bounded by every cut.
Open lessonLesson · Abstract algebra I: groups and symmetryCertify a subgroup efficiently
Use the one-step test and distinguish intersection from union.
Open lessonLesson · Directed random walks and stationary guaranteesCertify convergence and ranking under direct restart
Prove a contraction, construct its stationary distribution and convert a residual into a justified ranking certificate.
Open lessonLesson · Optimization: models, certificates and algorithmsCertify infeasibility by a separating vector
Prove the finite-dimensional alternative behind LP duality.
Open lessonLesson · Positive matrices, factors and energyCertify positive energy with Schur complements
Derive positive leading-minor and LDL* tests, then prove why semidefinite energy requires all principal minors.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsCertify repeated roots from coefficients
Connect squared root differences, derivative products and a quotient-algebra determinant.
Open lessonLesson · Matching completions, families and unequal sidesCertify stability when preference comparisons are incomplete
Use reachability in consistent comparison orders to determine whether a fixed matching stays stable under every independent completion.
Open lessonRefresher · Math refreshersChain rule and gradients
Practice the mathematics, then return to your lesson.
Open refresherLesson · Multivariable calculus: geometry, change and accumulationChain rule, gradients and directional derivatives
Compose derivative maps and distinguish a unit-direction rate from a parameter-dependent rate.
Open lessonLesson · Social studiesChaldiran and Qing expansion: victory, authority and control
Compare Ottoman–Safavid battle and occupation evidence with Qing transition, institutional adaptation and imperial instructions.
Open lessonLesson · Science · General Chemistry IChallenge a calorimetric inference with a missing heat path
Include cup, solution and external transfer before interpreting a precise temperature rise.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsChallenge a force inventory with independent acceleration
Choose the recipient before adding forces. A force describes an interaction on that recipient, with a direction and an agent. Velocity describes motion; acceleration describes a change of velocity. A point moving east can have a northward acceleration, and a point moving west can have an eastward acceleration. The force diagram is not a sketch of its velocity arrows.
Open lessonLesson · Science · General Chemistry IChallenge a lattice hypothesis with an independent spacing
Combine an explicitly assigned reflection with density rather than identifying a structure from one peak.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsChallenge a lens fit with distance-reference errors
One object-image pair can produce a focal-length estimate, but it cannot strongly test the thin-lens model or the distance references. Use several real-image observations with object and image distances measured from the same lens reference plane. Rearranging gives 1/i=1/f−1/o, so plotting 1/i vertically against 1/o horizontally predicts slope −1 and intercept 1/f.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsChallenge a period-based spring inference with an independent clock
A simple harmonic oscillator experiences a restoring force proportional to displacement from equilibrium: F=−kx. Its acceleration is −(k/m)x, always toward equilibrium. This ideal linear relation gives angular frequency ω=√(k/m) and period T=2π√(m/k). The amplitude sets excursion and energy, not the period within this model.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceChallenge a pressure response without erasing calibration ambiguity
Original indicated pressure records follow Pind=g[K/V+A/V²], with V expressed in litres, K in kPa L and A in kPa L². The revealed teaching generator uses K=12, while shared pressure gain g and correction A can vary. For the inverse task K is unknown to the analyst and is not independent calibration evidence. If an independently established K=12 were supplied instead, gain would be identifiable as Kfit/12; the nonidentifiability statement would no longer apply. The first two exact-volume records at 1 and 2 litres are calibration data; the 3-litre record is held out for a frozen prediction.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsChallenge a ray-shadow prediction with diffraction width
Diffraction is wave spreading around obstacles and through openings. A narrow aperture cannot always be represented by a simple ray shadow. Contributions from different parts of one slit interfere, so even a single opening produces a broad central feature and weaker side features. Spreading becomes especially important when aperture size is comparable with wavelength, although it is not identically absent for larger apertures.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceChallenge a unique explanation for heterozygote deficiency
Nonrandom mating changes which allele copies pair in individuals. Inbreeding increases the probability of identity by descent relative to a specified reference population. It is not synonymous with a rare allele, a low census count or a unique molecular defect. Population structure can also alter aggregate genotype proportions.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsChallenge agreement between two energy measurements before defending a capstone
Two methods agreeing is useful evidence, but it does not prove that either is correctly calibrated. The original packet compares energy inferred from capacitor voltage with energy inferred from a temperature rise. The hidden generator uses C=0.020 F and heat capacity H=0.10 J/K, with all stored capacitor energy entering that heat capacity. Hidden truth is disclosed to teach an inference failure; it is not an independent measurement available in a real investigation.
Open lessonLesson · Science · University Physics I with calculusChallenge an exact fit with a withheld trajectory point
Interpolation can hide a wrong functional model.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsChallenge an exact two-gap capacitance fit
For a vacuum parallel-plate source, geometric C = epsilon0 A/d.
Open lessonLesson · Science · Biology and living systemsChallenge an unlimited-growth prediction
Compare density-limited and exponential models under changed resource assumptions.
Open lessonLesson · Science · AP Chemistry preparationChallenge competing concentration–time laws
Calibrate three laws on the same two anchors, discriminate with later bounded observations and transfer half-life predictions to a fresh concentration.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceChallenge equal training fits with new evidence
A model can match every available training point yet fail outside that range. Fit quality describes agreement with data used in construction; prediction quality requires new evidence and a plausible mechanism. Environmental forecasts also need uncertainty, boundary conditions and a domain where their assumptions remain defensible.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceChallenge ideal doubling in a conceptual amplification record
Polymerase chain reaction conceptually repeats strand separation, primer recognition and polymerase extension to increase a selected DNA target. Primer specificity defines which sequence is preferentially amplified. The lesson describes this mechanism without supplying primers, reaction composition, operating temperatures or a physical procedure.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceChallenge the counterfactual behind a paired change contrast
A before–after change can reflect weather, background trends or measurement changes as well as an intervention. A comparison site helps estimate what might have happened without treatment. Pairing can reduce nuisance variation, but causal interpretation still depends on design, interference, measurement comparability and the relevant counterfactual.
Open lessonLesson · Course 2 · Learning without losing old skillsChange context or change parameters
Give identical support measurements to a context fit, exact-key memory and gradient-based parameter adaptation.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsChange coordinates without changing the vector
Use basis matrices and distinguish a vector from its coordinate list.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsChange force branches at a stopping event
A 2 kg object starts sliding right with speed v0 on a level plane. Gravity is 10 m/s², μk=0.3 and μs=0.5. A constant rightward drive F lies between 0 and 10 N. During rightward slip kinetic friction is 6 N leftward, so acceleration is (F−6)/2 m/s².
Open lessonLesson · Mathematics investigationChange one input while holding another fixed
How does a two-input function have more than one local rate?
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsChange system boundaries between capture and oscillation
Compound events require separate models for separate stages. Consider a horizontal spring mass that briefly captures a second object with zero horizontal velocity. During an idealized very short attachment, the spring’s horizontal impulse is negligible compared with the interaction impulse, so combined horizontal momentum is approximately conserved. Mechanical energy need not be conserved during this inelastic stage.
Open lessonAssessment · Triangle-free coloring certificatesChange the construction and audit the obstruction
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Statistics foundations: data, design and sampling variabilityChange the denominator deliberately
Build joint, marginal and conditional proportions from a two-way table.
Open lessonLesson · College compositionChange the genre without changing the facts
Adapt a supported claim to different academic genres while preserving its evidence, uncertainty and source trail.
Open lessonLesson · Write with purposeChange the sentence, change the emphasis
Revise sentence structure, reference and punctuation to clarify relationships while preserving intended meaning.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityChange units without changing the question accidentally
Track centers, spreads and multiplicative patterns under transformations.
Open lessonAssessment · Discrete synthesis: schedules, sampling and certificatesChanged constraints and random precedence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Picard construction and uniquenessChanged equations, inexact iterates and continuation
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Network incidence and the four spacesChanged networks, energy and constraint audits
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Sequence spaces and completenessChanged norms, compact families and continuous measurements
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Geometric definitions, motion witnesses and theorem proofsCharacterize parallelograms and rectangles
Derive quadrilateral properties and keep the hypotheses of converse criteria.
Open lessonLesson · Statistical investigations, simulation and probability decisionsCheck a normal model against raw observations
Estimate center/spread with technology, inspect fit and qualify a normal-area calculation.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeCheck absolute integrability before changing order
Derive Fubini by positive and negative parts.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsCheck accessible motion before reporting a speed
Use signed charge and the energy boundary to reject impossible trajectories.
Open lessonLesson · Science · Physics and physical systemsCheck entropy as well as energy
Find engine proposals that conserve energy but require an impossible total entropy decrease.
Open lessonLesson · Science · University Physics I with calculusCheck every direction before calling an equilibrium stable
For U=(kx x²+ky y²)/2, the negative gradient gives force(−kx x,−ky y). The origin has zero force for every coefficient pair, but that alone does not classify its stability. Inspect how potential changes under displacements in both independent directions.
Open lessonLesson · Science · Physics and physical systemsCheck flow regimes and dissipative pressure work
Test a laminar candidate before interpreting its energy account.
Open lessonLesson · Science · University Physics II with calculusCheck inductance against a field-volume integral
A flux through one turn is not the same as total flux linkage. In the long-solenoid approximation, turns per length n, length ell and area A give total turns N=n ell and internal field B=μ0nI. This packet uses ell=2 m, A=0.01 m² and a vacuum interior; external field and end corrections are neglected.
Open lessonLesson · Science · General Chemistry IICheck phase existence before imposing saturation
Leaving a pure solid out of a quotient does not supply unlimited material.
Open lessonLesson · Algebra foundations: expressions, equations and rootsCheck roots introduced by squaring
Respect the principal square root and use the original equation to remove extraneous candidates.
Open lessonLesson · Mathematical modeling and senior investigationCheck the computation before interpreting its output
Build an auditable numerical evidence package.
Open lessonLesson · Science · University Physics I with calculusCheck the motion before assigning zero work to a normal force
A normal force is perpendicular to a contact surface, but it need not be perpendicular to the contact point's velocity. A stationary ideal guide often has zero normal work because motion is tangent to it. A moving support can transfer energy in its normal direction.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityCheck trials before counting successes
Use binomial counts only when the trial mechanism supports them.
Open lessonLesson · Science · Engineering design and scientific computingCheck units and boundaries before trusting a model
Build a thermal resistance ledger and separate power, flux and energy.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsCheck whether a fourth-power flow prediction remains admissible
Viscosity describes resistance to relative motion between neighboring fluid layers. A Newtonian fluid has shear stress proportional to velocity gradient, with viscosity η in Pa s. Laminar flow has organized layers; turbulent flow includes irregular fluctuations and mixing. A smooth tube does not guarantee laminar flow at every speed.
Open lessonLesson · Mathematics investigationCheck whether conditioning changes a probability
Are independence and mutual exclusion the same?
Open lessonLesson · Science · Physics and physical systemsCheck whether rolling without slipping is possible
Use force and torque together to test a rolling assumption before trusting its speed prediction.
Open lessonCourse · ScienceChemistry and matter
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseLesson · Social studiesCherokee petition: consent, authority and forced removal
Read a complete petition body and compare its argument with a later military order.
Open lessonLesson · Social studiesChina: distinguish state institutions and party leadership
Keep the state's formal organizational chart separate from the party's leadership role. A diagram omitting either may misdescribe the written system, but the written system alone does not measure actual competition or accountability.
Open lessonLesson · Social studiesChinese exclusion in 1882: entry, documents and citizenship
Read a discriminatory immigration statute by provision and distinguish entry, documentation, naturalization and enforcement.
Open lessonLesson · Social studiesChinese revolution, rural mobilization and collective production
Distinguish rural revolutionary mobilization, land redistribution and collectivization, and evaluate the evidence each requires.
Open lessonLesson · Positive matrices, factors and energyCholesky factors, solves and energy certificates
Construct a triangular factor with positive diagonal, solve through it, and identify exactly which energy and least-squares conclusions depend on definiteness.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsChoose a branch before simplifying a radical
Use trigonometric identities to simplify quadratic radicals without dropping absolute values.
Open lessonLesson · Science · Chemistry and matterChoose a conductor using two explicit constraints
Compare conductivity and density under voltage-drop and mass limits.
Open lessonLesson · Linear Algebra II: operators and spectral structureChoose a conjugation convention before measuring angles
Develop real and complex inner products, norms and Cauchy–Schwarz with a consistent convention.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsChoose a conservation statement separately for each stage
A sequence of events does not inherit one conservation shortcut for every stage. In this virtual packet, an incoming point of mass m and speed u sticks to a stationary2 kg target during a short horizontal collision. The combined point then rises on an ideal frictionless guide. Rotation and external horizontal impulse during the collision are neglected.
Open lessonLesson · Science · AP Chemistry preparationChoose a defensible window for a reaction-rate estimate
Convert timed optical readings into stoichiometric rates, propagate separate reading bounds and distinguish shared blank cancellation from drift.
Open lessonLesson · Science · Engineering design and scientific computingChoose a discriminating reading within a budget
Spend two distinct probes on evidence that separates the remaining explanations.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityChoose a display that preserves the question
Read bins, density, quartiles and outlier conventions honestly.
Open lessonLesson · Science · Physics and physical systemsChoose a gravitational energy reference
Change the zero of potential energy without changing the physical descent.
Open lessonLesson · Differential equations: models, methods and dynamicsChoose a particular solution that does not duplicate the homogeneous basis
Use undetermined coefficients and account for overlap.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsChoose a principal inverse branch
Recover angles without confusing one principal answer with every possible angle.
Open lessonLesson · Social studiesChoose a projection for the question
Can a visually larger country have a smaller measured area?
Open lessonLesson · Algebra reasoning, constraints and solution evidenceChoose a quadratic method and derive the formula
Turn a square into a solution argument rather than a memorized rule.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsChoose a system and locate its center of mass
A system is the collection of objects included in an analysis. A force between two members is internal; a force exerted by something outside the boundary is external. These words depend on your chosen boundary, not on the kind of force. String tension can be external for one cart but internal for a two-cart system that includes the string.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsChoose a system boundary for thermal equilibration
A body can conserve its mass while gaining or losing internal energy. A closed subsystem is therefore not automatically isolated. Here the hot body, cold body and cup exchange energy internally, while their combined boundary is stipulated to exchange neither heat nor work with the surroundings.
Open lessonExperiment · Design and product developmentChoose a useful next increment
Which work fits the available days and addresses observed failures?
Open experimentLesson · Abstract algebra II: rings, fields and Galois foundationsChoose a valid test and respect its limits
Use rational roots, modular reduction and Eisenstein.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceChoose aggregation weights that match the environmental quantity
A reported average hides a choice of weights. The mean concentration experienced at a randomly chosen time differs from the mean concentration attached to a randomly chosen litre of delivered water when flow changes. Neither mean is universally correct; each answers a different question. Define the sampling unit and target quantity before selecting an averaging rule.
Open lessonLesson · Mathematical statistics: derivation and model criticismChoose an action by posterior expected loss
Derive conjugate updating and separate prior-sensitive decisions from sampling guarantees.
Open lessonLesson · Partial differential equations and Fourier methodsChoose an admissible conservation-law continuation
Solve Burgers Riemann problems and check entropy.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceChoose an enzyme-rate window before comparing treatments
An enzyme-rate claim concerns change per unit time, not merely the amount of product at an arbitrary endpoint. A product curve can bend as substrate is depleted, product accumulates or active enzyme changes. Comparing endpoints at different times can therefore mix a catalytic effect with a changing measurement window.
Open lessonAssessment · Inference workflows: calculate, check and concludeChoose and defend an unfamiliar inference workflow
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Sections, solid measures and geometric designChoose and test a model for a measured object
Compare idealized solid predictions against synthetic measurements.
Open lessonLesson · Mathematics investigationChoose between cost and quality
Which options can be rejected without choosing a preference weight?
Open lessonLesson · Course 1 · How does a network learn?Choose complexity using development evidence
Compare model capacity, weight penalties, dropout and training duration using saved classifier candidates before opening the test set.
Open lessonLesson · Course 2 · Geometry and control foundationsChoose continuous actions
Train a small force controller, then compare continuous outputs, discrete choices, noise and saturation.
Open lessonLesson · Science · Scientific inquiry and measurementChoose controls and independent repetitions
Distinguish a useful control from a repeated biased comparison.
Open lessonLesson · Euclidean and coordinate geometry foundationsChoose coordinates that expose the geometry
Translate geometric assumptions into algebra while preserving generality and exceptional cases.
Open lessonLesson · Science · University Physics I with calculusChoose derivative spacing using an error budget
A smaller finite difference can amplify measurement error even while its truncation error falls.
Open lessonLesson · Science · Introductory MicrobiologyChoose discriminating evidence without mistaking fit for truth
A useful virtual control plan distinguishes explanations by predicting what additional evidence would look like. A good fit to already inspected observations is not the same as successful prediction on an unused record. Calibration failure, recovery bias, population heterogeneity and genuine response changes can sometimes fit the same initial pattern.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsChoose enough directions without redundancy
Construct bases for coordinate, solution and polynomial spaces.
Open lessonLesson · Social studiesChoose evidence beyond GDP
Test paid care, disaster repair and population growth before using GDP to judge living conditions.
Open lessonExperiment · Design and product developmentChoose evidence for a product assumption
Is stated interest enough to justify building the whole product?
Open experimentLesson · Science · Scientific inquiry and measurementChoose evidence that separates explanations
Design a useful test under instrument limits.
Open lessonLesson · Course 1 · How does AI process language?Choose how to sample
Compare greedy, temperature, top-k and top-p decoding on trained checkpoints with fixed random draws and separate quality diagnostics.
Open lessonLesson · Mathematical modeling and senior investigationChoose observations that distinguish competing explanations
Use rank, precision and intervention limits.
Open lessonLesson · Social studiesChoose plant scale and distinguish industry cost patterns
Does the cheapest small plant remain cheapest at every output?
Open lessonLesson · Functions in graphs, measurements and growth decisionsChoose quantities, units and honest axes
Define a useful comparison quantity and its denominator.
Open lessonLesson · Euclidean and coordinate geometry foundationsChoose ratios relative to an angle
Use similarity to explain why sine, cosine and tangent depend on an acute angle.
Open lessonLesson · Science · University Physics II with calculusChoose reference levels before taking an infinite-source limit
A potential reference valid for a finite localized source need not survive an infinite-source limit. For a uniform finite rod of half-length L and fixed λ=1 nC/m, the perpendicular-bisector potential with zero at infinity is the integral Kλ ds/√(s²+y²). It evaluates to 2Kλ asinh(L/y)=18 asinh(L/y) V.
Open lessonLesson · Statistical PCA: coordinates, reconstruction and scalingChoose scales and diagnose unidentified directions
Compare covariance and correlation PCA without hiding zero variance or tied eigenvalues.
Open lessonLesson · Research, editing and publishingChoose sources for the claims they can support
Evaluate provenance, purpose, method and relevance rather than assigning a source a single blanket credibility score.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsChoose subsets compatible with multiplication
Prove quotient multiplication is well-defined exactly for ideals.
Open lessonLesson · Science · Earth and space scienceChoose the boundary before counting water flows
Separate internal transfers from boundary fluxes and challenge a budget that balances for the wrong reason.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsChoose the boundary before predicting dielectric insertion
A dielectric has bound charges that shift or orient under an electric field rather than moving freely through it as carriers do in a conductor.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsChoose the correct Amperian path for slabs and solenoids
An infinite slab occupies −a ≤ x ≤ a with current density J along +z.
Open lessonLesson · Algebra II and precalculus: functions and structureChoose the correct branch at a boundary
Build and interpret rules whose formulas depend on input conditions.
Open lessonLesson · Science · University Physics II with calculusChoose the correct branch of enclosed cylindrical charge
For an infinite cylindrical source, symmetry makes the field radial and constant on a coaxial cylindrical side. The end caps have normals along the axis, perpendicular to the field, so they contribute no flux. This reasoning depends on the infinite-source idealization and absence of an external field; a finite tube near an end need not behave this way.
Open lessonLesson · Social studiesChoose the correct monetary implementation framework
Will adding reserves always lower the overnight rate?
Open lessonAssessment · Statistics: displays, designs and differencesChoose the evidence and justify the comparison
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Course 1 · How does AI process language?Choose the pieces a model predicts
Fit character, word and subword tokenizers; inspect unknown text and a count-based next-token baseline.
Open lessonLesson · Science · AP Chemistry preparationChoose the property that answers the bonding question
Compare electronegativity and affinity without equating their scales or chemical meanings.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsChoose the question before the formula
Match outcome, design and target to an inferential procedure.
Open lessonExperiment · Scientific reasoning and experimental designChoose the test that could prove you wrong
Where should you measure next when two explanations fit the existing observations?
Open experimentLesson · Fitted regression: coefficients, uncertainty and validationChoose the uncertainty target
Derive conditional mean and new-response intervals, including rank and zero-spread boundaries.
Open lessonLesson · Course 2 · Learning without losing old skillsChoose what experience to retain
Compare reservoir, task-stratified and recent replay under a frequency shift, from the same neural checkpoint.
Open lessonLesson · Course 2 · Connecting tools and specialistsChoose where a connected system learns
Compare frozen, adapter-only, partial, and joint training while checking an existing consumer for regressions.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityChoose who can enter the sample
Separate the target population, frame, selection and response processes.
Open lessonLesson · Social studiesChoosing senators: Progressive reform and the Seventeenth Amendment
Analyze a 1911 cartoon, compare constitutional selection rules, and trace reform from advocacy to ratification and election.
Open lessonModule · Course 2Choosing what to do next
Coordinate models, tools, and plans under a limited budget.
Open moduleLesson · Finite Fourier bases, convolution and FFTCircular convolution, modes and wraparound
Track cyclic indices through the Fourier transform, identify circulant eigenmodes and determine when padding turns a circular computation into an ordinary convolution.
Open lessonCourse · Social studiesCivics and United States government — AP preparation practice
Hands-on social studies course and project
Open courseCourse · Social studiesCivics and United States government — High school
Hands-on social studies course and project
Open courseCourse · Social studiesCivics and United States government — Introductory college
Hands-on social studies course and project
Open courseLesson · Social studiesCivil War, emancipation and Reconstruction
Trace emancipation from wartime policy to constitutional change and contested enforcement.
Open lessonLesson · Social studiesClassical conditioning: what does the cue predict?
Identify stimulus–stimulus learning and distinguish acquisition, extinction, generalization and discrimination.
Open lessonStudio · Optional P03 implementation studioClassical ML pipeline studio
Python logistic prediction, library tree baselines, fold-local preprocessing, grouped cross-validation and frozen recipe replay.
Open studioLesson · Differential equations: models, methods and dynamicsClassify a whole parameter family, including its boundaries
Use trace, determinant and discriminant together, then resolve the exceptional cases by structure.
Open lessonLesson · Algebra foundations: expressions, equations and rootsClassify and solve quadratic equations
Derive the formula by completing the square and distinguish repeated, irrational and complex roots.
Open lessonLesson · Homotopy, fundamental groups and covering spacesClassify connected covers
Construct path-class neighborhoods for a universal cover.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryClassify Euclidean isometries algebraically
Derive rigid motions from distances rather than assuming their form.
Open lessonLesson · Abstract algebra I: groups and symmetryClassify every group of order eight
Prove exhaustion, existence and distinction of all five types.
Open lessonLesson · Abstract algebra I: groups and symmetryClassify finite plane symmetry groups
Connect orthogonal matrices, circle rotations and the fixed point of a finite isometry group.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryClassify hyperbolic Möbius motions
Relate fixed points to elliptic, parabolic and hyperbolic types.
Open lessonLesson · Social studiesClassify international flows before assigning signs
Is a foreign purchase of a bond an export of a good?
Open lessonLesson · Differential equations: models, methods and dynamicsClassify linear motion beyond two distinct real modes
Handle complex and defective matrices while distinguishing local geometry from formulas.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceClassify mirror images before placing a screen
A spherical mirror is described here by a paraxial focal length, meaning rays remain close enough to the axis that the simplified imaging equation applies. A concave converging mirror has positive f and a convex diverging mirror negative f. The real object distance o is positive in front of the mirror. A positive image distance i lies in front of the mirror, while negative i locates a virtual image behind it.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsClassify operators with companion blocks
Construct rational canonical form over any field without assuming eigenvalues split.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsClassify primitive right triangles
Derive the coprime opposite-parity parametrization.
Open lessonLesson · Science · Earth and space scienceClassify rocks by their formation
Connect rock categories to processes and distinguish composition from texture.
Open lessonLesson · Optimization: models, certificates and algorithmsClassify stationary points without overclaiming
Derive necessary and sufficient local conditions.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsClose a branching flow balance without guessing the split
Volume flow rate Q is volume transported per time, not the speed of an individual fluid element. For a section of area A and mean normal speed v, Q=Av. One litre is 0.001 m³ and one minute is 60 s, so 6 L/min is 0.0001 m³/s. Converting area requires squaring the length conversion: 1 cm² is 0.0001 m².
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceClose a dielectric-change energy ledger under two constraints
A dielectric can polarize without conducting free charge across the gap. In an ideal linear material filling fixed parallel plates, capacitance increases from C0 to κC0. Before applying an energy formula, decide whether free plate charge or source voltage is held fixed. Both cannot generally remain fixed while capacitance changes.
Open lessonLesson · Science · General Chemistry IIClose a finite reaction without counting heat and work twice
A favorable reaction is not a promise of full conversion, rapid approach or delivered electrical energy.
Open lessonLesson · Science · University Physics II with calculusClose a finite-mass motional-induction energy account
A sliding rod can induce a current that opposes the mechanical drive, but its speed need not instantly equal a terminal value. In this original ideal model mass, rod length, field magnitude and total resistance are each one in SI units. A constant force F drives it from rest. Friction, loop inductance and radiation are neglected explicitly.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceClose a photon recoil energy and momentum ledger
A photon has momentum magnitude E/c along its propagation direction. In the displayed plane the incident photon travels along positive x; the scattered photon leaves at angle θ. Conservation therefore gives electron recoil components px=E−Eprime cosθ and py=−Eprime sinθ when momentum is expressed in keV/c and photon energy in keV. Subtracting scalar momentum magnitudes would discard the direction that makes this account meaningful.
Open lessonLesson · Science · Introductory AstronomyClose a planet’s two-way radiative exchange
A spherical planet intercepts starlight over a disk but emits over a sphere. Uniform global redistribution therefore gives absorbed mean flux S(1−A)/4, not S itself. Albedo A describes reflected incoming radiation; longwave emissivity describes thermal absorption and emission and is a different property.
Open lessonLesson · Science · Environmental science and sustainabilityClose a pollutant pulse account
Follow parent mass through export and transformation without confusing their destinations.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsClose a renal solute ledger along the nephron route
Blood reaches glomerular capillaries through an afferent arteriole and leaves through an efferent arteriole toward peritubular or medullary vascular pathways. Filtrate enters Bowman’s space, then passes through proximal tubule, loop segments, distal tubule and collecting pathways. The tubular route and surrounding vascular route are connected by selective transport, not by one continuous open pipe.
Open lessonLesson · Science · General Chemistry IIClose a shared inventory across two equilibrium relations
A shared intermediate belongs to one conserved population, not to independent reaction batches.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsClose a source, internal-loss and load-power ledger
An ideal emf E supplies energy per charge inside the source.
Open lessonLesson · Science · Introductory MicrobiologyClose a spatial oxygen-like budget without identifying a metabolism from one profile
Oxygen availability can vary across a community or layer because diffusion supplies it while local processes consume it. An oxygen-rich outer boundary does not imply that every interior location has the same concentration. Obligate aerobes, obligate anaerobes, facultative anaerobes, aerotolerant organisms and microaerophiles differ in how oxygen relates to metabolism and tolerance; concentration alone does not identify which group is present.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsClose a thermal ledger at the actual stopping point
A 2 kg point begins at 4 m/s on a level surface, so its kinetic energy is 16 J. A constant kinetic friction magnitude f opposes motion. For the point alone, friction work over actual forward distance d is−fd, giving K=16−fd. This equation is valid only while that forward sliding branch exists.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsClose a two-dimensional photon–electron momentum ledger
Compton scattering couples photon energy and momentum. An initially stationary free electron recoils when a photon changes direction and wavelength. The outgoing photon generally has a longer wavelength and lower energy. With scattering angle θ measured from the incoming +x direction, the shift is Δλ=h(1−cosθ)/(me c). It depends on angle, not directly on the incident wavelength.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsClose a two-leg survey without confusing distance and displacement
A two-leg route can cover substantial ground while ending near its starting point. Distance adds lengths of the traveled legs. Displacement connects the starting and ending positions and carries direction. Use one shared east–north coordinate system to combine the two legs; do not add their magnitudes unless their directions justify that special case.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceClose an internal heat transfer without inventing extra forcing
The climate system stores heat in reservoirs with different capacities and exchange rates. A surface temperature record alone does not describe all stored energy. Transfer between a surface and deep reservoir can alter their separate temperature rates while conserving their combined heat budget.
Open lessonLesson · Science · Introductory AstronomyClose both component orbits around a common center
Both members of a binary respond to the same mutual gravitational force, but their accelerations differ inversely with mass. Neither body must remain fixed. In the center-of-mass frame their position vectors are opposite and obey M₁r₁=M₂r₂. The more massive component follows the smaller orbit, while both complete a cycle in the same period.
Open lessonLesson · Science · Introductory MicrobiologyClose carbon across a donor and a receiving partner
Cross-feeding occurs when material released by one population becomes available to another. A metabolic product can be another organism’s substrate, but a shared metabolite does not guarantee mutual benefit. The donor, receiver and whole-community boundaries must be distinguished, as must atoms, molecules and energy.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceClose energy pathways while distinguishing cycling elements
Food webs include grazing and detrital pathways. Gross primary production is the rate of energy fixed into organic material; subtracting producer respiration gives net primary production. Standing biomass is a stock at a time, not that production rate, so a small rapidly renewing producer stock can support a substantial consumer flux.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceClose node current and loop energy in a mixed network
A circuit diagram specifies which terminals share a node, not where components happen to be drawn on a page. Here a 12 V ideal source feeds one 6 Ω resistor, followed by two branches that reunite at the source return. The branch resistors are parallel because their two endpoint nodes coincide. Their currents need not be equal, but their potential differences must match under the ideal-wire assumption.
Open lessonLesson · Science · Introductory AstronomyClose photon collection and detector uncertainty separately
A telescope collects radiation over an aperture, an instrument selects or disperses it, and a detector records events. For an unobstructed circular aperture, collecting area is πD²/4. Doubling diameter quadruples area, but angular resolution and count statistics follow different relations; a larger display image alone improves neither collection nor the original information content.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsClose pressure-work, kinetic and gravitational energy ledgers
A flowing parcel gains or loses energy through pressure forces as surrounding fluid pushes it. If volumeΔV moves through an inlet, pressure work there isP1ΔV; pushing fluid out at the outlet requiresP2ΔV. The difference supplies changes in kinetic and gravitational energy when viscous losses and machinery are absent. This connects fluid motion to the work-energy reasoning already used for particles.
Open lessonLesson · Science · Introductory AstronomyClose primordial baryon bookkeeping without claiming a full network
A hot expanding early universe permits nuclear reactions only during a changing interval of temperature and density. Light-element abundances therefore depend on expansion, reaction rates, baryon density and neutron survival. Helium, deuterium and other species provide different constraints; a single endpoint count is not a complete nucleosynthesis calculation.
Open lessonLesson · Science · University Physics II with calculusClose six faces without inventing a local field
Gauss law relates a closed-surface flux to net enclosed charge. It does not say every field on that surface is uniform, nor that zero net charge means zero field. A useful test case is an affine field whose constant offsets affect individual faces but cancel from their closed sum.
Open lessonLesson · Compactness of function familiesClose the family before claiming attainment
Prove the converse compactness conditions, pass to a compact closure, and justify optimization over closed function constraints.
Open lessonLesson · Science · Physics and physical systemsClose the first-law energy account
Distinguish path-dependent heat and work from endpoint-dependent internal energy.
Open lessonLesson · Science · Introductory MicrobiologyClose the surface and volume of a rounded microbial shape
Microbial shapes include cocci, rods, curved rods, spirals and branching or filamentous forms. Arrangements can reflect division planes and whether cells remain attached, but morphology alone rarely supplies a complete identification. Geometry can help frame transport constraints without replacing membrane chemistry, metabolic capacity or ecological evidence.
Open lessonLesson · Science · Introductory AstronomyClose the torque and work boundary of an orbital change
Conservation applies to a defined system. An orbiting body subject to an external torque can gain orbital angular momentum; that does not violate conservation when the driver is included. Similarly, orbital energy can increase through work. A mechanism such as interaction with another body or a disk needs a complete exchange ledger rather than the assertion that every orbit must conserve its own quantities while being driven.
Open lessonLesson · Science · Physical Geology and Earth SystemsClose weathering and carbonate burial across a geological carbon boundary
Weathering links exposed rocks, water, atmosphere and sediment burial. Dissolved products can travel far beyond the original outcrop, so interpreting a carbon effect requires a boundary that follows those products. An initial uptake during dissolution is not necessarily equal to persistent net removal after downstream reactions.
Open lessonLesson · Differential geometry of curves and surfacesClosed curves, turning and area bounds
Connect local curvature to global restrictions.
Open lessonLesson · Point-set topology: spaces, constructions and separationClosure, interior and boundary
Compute closure and interior from neighborhood definitions.
Open lessonLesson · Differential geometry of curves and surfacesCodazzi and compatible surface data
Identify the compatibility conditions for a metric and bending form.
Open lessonCoding · Optional codingCode a small idea
Variables, arrays, conditions, loops, functions, and gradient descent.
Open codingLesson · Determinants from permutations to certificatesCofactors, adjugates and exact systems
Derive cofactor expansion and the adjugate identities, then distinguish inverse formulas from the information retained at singular matrices.
Open lessonConcept · GlossaryCold start
A user or item lacks the history a method needs. Content, onboarding or explicit fallbacks may help; absent feedback is not a verified preference.
Open conceptLesson · Social studiesCold War crisis: signals, deterrence and missing voices
Compare diplomatic evidence across time and explain why successful de-escalation does not prove a safe strategy.
Open lessonLesson · Social studiesCold War power and movements for rights
Connect movement organizing, policy decisions and actual enforcement.
Open lessonConcept · GlossaryCollaborative filtering
Estimate preferences using patterns of observed feedback across users and items. Missing entries are not automatically negative ratings; learned factors need evidence and cold-start handling.
Open conceptLesson · Collisions, collection and waiting timesCollect coupons by decomposing waiting stages
Derive expected collection time and deadline bounds while tracking the assumptions behind uniform sampling.
Open lessonLesson · Course 2 · Geometry and control foundationsCollect useful robot demonstrations
Define observation/action timing, resets and whole-trajectory splits.
Open lessonCourse · English language artsCollege composition
Develop a qualified position through inquiry, source synthesis, purposeful organization and substantive revision, with an inspectable research trail.
Open courseCourse · ScienceCollege Physics I — algebra-based measurement and mechanics
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceCollege Physics II — algebra-based fields and evidence
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseLesson · Collisions, collection and waiting timesCollision products and birthday bounds
Count repeated labels exactly, distinguish collision counts from collision events, and certify a controlled exponential approximation.
Open lessonCourse · Mathematics program unitCollisions, collection and waiting times
Build probability estimates from the sampling mechanism, distinguish events from counts, and turn average waiting times into carefully justified deadline bounds.
Open courseLesson · Social studiesColonial labor: a law, a claim and an outcome
Distinguish enslavement, encomienda and tribute while tracing the gap between a legal rule and a lived outcome.
Open lessonLesson · Social studiesColonial religion: protection, conditions and exclusion
Compare Maryland’s 1649 act with Rhode Island’s 1663 charter without equating toleration with universal equality.
Open lessonLesson · Social studiesColonial research archives: authority to describe and share
An archive's catalog reflects choices made by collectors and institutions. Contemporary community knowledge can correct interpretation and identify responsibilities absent from an old catalog. Online access does not mean every reuse is culturally appropriate or authorized.
Open lessonLesson · Social studiesColonial status laws: liberty, exceptions and inherited bondage
Compare Massachusetts and Virginia legal provisions without turning a rights title into a claim of universal freedom.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesColor graphs with structural orderings
Prove greedy and degeneracy coloring bounds.
Open lessonCourse · Mathematics program unitCombinatorics and graph theory: enumeration, structure and certificates
A proof-based undergraduate sequence extending discrete foundations through enumerative methods, structural graph theory, algorithms and probabilistic existence. Public practice and reviewed assessment remain distinct evidence.
Open courseAssessment · Combinatorics and graph theory: enumeration, structure and certificatesCombinatorics: constructions, bounds and certificates
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · Physical Geology and Earth SystemsCombine calibrated fossil ranges without ignoring conflicting evidence
Fossil succession can correlate strata because organisms appeared, changed and disappeared through time. Numerical ages for ranges require independent calibration. An observed fossil range is not automatically a globally synchronous biological boundary: sampling, environment, preservation and reworking affect what is found.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCombine congruences and certify integer solutions
Use coprimality for unique residues and gcd conditions for linear equations.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsCombine congruences without losing compatibility
Construct CRT solutions and prove uniqueness modulo the correct common period.
Open lessonLesson · Science · Physics and physical systemsCombine convection and radiation at a surface
Use signed rates to determine the thermal power needed to hold a surface temperature.
Open lessonLesson · Science · Earth and space scienceCombine daylight duration and solar angle
Read daily solar-input curves and handle polar day, polar night and the exact-pole equinox limit.
Open lessonLesson · Science · Earth and space scienceCombine deformation with seismic evidence
Compare actual summary windows and complementary measurements without inventing exact counts.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceCombine fast proposals with a reliable bracket
Derive secant steps, explain superlinear order and specify a genuinely contracting safeguard.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceCombine fossils, homology and molecular evidence
Common ancestry predicts nested similarities across independent lines of evidence.
Open lessonLesson · Abstract algebra I: groups and symmetryCombine independent coordinates
Compute orders and distinguish products from cyclic groups.
Open lessonLesson · Abstract algebra I: groups and symmetryCombine invariants, actions and presentations
Classify all groups of order six and distinguish order-eight examples.
Open lessonLesson · Course 1 · Simple models and useful baselinesCombine many imperfect trees
Fit bootstrap trees, inspect their votes and out-of-bag coverage, and test whether shared mistakes survive a measurement shift.
Open lessonLesson · Functions in graphs, measurements and growth decisionsCombine meaningful functions into a quadratic decision
Build products and sums from units, then compare factored and vertex forms on a contextual domain.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCombine osmotic and pressure effects before predicting water flow
Osmosis concerns water movement across a selectively permeable boundary. For the stated ideal non-dissociating (van’t Hoff i=1), nonpenetrating solute, the solute contribution to water potential is−RTc. Pressure adds a separate contribution, so totalΨ=P−RTc in this simplified comparison. Water tends to move from higher toward lower total water potential through a permeable route.
Open lessonLesson · Science · Physical Geology and Earth SystemsCombine petroleum-system history with a bounded pore inventory
Organic-rich source material can generate hydrocarbons during burial and heating. Migration, porous reservoir rock, a sufficiently effective seal and a trapping geometry must align in time for accumulation. Coal represents a different concentrated organic deposit history; not every fossil fuel occupies a conventional porous trap.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCombine positive and negative regulation in an operon model
An operon groups genes whose transcription can be coordinated from shared regulatory DNA. A repressor can reduce transcription through a compatible operator, while an activator can improve productive transcription under another condition. An inducer or corepressor changes regulator behavior; it is not simply another name for the structural gene being expressed.
Open lessonLesson · Science · Introductory AstronomyCombine size and reflex motion without dropping finite mass
A transit constrains a size ratio under photometric and geometric assumptions, while periodic stellar reflex motion constrains masses with an inclination dependence. Combining independent signals can constrain density, but only after checking that they describe the same physical system and accounting for stellar properties.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceCombine slopes to cancel leading error terms
Derive second-order stages and the classical RK4 update while distinguishing order from stability.
Open lessonLesson · Science · Introductory AstronomyCombine solar height and day duration in a seasonal energy argument
Axial tilt changes how solar rays meet a surface and how long the Sun remains above its horizon. Opposite hemispheres therefore have opposite seasonal geometry at the same moment. Orbital distance alone cannot explain that simultaneous reversal; this controlled model holds the normal-ray input fixed to isolate orientation.
Open lessonLesson · Mathematical modeling and senior investigationCombine symbolic arguments with bounded search
Make computational evidence logically sufficient for its claim.
Open lessonLesson · Science · Physics and physical systemsCombine thermal resistances in series
Explain how layers share a temperature difference without splitting the heat rate.
Open lessonLesson · Science · Physics and physical systemsCombine translation and rotation in rolling
Compare how mass distribution divides gravitational energy between forward motion and rotation.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCombine voltage and pH contributions to chemiosmotic coupling
A proton gradient has both chemical and electrical aspects. A pH difference represents a hydrogen-ion concentration ratio, while a membrane voltage changes the electrical energy of a charged proton. Depending on direction and sign, the two terms can reinforce or oppose one another; a pH reading alone does not describe the complete driving force.
Open lessonSubject · Experiment collectionCommunication and evidence-based writing
Choose honest denominators and assemble evidence for a concise recommendation.
Open subjectLesson · Complex analysis: holomorphic functions, contours and mappingsCompact homotopies: prove endpoint-germ independence
Build finite compatible strips, then prove the global monodromy conclusion and its limits.
Open lessonLesson · Point-set topology: spaces, constructions and separationCompact images, closedness and inverse maps
Prove continuous images and closed subspaces of compact spaces are compact.
Open lessonLesson · Point-set topology: spaces, constructions and separationCompact products, ultrafilters and choice
Prove the ultrafilter characterization of compactness.
Open lessonLesson · Point-set topology: spaces, constructions and separationCompactifications and the Stone–Čech construction
Define a compactification as a dense embedding into a compact Hausdorff space.
Open lessonLesson · Point-set topology: spaces, constructions and separationCompactness and open covers
Prove the finite-intersection characterization of compactness.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusCompactness and the Heine–Borel theorem
Connect finite subcovers, sequential compactness and closed bounded subsets of R.
Open lessonCourse · Mathematics program unitCompactness of function families
Turn common continuity control into uniformly convergent subsequences, distinguish a family from its closure, and justify optimization over functions.
Open courseLesson · Real Analysis II: uniform limits, metric spaces and local calculusCompactness, completeness and total boundedness
Prove the metric compactness equivalences rather than importing Heine–Borel blindly.
Open lessonCourse · Social studiesComparative politics — AP preparation practice
Hands-on social studies course and project
Open courseCourse · Social studiesComparative politics — High school
Hands-on social studies course and project
Open courseCourse · Social studiesComparative politics — Introductory college
Hands-on social studies course and project
Open courseLesson · Statistical inference: estimates, tests and defensible conclusionsCompare a joint table with independence
Calculate expected counts and nonnegative discrepancy contributions.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceCompare a lake equilibrium with its response timescale
A receiving-water concentration depends on external loading, water exchange and internal processing. A lower incoming concentration is not the same as a lower mass load when flow changes. A lake can also release previously stored nutrients, so a loading reduction need not produce immediate ecological recovery.
Open lessonLesson · Science · Introductory AstronomyCompare a larger fuel supply with a faster luminosity demand
Main-sequence stars sustain central hydrogen fusion. More mass generally means more potential fuel, but it can also produce a much larger luminosity through a different internal structure. The ratio of usable energy to energy output, not mass alone, determines a fuel-consumption timescale.
Open lessonLesson · Science · Earth and space scienceCompare a lifted parcel with its environment
Track idealized parcel cooling separately from the surrounding temperature profile.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingCompare a local force derivative with a secant stiffness
Derive restoring direction and linear limits without treating every elastic relation as one constant spring.
Open lessonExperiment · Media literacy and civic reasoningCompare a policy’s distribution of benefits
Can equal total benefits reach different groups?
Open experimentLesson · Science · Environmental Science: ecological systems and evidenceCompare abundance trends without assuming constant detection
An observed count is affected by abundance and by how likely individuals are to be detected. Observer skill, vegetation, weather, activity and survey effort can change that probability. A repeated count protocol is useful, but unchanged written instructions do not prove unchanged detection.
Open lessonLesson · Science · AP Environmental ScienceCompare agricultural yield with land and nutrient burdens
Connect Green Revolution practices, erosion mechanisms and conservation choices.
Open lessonLesson · Differential calculus: techniques and applicationsCompare all candidates before choosing a design
Use existence hypotheses, boundaries and contextual constraints in optimization.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeCompare almost uniform and nearly continuous behavior
Prove Egoroff’s theorem on a finite measure space.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsCompare angular rates with radius-dependent point motion
A planar rigid body rotating about one fixed axis shares angular velocity and angular acceleration across its points. That does not mean every point has the same linear speed. At perpendicular radius r, speed magnitude is r|ω|, tangential acceleration is rα and inward acceleration magnitude is rω².
Open lessonLesson · Science · Earth and space scienceCompare atmospheric models without overclaiming
Hold base pressure fixed while testing temperature effects and pressure-report definitions.
Open lessonExperiment · Money, economics and businessCompare benefits at the same date
Can a project repay its cost yet have negative discounted value?
Open experimentLesson · Science · AP Environmental ScienceCompare biodiversity protection across area, edges and connections
Evaluate habitat fragmentation and solution tradeoffs with explicit geometry.
Open lessonLesson · Science · AP Chemistry preparationCompare bond distances without hiding unmatched evidence
Make a signed numerical prediction, then audit whether the reference pair isolates the proposed explanation.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceCompare carbon stocks on fixed-depth and equivalent-mass bases
Soil carbon concentration is a mass fraction, not an areal stock. To estimate stock, multiply the relevant fine-earth mass per area by its carbon fraction. Layer thickness, bulk-density definition and coarse fragments determine that mass. Weathering, organic inputs, erosion and compaction can change different parts of this calculation.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsCompare categorical distributions across populations
Keep the same table arithmetic while changing the sampling question.
Open lessonLesson · Social studiesCompare causes without rerunning the past
Build a causal explanation that distinguishes conditions, triggers and rival explanations.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceCompare cell exchange geometry without mistaking area for flux
Use equal-volume shapes, uniform scaling and a corrugated surface to test conditional exchange arguments.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare cellular organization without identifying organisms by size alone
Microbes are not one evolutionary clade. Bacteria and Archaea are cellular lineages lacking a membrane-bound nucleus, while many eukaryotes are microscopic. The historical label protist groups diverse eukaryotes rather than one exclusive natural clade. Photosynthetic, ingestive and absorptive lifestyles occur in different microbial lineages.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCompare chains and antichains
Prove the LYM inequality by chain counting.
Open lessonLesson · Science · Earth and space scienceCompare climate indicators with their metadata
Preserve temperature and CO₂ units, geographic scopes, coverage and source uncertainty.
Open lessonLesson · Science · Earth and space scienceCompare climate windows and trend summaries
Distinguish endpoint slopes from fitted slopes while retaining the full annual context.
Open lessonLesson · Science · Environmental science and sustainabilityCompare competing uses of a fixed area
Keep food, irrigation, nutrient export and habitat indicators visible.
Open lessonLesson · Mathematical statistics: derivation and model criticismCompare constrained and unconstrained fits
Derive a likelihood-ratio statistic and mark the limits of Wilks calibration.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeCompare convergence modes using event bounds
Prove both Borel–Cantelli lemmas under stated hypotheses.
Open lessonLesson · Course 1 · What has the network learned?Compare convolution with image attention
Compare local filters with patch attention, then test which changes in image arrangement remain visible.
Open lessonLesson · Science · Physics and physical systemsCompare cooling and heating performance
Use coefficients of performance and temperature lift without confusing heat movement with energy creation.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceCompare cost efficiency without losing total-service constraints
An intervention comparison needs a shared service definition, duration and cost boundary. A low cost per unit of nominal improvement may not deliver enough total improvement, and nominal performance may not persist under actual conditions. Costs alone also omit ecological side effects and distributional consequences.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryCompare curvature through triangle area
Separate exact curvature identities from small-scale approximations.
Open lessonLesson · Mathematical modeling and senior investigationCompare decisions using feasible bounds
Turn a model into a justified optimization problem.
Open lessonLesson · Social studiesCompare development measures and competing explanations
Can higher average income coexist with worse measured health?
Open lessonLesson · Social studiesCompare development without ranking people
Development indicators summarize different aspects of social and economic life.
Open lessonLesson · Science · University Physics II with calculusCompare direct shell integration with Gauss symmetry
A spherical shell provides a useful independent benchmark because its field can be found both by a direct source integral and by symmetry with Gauss law. The original shell has radius R=1 m and total charge Q=2 nC. Observers lie on the positive z axis, strictly away from the charged surface. The model does not assign an on-surface self-field.
Open lessonLesson · Science · General Chemistry ICompare dispersion contacts without a mass-only shortcut
Separate polarizability, distance and contact multiplicity in a declared attractive term.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare diversity observations without equating absence of detection and absence
Biodiversity includes genetic differences, species composition, evolutionary history and ecosystem variation. A species list measures only part of that diversity. Taxonomic classification organizes names, but unequal sampling and changing species delimitations can alter reported richness without an equivalent biological gain or loss.
Open lessonLesson · Science · AP Environmental ScienceCompare energy demand, access and resource distribution
Distinguish renewable resources, accounting bases and per-person versus total demand.
Open lessonLesson · Science · AP Environmental ScienceCompare environmental health evidence without causal overclaiming
Calculate incidence proportions and connect chemical, biological and physical hazards to distinct prevention pathways.
Open lessonLesson · Science · Introductory MicrobiologyCompare environmental records without hiding interaction or confounding
Temperature, pH, water availability and oxygen can affect microbial processes through enzyme behavior, membrane properties, chemical gradients and damage or repair. A condition that permits persistence is not necessarily one that supports net multiplication. Broad labels such as psychrophile, mesophile, thermophile, acidophile or alkaliphile summarize comparative response patterns rather than providing a universal operating recipe.
Open lessonLesson · Mathematical statistics: derivation and model criticismCompare estimators through expected loss
Derive MSE and distinguish pointwise risk from universal preference.
Open lessonLesson · Science · University Physics II with calculusCompare exact dipole geometry with a far-field expansion
Two equal and opposite charges define a dipole moment p=2qd pointing from negative to positive charge when their positions are −d and +d on the x axis. Holding p fixed while varying d requires changing q. Equal moment therefore does not imply equal individual charges or identical near fields.
Open lessonLesson · Social studiesCompare executive accountability
In a parliamentary arrangement an executive generally depends on legislative confidence; in a presidential arrangement separately elected branches have distinct mandates and terms.
Open lessonLesson · Probability protocols and expected workCompare expected work in pooled factory inspection
Derive inspection costs from an explicit branching protocol, bound a rare-defect approximation, and change the sampling and recursion assumptions.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceCompare exponential and density-limited growth models
A population changes through births, deaths, immigration and emigration.
Open lessonLesson · Science · Earth and space scienceCompare exposed area after fragmentation
Use ideal cube geometry to separate exposed surface from conserved volume and reaction rate.
Open lessonLesson · Course 2 · Planning and reasoningCompare families of world models
Separate generated observations, latent predictions, action conditioning and policy learning.
Open lessonLesson · Science · Engineering design and scientific computingCompare feasible designs before ranking preferences
Use scenario-specific cost and energy comparisons without hiding failed constraints.
Open lessonLesson · Social studiesCompare fiscal multipliers and automatic responses
Do equal spending and tax changes cancel in the simplest model?
Open lessonLesson · Science · Physics and physical systemsCompare fission and fusion without confusing balance with feasibility
Compare selected fusion and fission reaction ledgers and distinguish conservation, energy release and rate.
Open lessonLesson · Statistical investigations, simulation and probability decisionsCompare fitted models and plotted residuals
Fit three candidate families, compare errors on the same response scale and inspect residual structure.
Open lessonLesson · Science · Physics and physical systemsCompare force pulses with the same impulse
Separate average force, peak force, duration and intermediate motion.
Open lessonLesson · Science · Earth and space scienceCompare fuel inventory with average power
Use reservoir and luminosity comparisons to interpret a conditional support duration.
Open lessonLesson · Science · AP Environmental ScienceCompare fuel pathways per useful energy delivered
Connect fuel formation and uses with efficiency, emissions and extraction impacts.
Open lessonLesson · Functions in graphs, measurements and growth decisionsCompare functions across representations
Read contextual signs, extrema and domain from supplied graphs.
Open lessonLesson · Science · Engineering design and scientific computingCompare gain without assuming bigger is better
Use nominal and offset cases to distinguish tracking, clipping and persistent error.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCompare genotype effects across environments without universal rankings
A phenotype reflects gene products acting in a developmental and environmental context. Many traits involve several loci and environmental influences rather than a single dominant/recessive category. An observed group difference cannot be assigned uniquely to genotype without knowing how environments, sampling and other causal factors compare.
Open lessonLesson · Science · AP Environmental ScienceCompare greenhouse gas sources without confusing stocks and metrics
Separate actual gas masses, time horizons and carbon-equivalent accounting.
Open lessonLesson · Science · Earth and space scienceCompare greenhouse parameter sensitivities
Distinguish surface, layer and effective emitting temperatures across prescribed equilibria.
Open lessonLesson · Growth estimates and controlled approximationCompare growth and distinguish absolute from relative error
Use eventual inequalities and ratio limits to justify O, Θ, little-o and asymptotic equivalence, including oscillation and cancellation boundaries.
Open lessonLesson · Rhetoric, media and public voiceCompare how messages direct attention
Analyze credibility, emotion and reasoning while checking what a message emphasizes or omits.
Open lessonLesson · Science · AP Environmental ScienceCompare hydroelectric and geothermal mechanisms
Connect hydraulic power and subsurface heat with site-specific ecological limits.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsCompare independent means without assuming equal variances
Use separate spread estimates and an appropriate t reference.
Open lessonLesson · Science · General Chemistry IICompare integrated laws without confusing units or depletion
The single-reactant irreversible models use disappearance rate k[A]^n at fixed volume.
Open lessonLesson · Science · University Physics I with calculusCompare integration rules with an exact displacement
Sampling and the weighting rule are both parts of a numerical integral.
Open lessonLesson · Science · Introductory AstronomyCompare internal production with surface heat loss
Accretion and differentiation can supply early heat, radioactive decay supplies continuing internal energy, and tidal deformation can dissipate orbital or rotational energy. Conduction, convection and volcanism redistribute or remove that heat. A planet’s present heat output need not equal its present radioactive production because stored energy can change.
Open lessonLesson · Science · Environmental science and sustainabilityCompare interventions beyond the local store
Keep downstream and product obligations visible when parent concentration declines.
Open lessonLesson · Abstract algebra I: groups and symmetryCompare irreducible actions with intertwiners
Prove Schur's lemma and distinguish its field and multiplicity hypotheses.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsCompare kinetic energy without losing the frame
Translational kinetic energy is K=½mv² for a nonrelativistic point-object model. It is a scalar and cannot be negative, because mass is positive and speed is squared. A negative velocity does not imply negative kinetic energy. The unit kg m²/s² is a joule, J.
Open lessonLesson · Social studiesCompare land-use choices and their constraints
Does the highest yield determine the best crop?
Open lessonLesson · Social studiesCompare language recognition with everyday use
Can an official-language law show how many families use a language at home?
Open lessonLesson · Science · Environmental science and sustainabilityCompare lifecycle burdens at matched service
Track fixed and operating contributions without hiding boundaries.
Open lessonLesson · Precalculus: local rates and polynomial evidenceCompare local change at two inputs
Use nearby interval slopes to compare change while keeping output sign, rate sign and rate trend distinct.
Open lessonLesson · Course 1 · How does a network learn?Compare manual and automatic gradients
Build a small computation graph, inspect each backward contribution and compare automatic gradients with manual formulas and finite differences.
Open lessonLesson · Science · Physics and physical systemsCompare matter wavelength with an observational scale
Calculate de Broglie wavelength from nonrelativistic kinetic energy and test when a wave description matters.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceCompare maximum load power with efficient transfer
Electromotive force is energy supplied per unit charge, measured in volts despite its name. It is not a mechanical force. The supplied source has fixed emf 9 V and series internal resistance r. With load R, current is 9/(r+R), and terminal voltage is 9−Ir=IR. The open-circuit limit approaches the emf because current and internal drop approach zero.
Open lessonLesson · Science · AP Environmental ScienceCompare meat-production systems using honest denominators
Connect feed accounting, manure, grazing and displaced impacts.
Open lessonLesson · Literary inquiry and interpretationCompare mechanisms, not just themes
Develop a comparative claim about how two works construct private experience and public interpretation through different formal choices.
Open lessonLesson · Science · Introductory MicrobiologyCompare microbial envelopes without reducing them to a stain label
A cytoplasmic membrane is a selective boundary, while an external wall or layer can contribute mechanical and chemical functions. Typical bacterial walls contain peptidoglycan; many bacteria with a Gram-negative envelope also have an outer membrane. Gram-positive and Gram-negative patterns summarize important differences but are not exhaustive categories for every organism or every observed staining result.
Open lessonLesson · Science · AP Chemistry preparationCompare molecular speed distributions and threshold fractions
Predict changes in a whole distribution and distinguish density, tail probability and three speed averages.
Open lessonLesson · Science · General Chemistry ICompare molecular speed without confusing it with bulk transport
Use temperature and molar mass to distinguish characteristic speed, energy and a conditional effusion ratio .
Open lessonLesson · Social studiesCompare money returns with purchasing power
Does a larger balance always buy more?
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsCompare muscle and nervous tissue without confusing fields and populations
Muscle tissue generates force through organized contractile machinery; nervous tissue receives, integrates and transmits signals. Both can be electrically excitable, so excitability alone does not identify one tissue class. Skeletal muscle typically has long striated multinucleate fibers, cardiac muscle has striated interconnected cells with central nuclei and specialized junctions, and smooth muscle has spindle-shaped cells without visible cross-striations. Involuntary regulation does not mean a tissue lacks electrical or chemical control.
Open lessonLesson · Science · AP Environmental ScienceCompare niche breadth and reproductive tradeoffs
Distinguish niche breadth from life-history traits and compare expected recruitment.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare nitrogen elimination without equating habitat with mechanism
Amino-acid metabolism creates nitrogen-containing products that must be managed. Animals differ in how much nitrogen they eliminate as ammonia, urea or uric-acid-related material. High water availability can support dilution and removal of ammonia; conversion to other forms can reduce water demand while requiring metabolic investment. Developmental stage and lineage constrain the available mechanisms.
Open lessonLesson · Science · Biology and living systemsCompare observational groups without inventing causation
Separate island and species composition and interpret unrepresented combinations.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceCompare observed richness at a common number of individuals
Observed species richness usually increases as more individuals are examined, so unequal effort can confound comparisons. Individual-based rarefaction asks how many species would be expected in a random smaller subsample from the observed pool. It standardizes one aspect of effort rather than estimating every species in the ecosystem.
Open lessonLesson · Science · Earth and space scienceCompare observed warming with forcing counterfactuals
Inspect archived human-plus-natural and natural-only simulations alongside compatible observations.
Open lessonLesson · Science · Environmental science and sustainabilityCompare occupancy and empty travel
Trace a conditional ranking reversal and audit historical ratio definitions.
Open lessonLesson · Science · Biology and living systemsCompare offspring counts with an inheritance model
Distinguish per-offspring probabilities from finite sample proportions.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare osmoregulatory organs through directional solute transfers
Osmoregulation controls water and solute balance, whereas excretion removes metabolic products; these tasks overlap but are not identical. Freshwater and marine environments impose different osmotic challenges, and some animals regulate internal concentration more strongly than others. An animal’s external habitat does not by itself reveal every internal transport mechanism.
Open lessonLesson · Mathematics for AI · M10Compare paired experiments and interactions
Keep task identity when comparing methods and enumerate a small bootstrap.
Open lessonLesson · Mathematical statistics: derivation and model criticismCompare parameter values for fixed data
Use likelihood ratios and distinguish observationally equivalent parameters.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsCompare passive spread with thresholded signal regeneration
A local membrane voltage change spreads through electrical properties of the membrane and cytoplasm, but passive spread diminishes with distance. An action potential is regenerated through voltage-dependent channel activity rather than being a single unchanged charge packet traveling intact along the axon. The distinction explains why a passive signal and a propagated spike require different models.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsCompare path integrals with endpoint energy
Integrate field components along each displacement and test path independence.
Open lessonLesson · Science · AP Environmental ScienceCompare phosphorus stocks turnover and limitation
Explain phosphorus reservoirs and distinguish balanced stocks from nutrient limitation evidence.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsCompare photon and matter wavelengths without conflating their energies
A diffraction pattern can reveal a wavelength for electrons as well as for light. For a photon, momentum magnitude is p=h/λ and energy is E=pc=hf. For a nonrelativistic electron, the same de Broglie wavelength relation λ=h/p applies, but kinetic energy is p²/(2m), not pc. Equal wavelength therefore means equal momentum magnitude, not equal kinetic energy. The particle and wave descriptions constrain different observations rather than describing a tiny classical ball carrying a literal transverse wiggle.
Open lessonLesson · Science · AP Environmental ScienceCompare pollutant-specific emission controls
Apply sequential removal while tracing captured wastes and residual emissions.
Open lessonLesson · Science · Earth and space scienceCompare possible stellar outcomes
Use a compact-object fact card and conditional pathways to examine remnant claims.
Open lessonLesson · Science · Earth and space scienceCompare prescribed planetary albedos
Separate fixed-temperature energy differences from equilibrium comparisons and feedback claims.
Open lessonLesson · Science · AP Environmental ScienceCompare prevention, recovery and disposal boundaries
Close a waste ledger while explaining landfill, combustion, recycling and composting tradeoffs.
Open lessonLesson · Social studiesCompare pricing rules without confusing revenue and welfare
Can a pricing change alter distribution even when quantity stays fixed?
Open lessonLesson · Science · Earth and space scienceCompare rainfall timing with water entry
Contrast equal-total storms under explicit entry and storage limits.
Open lessonLesson · Social studiesCompare real monthly and annual economic measures
Can a price index fall for one month yet remain higher over a year?
Open lessonLesson · Science · Earth and space scienceCompare redshift with independent distance
Analyze the complete historical observed panel before contrasting a controlled fictional sample.
Open lessonLesson · Social studiesCompare reform without assuming agreement
Can two religious reform arguments criticize abuse while disagreeing about authority?
Open lessonLesson · Social studiesCompare relative costs and the distribution of gains
Can trade create gains without making everyone better off?
Open lessonLesson · Science · Earth and space scienceCompare reported and calculated weather humidity
Use recorded temperature and dew point to reconstruct a humidity estimate while retaining the API-reported value separately.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare reproductive barriers using conditional denominators
Speciation involves divergence and barriers to gene exchange, but no single species definition works equally well for every organism and evidence type. Biological-species reasoning emphasizes reproductive compatibility in natural contexts; morphology, phylogenetic history and ecological differences supply other evidence, especially for fossils and asexual lineages.
Open lessonLesson · Science · Environmental science and sustainabilityCompare reservoir allocation policies
Hold inputs fixed while comparing requested and fulfilled allocations.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare respiratory architectures through local exchange gradients
Gas exchange requires a thin effective interface and maintained gradients. Gills exchange with water, lungs exchange with air through internal surfaces, and insect tracheal systems deliver gases through branching air pathways near tissues. Ventilation moves the external medium; perfusion moves blood or other transport fluid. These are not interchangeable with diffusion across an interface.
Open lessonLesson · Social studiesCompare revenue, cost and restricted output
Why might one seller withhold a beneficial unit?
Open lessonLesson · Course 2 · Testing whether improvement is realCompare robotic foundation models
Compare demonstration provenance, action representation, embodiment and reported generalization.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsCompare route-dependent heat with endpoint-dependent internal energy
Choose the sign convention before balancing thermal energy: heat into the gas is positive and work done by the gas is positive, so ΔU=Q−W. For a quasistatic constant-pressure expansion, W=pΔV; a fixed-volume segment has zero boundary work. On a pressure-volume diagram, the area under the path represents this boundary work.
Open lessonLesson · Social studiesCompare rural land patterns and the institutions behind them
Why can two farming landscapes have different parcel shapes?
Open lessonLesson · Social studiesCompare sacred landscapes across scale and time
Can a heritage description tell you every resident’s religious identity?
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCompare sequence-read hypotheses without calling a genotype by fraction alone
A sequence read is a measurement with orientation, alignment and error assumptions, not a direct declaration of an organism’s genotype. At a candidate position, alternate-supporting reads can reflect a genuine variant, sequencing error, mapping to a similar region, contamination or a mixed sample. The surrounding evidence and quality controls matter as much as the observed fraction.
Open lessonLesson · Statistical investigations, simulation and probability decisionsCompare shapes before choosing summaries
Use supplied observations and constructed displays to compare skew, center, spread and unusual values in context.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCompare sizes by pairing objects
Use bijections for finite counting and distinguish countable from uncountable sets.
Open lessonLesson · Science · AP Environmental ScienceCompare soil formation with erosion and horizon evidence
Explain soil-forming factors and audit a thickness budget without treating remaining soil as sustainable.
Open lessonLesson · Science · AP Environmental ScienceCompare solar and wind energy beyond nameplate capacity
Explain renewable conversion, annual generation and time-matched reliability.
Open lessonLesson · Science · General Chemistry IICompare solubilities only after reconstructing ion inventories
The solubility product and formula-unit solubility are not interchangeable quantities.
Open lessonLesson · Science · AP Chemistry preparationCompare solubility without confusing amount, rate and interactions
Infer bounded saturation evidence from mass records and test an interaction-based solvent comparison.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare species counts with represented evolutionary branches
Classification and diversity measures answer different questions. Domain, kingdom and other taxonomic ranks are naming conventions rather than equal units of evolutionary time or ecological function. A phylogenetic comparison should retain the actual branching hypothesis rather than treat every named group as equally distant.
Open lessonLesson · Science · General Chemistry IICompare splitting and pairing while preserving a degenerate boundary
An orbital energy model can constrain spin occupation without uniquely predicting observed color.
Open lessonLesson · Science · AP Environmental ScienceCompare succession pathways and disturbance legacies
Distinguish primary and secondary succession without assuming a predetermined final community.
Open lessonLesson · Science · Earth and space scienceCompare temperature and salinity contributions
Use a stated linear teaching rule to construct reinforcing and compensating density changes.
Open lessonLesson · Social studiesCompare territorial authority in India and France
Does a decentralized state necessarily have the same constitutional structure as a federation?
Open lessonLesson · Calculus II: series, curves and applicationsCompare the tail to a known benchmark
Choose an inequality in the useful direction and apply limit comparison to positive terms.
Open lessonExperiment · Environment and sustainabilityCompare the trip, not the empty seat
How do occupancy and whole-vehicle trips change an energy comparison?
Open experimentLesson · Science · AP Chemistry preparationCompare thermal and mechanical changes in equilibrium
Combine evaluated temperature-specific formation constants, predict signed extent and distinguish fixed-volume from fixed-pressure thermal responses.
Open lessonLesson · Science · Engineering design and scientific computingCompare thermal prototypes under matched conditions
Separate hardware effects from controller settings while retaining every requirement.
Open lessonLesson · Science · AP Environmental ScienceCompare timber accounting with forest sustainability
Explain clearcutting, selective harvest, regeneration and landscape evidence.
Open lessonLesson · Optimization: models, certificates and algorithmsCompare tradeoffs without inventing one universal score
Analyze domination, weighted sum s and unsupported efficient points.
Open lessonLesson · Social studiesCompare trait patterns with situation effects
Use repeated observations to distinguish stable differences from context-dependent behavior.
Open lessonLesson · Social studiesCompare transport costs with production networks
Is the cheapest factory location always nearest the market?
Open lessonLesson · Science · Biology and living systemsCompare tree hypotheses with character evidence
Score alternative branching patterns and recognize ties and conflicting evidence.
Open lessonLesson · Mathematics · Functions and graphsCompare two options at the right scale
Find where costs match and decide which option is cheaper for an allowed input.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceCompare two spectral fits with equal data budgets and a new observation
A pair of level indications can be fitted by more than one family. Compare a quadratic model y=A n²+B with an affine-in-n model y=C n+D. Both have two unknown parameters. Give both only the level labels n=1,2 and their indicated energies y1,y2. The generating coefficient and offset are revealed teaching controls, not independent inputs to either inverse fit.
Open lessonLesson · Statistical investigations, simulation and probability decisionsCompare uncertain strategies and consequences
Compare expected consequences while preserving uncertainty, thresholds and risk preferences.
Open lessonLesson · Social studiesCompare urban hierarchy with neighborhood models
Does one city-size pattern explain every city’s internal layout?
Open lessonLesson · Sections, solid measures and geometric designCompare volumes through complete section rules
Connect base times height to tiling, extrusion and Cavalieri comparison.
Open lessonLesson · Science · AP Environmental ScienceCompare watershed event volume without inventing a flood peak
Use drainage boundaries and area conversions to distinguish event runoff from peak flow.
Open lessonLesson · Course 2 · Testing whether improvement is realCompare world-model research approaches
Compare dated prediction targets and control mechanisms without ranking incompatible evaluations.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCompare xylem conduit geometry with loss of functional paths
Transpiration removes water at leaf surfaces, and cohesion within water columns can transmit tension through xylem. Adhesion, conduit structure, gravity and resistance affect the path. This mechanism is not a pump in each mature xylem cell; many conducting cells are dead at maturity.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceCompare zero torque with stable and unstable alignment
A loop normal is oriented by the right-hand convention around conventional current. Its magnetic moment is μ=NIA normal_vector, not an arrow lying in the loop plane. For angle θ between that normal and a uniform field, torque magnitude is NIABsinθ. Confusing a plane angle with a normal angle exchanges sine and cosine and can reverse which orientation is called maximum.
Open lessonLesson · Course 1 · How does AI process language?Compare, normalize, combine
Edit queries, keys, and values, and trace the weighted sum that lets tokens share context.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusComplete metric spaces and completion boundaries
Separate pairwise tail control from existence of a limit in the chosen space.
Open lessonLesson · Inference workflows: calculate, check and concludeComplete one- and two-proportion analyses
Keep interval and test standard errors separate while calculating reference probabilities and contextual conclusions.
Open lessonLesson · Inference workflows: calculate, check and concludeComplete one-mean, paired and Welch analyses
Choose the independent unit before calculating t probabilities and confidence intervals from summary statistics.
Open lessonLesson · Sequence spaces and completenessComplete sequence spaces and approximate by finite support
Construct norm limits from coordinates, control all tails, and identify exactly which sequences finite records approximate.
Open lessonLesson · Calculus synthesis: regions, motion and approximationComplete the square before integrating
Turn a quadratic denominator into a scaled inverse-trigonometric integral and verify the result.
Open lessonLesson · Differential geometry of curves and surfacesCompleteness and global shortest paths
Separate local geodesic existence from global continuation.
Open lessonAssessment · Complex analysis: holomorphic functions, contours and mappingsComplex analysis: domains, contours and proof obligations
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitComplex analysis: holomorphic functions, contours and mappings
Proof-centered lessons on complex differentiability, contour theorems, analytic and Laurent series, residues, root counts, conformal maps and harmonic boundary values. Written arguments require mathematical review. The finite monodromy proof includes stable analytic-element chains and compact homotopy strips. General homological Cauchy remains a stated theorem dependency; the Riemann mapping theorem, Picard theorems and Riemann surfaces remain further study. Guided work is not independently assessed course equivalence.
Open courseLesson · Complex analysis: holomorphic functions, contours and mappingsComplex exponential and trigonometric functions
Derive the exponential addition law from absolutely convergent series.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsComplex geometry and modulus
Connect multiplication with rotation and scaling.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsComplex line integrals and estimates
Compute contour integrals through piecewise smooth parametrizations.
Open lessonLesson · Real skew-adjoint geometry and complex structuresComplex structures and compatible metrics
Turn a real square root of minus the identity into complex scalar multiplication, identify the compatible geometry, and normalize skew rotation blocks without hiding the kernel.
Open lessonAssessment · Relation closure and directed reachabilityCompose the rules and prove the reachability claim
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Course 1 · Generative models beyond textCompress and reconstruct
Train a tiny shape autoencoder, inspect its bottleneck and test why reconstruction does not guarantee useful generation.
Open lessonLesson · Mathematical statistics: derivation and model criticismCompress data without losing model information
Use conditional distributions and dominated factorization.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsCompute a correspondence and test its boundaries
Compute biquadratic and complete S₃ correspondences, distinguishing fixed fields from normal extensions.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCompute a gcd while preserving its meaning
Use remainder reduction and back-substitution to construct integer certificates.
Open lessonLesson · Science · Physical Geology and Earth SystemsCompute a source moment without predicting local shaking
Scalar seismic moment combines rigidity, ruptured fault area and mean displacement: M0=μAD. It captures source deformation on a fault. Moment magnitude maps this large dimensional quantity onto a logarithmic scale; it is not obtained by averaging arbitrary local shaking intensities.
Open lessonLesson · Abstract algebra I: groups and symmetryCompute compositions in a declared order
Use disjoint cycles, inverses and orders.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsCompute expectations without deriving every transformed law
Use sums, integrals and integrability conditions.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsCompute in an eigenvector basis when one exists
Distinguish diagonalizability from merely having characteristic roots.
Open lessonLesson · Calculus II: series, curves and applicationsCompute length when radius and direction both change
Derive polar speed from Cartesian derivatives and handle zero-radius points.
Open lessonLesson · Mathematics for AI · M05Compute probabilities without overflow
Use exponential ratios, logarithms, cross entropy and stable arithmetic.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsCompute square roots with a shrinking order invariant
Derive the shortcut for primes congruent to three modulo four.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceCompute state entropy without inventing an actual reversible process
Entropy is a state function defined for equilibrium states. Our ideal-gas packet has total Cv=12 J/K and nR=8 J/K, initially 300 K and 1 litre. Only the final temperature and volume are specified. A reversible reference path is a mathematical way to evaluate the state difference; it is not a description of an unknown actual history.
Open lessonLesson · Homotopy, fundamental groups and covering spacesCompute the fundamental group of the circle
Associate an integer displacement to every based circle loop.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceCondition a genotype inference on how the offspring was selected
A Punnett square gives probabilities conditional on parental genotypes and a specified inheritance model. In an Aa×Aa cross, the prior genotype probabilities are 1/4 AA, 1/2 Aa and 1/4 aa under equal gamete transmission and viability. If only phenotype-negative offspring are selected for further reasoning, the relevant denominator changes.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsCondition on a coordinate using a density ratio
Normalize each slice and avoid a zero-probability event ratio.
Open lessonLesson · Probability protocols and expected workCondition on the observation protocol, not just its words
Count hands and reveal outcomes separately, derive size-weighted posteriors, and track how a reporting policy changes the evidence.
Open lessonConcept · GlossaryConditional computation
Activating different parts of a computation for different inputs, such as routing tokens to selected experts. Inactive parameters can still occupy storage.
Open conceptConcept · GlossaryConditional independence
After a specified condition is known, learning one variable supplies no further information about the other. Conditional independence does not generally imply independence in the mixed population.
Open conceptLesson · Calculus-based probability: laws, dependence and limitsConditional laws and joint independence
Separate conditioning, disjointness and independence of collections.
Open lessonLesson · Social studiesConfederation decisions: votes, consent and resources
Test state voting thresholds and separate approval from money actually supplied.
Open lessonLesson · Social studiesConflicting reports may measure different things
Corroboration compares evidence with attention to timing, definitions, access and independence.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsConformal maps and local geometry
Prove angle preservation from a nonzero complex derivative.
Open lessonConcept · GlossaryConfounding
A common cause influences both an exposure or assignment and its outcome, so an observed association may differ from the effect of intervening. Adjustment requires appropriate causal assumptions, measurements and overlap.
Open conceptConcept · GlossaryConfusion matrix
A table that separates predictions by their observed and predicted classes, revealing different kinds of classification mistakes.
Open conceptLesson · Axiomatic Euclidean and non-Euclidean geometryCongruence axioms and SAS
Separate equality of measurements from geometric transport.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryConics as homogeneous quadratic geometry
Use quadratic forms to explain projective conic transformations.
Open lessonLesson · Science · Biology and living systemsConnect a coding sequence to a translation outcome
Trace a DNA coding triplet to mRNA and its standard-code meaning.
Open lessonLesson · Social studiesConnect a firm’s cost table to a price-taking decision
Can a firm produce while making a total loss?
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceConnect a kinetic-energy inference to an allowed diffraction direction
De Broglie wavelength is lambda=h/p, where momentum must match the kinetic-energy regime. For an electron with kinetic energy K, total energy is K+mc². Relativistic energy-momentum conservation gives pc=sqrt[K(K+2mc²)], whereas the nonrelativistic approximation is pc≈sqrt(2mc²K). The omitted K² term is a controlled approximation, not a separate particle identity.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceConnect a local conduction law to a global entropy balance
A steady slab has conductivity k=2 W/(m K), area 0.5 m² and length L, with hot boundary Th and cold boundary 300 K. There is no volumetric heating, material motion or time-dependent storage. Under the supplied constant-conductivity/local-equilibrium assumptions, T(x)=Th+(300−Th)x/L.
Open lessonLesson · Science · Chemistry and matterConnect a measured mass to a particle amount
Compare two samples using mass, moles and explicitly named entities.
Open lessonLesson · Social studiesConnect a seed innovation with conditions of adoption
Does a high-yielding variety alone explain every farm’s outcome?
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConnect a shared rotation to different point motions
Choose two marks on a rigid rotating surface at different distances from a fixed axis. Each mark completes a turn in the same time, so both have the same angular velocity. Their arc lengths differ: s=rθ for angle in radians. Dividing by time gives tangential speed v=r|ω|. A larger radius gives more distance per turn, not more turns per second.
Open lessonLesson · Science · Physics and physical systemsConnect acceleration, velocity and position
Use slopes and signed areas to link three descriptions of motion.
Open lessonLesson · Science · Earth and space scienceConnect air pressure to the mass above
Interpret pressure, density and column mass without confusing their units.
Open lessonLesson · Course 2 · Assemble an experimental agentConnect an agent’s decisions, actions, and learning
Trace a proposal through a controller, checked execution, episode memory, and a separately updated world model.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceConnect an electron-donor inventory to oxygen reduction
Reduced carriers transfer electrons through respiratory components toward a terminal acceptor. In aerobic respiration, molecular oxygen is reduced to water at the end of the chain. Oxygen is not the source of the carbon in respiratory CO2; that carbon comes from oxidized organic material.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsConnect an ideal-gas state to a molecular energy distribution
The ideal gas law pV=nRT connects equilibrium pressure, volume, amount and absolute temperature. Pressure must be absolute, volume must use compatible units and T must be in kelvins. The model neglects molecular volume and interactions except during idealized collisions; it is not a universal equation for dense gas or a liquid.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceConnect an image plane to a separately specified photon readout
A sensing chain can combine correct component formulas into an incorrect overall conclusion if the connections between them are not justified. Here a thin converging lens with focal length10 cm images an object20–40 cm before it. The lens relation determines a real image plane and signed magnification. It does not determine how much optical power reaches the detector: source radiance, aperture, loss and geometry would be needed. We therefore place the virtual detector at the calculated plane and independently stipulate illumination2 nW.
Open lessonAssessment · Arrival processes, absorption and honest simulationConnect arrivals, route outcomes and incomplete evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesConnect biological mechanisms with evidence
Neurons communicate through electrical activity and chemical signaling; nervous and endocrine systems coordinate behavior on different timescales.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConnect bone architecture to a fixed-material geometric comparison
Bone is living connective tissue, not merely mineral scaffolding. Collagen-rich organic matrix and mineral contribute different material properties; cells maintain and remodel the tissue. Compact cortex and trabecular networks organize that material at larger scales. Marrow spaces, vessels and nerves have functions that are absent from a simple strength analogy.
Open lessonLesson · Social studiesConnect capital flows with a currency-market graph
How can higher demand for domestic assets change the exchange rate?
Open lessonLesson · Science · Biology and living systemsConnect cell structures to the work they do
Distinguish a size observation from evidence about organelles and function.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceConnect cellular geometry to exchange constraints
A cell must exchange material across membranes while supporting processes throughout its volume. For a sphere, area is4πr² and volume is4πr³/3. Dividing gives area/volume=3/r. Doubling radius therefore halves this geometric ratio even though both area and volume increase.
Open lessonLesson · Science · Earth and space scienceConnect circular speed, period and position
Calculate a circular orbit’s period and distinguish accumulated angle from wrapped orientation.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConnect circulatory architecture to oxygen content and extraction
Open circulatory systems distribute hemolymph through body spaces; closed systems retain blood within vessels. Insects commonly rely on tracheae rather than blood as the principal oxygen-delivery route. Vertebrate single- and double-circuit arrangements differ in how respiratory and systemic paths connect. These comparisons describe transport architecture, not a universal ranking of animal performance.
Open lessonLesson · Science · Physics and physical systemsConnect compression, displacement and sound pressure
Use neighboring displacements to explain pressure changes and compare their phase with parcel motion.
Open lessonLesson · Science · Physics and physical systemsConnect current, voltage and resistance
Use charge flow and a voltage sweep to interpret an ohmic circuit.
Open lessonExperiment · Programming and software engineeringConnect data by identity
Why does joining rows by position send orders to the wrong person?
Open experimentLesson · Measure, Lebesgue integration and the probability bridgeConnect distributional limits to integral transforms
Use bounded continuous test functions to define weak convergence.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceConnect disturbance, nutrient cycles and evidence-based intervention
Human activities can change habitats, connectivity, species introductions and biogeochemical cycles.
Open lessonLesson · Mathematics for AI · M09Connect dynamics estimates and constraints
Step a feedback system and separate numerical accuracy from stability and measurement assumptions.
Open lessonLesson · Science · Physical Geology and Earth SystemsConnect elastic moduli to P and S wave support
An isotropic elastic medium resists volume change through its bulk modulus K and shape change through its shear modulus μ. Small-amplitude wave speeds obey Vp²=(K+4μ/3)/ρ and Vs²=μ/ρ. Modulus in pascals divided by density in kg/m³ produces squared speed in m²/s².
Open lessonLesson · Science · Chemistry and matterConnect electron arrangements to atomic structure
Read neutral-atom shell and subshell counts without mistaking them for electron paths.
Open lessonLesson · Science · Earth and space scienceConnect energy imbalance to temperature change
Use heat capacity to distinguish energy flux, stored energy and a temperature-change rate.
Open lessonLesson · Science · AP Environmental ScienceConnect eutrophication to a bounded oxygen budget
Explain nutrient enrichment, microbial respiration and oxygen constraints.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConnect extracellular matrix architecture to directional support
Connective tissues contain cells within an extracellular matrix whose ground substance and fibers strongly shape function. Fibroblasts maintain many connective matrices; adipocytes store lipid; cartilage and bone have specialized cells and matrices; blood has a fluid matrix. These are not simply gaps between more important tissues. Matrix composition, hydration, mineralization and organization can support, cushion, transport or transmit forces in different contexts.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConnect fiber direction and tendon leverage without inventing a safe load
Muscles transmit force through connective attachments. An agonist contributes to a specified action, an antagonist opposes it, and synergists or fixators assist or stabilize; these roles depend on the action rather than being permanent labels. Origins and insertions are useful anatomical descriptions, but which end is mechanically fixed depends on the task.
Open lessonLesson · Social studiesConnect firm profits with market entry
Why can short-run profit disappear after entry?
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsConnect force direction with constant-speed magnetic curvature
A moving charge in a magnetic field experiences magnitude F=|q|vB sinθ, where θ is the angle between velocity and field. The force is perpendicular to both directions. Use the right-hand orientation for positive charge and reverse it for negative charge. A stationary charge or a charge moving exactly parallel to the field has no magnetic force from that field alone.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceConnect functional groups to a biological atom ledger
Carbon-containing biological molecules are defined by connectivity and chemical context, not just a list of elements. Carbon commonly forms four covalent bonds; single and multiple bonds constrain geometry. A molecular formula reports atom counts but cannot distinguish every structural isomer, linkage orientation or three-dimensional arrangement.
Open lessonLesson · Algebra II and precalculus: functions and structureConnect functions through intermediate values
Track the order, units and domain of a two-stage calculation.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConnect fungal filament geometry to absorptive nutrition
Fungi are eukaryotic heterotrophs, not non-green plants. Many form branching hyphae whose network is a mycelium; yeasts represent unicellular forms. Fungal walls commonly contain chitin, and nutrition often involves extracellular breakdown followed by absorption of soluble products. The visible fruiting structure is not necessarily most of the organism’s body.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceConnect heritable variation to relative reproductive success
Natural selection requires variation among individuals, a heritable component of that variation, and differences in reproductive contribution associated with it.
Open lessonLesson · Science · Physics and physical systemsConnect impulse to signed momentum
Use force-time area to predict stopping and reversal without confusing momentum with energy.
Open lessonLesson · Course 1 · How does AI process language?Connect instruction tuning, preferences and verified rewards
Compare actual supervised, DPO-style and exact expected-reward updates on a finite response set; expose a weak verifier and inspect a fixed-reference penalty.
Open lessonLesson · Science · AP Chemistry preparationConnect ionic ordering to many-particle interactions
Audit a neutral three-dimensional patch, all pair energies and a constrained layer displacement.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceConnect lipid architecture to organization in water
Distinguish ester-linked storage lipids, amphipathic phospholipids and steroid structures.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConnect local height energy to inverse-square gravity
Potential energy belongs to a system of interacting objects and its configuration, not to an isolated object without its interaction partner. For two nonoverlapping spherical masses, U=−GMm/r when zero is chosen at infinite separation. Finite-separation values are negative because positive work is needed to separate the attractive pair to infinity quasistatically.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceConnect meiotic mechanisms to chromosome combinations
Sexual inheritance links chromosome segregation with the union of haploid complements. Homologous chromosomes carry corresponding loci but may carry different alleles. Sister chromatids arise by replication of one chromosome; they are not the two homologs. Confusing these relationships makes it difficult to explain why meiosis reduces chromosome-set number while mitosis ordinarily preserves it.
Open lessonLesson · Science · Chemistry and matterConnect metallic bonding to charge transport
Explain mobile electrons and distinguish a material property from a wire property.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceConnect molecular change to organism and population evidence
A cross-scale explanation needs justified links: a sequence difference can affect expression or protein function, which may alter a phenotype and reproductive contribution in a particular environment.
Open lessonLesson · Science · Physics and physical systemsConnect molecular motion to gas pressure
Use collision reasoning and statistical averages to explain an ideal-gas state.
Open lessonLesson · Science · Chemistry and matterConnect molecular shape to bond-polarity cancellation
Separate electron-region geometry from molecular shape and add equal bond vectors with a fixed drawing scale.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConnect muscle mechanism, support architecture and mechanical tradeoffs
Animals support movement using different structures. A fluid-supported hydrostatic skeleton couples muscle action to a constrained body volume; an exoskeleton provides an external framework with growth constraints; an endoskeleton provides internal support and joint attachments. None eliminates the need to transmit forces and coordinate antagonistic actions.
Open lessonLesson · Science · Physics and physical systemsConnect optical path difference to bright and dark fringes
Predict two-slit interference from phase and compare amplitude with intensity.
Open lessonLesson · Science · Earth and space scienceConnect orbital nodes to three-dimensional alignment
Separate moving along an inclined orbit from rotating its plane and measure centerline offset.
Open lessonLesson · Science · AP Environmental ScienceConnect ore grade, recovery and mining residuals
Relate material demand to mining methods, drainage and reclamation limits.
Open lessonLesson · Science · Physics and physical systemsConnect pendulum length, gravity and period
Use a geometric and force model to predict pendulum timing and bottom speed.
Open lessonLesson · Social studiesConnect personal experiences to institutions
Sociological explanations connect individual experiences with patterned social arrangements.
Open lessonLesson · Science · AP Chemistry preparationConnect photoelectron peaks to electron structure
Infer idealized subshell occupancy and separate binding energy from emitted-electron kinetic energy.
Open lessonLesson · Science · Physics and physical systemsConnect pipe geometry to viscous flow
Compare pressure-driven flows and distinguish radius, area and resistance.
Open lessonLesson · Science · AP Environmental ScienceConnect plate motion to boundary evidence
Distinguish divergent, convergent and transform mechanisms without predicting an earthquake date.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsConnect polynomial factors with multiplicative orders
Construct cyclotomic polynomials, prove irreducibility and identify their field automorphisms.
Open lessonLesson · Linear Algebra II: operators and spectral structureConnect positive energy with spectral square roots
Classify Hermitian forms, construct positive square roots, and distinguish congruence from similarity.
Open lessonLesson · Abstract algebra I: groups and symmetryConnect powers, divisibility and generators
Classify subgroups of a cyclic group and compute element orders.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConnect projectile components through one clock
With uniform gravity and negligible air resistance, horizontal acceleration is zero while vertical acceleration is −g for an upward-positive axis. This separates the component equations: x=vₓ₀t and y=vᵧ₀t−0.5gt² for an origin at launch. The components share one elapsed time; they are not two separate trips.
Open lessonLesson · Science · General Chemistry IIConnect proton transfer, conjugate stability and Lewis roles
A proton-transfer prediction requires matching conjugate pairs and compatible equilibrium conventions.
Open lessonLesson · Science · Chemistry and matterConnect proton transfer, water ions and pH
Interpret pH as a logarithmic measure while retaining water ions and charge balance.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsConnect radical formulas to solvable Galois groups
Prove both directions using cyclic resolvents and solvable Galois envelopes.
Open lessonLesson · Science · Introductory AstronomyConnect redshift to expansion ratios without a speed shortcut
Cosmological redshift is the ratio of observed to emitted wavelength minus one. In a homogeneous expanding background with present scale factor one, 1+z=1/a_emit. The scale factor relates large-scale comoving separations; it does not imply that every atom, star or bound galaxy expands with the same rule.
Open lessonLesson · Science · Physics and physical systemsConnect refraction to wave speed and wavelength
Compare transmitted rays and wave properties as refractive index changes.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConnect reproductive stages, development and conditional offspring survival
Animals can reproduce sexually and, in some lineages, asexually through budding, fission or other processes. Sexual reproduction combines gametes; asexual reproduction can still accumulate mutations and does not guarantee indefinite genetic identity. Internal versus external fertilization and egg retention versus live birth are additional distinctions, not synonyms for sexual versus asexual reproduction.
Open lessonLesson · Euclidean and coordinate geometry foundationsConnect right angles with squared lengths
Prove the Pythagorean relation and use its converse without assuming the desired right angle.
Open lessonLesson · Science · Biology and living systemsConnect roots, leaves and a water budget
Follow water uptake, transpiration and changing stores without inventing a negative amount.
Open lessonLesson · Mathematical statistics: derivation and model criticismConnect sample moments to population parameters
Use convergence arguments and identify weak points in moment matching.
Open lessonLesson · Science · AP Environmental ScienceConnect seasonal sunlight and mountain-climate mechanisms
Explain solar geometry, elevation and geographic climate controls without overextending a dry-parcel model.
Open lessonLesson · Science · Physical Geology and Earth SystemsConnect sedimentary texture to compaction and cementation
Clastic sedimentary rocks contain transported fragments. Gravel-sized clasts exceed about 2 mm; sand lies roughly between 1/16 and 2 mm, while finer particles include silt and clay-sized material. Size is not composition: clay-sized particles and clay minerals are related but distinct categories. Sorting and rounding reflect transport and source history, not a unique environment by themselves.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConnect sensory integration to an all-or-none signaling event
Sensory receptors transduce a physical or chemical input into cellular signaling. Neural pathways integrate such signals, while motor and autonomic outputs coordinate effectors. Distributed nerve nets and centralized nervous systems organize information differently; a threshold calculation is not a complete account of behavior, learning or consciousness.
Open lessonLesson · Science · Physical Geology and Earth SystemsConnect silicate structure to oxygen and charge accounting
A silicon–oxygen tetrahedron has four oxygen corners around one silicon. Minerals differ in how these tetrahedra connect and which cations balance their charges. Isolated units occur in olivine; chains in pyroxenes and amphiboles; sheets in micas and clays; frameworks in quartz and feldspars. These arrangements help explain cleavage and other properties, but are not merely decorative geometric labels.
Open lessonLesson · Science · Environmental science and sustainabilityConnect soil functions with food and water
Design a fair comparison and distinguish soil mechanisms from predicted yield.
Open lessonLesson · Science · Physics and physical systemsConnect string tension and mass density to wave speed
Test square-root tension scaling and distinguish wave speed from a string mode frequency.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConnect sugar loading to pressure-driven source–sink transport
Phloem transports organic solutes among sources and sinks. A mature photosynthetic leaf may be a source, while growing roots, fruits or storage tissues may be sinks. A storage organ can later become a source. These roles describe net loading and unloading, not permanent top and bottom positions.
Open lessonLesson · Science · Earth and space scienceConnect surface emission to total stellar power
Use temperature and emitting area to distinguish surface flux from luminosity.
Open lessonLesson · Science · Earth and space scienceConnect synthesis with released material
Use Cas A evidence to distinguish nuclear production, transport and ejection.
Open lessonLesson · Science · AP Environmental ScienceConnect temperature inversions to a mixing-volume constraint
Explain atmospheric stability without treating trapped pollution as newly created mass.
Open lessonLesson · Science · AP Environmental ScienceConnect the greenhouse mechanism to an energy imbalance
Distinguish reflected sunlight, outgoing infrared radiation and stored energy.
Open lessonLesson · Science · Physics and physical systemsConnect torque to force and lever arm
Change a force’s application point and compare its turning effect about a pivot.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConnect torque-time evidence to angular-momentum change
External torque transfers angular momentum to or from a chosen system. For a constant torque, angular impulse is τΔt. For a variable pulse it is the signed torque-time area. The angular impulse equals ΔL, with units N m s, equivalent to kg m²/s. This equality is different from the torque-angle work relation.
Open lessonLesson · Science · Physics and physical systemsConnect torque, rotational work and power
Account for energy during a constant-torque acceleration without confusing torque with power.
Open lessonLesson · Science · General Chemistry IIConnect two phase shifts to one effective particle account
In a dilute ideal approximation, several solvent properties respond to the concentration of dissolved species rather than their detailed identity.
Open lessonLesson · Science · AP Chemistry preparationConnect valence patterns to a complete energy argument
Predict analogous chloride ratios, then test why making a closed-shell ion is not a complete reaction explanation.
Open lessonLesson · Optimization: models, certificates and algorithmsConnect vertices with independent columns
Prove the standard-form vertex criterion.
Open lessonLesson · Course 2 · Geometry and control foundationsConnect vision language and action
Trace finite instruction features through a learned action policy.
Open lessonLesson · Science · Chemistry and matterConnect voltage, charge and reaction free energy
Compare energy per charge with standard free energy per mole of a written reaction.
Open lessonLesson · Science · Physics and physical systemsConnect voltage, interaction energy and electric work
Compare potential with energy and follow the sign of work for both charge signs.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceConnect water’s molecular interactions to biological evidence
Distinguish covalent bonds, hydrogen bonds, partial charge and selective solvation.
Open lessonLesson · Science · Earth and space scienceConnect winds, boundaries and surface currents
Separate moving water from wind and wave-pattern motion while tracing momentum transfer.
Open lessonLesson · Science · Physics and physical systemsConnect work to kinetic energy
Read signed force-position area and calculate a change in motion energy.
Open lessonLesson · Point-set topology: spaces, constructions and separationConnectedness and components
Prove connectedness is preserved by continuous images and closure.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusConnectedness and paths
Use separations and inverse images to prove global intermediate-value behavior.
Open lessonModule · Course 2Connecting tools and specialists
Give different components a useful way to communicate.
Open moduleLesson · Social studiesConquest and alliance: whose account?
Compare distinct Indigenous perspectives and investigate the dates, purposes and limits of conquest accounts.
Open lessonLesson · Social studiesConsent and reflexivity change the research
Fieldwork involves relationships, not simply collecting facts.
Open lessonLesson · Social studiesConsent, participation and legitimacy
Legitimacy concerns beliefs that authority is rightful.
Open lessonLesson · Network incidence and the four spacesConservation certificates and potentials
Use component balances to certify feasibility, then select a unique minimum-norm signed flow through vertex potentials.
Open lessonLesson · Science · General Chemistry IIConserve a ligand before claiming quantitative complexation
A large formation constant cannot supply ligand that is not present.
Open lessonLesson · Science · Physics and physical systemsConserve angular momentum when radius changes
Predict spin from mass distribution and the external-torque condition.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingConserve angular momentum while inertia changes
Distinguish an isolated angular-momentum budget from a kinetic-energy budget.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConserve angular momentum without assuming constant kinetic energy
For a system about the specified inertial reference, negligible net external torque makes total angular momentum constant. Choose the system large enough to include the internal exchange being studied. The torque one part exerts on another can change each part’s angular momentum while the opposite internal exchange cancels in the total.
Open lessonLesson · Science · Introductory MicrobiologyConserve carbon and reducing equivalents across supplied pathways
Microbial metabolism couples energy transfer to chemical transformations. Respiration uses an electron-transport system and a terminal acceptor, while fermentation regenerates carrier balance through organic product formation without that respiratory chain. Anaerobic respiration is not the same as fermentation: it can use terminal acceptors other than oxygen. Light or chemical reactions can supply energy, and organic or inorganic sources can supply carbon or electrons; these classifications describe different axes.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceConserve electron equivalents without inventing an ATP yield
Respiration transfers electrons through coupled reactions to a terminal acceptor. Oxygen is one possible acceptor, but some microbial pathways use others. Fermentation is distinct: it regenerates redox carriers using internally derived organic products rather than the respiratory arrangement with an external terminal acceptor.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceConserve four-momentum rather than a sum of rest masses
The original isolated collision joins a moving unit-rest-mass body to a stationary target of mass m2. All products remain in one composite, with no escaping radiation. Energy and momentum must be added in a common frame before the final invariant mass is computed.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConserve mass flow without inventing a branch split
In timeΔt, fluid moving at mean speedv through cross-sectional areaA transports volumeAvΔt. Its volume flow rate isQ=Av and mass flow rate isρAv. A steady control volume has no accumulating mass, so incoming mass flow equals total outgoing mass flow. For the same incompressible fluid, density cancels and the volume flows balance.
Open lessonLesson · Science · Physics and physical systemsConserve mass through a changing pipe area
Distinguish fluid speed from volume flow and use continuity to compare sections.
Open lessonLesson · Science · Physics and physical systemsConserve momentum across a two-object collision
Track equal-and-opposite impulses while keeping both objects inside the system.
Open lessonLesson · Science · Physics and physical systemsConserve nucleon number and charge in nuclear change
Complete nuclear reaction equations while distinguishing necessary conservation from energetic possibility.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConserve oxygen through unequal parallel lung pathways
Ventilation supplies gas to exchange regions and perfusion supplies blood. A region with little ventilation relative to its blood flow cannot be treated as equivalent to a well-ventilated region merely because total ventilation elsewhere is large. Conversely, ventilating a poorly perfused region does not automatically deliver its oxygen to the rest of the circulation.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingConserve planar momentum and test an inferred external impulse
Keep vector components through a sticking collision before interpreting speed or energy.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConserve tissue fluid while separating filtration and lymph return
Capillary walls exchange gases and dissolved substances through pathways that depend on endothelial structure, molecular properties and local conditions. Diffusion of a substance and bulk movement of water are not the same process. Continuous, fenestrated and discontinuous capillaries differ structurally, so a single transport coefficient cannot represent every organ.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConserve total energy when mechanical energy decreases
Mechanical energy is the sum of the kinetic and potential energies tracked in a mechanical model. It need not be conserved when friction or other dissipative interactions convert it into internal or thermal energy. Total energy conservation is broader: energy does not disappear merely because the mechanical subtotal falls.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConserve total hormone while separating carrier binding and target occupancy
Endocrine signals travel through circulation, but target responses depend on receptor expression and downstream machinery. Peptide hormones commonly bind membrane receptors, whereas many lipid-soluble hormones bind intracellular receptors; thyroid hormones are amines with intracellular signaling behavior. Hormone chemical class, transport, receptor location and response time are related but not interchangeable labels.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsConserve water and particles across a cellular boundary
Intracellular fluid lies inside cell membranes; extracellular fluid includes interstitial fluid and plasma. Plasma is not the same as whole blood: blood cells contain intracellular fluid. Capillary walls separate plasma from interstitial spaces, while cell membranes separate intracellular from extracellular compartments. A substance can cross one boundary readily but not another, so a diagram must name both the compartment and the barrier.
Open lessonConcept · GlossaryConsolidation
Transforming selected experience into more durable or reusable knowledge or skills while preserving scope and checking counterexamples.
Open conceptLesson · Social studiesConsolidation: sleep timing and memory after learning
Distinguish post-learning stabilization from initial encoding and retrieval alertness using a timed design.
Open lessonLesson · Abstract algebra I: groups and symmetryConstrain prime-power subgroups and prove the counting mechanism
Use existence, conjugacy and number congruences with their hypotheses.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationConstrained extrema and Lagrange multipliers
Find regular constrained candidates and inspect exceptional points and global bounds.
Open lessonConcept · GlossaryConstraint satisfaction
Choose one value for each variable from its allowed domain so every stated relationship holds. Feasibility alone does not optimize preferences or prove the requirements describe reality.
Open conceptLesson · Science · AP Chemistry preparationConstruct a catalytic surface without double-counting its sites
Build a bound-reactant layer, propose a neighboring pair and check regenerated sites, blocked occupancy and overlapping reaction opportunities.
Open lessonLesson · Science · AP Chemistry preparationConstruct a composition that satisfies both equilibrium and conservation
Build independent A and B concentrations, distinguish reaction events from species changes, and reject a rate-balanced mixture with the wrong conserved inventory.
Open lessonLesson · Social studiesConstruct a continuous entry equilibrium
Can zero economic profit coexist with price above marginal cost?
Open lessonLesson · Science · AP Chemistry preparationConstruct a Hess pathway without losing species or physical states
Choose signed reaction multipliers, inspect cancellation and compare graphite with atomic gas carbon before assigning a target enthalpy.
Open lessonLesson · Science · AP Chemistry preparationConstruct a material representation without overclaiming evidence
Build individual substitution sites, preserve composition while changing arrangement, and identify what empirical properties cannot determine.
Open lessonLesson · Encodings, diagonal arguments and limits of algorithmsConstruct a missing sequence and check its domain
Prove a diagonal obstruction for infinite binary sequences, transfer it to subsets and languages, and diagnose failed domain and computability arguments.
Open lessonLesson · Science · AP Chemistry preparationConstruct a net ionic change without deleting matter
Cancel identical species portions, retain the complete inventory and independently check elements, charge and state.
Open lessonLesson · Euclidean and coordinate geometry foundationsConstruct a point and justify why it works
Separate a permitted construction procedure from its mathematical correctness.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsConstruct a similarity and derive AA
Construct a similarity as motions plus a positive dilation, then derive AA by ray intersection.
Open lessonLesson · Science · General Chemistry IIConstruct a standard free-energy comparison with explicit states
Standard reaction entropies come from a stoichiometric property ledger, not a visual disorder slogan.
Open lessonLesson · Science · AP Chemistry preparationConstruct a standard reaction entropy ledger
Calculate reaction entropy using balanced coefficients, direction and extent.
Open lessonLesson · Science · General Chemistry IConstruct allowed quantum labels and node counts
Use label constraints to count spatial orbitals and distinguish orbital identity from spin occupancy.
Open lessonLesson · Science · AP Chemistry preparationConstruct an acid or base catalytic proton-transfer cycle
Repair proton partners in two local steps and distinguish catalyst regeneration from intermediate cancellation.
Open lessonLesson · Science · AP Chemistry preparationConstruct and challenge an equilibrium quotient
Build signed species exponents, simplify pure-phase factors and test numerical coincidences and zero-activity boundaries.
Open lessonLesson · Matching completions, families and unequal sidesConstruct and count independent stable-matching choices
Use strict preference barriers to combine local outcomes, exclude cross-block matches and prove an exact exponential family count.
Open lessonLesson · Precalculus: exponential and logarithmic evidenceConstruct and validate logarithmic responses
Fit a log-input response and distinguish its original-scale residual evidence from other transformations.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsConstruct and verify a motion witness
Give an executable sequence or a precise obstruction.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociConstruct both tangents from an external point
Build a diameter circle to find contact points and prove completeness of the construction.
Open lessonLesson · Science · AP Chemistry preparationConstruct buffer reaction inventories before equilibrium
AP scope: numerical post-addition pH and Henderson–Hasselbalch derivation are optional enrichment; qualitative capacity and initial buffer pH remain required. Build unequal conjugate-component inventories after acid/base challenges, retain excess reagent and test where the nominal buffer shortcut stops applying.
Open lessonLesson · Precalculus: polar curves and signed-radius evidenceConstruct complete signed-radius curves
Trace noncircle polar curves with exact sign intervals and meaningful angular restrictions.
Open lessonLesson · Science · AP Chemistry preparationConstruct conjugate-salt constants and species ledgers
Derive a hydrolysis constant from its parent, distinguish ion charge from acid/base direction and keep missing numerical evidence explicit.
Open lessonLesson · Science · AP Chemistry preparationConstruct conjugates without confusing H⁺ with H
Build donor and acceptor products, test charge changes and distinguish globally balanced candidates from a proton transfer.
Open lessonLesson · Functions in graphs, measurements and growth decisionsConstruct cube-root and absolute-value graphs
Build cube-root graphs from exact landmark coordinates.
Open lessonLesson · Partial differential equations and Fourier methodsConstruct diffusion on the whole line
Verify the Gaussian fundamental solution and its initial limit.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsConstruct division without losing equality
Build Frac(R) from a domain and prove its universal property.
Open lessonLesson · Science · AP Chemistry preparationConstruct electron-matched halves and recognize dual roles
Choose whole-half multipliers, retain residual electrons honestly and inspect acidic-medium and disproportionation transfers.
Open lessonLesson · Boolean specifications, certificates and circuitsConstruct formulas from complete truth tables
Supporting depth: build canonical DNF and CNF, prove their meaning row by row, and separate finite decidability from efficient representation.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsConstruct inverses and nontrivial ideal classes
Use exact products to prove invertibility and expose failure in a nonmaximal order.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeConstruct limits of Cauchy sequences
Prove completeness with a summable-difference subsequence.
Open lessonLesson · Optimization: models, certificates and algorithmsConstruct lower bounds before solving the primal
Derive dual functions, strong-duality conditions and nonattainment examples.
Open lessonLesson · Exact factorization and repeated solvesConstruct LU and know when it exists
Prove the exact no-pivot criterion, recover normalized LDU factors, and diagnose singular and rectangular boundaries.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociConstruct perpendicular and parallel lines
Build through-point lines with compass arcs, then justify a second method using folds.
Open lessonLesson · Functions in graphs, measurements and growth decisionsConstruct polynomial and rational graphs
Use factors, signs and domain exclusions to build branches rather than isolated plotted points.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceConstruct rational models from rates
Define rates, combine their effects and reject impossible algebraic candidates.
Open lessonLesson · Real skew-adjoint geometry and complex structuresConstruct real rotation planes for a skew-adjoint map
Prove the real skew-adjoint block theorem by positive squares, then derive rotations, complex eigenvalues and real-coordinate limits.
Open lessonLesson · Precalculus: periodic and reciprocal graph evidenceConstruct secant and cosecant graphs
Use reciprocal geometry to recover disconnected ranges, branch vertices and direction.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsConstruct separate component ledgers for a planar encounter
Momentum is a vector, so conservation applies separately to each component in a fixed inertial frame. A northward contribution cannot be added to an eastward contribution as though both were positive numbers on one axis. Draw components, sum them by direction, and only then calculate a resultant magnitude if needed.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeConstruct size from countable covers
Prove an interval-cover outer measure is subadditive.
Open lessonLesson · Precalculus: periodic and reciprocal graph evidenceConstruct tangent and cotangent branches
Track transformed asymptotes and solve inequalities across all admissible intervals.
Open lessonLesson · Calculus synthesis: regions, motion and approximationConstruct Taylor models from derivative evidence
Use a derivative table and a separately justified remainder bound, then transfer known series through substitution.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociConstruct the circumcircle and incircle
Use equidistance loci to construct triangle centers and prove the circles actually touch the intended objects.
Open lessonAssessment · Tail limits and root testsConstruct the cluster set and audit a sparse power series
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Determinants from permutations to certificatesConstruct the determinant from permutations
Build the signed permutation sum and prove its alternating multilinear characterization without assuming determinant identities.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceConstruct the endpoint events that actually measure length
A rod length in a frame is the distance between its endpoints measured simultaneously in that frame. The proper length L0 uses the rod’s rest frame S′. A laboratory S sees the rod move at +β, so its simultaneous endpoint events are not the same pair as the rod-frame simultaneous events.
Open lessonAssessment · Infinite cardinality through explicit constructionsConstruct the map and verify its inverse
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Abstract algebra I: groups and symmetryConstruct the three polyhedral rotation groups
Verify solid geometry, supporting faces and symmetry orders from exact coordinates.
Open lessonLesson · Statistical investigations, simulation and probability decisionsConstruct three views of one dataset
Build a dotplot, histogram and boxplot and document the choices that make their meanings reproducible.
Open lessonLesson · Science · AP Chemistry preparationConstruct transition states from local energy barriers
Repair a two-step energy diagram, compare directional barriers and separate a common datum shift from a changed pathway.
Open lessonLesson · Science · AP Chemistry preparationConstruct water-ion evidence without assuming neutral means pH 7
Build independent pH and pOH proposals, repair their product and compare temperature-dependent neutral references.
Open lessonAssessment · Geometric definitions, motion witnesses and theorem proofsConstruct witnesses and justify geometric conclusions
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Geometric constructions, circle proofs and coordinate lociConstruct, prove and measure unfamiliar configurations
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Axiomatic Euclidean and non-Euclidean geometryConstructible coordinates and quadratic extensions
Translate ruler-and-compass steps into field operations.
Open lessonCourse · Mathematics program unitConstructing the real numbers
Build an ordered complete field from rational Cauchy sequences, proving arithmetic, order and completeness without assuming real limits in advance.
Open courseLesson · Structural induction: constructions, recursion and parsingConstructors and tree invariants
Define finite ordered full binary trees, compute counts recursively and prove a statement for every permitted construction.
Open lessonExperiment · Cybersecurity and digital literacyContain a fictional incident
How does a delay change affected devices and interrupted service?
Open experimentConcept · GlossaryContextual bandit
A decision process that uses the current task’s features to choose an action and learns from its observed outcome, without directly optimizing a long sequence of state changes.
Open conceptLesson · Single-variable calculus foundationsContinuity and guaranteed crossings
Check the three conditions for continuity and distinguish existence guarantees from locating or counting roots.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryContinuity and the coordinate field
Expose the existence assumptions behind geometric constructions.
Open lessonLesson · Point-set topology: spaces, constructions and separationContinuity through inverse images
Relate global open-preimage continuity to the pointwise neighborhood condition.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsContinuous families and their mechanisms
Connect interval length, waiting-time survival and Gaussian standardization.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusContraction mappings and fixed points
Prove convergence, existence, uniqueness and a computable tail bound under contraction hypotheses.
Open lessonLesson · Proof foundations: from claims to argumentsContradict the exact assumption
Use a constructed smaller object to refute existence of a least positive rational.
Open lessonConcept · GlossaryContrastive learning
Learning representations by favoring matched examples over contrasting alternatives. The chosen comparisons determine which distinctions the objective rewards.
Open conceptLesson · Numerical analysis: algorithms, errors and evidenceControl elimination while tracking its backward error
Derive pivoted LU, explain growth and connect residuals with componentwise or normwise errors.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeControl limits with an integrable envelope
Define signed integrals without infinity minus infinity.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeControl products with conjugate exponents
Define Lp classes and essential bounds.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsControl roots and quotient coordinates
Use leading coefficients, remainders and degree bounds.
Open lessonLesson · Course 1 · Generative models beyond textControl the generated result
Train a shape-conditioned flow and compare guidance, validity and coverage at equal sampling work.
Open lessonLesson · Science · University Physics I with calculusConverge a numerical method without confusing it with validation
The numerical model assumes dv/dt=−v with initial10 m/s, evaluated at2 s. A separate synthetic observation comes from dv/dt=−lambdaTrue v. Keeping the exact assumed-model endpoint and independent observed endpoint distinct lets us separate numerical and physical-model discrepancies.
Open lessonConcept · GlossaryConvergence
Approaching a stable value or state under an update process. Stability alone does not establish task correctness.
Open conceptLesson · Abstract algebra II: rings, fields and Galois foundationsConvert algebraic relations into a quotient field
Prove irreducibility, divisibility and basis consequences.
Open lessonLesson · Science · General Chemistry IIConvert charge to a product only after electron and efficiency accounting
Passed charge counts electron equivalents, not automatically the desired product.
Open lessonLesson · Science · Chemistry and matterConvert electrical current into reaction amount
Use a fixed current and elapsed duration to calculate charge, electron amount and copper deposition.
Open lessonLesson · Science · Physical Geology and Earth SystemsConvert sediment storage into a porous-bed elevation change
Water discharge and sediment transport are related but distinct fluxes. A river can carry dissolved material, suspended particles and bed load. Competence concerns sizes that flow can move, whereas capacity concerns the amount transported under stated conditions. Cohesion, grain size, sorting and available supply complicate a simple faster-flow narrative.
Open lessonLesson · Mathematics investigationConvert units before buying materials
Why does converting area require two factors of 100?
Open lessonLesson · Science · Physics and physical systemsCool an expanding gas without removing heat
Distinguish reversible adiabatic expansion, isothermal expansion and free expansion.
Open lessonLesson · Course 2 · Learning without losing old skillsCoordinate agents with limited information
Compare centralized observations, local missing information and explicit communication in a synchronous two-agent grid.
Open lessonLesson · Course 2 · Choosing what to do nextCoordinate dependencies and reusable skills
Schedule dependent tasks, repair a stale-file conflict, and execute a bounded verification skill.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsCoordinate displacement, velocity and acceleration phases
For release from the positive endpoint at t=0, write x=A cos(ωt). The velocity is v=−Aω sin(ωt) and acceleration is a=−Aω² cos(ωt)=−ω²x. You can interpret these without differentiating: velocity is the slope of the position graph, and acceleration tells how that slope changes.
Open lessonConcept · GlossaryCoordinate frame
An origin and directed axes used to describe a position or orientation. The same physical point has different coordinates in different frames; transformations require named source/destination frames and consistent units.
Open conceptLesson · Course 2 · Learning through actionsCopy a demonstrated action
Train a puzzle policy from exact expert targets, inspect legal masks and compare proposals after a non-expert move.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociCopy lengths and angles with a proof
Turn compass-and-straightedge instructions into justified constructions, including angle bisectors and a folding alternative.
Open lessonLesson · Rhetoric, media and public voiceCorrect a message without hiding the correction
Trace public claims to their evidence and issue a visible correction when wording overstates that evidence.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsCorrect a temperature record before comparing expansion
Temperature describes thermal state, not the amount of energy stored in an object. Two objects in thermal equilibrium have the same temperature; the zeroth-law relation allows a third system to serve as a thermometer. Equal temperature does not mean equal total internal energy, because mass, material and microscopic degrees of freedom can differ.
Open lessonLesson · Science · Earth and space scienceCorrect an instrument offset with separate evidence
Use independent standards to restore a wavelength comparison without changing raw observations.
Open lessonLesson · Science · Scientific inquiry and measurementCorrect an offset without hiding uncertainty
Use a reference to distinguish accuracy from repeatability.
Open lessonLesson · Course 2 · Geometry and control foundationsCorrect errors using feedback
Tune an ideal cart controller and inspect overshoot, velocity damping and delayed measurements.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCorrect multiple counting systematically
Count unions of overlapping sets and explain alternating corrections.
Open lessonLesson · Course 1 · Simple models and useful baselinesCorrect residual errors with boosting
Fit small trees to the mistakes left by earlier predictions, inspect each correction and choose complexity using separate development evidence.
Open lessonAssessment · Algorithm reasoning: invariants, termination and costCorrectness and cost on changed processes
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesCorroboration without counting copies
Distinguish independent corroboration from repeated reporting and explain archival silence.
Open lessonConcept · GlossaryCosine similarity
A comparison of directions: the dot product divided by the two vector lengths. It is undefined for a zero vector.
Open conceptLesson · Science · General Chemistry IICount a disclosed entropy contribution without equating it to disorder
A macrostate probability depends on how many allowed microscopic arrangements realize it.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceCount alleles without confusing a census and a sampling model
Evolutionary population change concerns inherited variation across generations, not an individual changing its genotype because conditions demand it. A population must be defined in space, time and reproductive context before a frequency has a useful denominator. Samples from different life stages or locations may represent different populations rather than a temporal evolutionary trajectory.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount allocations with bounds
Translate allocation restrictions into shifted variables.
Open lessonLesson · Linear algebra over finite fieldsCount bases, subspaces and invertible matrices
Count independent vectors sequentially, then remove the exact multiplicity of bases to count subspaces over a finite field.
Open lessonLesson · Science · Biology and living systemsCount chromosome sets separately from DNA copies
Follow replication and two divisions without confusing homologs and sister chromatids.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount colorings with a recurrence
Prove deletion–contraction.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount constrained paths and recursive trees
Prove a reflection bijection for bad paths.
Open lessonLesson · Discrete synthesis: schedules, sampling and certificatesCount dependent events and prove a tail bound
Build a finite probability model, add indicator expectations without independence, and sharpen a tail bound using pairs of events.
Open lessonLesson · Orders, graph representations and certificatesCount directed walks with matrix powers
Prove what each matrix entry counts, distinguish fixed-length walks from reachability, and justify a finite reachability test.
Open lessonLesson · Domains, invariants and structural countingCount distinct labels by checking constructor assumptions
Separate vertex occurrences from label sets, locate overlap losses, and repair structural induction using disjointness and freshness.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsCount divisors and units through prime powers
Develop multiplicative arithmetic functions and divisor-sum identities.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsCount embeddings without assuming every root stays inside
Connect splitting fields, normality and automorphism bounds.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsCount every solution to a modular linear equation
Derive solvability and solution counts for ax≡b modulo m.
Open lessonLesson · Social studiesCount facilities, then investigate access
Does having more facilities mean better service?
Open lessonLesson · Abstract algebra I: groups and symmetryCount groups by equal-sized pieces
Prove the partition theorem before using divisibility.
Open lessonLesson · Mathematics investigationCount how work grows
Does doubling input double the work of every algorithm?
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCount incidences instead of guessing edges
Define simple graphs and prove degree identities and parity constraints.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount labeled products with exponential series
Derive the binomial convolution product rule.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount objects up to symmetry
Prove Burnside’s lemma by double counting.
Open lessonLesson · Mathematics investigationCount only after naming the outcomes
How do order and reuse change the number of possibilities?
Open lessonLesson · Statistical investigations, simulation and probability decisionsCount ordered and unordered selections
Define equally likely outcomes before choosing permutations or combinations.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsCount outcomes without overlaps or omissions
Separate disjoint alternatives from sequential choices and use complements.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount partitions through diagrams and products
Use conjugation to exchange partition restrictions.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceCount production needed for a consumed-food service
A kilogram harvested is not always a kilogram consumed. Storage, handling and distribution losses separate production from delivered food service. A fair comparison fixes consumed edible quantity and relevant nutritional quality before assigning upstream burdens or evaluating an added preservation process.
Open lessonLesson · Mathematics for AI · M01Count rates and price mistakes
Move between counts, fractions, percentages and expected error costs.
Open lessonLesson · Science · General Chemistry ICount shared lattice sites before calculating density
Connect conventional-cell bookkeeping, geometric radius and bulk mass per volume.
Open lessonLesson · Science · AP Chemistry preparationCount sigma and pi bonds without miscounting electron regions
Audit supplied molecular graphs and diagnose local counting rules with multiple bonds and lone pairs.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount spanning trees with a determinant
Relate oriented incidence matrices to Laplacians.
Open lessonExperiment · Systems thinking and operationsCount stock that has not arrived yet
Why can empty shelves trigger too many replacement orders?
Open experimentLesson · Abstract algebra I: groups and symmetryCount symmetry inside the group itself
Use centralizers to measure conjugacy classes.
Open lessonLesson · Mathematics investigationCount teams and ordered roles
When does order create a different outcome?
Open lessonLesson · Integral calculus: accumulation, methods and applicationsCount thin cylindrical layers
Choose a slicing direction that avoids unnecessary inverse branches.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesCount unlabeled blocks of labeled objects
Derive the second-kind Stirling recurrence.
Open lessonLesson · Point-set topology: spaces, constructions and separationCountability axioms, density and Lindelöfness
Distinguish first countability, second countability and separability.
Open lessonConcept · GlossaryCounterfactual
An outcome under an alternative intervention for the same unit. Population observational and intervention rates may leave the pairing of individual potential outcomes undetermined without additional structural assumptions.
Open conceptLesson · Science · University Physics I with calculusCouple buoyancy stiffness to a bounded oscillation model
A partially immersed body can gain an incremental restoring force from buoyancy. The supplied 2 kg vertical cylinder has constant cross-section 0.01 m², height 0.3 m and equilibrium immersion depth 0.15 m. An ideal 20 N/m spring has a suitable static preload; displacement is measured upward from that equilibrium.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingCouple translation and rotation through a massive pulley
Recover unequal tensions from the rotational equation instead of assuming an ideal massless pulley.
Open lessonLesson · Proof foundations: from claims to argumentsCover every case and both directions
Organize parity arguments and understand what an if-and-only-if claim requires.
Open lessonLesson · Point-set topology: spaces, constructions and separationCovering dimension and finite-dimensional embeddings
Define covering dimension by finite open refinements and multiplicity.
Open lessonLesson · Homotopy, fundamental groups and covering spacesCovering maps and local sheets
Check the full evenly-covered-neighborhood definition.
Open lessonExperiment · Communication and evidence-based writingCreate an honest chart specification
How does an axis floor change the apparent size of a difference?
Open experimentConcept · GlossaryCredit assignment
Determining how actions or components contributed to an outcome, including interactions that make local activity a poor measure of joint success.
Open conceptConcept · GlossaryCritical path
A longest chain of dependent tasks. Its total duration places a lower bound on completion time under the stated timing assumptions.
Open conceptLesson · Multivariable calculus: geometry, change and accumulationCritical points, Hessians and boundary checks
Classify interior critical points and distinguish local tests from global optimization.
Open lessonLesson · Science · General Chemistry IICross a titration equivalence point without changing conservation laws
One charge balance can describe initial acid, buffer, equivalence and excess base.
Open lessonConcept · GlossaryCross-entropy
A prediction loss that penalizes assigning little probability to the correct outcome.
Open conceptLesson · Social studiesCuba 1962: assurance, dismantling and verification
A crisis bargain can connect reciprocal assurances and observable steps. Each party may frame its actions differently, and a public message may omit other negotiations. Analyze what is offered, conditional, claimed and verifiable.
Open lessonCourse · Social studiesCultural, linguistic and archaeological inquiry — High school
Hands-on social studies course and project
Open courseCourse · Social studiesCultural, linguistic and archaeological inquiry — Introductory college
Hands-on social studies course and project
Open courseConcept · GlossaryCurriculum
A rule or strategy for selecting which tasks or examples to practice, often based on difficulty or current performance.
Open conceptSubject · Experiment collectionCybersecurity and digital literacy
Repair permissions and test recovery goals in safe, fictional systems.
Open subjectProject · Foundation projectData audit project and assessment
Capture evidence, write a guided or decision report, submit implementation artifacts and solve independent transfer cases.
Open projectModule · Course 1Data audits and training diagnosis
Repair an evaluation pipeline and diagnose a training failure, with separate project assessments.
Open moduleSubject · Experiment collectionData literacy and statistics
Investigate whose data you see and what a summary hides.
Open subjectLab · Foundation lessonData quality and leakage lab
Iris lesson playground: missing values, units, grouping, train-only preprocessing and nearest neighbors.
Open labLesson · Science · Chemistry and matterDecide what a coarse density measurement can establish
Propagate two reading bounds and avoid false certainty about material identity.
Open lessonLesson · Statistics: displays, designs and differencesDecide what a study must answer before collecting data
Translate a decision into a population, measurable outcome, comparison and collection plan.
Open lessonLesson · Science · AP Chemistry preparationDecide what elemental composition can identify
Connect masses, empirical formulas and mixture constraints while retaining contradictory or nonunique results.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsDecide what makes outcomes different
Derive ordered and unordered selection counts and account for repeated symbols.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityDecide what the evidence can support next
Audit design, independence, model conditions and uncertainty before selecting inference.
Open lessonLesson · Abstract algebra I: groups and symmetryDecide when an extension splits
Use complements and actions to classify groups of order pq.
Open lessonLesson · Course 1 · Data audits and training diagnosisDecide where AI belongs
Compare a rule, supplied candidate predictions and simulated human review, including costs and unfinished work.
Open lessonLesson · Course 2 · Choosing what to do nextDecide whether another computation is worth it
Compare acting now with gathering evidence, then follow each observation to a different next action.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDecide whether static friction can maintain rest
Static friction prevents relative sliding between surfaces when the needed tangential force is within a limiting magnitude. The rule is |fₛ|≤μₛN, not fₛ=μₛN in every resting situation. A small push can be balanced by small static friction; no horizontal push may require no friction at all.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsDecide which integers are sums of two squares
Prove the obstruction from primes congruent to three modulo four.
Open lessonConcept · GlossaryDecision tree
A predictor that follows learned input tests to a leaf. Training chooses splits; leaf predictions summarize training examples. More depth increases flexibility, not guaranteed generalization.
Open conceptLesson · Social studiesDeclaration and Gettysburg: an ideal in two settings
Political texts can reuse an ideal while serving different purposes.
Open lessonLesson · Social studiesDecolonization, Cold War and global connections
Distinguish independence, development and rights through comparative evidence.
Open lessonLesson · Social studiesDecolonization: sovereignty, partition and nonalignment
Compare routes to independence and distinguish sovereignty from economic and political outcomes.
Open lessonLesson · Euclidean and coordinate geometry foundationsDecompose a region without counting it twice
Justify polygon angle sums and area formulas by partitioning into simpler shapes.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsDecompose uncertainty within and between groups
Connect total variance with the best squared-error predictor.
Open lessonLesson · Science · Earth and space scienceDefend a bounded geological interpretation
Combine complementary records with alternatives and explicit interpretation limits.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDefend a charge-sharing transient across model boundaries
Two capacitors are prepared at voltages 6 V and 0 V, then every source is removed before the modeled sharing interval.
Open lessonLesson · Science · Scientific inquiry and measurementDefend a claim and revise its limits
Build an evidence-based conclusion that can change.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDefend a composed optical, quantum and thermal energy account
A cross-category claim is only as strong as the links between its evidence stages. A virtual monochromatic pulse at500 nm lasts10 s. Two independent power standards20 and60 W calibrate an affine instrument indication y=gP+B. Their supplied readings19 and51 determine gain0.8 indication units/W and offset3. The pulse reading is then corrected using that calibration; neither the generating power nor the gain is silently supplied to the inverse calculation.
Open lessonLesson · Science · Physical Geology and Earth SystemsDefend a conditional site alternative without hiding constraints in a score
An environmental decision requires more than choosing the largest resource estimate or the smallest average hazard. The decision-maker must state which outcomes are constraints, which can be compared, which scenarios are credible and whose impacts remain unrepresented. A numerical ranking cannot silently decide those ethical and practical boundaries.
Open lessonLesson · Science · Engineering design and scientific computingDefend a constrained design choice with evidence
Connect requirements, alternatives and limits in a reviewable decision brief.
Open lessonProject · Model, constrain, control and troubleshootDefend a design and troubleshoot its failure
Choose a virtual motion, power or heating design and specify a target plus a binding constraint. Compare alternatives, deliberately introduce a failure or disturbance, and justify a diagnostic or control change using captures from at least two experiments. This is a model-based design review, not authorization to operate physical hardware.
Open projectLesson · Science · Environmental Science: ecological systems and evidenceDefend a district decision across services, budgets and stress
Integrate ecological evidence, resource services and explicit constraints without turning a favorable aggregate score into permission to ignore a failed requirement.
Open lessonProject · Optimization: feasible choices and competing goalsDefend a feasible resource decision
Formulate a small planning problem with an explicit feasible set, compare exact or enumerated and iterative solutions, test a resource change and explain competing goals. Supply an artifact that a reviewer can recompute.
Open projectLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDefend a molecular mechanism across biological and measurement layers
A course-wide explanation must connect molecular state, cellular context and the observation process without treating them as interchangeable. A DNA variant may change an RNA, an RNA change may alter a protein, and protein amount or activity may alter a metabolic output. Each arrow has assumptions and possible alternatives: unchanged transcript amount does not prove unchanged catalytic function, and altered fluorescence does not prove altered metabolism.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDefend a population-change explanation with an auditable evidence chain
The course culminates in an argument that links population change to organismal and environmental evidence without overstating what was measured. Begin with the sampling frame, record identities and biological stage. Then separate source observations, justified corrections, model calculations and causal interpretation. A corrected numerical result is not automatically a complete mechanism.
Open lessonExperiment · Communication and evidence-based writingDefend a proposal with trade-offs
When does a faster process violate the service requirement?
Open experimentLesson · Poetry, drama and performanceDefend a reading, then change one choice
Create an integrated performance score and revise it in response to a plausible alternative interpretation.
Open lessonPath · Students, software engineers and managersDefend a real-data model comparison
Compare a simple reference and fitted candidate, justify validation design and report a frozen real-data experiment with error analysis.
Open pathLesson · Science · Human Anatomy and Physiology FoundationsDefend a regional delivery budget before declaring systemic balance
An organ is not supplied merely because a whole-body total looks adequate. Arterial flow divides among parallel tissue beds, while venous return mixes after exchange. Epithelial, vascular, muscle and neural structures contribute different parts of this chain. A complete explanation identifies where material moves, where work or transport occurs and which signals regulate the pathway rather than treating the body as a single uniform container.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDefend a resource portfolio across explicit stress scenarios
Energy, agriculture and water choices supply multiple services under uncertain conditions. A portfolio mixes alternatives; a tradeoff compares outcomes without pretending all values have one natural unit. Before optimizing, declare minimum service needs, relevant environmental limits, affected groups and the scenarios being considered.
Open lessonLesson · Science · Environmental science and sustainabilityDefend a water decision with trade-offs
Compare indicators and revise a recommendation after a supplied demand stress.
Open lessonLesson · Science · AP Environmental ScienceDefend a watershed claim across competing evidence
Distinguish concentration, load and causal evidence before selecting an intervention.
Open lessonLesson · Science · Introductory MicrobiologyDefend a whole-course inference with source, timing and validation intact
A final report should preserve what was observed, what was assumed and what remains unestablished. Microbial mechanisms can make an explanation plausible, but evidence must still distinguish it from measurement, selection and timing alternatives. This destination connects host/population interpretation to control specificity and the earlier source/measurement modules.
Open lessonPath · Students, engineers and managers · cumulative projectDefend an AI decision
Connect task framing, data repair, review costs, uncertainty and supported claims in one evidence brief.
Open pathLesson · Science · Introductory AstronomyDefend an astronomical explanation with actual source provenance
A full astronomical argument must distinguish actual source records, source-derived products, original model assumptions and a learner’s independently retained reasoning. A numerical fit is not enough if the calibration, selection, uncertainty or prior exposure is hidden. The course’s actual NASA rendering/ephemeris and SDSS spectrum have concrete but different evidentiary roles.
Open lessonLesson · Science · Earth and space scienceDefend an evolutionary interpretation
Construct a bounded claim with alternatives and a proposed independent constraint.
Open lessonLesson · Science · General Chemistry IDefend and revise a quantitative chemical explanation
Reconcile analytical amount, heat paths, reaction states and independent review in a course-wide evidence defense.
Open lessonLesson · Course 2 · Testing whether improvement is realDefend and revise a result
Record a critique, a bounded claim revision and a counterexample test without rewriting earlier evidence.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDefend calculus and conservation across a changing system
Integrate a force history, change the system at capture and retain a new oscillator initial state.
Open lessonLesson · Mathematical modeling and senior investigationDefend conclusions and preserve the limits of the evidence
Assemble a report, reproducible artifacts and a response to critique.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDefend conclusions, identify gaps and plan next evidence
A final evidence defense compares claims with retained records and acknowledges what remains unknown.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDefend independent evidence rather than shared model agreement
Compare methods, identify a shared error and retain a discriminating evidence plan.
Open lessonLesson · Science · University Physics II with calculusDefend independent spatial and temporal calibration in one capacitor dossier
A long coaxial capacitor has inner radius a0, outer radius s a0 and radial relative permittivity κ(r)=1+r/a0. Gauss law gives radial displacement D=Q/(2π ell r); integrating E=D/(epsilon0 κ) gives V=Q F(s)/(2π epsilon0 ell), where F(s)=ln[2s/(1+s)]. Thus C=A/F(s), with A=2π epsilon0 ell expressed in nF. This is a prescribed linear dielectric profile and long-device approximation, not a unique material reconstruction.
Open lessonLesson · Science · Earth and space scienceDefend limits and transfer
Repair claims that confuse cooling, local pressure tendency and a regional forecast.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDefend original evidence, critique and an unfamiliar transfer
Audit a claimed concentration improvement across flow, load, energy and evidence provenance, then submit the full-course retained project and mixed written assessment.
Open lessonLesson · Science · Physical Geology and Earth SystemsDefend the full geological evidence portfolio and its remaining gates
A defensible environmental conclusion connects geological history, pathways and consequences without collapsing them into one number. The fictional site report claims that a lower downstream concentration proves lower tracer transport. Test that statement with an explicit confluence boundary rather than importing an unrelated actual stream measurement or relying on a qualitative dilution story.
Open lessonAssessment · Finite sampling guarantees and dependenceDefend the sampling guarantee
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Matching completions, families and unequal sidesDeferred acceptance when the two sides have different sizes
Allow exhausted proposers, prove stability and saturation with complete acceptable lists, and locate the boundary when lists become incomplete.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDefine a character before treating similarity as shared ancestry
Systematics reconstructs relationships using evidence; taxonomy names and organizes groups. A phylogenetic hypothesis should explain patterns of inherited characters, not simply group organisms that look useful or familiar. The same comparison can be homologous at one structural level and analogous at another functional level.
Open lessonLesson · Course 2 · Connecting tools and specialistsDefine an assistant’s job
Specify an assistant’s inputs, scope, permissions, evidence and stopping rules before choosing a model or adding tools.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeDefine conditional expectation by integral tests
Construct conditional expectation using Radon–Nikodym.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsDefine each infinite or singular contribution by a limit
Separate convergent improper integrals from misleading cancellations.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDefine flux with an area normal before predicting induction
Magnetic flux combines field magnitude, area and orientation. For a uniform field through a flat surface, Φ=BA cosθ, where θ is measured between B and the chosen surface normal, not the surface itself. A normal is a perpendicular arrow assigned to orient the surface; either choice is allowed if the subsequent signs remain consistent.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeDefine integration from simple functions
Prove representation independence for simple integrals.
Open lessonLesson · Structural induction: constructions, recursion and parsingDefine recursive functions and strengthen the claim that proves them
Trace finite tree recursion, prove a mirror law and a sharp leaf bound, and generalize a word-reversal invariant when a weak induction hypothesis fails.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDefine the target population before expanding a survey
A target population is the set of units about which a study intends to infer. The sampling frame is the operational list or spatial representation from which units can actually be selected. They can differ because of access restrictions, omitted habitats, outdated maps or ambiguous eligibility. A precise estimate for the frame need not represent the target.
Open lessonLesson · Science · Environmental science and sustainabilityDefine transport service before comparing vehicles
Separate requests, served travel, capacity and vehicle movement.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsDefine triangle congruence through a witness
Connect side and angle correspondence to a whole-triangle rigid map.
Open lessonLesson · Homotopy, fundamental groups and covering spacesDeformation retracts and homotopy equivalence
Verify a strong deformation retraction by formula.
Open lessonLesson · Discontinuities and Riemann integrationDense exceptions and integral values
Construct an integrable function with dense discontinuities and identify the exact limits of null-set modification arguments.
Open lessonAssessment · Discontinuities and Riemann integrationDense jumps, a Cantor set and cancellation
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Real Analysis I: completeness, limits and rigorous calculusDensity and nested intervals
Construct rational and irrational points between reals and prove a nested-interval result.
Open lessonConcept · GlossaryDependency graph
Tasks and directed prerequisite relationships specifying which outputs must be available before other tasks start.
Open conceptLesson · Course 1 · Why does AI need computation?Deploy with a reversible plan
Compare local and service boundaries, inspect shadow and staged traffic, and restore stable routing without erasing history.
Open lessonLesson · Social studiesDepression, New Deal and wartime mobilization
Distinguish relief, recovery and civil liberties during crisis.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusDerivative intermediate values and Taylor control
Separate continuity of a derivative from its intermediate-value property and justify remainders.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusDerivatives, Rolle and the mean value theorem
Derive global comparison results from local derivatives with endpoint hypotheses.
Open lessonLesson · Science · Physical Geology and Earth SystemsDerive a crustal root from column mass balance
Isostasy is a gravitational balance, not the claim that crust rests on a globally liquid mantle. Mostly solid mantle can deform slowly enough for loads to be redistributed over geological time. An ideal local Airy model compares equal-area columns above a shared compensation depth with equal density in the reference crust and mantle.
Open lessonLesson · Science · University Physics II with calculusDerive a curvature correction to capacitance
Capacitance is a relation between conductor charge and a potential difference for a specified geometry and material model. The spherical capacitor has concentric surfaces at a and b=a+d, with equal and opposite charges. The vacuum field between them is radial, E=KQ/r², and the voltage must be integrated across the gap rather than approximated as one field value times d.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive a finite phasor sum and remove its singularities
For N equally illuminated narrow slits, adjacent paths differ by q wavelengths and adjacent phases by 2πq. The total field in single-slit units is A=Σ from j=0 to N−1 exp(i2πjq). This finite sum is always defined, including when every phasor points in the same direction.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDerive a finite-line field and compare distinct limits
Use pairwise cancellation while retaining the full source distance.
Open lessonLesson · Science · University Physics I with calculusDerive a pipe profile and then test its admissibility
For steady fully developed flow through a straight circular tube, an axial pressure drop supplies force on a cylindrical fluid core. Balancing it with shear around the core gives shear magnitude Δp r/(2L). Newtonian stress relates that shear to the radial velocity gradient.
Open lessonLesson · Science · University Physics I with calculusDerive a radial barrier from angular momentum
In a central force, angular momentum per mass h = r²θdot is conserved. Tangential speed is h/r, so its kinetic energy per mass is h²/(2r²). Combining this with gravitational potential −μ/r gives the radial effective potential Ueff = h²/(2r²) − μ/r.
Open lessonLesson · Finite Fourier bases, convolution and FFTDerive a radix-two FFT and count its work
Split a negative-exponent Fourier transform into even and odd subsequences, prove the recursive index contract, and audit exact work and numerical checks.
Open lessonLesson · Science · General Chemistry IIDerive a saturating rate from bound-catalyst conservation
The supplied candidate has rapid reversible binding C+A⇌CA followed by slower CA→C+P.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive a sequential population account with a removable equal-rate limit
A parent population N1 converts to a daughter N2, which converts to a stable product, with branching fraction one. Initially N1=N0 and N2=0. The mean equations are dN1/dt=−lambda1 N1 and dN2/dt=lambda1 N1−lambda2 N2. Solving the first gives N1=N0 exp(−lambda1 t).
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDerive a slab field and challenge a symmetry shortcut
Separate what Gauss law guarantees from what source symmetry supplies.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDerive a vector trajectory from recipient-specific forces
Apply an inertial net-force law before integrating with both initial values.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive a wavelength shift from vector momentum and total energy
A photon scatters from a free electron initially at rest. Let incoming photon energy be E, outgoing energy Eprime and deflection angle theta. Photon momentum magnitude is E/c. Resolve recoil momentum: px c=E−Eprime cos(theta), py c=−Eprime sin(theta). Vector subtraction is essential; subtracting momentum magnitudes would miss transverse recoil.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive a work limit when both source temperatures change
Two finite bodies with constant total capacities Ch and Cc=4 J/K start at Th and 300 K. A cyclic reversible engine can operate between them in infinitesimal steps until their temperatures meet. Unlike ideal infinite reservoirs, these bodies cool and warm; their initial Carnot efficiency cannot be used unchanged for all later heat.
Open lessonLesson · Science · University Physics I with calculusDerive allowed modes without mistaking harmonic ratios for calibration
A string fixed at both ends must have zero displacement at x = 0 and x = L. A separated spatial solution proportional to sin(kx) meets the first condition; the second requires kL = nπ for positive integer n. Combining with the wave equation gives fn = nv/(2L).
Open lessonLesson · Science · University Physics I with calculusDerive an axis-dependent inertia from the actual density
Moment of inertia weights each mass element by squared perpendicular distance to a specified axis: I = integral(rperp² dm). For a thin rod along x and a perpendicular axis through x = d, that distance is |x − d|. A formula about the left end cannot be reused about the mass center without accounting for the translation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive an image plane by composing ray transformations
A paraxial ray state is the column (y,θ), where y is signed height and θ is a small signed angle. Free propagation through length l maps y to y+lθ and keeps θ, giving P(l)=[[1,l],[0,1]]. A thin lens of focal length f changes angle to θ−y/f, giving L=[[1,0],[−1/f,1]]. Units matter: the off-diagonal entries are length and inverse length.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDerive and test a no-slip rolling constraint
Combine translation and rotation, then check whether static friction can enforce the constraint.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsDerive and use addition and double-angle formulas
Connect angle addition to rotations and retain both cross terms.
Open lessonLesson · Precalculus: local rates and polynomial evidenceDerive and use quadratic rate patterns
Explain constant changes in secant rates algebraically, then keep restricted-domain extrema distinct from whole-polynomial end behavior.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociDerive arc scaling and sector area
Use similarity and angular additivity to derive arc and sector formulas, including a school-level limiting argument.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociDerive both branches of a hyperbola
Use an absolute distance difference to derive two branches and keep the sign restrictions through squaring.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDerive braking motion and its heat ledger
A virtual rod of length l slides on ideal rails to close a resistive loop in a uniform perpendicular field.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDerive buoyancy and distinguish support from free motion
A submerged object receives pressure forces on all wetted surfaces. In a uniform static fluid, pressure on a horizontal lower face exceeds pressure on an equal upper face because the lower face is deeper. Horizontal forces on opposite vertical sides cancel for a simple symmetric block. The remaining upward resultant is buoyancy, not an extra unrelated force to add on top of the same pressure resultant.
Open lessonLesson · Science · Physics and physical systemsDerive buoyancy from displaced liquid
Connect pressure differences across a submerged object to the weight of displaced liquid.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDerive capacitance and bound a plate trajectory
One ideal infinite sheet has field magnitude sigma/(2epsilon0) on either side.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDerive capacitor combinations from charge and voltage
Capacitor combination rules follow charge and potential constraints, not a memorized resemblance to resistor rules. Parallel capacitors share the same two nodes and voltage. Their source-separated charges add, giving Ceq=C₁+C₂. The equivalent capacitance is larger than either individual capacitance.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDerive center and relative energy for an internal release
Explain how translational kinetic energy can increase without an external momentum impulse.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDerive collision endpoints from momentum and relative motion
Separate conservation from the extra physical condition needed to choose one collision outcome.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeDerive continuity from countable additivity
Prove continuity from below and qualified continuity from above.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDerive cross probabilities and distinguish expectation from outcome
An allele is one version of a locus.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDerive elastic energy from a force-area construction
Loading an ideal spring slowly from equilibrium requires an external holding force whose magnitude rises from 0 to kx. The force-versus-extension region is a triangle. Its area is½×x×kx=½kx², the stored elastic potential energy when zero is chosen at equilibrium. The factor½ follows from the increasing force; using final force times displacement would overcount.
Open lessonLesson · Mathematical modeling and senior investigationDerive equations from inventories and flows
Choose a boundary and account for every transfer.
Open lessonLesson · Science · General Chemistry IDerive exact density bounds and conditional identification
Use monotonicity of a positive ratio to build a feasible interval instead of reporting unsupported precision.
Open lessonLesson · Mathematical statistics: derivation and model criticismDerive exact normal-model inference
Use orthogonal decomposition to prove t and chi-square pivots.
Open lessonLesson · Science · University Physics I with calculusDerive independent modes before interpreting energy exchange
Two equal 1 kg masses each connect to a fixed wall with a 4 N/m spring and to each other with a spring of stiffness kc. Their equations are x1ddot = −4x1 − kc(x1 − x2) and x2ddot = −4x2 − kc(x2 − x1). The coupling acts on both recipients with opposite forces.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDerive inductance from flux linkage and energy
A current produces magnetic flux linked to a circuit.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDerive kinetic-energy change from Newtonian dynamics
Connect a vector rate equation with a scalar integral without losing sign or frame information.
Open lessonLesson · Social studiesDerive labor demand from additional output
How does a product price change alter a hiring calculation?
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive local intensity from complex field addition
A complex amplitude records two real pieces of information: magnitude and phase relative to a reference. For equally polarized waves of the same frequency in the same medium, the detector intensity is proportional to the squared magnitude of the summed field. The common proportionality constant is absorbed into the original normalized units here.
Open lessonLesson · Science · University Physics II with calculusDerive local uniformity instead of matching one amplitude
An on-axis loop integral cancels transverse components while retaining the common axial projection. One loop of radius R and current I gives Bz=μ0IR²/[2(R²+z²)^(3/2)]. Two identical, same-current coaxial loops centered at ±d/2 add shifted copies of this result. In the workbench R=1 m and I=2 A.
Open lessonLesson · Science · General Chemistry IDerive matter-wave scaling from momentum rather than photon formulas
Combine a nonrelativistic energy–momentum relation with de Broglie wavelength and check its domain.
Open lessonLesson · Optimization: models, certificates and algorithmsDerive multipliers from tangent directions
Connect constrained stationarity to the implicit-function theorem.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDerive network capacitance from a floating-node inventory
Capacitors in parallel share voltage, and their plate charges add: Ceq = C1 + C2.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive photon energy with the emitter recoil retained
Consider an original initially stationary emitter of invariant mass M. It ends as one body of rest mass m and one photon moving in the positive x direction. The model is a formal conservation packet, not a claim about an actual atomic or nuclear transition or a prescription for operating a source.
Open lessonLesson · Mathematical statistics: derivation and model criticismDerive regression inference from projections
Separate fixed-design algebra from normal-error inference.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDerive spring period and distinguish frequency measures
One cycle returns the oscillator to the same position and direction of motion. The period T is time per cycle; frequency f=1/T counts cycles per second. Angular frequency ω=2πf measures phase advance in radians per second. Confusing f and ω introduces a factor of 2π, even though both describe the same repetition.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDerive spring topology and shift the equilibrium coordinate
Separate an equilibrium offset from the stiffness that sets harmonic frequency.
Open lessonLesson · Science · University Physics I with calculusDerive terminal motion and its approach time separately
A horizontal point starts from rest under constant drive F=10 N and linear resistance−b v relative to a stationary medium. Newton's law gives m dv/dt=F−b v. The coefficient b has units kg/s. A terminal state solves force balance, but a force-balance value alone says nothing about how quickly it is approached.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive the aperture transform and its central limit
Fraunhofer diffraction adds contributions from every point of an aperture with position-dependent phase. For uniform monochromatic illumination and a scalar far-field approximation, a point x across a slit contributes exp(i2πux), where u=sinθ/λ. This spatial frequency has inverse-length units; it is not automatically an angle.
Open lessonLesson · Science · University Physics I with calculusDerive the coordinate constraint before eliminating tension
Two virtual hanging masses share a taut inextensible cord over an ideal massless pulley. Choose upward for the first mass and downward for the second as positive directions. Fixed cord length then requires equal signed displacement magnitudes, velocities and accelerations in these paired coordinates; it does not mean their upward Cartesian accelerations have the same sign.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociDerive the ellipse from its two foci
Turn the two-focus definition into a standard equation and prove the reverse implication with explicit signs.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDerive the equations and test their assumptions
Constant acceleration means equal time intervals produce equal velocity changes. Starting from a=(v−v₀)/t gives v=v₀+at. The velocity graph is then a straight line, so its average height is (v₀+v)/2. Multiplying that height by time gives displacement Δx=(v₀+v)t/2.
Open lessonLesson · Partial differential equations and Fourier methodsDerive the heat time-step restriction
Connect stencil weights, Fourier amplification and convergence.
Open lessonLesson · Sections, solid measures and geometric designDerive the one-third volume factor
Use similarity and finite slab bounds to justify pyramid and cone volume.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDerive the pendulum approximation and test its boundary
A pendulum bob constrained to a circular arc has a tangential gravitational force −mg sinθ toward the bottom, with θ measured from downward vertical. Tension is radial and does not supply that tangential restoring component. The arc displacement is s=Lθ when θ is in radians; the support length, not the bob’s mass, sets the geometrical conversion.
Open lessonLesson · Science · University Physics I with calculusDerive the period of an extended pendulum from its torque
An extended pendulum rotates about a pivot, so its inertia is not generally Md² even when its center lies distance d below that pivot. For a uniform rod of length 2 m and mass 3 kg, the center inertia is 1 kg m² and the parallel-axis result is Ipivot = 1 + 3d².
Open lessonLesson · Mathematical statistics: derivation and model criticismDerive the regular large-sample approximation
Use score expansion and distinguish a theorem from a diagnostic.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive the speed bias of a flux-selected sample
A bulk molecule selected uniformly from an equilibrium reservoir and a molecule selected because it crosses a small aperture are sampled differently. Faster molecules reach the opening more often. Assume a dilute gas, an aperture smaller than the mean free path, negligible depletion and speed-independent detection. This is a virtual kinetic-theory model, not an instruction to operate a vessel.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsDerive the three triangle criteria
Prove sufficient conditions by aligning a side and locating the third vertex.
Open lessonLesson · Science · University Physics I with calculusDerive the unstable mode of an otherwise decaying equation
The exact equation dv/dt=−v/tau with tau1 s and v0=10 m/s has positive exponential solution10exp(−t). Forward Euler gives vn+1=vn−dt vn/tau=(1−r)vn, where r=dt/tau. Repeating a scalar multiplication makes its numerical behavior transparent.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociDerive triangle laws and measure inaccessible distances
Derive area and sine relations across acute, right and obtuse triangles, then audit measurement ambiguity.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDerive velocity addition from transformed differentials
Velocity is a ratio of spatial and time increments in one frame. A transformation must change both increments before the ratio is taken. Frame S′ moves at +β relative to S, and a particle has primed speed u′. The inverse Lorentz map gives dt=γ(dt′+βdx′) and dx=γ(dx′+βdt′).
Open lessonLesson · Partial differential equations and Fourier methodsDerive wave equations from stationary action
Close the course by connecting equations, boundaries and energy.
Open lessonLesson · Mathematics investigationDescend without leaving the allowed region
Can a constrained minimum have a nonzero derivative?
Open lessonLesson · Statistics foundations: data, design and sampling variabilityDescribe a relationship before fitting it
Separate direction, form, strength and unusual observations.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeDescribe information with sigma-algebras
Construct generated sigma-algebras from observations.
Open lessonLesson · Science · Physics and physical systemsDescribe motion from a moving observer
Change the observer without changing the prescribed ground trajectory.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationDescribe regions and reverse integration order
Reconstruct a region before changing bounds or evaluating an integral.
Open lessonLesson · Linear Algebra II: operators and spectral structureDescribe spaces through linear measurements
Build the dual basis, annihilators and the pullback of a linear map.
Open lessonLesson · Mathematics investigationDescribe the shape beyond one average
Which summaries change when a tail observation grows?
Open lessonLesson · Mathematics for AI · M02Describe variation without inventing evidence
Inspect mean, variance, covariance and the assumptions behind uncertainty summaries.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsDescribe what a linear map loses and reaches
Connect abstract maps with matrices, kernels and images.
Open lessonLesson · Course 2 · Geometry and control foundationsDescribe where the robot is
Move a two-link arm, express points in named frames and distinguish ideal reach from verified motion.
Open lessonAssessment · Encodings, diagonal arguments and limits of algorithmsDescriptions, domains and impossible decision procedures
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · AP Chemistry preparationDesign a buffer that meets challenge and budget constraints
AP scope: numerical post-addition pH and Henderson–Hasselbalch derivation are optional enrichment; qualitative capacity and initial buffer pH remain required. Choose both conjugate-component amounts, test baseline and opposing challenges separately, and distinguish pH ratio from finite capacity.
Open lessonLesson · Science · General Chemistry IDesign a dilution with final-volume and uncertainty accounts
Conserve amount while propagating transfer and calibration bounds.
Open lessonExperiment · Data literacy and statisticsDesign a fair comparison
Can easier tasks make a weak treatment look better?
Open experimentLesson · Science · AP Physics C: Mechanics — calculus-based modelingDesign a sampling interval with competing error sources
Separate exact-function truncation, reading bounds and untested calibration assumptions.
Open lessonProject · Geometry and trigonometryDesign a scaled layout and an angled reach
Create a geometric design for a model display, storage layout or virtual reach mechanism. Specify coordinates, a uniform scale and an angled segment. Verify its dimensions, area or volume, and endpoint independently. Use at least two investigation mechanisms; a plausible drawing alone is insufficient.
Open projectLesson · Science · Chemistry and matterDesign a separation and account for every fraction
Track a dissolved component through filtration and evaporation, including material in the surroundings.
Open lessonLesson · Science · Introductory AstronomyDesign an accessible comparison using source-labeled lunar renderings
The source packet contains five unchanged NASA Scientific Visualization Studio frames at 00 UT on 1,8,15,22 and 29 January 2026, selected at a fixed seven-day cadence. These are scientific renderings based on lunar terrain/mosaic inputs and an ephemeris, not photographs exposed at those dates. Their provenance is useful evidence precisely because it identifies how the images and numbers were generated.
Open lessonLesson · Science · General Chemistry IIDesign an independent constraint before spending the sample
Two apparently different assays can share the same calibration ambiguity.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDesign an intervention that distinguishes competing mechanisms
Two mechanisms can reproduce the same baseline observation. Matching one endpoint therefore does not uniquely identify the process that generated it. A useful investigation seeks conditions under which the candidate explanations predict different outcomes, while preserving controls that keep the comparison interpretable.
Open lessonLesson · Statistical investigations, simulation and probability decisionsDesign and audit a fair random allocation
Design equal-chance procedures and account for rejection, finite limits and implementation bias.
Open lessonProject · Design evidence that can change your mindDesign and challenge an intervention study
Investigate whether a proposed study or process improvement produces a meaningful benefit. Use contrasting captures from at least two experiments, then specify a new investigation whose result could contradict your favored explanation. Deliver a protocol and a qualified conclusion; this workbench does not execute a real human study or certify its causal validity.
Open projectSubject · Experiment collectionDesign and product development
Test improvements against task completion and inspect who gets excluded.
Open subjectLesson · Science · Biology and living systemsDesign evidence that separates enzyme explanations
Use a controlled rate comparison and identify what the model leaves out.
Open lessonLesson · Science · AP Chemistry preparationDesign rate comparisons that distinguish competing orders
Compare controlled initial rates, intersect candidate k intervals and diagnose confounding, underdetermination and inconsistent data.
Open lessonLesson · Partial differential equations and Fourier methodsDetect missing constants and incompatible flux
Solve the structural obstacles in Neumann problems.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsDetect redundancy using the zero combination
Separate a dependence relation from a representation of a nonzero target.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsDetect repeated roots through derivatives
Distinguish perfect fields from characteristic-p counterexamples.
Open lessonCourse · Mathematics program unitDeterminants from permutations to certificates
Prove determinant identities over fields, derive cofactor certificates and test exact and numerical interpretations. Computation supports the proofs rather than replacing their hypotheses.
Open courseLesson · Abstract algebra II: rings, fields and Galois foundationsDetermine a quartic group from prime certificates
Combine proved modular cycles with a subgroup argument to identify S4.
Open lessonLesson · Science · General Chemistry IDetermine molecular polarity from vector evidence
Combine declared bond vectors and test how asymmetric contributions break cancellation.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsDetermine the full ring of quadratic integers
Prove integrality criteria and derive the correct quadratic integral basis.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDetermine the sign of work from force and displacement
Work is an energy transfer associated with a force acting through displacement. For a constant force and straight displacement, W=Fd cosθ, where θ is the angle between the force and displacement vectors. The product has units N m=J. Time does not appear in this expression; how rapidly work occurs is a separate power question.
Open lessonLesson · Science · General Chemistry IDetermine what an isotope mean leaves unidentified
Use normalization and mean mass as separate constraints, then exhibit multiple compositions that fit the same observation.
Open lessonLesson · Statistical investigations, simulation and probability decisionsDevelop a margin from repeated samples
Calibrate an empirical error margin with repeated random samples and test it on separate trials.
Open lessonLesson · Social studiesDeviance: separate behavior, detection and labeling
Deviance concerns departures from social norms; crime concerns legal prohibitions. Labeling and conflict perspectives ask how reactions and enforcement shape recorded outcomes, while strain theories investigate pressures and opportunities.
Open lessonLesson · Science · Earth and space scienceDiagnose a disturbed isotope clock
Test a hidden closure assumption and separate a model bias from evidence of a real disturbance.
Open lessonLesson · Science · AP Chemistry preparationDiagnose an atom or ion electron configuration
Use independent charge, orbital and ground-state checks instead of accepting a correct electron total.
Open lessonLesson · Science · Physical Geology and Earth SystemsDiagnose clock disagreement without averaging away an open system
A radiometric age is conditional on isotope retention and the geological event represented by closure. If daughter escapes after accumulating, a simple daughter/parent calculation can underestimate elapsed time. A younger apparent age need not mean radioactive decay slowed or reversed.
Open lessonLesson · Science · Chemistry and matterDiagnose corrosion and protection at a damaged surface
Trace anodic, cathodic, electronic and ionic paths when a coating is damaged.
Open lessonLesson · Science · AP Chemistry preparationDiagnose electrode roles and broken current paths
Identify oxidation, reduction, external electrons and ionic charge balance in both cell modes.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDiagnose numerical overshoot in a stable physical model
Compare discretization error and stability against an exact same-time solution.
Open lessonLesson · Science · Chemistry and matterDiagnose product yield and impure isolates
Use stoichiometry, balance readings and composition evidence to explain a product-yield claim.
Open lessonLesson · Science · Earth and space scienceDiagnose response curves with independent clues
Use paired cases to distinguish capacity and damping while exposing unknown-forcing ambiguity.
Open lessonLesson · Fitted regression: coefficients, uncertainty and validationDiagnose, freeze and evaluate a revision
Use residual and held-out evidence without turning fit agreement into a model-validation claim.
Open lessonExperiment · Media literacy and civic reasoningDid opinions change—or respondents?
Can a headline move while every group holds the same views?
Open experimentLesson · Course 1 · Data audits and training diagnosisDifferent mistakes have different costs
Move a threshold, count different errors, and defend a decision using precision, recall and explicit costs.
Open lessonAssessment · Differential calculus: techniques and applicationsDifferential calculus: mixed methods assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitDifferential calculus: techniques and applications
Twelve lessons extending calculus foundations through derivative rules, motion, theorem-based analysis and optimization. This is a differential calculus unit, not complete Calculus I or AP AB/BC preparation. Written arguments require rubric-based review.
Open courseCourse · Mathematics program unitDifferential equations: models, methods and dynamics
Twenty-five lessons from first-order rate models through linear systems, Laplace methods and nonlinear local analysis. Written reasoning and implementation need review; this unit alone does not establish full undergraduate ODE proficiency or examination readiness.
Open courseAssessment · Differential equations: models, methods and dynamicsDifferential equations: models, methods and verification
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitDifferential geometry of curves and surfaces
Thirty undergraduate lessons develop curvature, surface metrics and shape, connections, geodesics, Gauss–Codazzi and Gauss–Bonnet through original calculations and proofs. The convention S=−dN is used throughout. Global ODE, integrability, triangulation and topological theorem dependencies are stated explicitly; saved practice is not certification of proof quality or university-course equivalence.
Open courseAssessment · Differential geometry of curves and surfacesDifferential geometry: frames, compatibility and global obstructions
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Calculus II: series, curves and applicationsDifferentiate a curve through its parameter
Distinguish parameter velocity from geometric slope and handle zero horizontal speed.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDifferentiate a moving-origin transformation
Distinguish a constant offset, a constant observer velocity and an accelerating observer.
Open lessonLesson · Science · University Physics II with calculusDifferentiate a projected ring field
Every element of a uniform ring has the same distance to an axial observer, but only its axial projection survives the vector sum. With radius R and axial coordinate z, that projection contributes an additional z/√(R²+z²) factor. The axial field is KQz/(R²+z²)^(3/2), not simply KQ/(R²+z²).
Open lessonLesson · Differential calculus: techniques and applicationsDifferentiate a relationship without solving it
Track dependent variables and distinguish horizontal, vertical and singular points.
Open lessonLesson · Science · University Physics I with calculusDifferentiate a trajectory without losing its units
A derivative describes local change, not simply a difference between arbitrary endpoints.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDifferentiate a vector trajectory and distinguish its directions
Use fixed-basis components to separate position, velocity, acceleration and scalar speed.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsDifferentiate an accumulated quantity
Understand the first fundamental theorem through small added intervals and variable bounds.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDifferentiate an unwrapped angular record
Use angular derivatives and sign conventions without confusing an angle with its rate.
Open lessonLesson · Partial differential equations and Fourier methodsDifferentiate jumps without inventing point values
Define weak derivatives through test functions.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDifferentiate particle angular momentum about a fixed origin
Use a cross product for a moving particle and track an origin change consistently.
Open lessonLesson · Single-variable calculus foundationsDifferentiate powers and combinations
Use established derivative rules without confusing differentiation with evaluation or multiplication of functions.
Open lessonLesson · Science · University Physics II with calculusDifferentiate ring potential without losing the reference distinction
Potential contributions add as scalars, which can make source integration simpler than vector field integration. Every element of a uniform ring is the same distance from an axial point. With Q=2 nC, V(z)=18/√(R²+z²) V when the reference is zero at infinity. This finite-source reference is legitimate because the potential tends to zero at large distance.
Open lessonLesson · Science · University Physics I with calculusDifferentiate the basis vectors of circular motion
Ignoring changing directions loses an acceleration term.
Open lessonLesson · Science · University Physics I with calculusDifferentiate the observer transformation
The same events can have different coordinate velocities without different accelerations.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDifferentiate the radial basis of a rigid rotation
Derive inward and tangential acceleration from changing direction as well as changing speed.
Open lessonLesson · Calculus II: series, curves and applicationsDifferentiate, integrate and approximate inside the radius
Use power-series operations with interval restrictions and separate endpoint justification.
Open lessonLesson · Science · Chemistry and matterDilute a stock solution without losing the solute ledger
Choose an aliquot and final volume, then diagnose an incorrect preparation.
Open lessonLesson · Science · General Chemistry IIDilute both packets before deciding how much can precipitate
Supersaturation is a quotient comparison; precipitated amount additionally needs finite inventories.
Open lessonCourse · Mathematics program unitDirected random walks and stationary guarantees
Specify graph transition policies, separate stationary balance from convergence, and prove what a direct restart guarantees.
Open courseCourse · Mathematics program unitDiscontinuities and Riemann integration
Prove how one-sided limits, oscillation and small interval covers determine integrability, then test the theorem on dense and uncountable exceptional sets.
Open courseCourse · MathematicsDiscrete mathematics: logic, proof and algorithms
Define finite structures, count outcomes, construct proofs and counterexamples, and analyze graph and algorithm behavior.
Open courseAssessment · Discrete structures: counting, graphs and algorithmsDiscrete structures: count, construct and prove
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitDiscrete structures: counting, graphs and algorithms
Eighteen undergraduate lessons extending proof foundations through cardinality, counting, recurrences, graph theory, algorithm reasoning and elementary number theory. This course does not replace advanced combinatorics, graph theory or number theory; written arguments require review.
Open courseCourse · Mathematics program unitDiscrete synthesis: schedules, sampling and certificates
Connect precedence proofs, combinatorial sample spaces and dependent indicators. Use targeted returns to matching, orders, planarity and modular arithmetic without requiring whole advanced courses.
Open courseLesson · Science · Physical Geology and Earth SystemsDiscriminate a faulted site from an erosional channel using independent cores
A site reconstruction combines observations that constrain different parts of a history. Surface relief and variable gravel thickness can be compatible with several mechanisms. A correlated subsurface marker provides a different test: it can reveal whether displacement extends below a shallow erosional surface.
Open lessonLesson · Science · Physical Geology and Earth SystemsDiscriminate competing rock histories using relationships
A mineral inventory can be consistent with several geological histories. Quartz and feldspar may occur in an igneous source and in sediment derived from it. To order processes, examine relationships: inclusions, cross-cutting contacts, baked zones, depositional structures and later deformation. Each observation needs a location and an explicit interpretation.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDiscriminate DNA replication hypotheses using successive evidence
DNA stores sequence information in a polymer of nucleotides. Complementary A–T and G–C pairing relates its two antiparallel strands, while the covalent sugar–phosphate backbones give each strand a direction. Separating the strands exposes complementary templates, suggesting a copying mechanism without proving how old and new material are distributed among daughter duplexes.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDiscriminate endocrine feedback, production and clearance
Endocrine signals travel to cells bearing suitable receptors, so a circulating molecule does not impose an identical response on every tissue. Peptide signals commonly use membrane receptors and intracellular signaling, while lipid-soluble signals can act through intracellular receptors. Receptor availability and downstream response are distinct from the concentration of the signal itself.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDiscriminate equal steady outputs using an independent virtual transient
An endocrine concentration is an observed state, not a direct label for its cause. Increased secretion and reduced clearance can each raise a steady concentration; changes in volume or carrier binding can alter transient or measured-total behavior. A convincing mechanism explanation uses independent quantities rather than assigning a gland disorder from one number.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDiscriminate inhibition from loss of catalytic capacity
An enzyme lowers an activation barrier by providing a catalytic pathway. It does not change the free-energy difference between the same initial and final chemical states or make an unfavorable uncoupled reaction favorable. Binding specificity and transition-state stabilization explain why catalytic effects can depend strongly on molecular structure.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDiscriminate mechanisms and defend the full anatomical evidence portfolio
Anatomical reasoning links a visible structure to a mechanism and then to a bounded functional consequence. Similar endpoints need not imply the same mechanism. A tissue image can support recognition of a structural pattern without proving the functional state of an individual; an endpoint response can support an amount relation without revealing its production, receptor or clearance history.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDiscriminate mechanisms that fit the same allele-frequency endpoint
A model can reproduce an observation while giving the wrong causal explanation. Population frequencies can change through selection, migration, mutation and drift, often together. An exact fit to one endpoint is therefore a starting point for discrimination, not a proof of mechanism.
Open lessonConcept · GlossaryDistillation
Training a model to imitate targets from a teacher or a more expensive process such as search. Teacher errors and coverage limits can transfer too.
Open conceptLesson · Social studiesDistinguish a binding price rule from a market shift
Does a low legal price guarantee everyone can buy?
Open lessonLesson · Science · General Chemistry IIDistinguish a catalytic time scale from an equilibrium ratio
Dynamic equilibrium has equal opposing rates, not zero molecular transformation.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish a clone sample from population-wide absence
A clone descends from an ancestral cell, but later mutations and transfer can create variation within its descendants. A population can also contain several lineages. Counting many sequence reads from one cell or many descendants from one clone does not automatically provide the same independent information as sampling separate cells from a population.
Open lessonLesson · Social studiesDistinguish a commodity forecast from a completed harvest
Can a production forecast establish farmers’ final income?
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish a compensated pressure from restored fluid inventory
Physiological regulation coordinates processes on different timescales. Autonomic changes in vascular resistance and cardiac activity can alter pressure relatively rapidly, while renal changes in net fluid and solute handling alter inventories over longer intervals. A restored observed pressure therefore need not prove that the original circulating volume or initiating cause has been restored.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish a corrected ratio from replacement of a missing pool
Buffers moderate pH changes by distributing added acid or base among chemical forms, while respiration and renal handling can change material inventories. Carbon dioxide removal affects the volatile-acid system, and kidneys conserve or generate bicarbonate while excreting acid through multiple pathways. These mechanisms differ in timing and cannot all be reduced to instantaneous neutralization.
Open lessonLesson · Science · Engineering design and scientific computingDistinguish a design change from calculation error
Compare hardware and gain effects with a separately labeled numerical approximation.
Open lessonLesson · Social studiesDistinguish a displacement stock from migration flows
Does the world’s displaced population equal the number who crossed borders that year?
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDistinguish a displacement vector from a route
A displacement connects initial and final positions. Distance records the length of the route actually traveled. On a straight axis, opposite displacement components carry opposite signs; adding distances instead uses nonnegative lengths. A round trip can have zero displacement and nonzero distance. A scalar has magnitude without direction: distance and speed are examples. A vector has magnitude and direction: displacement, velocity and acceleration are examples. An arrow above a symbol denotes the whole vector; a signed x or y component does not need that arrow because the axis and sign specify its direction.
Open lessonLesson · Science · Chemistry and matterDistinguish a dissolved mixture from a pure substance
Compare total solute with the amount that can remain dissolved under fixed conditions.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceDistinguish a field null from a zero potential
For two positive point charges on a line, fields point away from each source. Between them the contributions oppose; outside them they point in the same direction. Therefore the finite cancellation point lies between the sources, not beyond an endpoint. Source positions themselves are singular and cannot be substituted as candidate nulls.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDistinguish a homeostatic target from effector capacity
Homeostasis is regulated dynamical stability, not the absence of change. A sensor, integrating pathway and effector can oppose a disturbance, but the response depends on available capacity and time. Endothermy describes substantial internal heat generation; ectothermy emphasizes environmental heat sources. Neither term alone guarantees a constant body temperature.
Open lessonLesson · Social studiesDistinguish a housing offer from neighborhood exposure
Compare the offer of a housing voucher with its use and examine how age and follow-up change the question.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish a larger image from a resolved structure
Magnification changes apparent image size; resolution concerns distinguishing nearby structures. Contrast is a further issue: a feature may be optically separable but difficult to detect against background. Bright-field, phase-contrast, differential interference contrast, fluorescence and electron microscopy form contrast through different interactions and preparation conditions.
Open lessonLesson · Science · Introductory AstronomyDistinguish a luminous population from most of a galaxy’s mass
A galaxy’s appearance depends on which stellar populations and gas components dominate the observed band. Young luminous stars can strongly affect blue or ultraviolet light while older stars contain much of the stellar mass. Dust and an active nucleus can further change the observed distribution without changing mass in the same proportion.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish a maintained compartment gradient from equilibrium
A compartment can maintain a stable composition while material continuously enters and leaves. This steady state differs from equilibrium: an energy-coupled pump can oppose a leak so that net accumulation is zero even though each one-way flux is nonzero. A constant pH reading therefore does not prove that transport has stopped.
Open lessonLesson · Science · General Chemistry IIDistinguish a pH ratio from a differential amount reserve
Equal buffer ratios need not provide equal resistance to an added amount.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish a physical barrier from a zero-material assumption
Physical defenses include epithelial barriers and mechanical movement of material. Mucus can retain particles while ciliary motion and fluid flow help remove them; intact junctions can restrict passage between cells. Host-associated communities can also compete for sites and resources. These processes interact with chemical and cellular defenses but are not identical to antigen-specific immune memory.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsDistinguish a rate from an amount of energy
Energy tells how much is transferred; power tells how quickly. Average power over an interval is transferred energy divided by elapsed time. Instantaneous mechanical power for a force and velocity is their dot product. These quantities agree only for a constant-power interval, not merely because the force is constant.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDistinguish a residual recharge estimate from an independent observation
A water budget relates inflows, outflows and storage change within a declared boundary. Precipitation can leave as evapotranspiration or surface export, remain in shallow storage, or cross a lower boundary as recharge. Recharge is an internal transfer if the aquifer is included in the larger system, but an output from the shallow-soil control volume used here.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceDistinguish a resistance ratio from a nonlinear response
Ohmic behavior is a physical model: voltage is proportional to current under specified conditions, with constant resistance. Writing R=V/I at one point does not establish that proportionality across a range. The supplied invented device instead follows V=10I+βI³. It is passive in the displayed nonnegative-current range, but is not claimed to be a particular lamp or semiconductor.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish a solute clearance from the urine volume
Renal clearance expresses the virtual plasma volume from which a solute amount is removed per time. If urine concentration is U, urine volume rate V and plasma concentration P, clearance is UV/P. It has volume/time units but is not necessarily the actual urine volume. The numerator is the excreted amount rate.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDistinguish a stable count from absence of cellular turnover
Binary fission duplicates and partitions cellular material into daughter cells. Each completed division replaces one parent cell with two daughters, so it adds one net cell to a count. Daughter-cell production and net population increase are different quantities. Cell size, biomass and viable fraction can also change without a proportional count change.
Open lessonLesson · Social studiesDistinguish a storage claim from a retrieval result
Compare free recall, cues and recognition without treating memory as a perfect recording.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish a transient peak from a regulated steady state
Negative feedback occurs when a response contributes to reducing its own upstream drive or production. A delayed inhibitor can allow an initial rise before suppressing further accumulation. Positive feedback instead reinforces a change and can help create switch-like transitions; neither label alone predicts an entire time course without the network and rates.
Open lessonLesson · Science · General Chemistry IIDistinguish a van’t Hoff inference from an activation barrier
Equilibrium temperature dependence and rate temperature dependence encode different information.
Open lessonLesson · Science · Physics and physical systemsDistinguish a virtual image from the focal-plane limit
Move the object through the focal plane and interpret backward extensions.
Open lessonLesson · Science · Physics and physical systemsDistinguish a wave snapshot from a point history
Read wavelength from a spatial graph and period from a time graph without confusing their axes.
Open lessonLesson · Science · AP Environmental ScienceDistinguish acid deposition, pH and buffering
Connect sulfur/nitrogen transformations with logarithmic concentration and ecosystem response.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish active tension, passive tension and external work
Muscle force can include active crossbridge contribution and passive resistance from stretched elastic structures. Length changes the arrangement of contractile and connective components, so force is not determined by activation alone. The same total force can combine active and passive terms in different proportions.
Open lessonLesson · Science · Physics and physical systemsDistinguish activity from remaining inventory and dose
Calculate activity from survivor number and half-life and explain why activity alone does not determine dose.
Open lessonLesson · Science · Engineering design and scientific computingDistinguish actuator capacity from an energy criterion
Explain a heating shortfall that gain alone cannot repair.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsDistinguish additive and multiplicative structure
Test ring axioms, units and zero divisors under explicit conventions.
Open lessonLesson · Science · AP Environmental ScienceDistinguish air emissions from concentration and exposure
Classify primary and secondary pollutants and connect inventories to monitoring boundaries.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDistinguish allocation conventions from avoided emissions
A process can supply more than one useful output. Assigning its whole burden to each output would double count; dividing its burden requires an explicit convention or a justified expanded-system comparison. Electricity and low-temperature heat carry different service quality, so equal energy is not the only relevant measure.
Open lessonLesson · Science · University Physics I with calculusDistinguish allowed positions from a finite-time crossing
The unit-mass conservative model U=x⁴/4−x² has two minima at±sqrt2 with energy−1 J and a central barrier U = 0=0. Mechanical energy gives K=E−U and speed magnitude sqrt(2(E−U)). Any position with U>E is forbidden in this model, but an allowed-position set need not be one connected interval.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDistinguish allowed positions from reachable motion
Use energy geometry while preserving initial direction, disconnected regions and time boundaries.
Open lessonLesson · Science · AP Chemistry preparationDistinguish alloy composition from a geometric fit
Compare replacement and insertion in a declared cage, then test what the radius calculation can establish.
Open lessonLesson · Science · Introductory AstronomyDistinguish an absorption line’s depth from its integrated deficit
Spectral lines arise when matter exchanges photons at particular transition energies. A hot dense source can supply a continuum, while excited low-density material can add emission lines and intervening material can remove light at selected wavelengths. The observed result depends on the source and viewing geometry, not merely on the presence of an element somewhere along the path.
Open lessonLesson · Science · Introductory AstronomyDistinguish an evolving core from its cool expanded envelope
After central hydrogen is depleted, a contracting core and hydrogen-burning shell can coexist with an expanding envelope. The photosphere can cool while total luminosity rises because emitting area grows. The core temperature and photospheric effective temperature are not interchangeable measurements.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDistinguish an exact fit from independent electromagnetic evidence
For an initially zero-current LR circuit, true current is (E/R)[1 − exp(−tR/L)].
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish an expected count from an outcome probability
Genetic segregation probabilities describe repeated outcomes under a model, not a schedule that each small family must obey. If a target outcome has probability p in each of n independent offspring, the expected count is np. An expectation need not be an integer and does not guarantee that any one cohort will contain exactly that number.
Open lessonLesson · Science · University Physics II with calculusDistinguish an instantaneous balance from stored charge
A node can temporarily accumulate charge. The steady junction rule is the zero-accumulation limit of dq/dt=Iin−Iout, not permission to erase an imbalance. Our ideal current sources prescribe inflow 3 mA and outflow (1+βt) mA, starting with zero stored charge. The sources are stipulated to maintain these histories independently of voltage.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsDistinguish an origin error from a failed projectile model
A model prediction and a sensor readout can disagree for different reasons. A constant horizontal origin error shifts every position by the same amount. A change in horizontal acceleration instead changes curvature over time. The supplied comparison is designed to distinguish those two explanations under explicitly exact timing and fixed spatial scale.
Open lessonLesson · Science · General Chemistry IIDistinguish analytical concentration from thermodynamic activity
Analytical concentration describes how much solute is assigned per solution volume.
Open lessonLesson · Science · AP Environmental ScienceDistinguish aquatic zones and optical evidence
Explain aquatic habitat constraints using salinity, flow, light and nutrient evidence.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish arterial storage from mean perfusion
Elastic arteries store some ejected volume as their walls expand and return it during the interval between ventricular ejections. Resistance and compliance are distinct properties: resistance links pressure drop to flow, while compliance links stored volume change to pressure change. A reservoir description can explain smoothing of pulsatile input without resolving every traveling wave in the arterial tree.
Open lessonLesson · Science · Chemistry and matterDistinguish atoms, isotopes and ions
Use proton, neutron and electron counts to distinguish element identity, isotope and charge.
Open lessonLesson · Science · General Chemistry IDistinguish average bond energies from a reaction-specific enthalpy
Count bonds and test a gas-phase approximation without treating every dissociation as identical.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDistinguish balanced mutation flows from no mutation
Mutation creates sequence changes without being directed toward an organism’s future needs. Some changes are neutral in a particular context; others affect contribution to future generations. Mutation and selection are distinct processes, so a useful mutation’s spread does not establish that environmental need caused its origin.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish balanced remodeling totals from a transient material deficit
Osteoblasts form osteoid and support mineral deposition, osteoclasts resorb bone, and osteocytes participate in sensing and coordinating the tissue environment. These roles differ from simply adding or subtracting calcium in a container. Bone stores minerals, supports loads and houses marrow, so a remodeling explanation should state which quantity is being measured.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceDistinguish beat recurrence from carrier oscillation
Two nearby frequencies produce a changing relative phase, even when both sources start in phase. The supplied signals have frequencies 200 Hz and 200+Δ Hz with peak amplitudes 1 Pa and b Pa. Their relative phase advances by 2πΔt, so the slowly varying resultant amplitude is √[1+b²+2b cos(2πΔt)]. This envelope is not the rapid pressure oscillation itself.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish binding specificity from a large mixed signal
Antibodies contain variable regions that contribute to antigen recognition and constant regions that contribute to downstream functions. Different antibody classes have different distributions and effector roles; IgA is important at mucosal surfaces, while other classes support distinct systemic or cellular interactions. Binding, neutralization, recruitment of other defenses and long-term memory are different outcomes.
Open lessonLesson · Science · General Chemistry IDistinguish bond depth, equilibrium and restoring slope
Use a supplied energy landscape without turning its shape into an unverified molecular measurement.
Open lessonLesson · Science · Introductory AstronomyDistinguish bound cloud energy from actual star formation
Star formation begins in gas whose gravity can overcome relevant support and whose energy can be redistributed or radiated. Cooling permits contraction without simply maintaining enough thermal pressure to halt it. Rotation, magnetic fields, turbulence, external pressure and feedback can matter; a dense cloud is not guaranteed to become one star.
Open lessonLesson · Science · AP Environmental ScienceDistinguish carbon dioxide, carbon monoxide and particles
Track a carbon-only balance without inferring particle exposure or toxicity.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish carbon fixation, acceptor regeneration and net export
The Calvin cycle fixes CO2 into organic intermediates in the chloroplast stroma. For a three-CO2 accounting unit, three five-carbon RuBP acceptors combine with incoming carbon and yield six three-carbon intermediates after reduction. Five trioses regenerate the three RuBP molecules; only one triose is net export in the ideal ledger.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish cargo transport changes from a smaller starting pulse
Analyze synchronous compartment transfers, normalized release and ambiguous endpoints.
Open lessonLesson · Science · General Chemistry IIDistinguish cell charge capacity, loaded energy and internal conversion
An ampere-hour is a charge quantity, not an energy specification.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish central organization, peripheral pathways and parallel outputs
The central nervous system comprises brain and spinal cord; peripheral nerves and ganglia connect it to receptors and effectors. Afferent pathways carry information toward central processing, while efferent pathways carry commands toward targets. These functional directions do not mean every axon in a named peripheral nerve has the same role: many nerves contain mixed fibers.
Open lessonLesson · Science · Physical Geology and Earth SystemsDistinguish chemical transformation from mechanical fragmentation
Mechanical weathering breaks material into smaller pieces, potentially increasing exposed area without requiring a change in mineral composition. Chemical weathering changes dissolved species or minerals through reactions. Hydrolysis can transform feldspars toward clay minerals; oxidation changes iron-bearing components; dissolution transfers constituents into water.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish cleavage from growth and lineage differentiation
Fertilization combines gametic contributions and initiates a developmental sequence, including changes that help prevent polyspermy. Early cleavage divides existing cellular material into blastomeres as the conceptus moves toward the uterus. Increasing cell number during this phase need not imply proportional increase in the aggregate volume of the cells.
Open lessonLesson · Science · Chemistry and matterDistinguish collection from usable material recovery
Follow a polymer stream through collection, processing and an explicitly assumed replacement use.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish concentration compensation from complete fluid restoration
Fluid regulation concerns amounts, volumes and concentrations, which are related but not interchangeable. Osmotic signals respond to concentration-related conditions, while volume and pressure signals reflect other aspects of the circulation. A response that normalizes one sensed variable can leave another quantity changed.
Open lessonLesson · Science · University Physics I with calculusDistinguish conservation from independence of origin
Angular momentum of a point particle is r × p about a specified origin. Conservation means that this vector is constant in time under the applicable zero-torque condition. It does not mean that every origin assigns the same constant vector.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDistinguish current growth from coexistence under competition
Competition is a negative effect associated with shared limiting resources, but its strength can differ in each direction. Resource partitioning changes overlap, while predation, facilitation and environmental variation can modify outcomes. An ecological niche includes conditions and resource use; it is not simply a geographic address.
Open lessonLesson · Social studiesDistinguish demand and supply shocks on a graph
Can falling output come with either rising or falling prices?
Open lessonLesson · Science · Introductory AstronomyDistinguish detected planets from intrinsic occurrence
A detected sample is filtered by geometry, time coverage, signal strength and analysis choices. Transit probability is larger for close orbits when other quantities are fixed, and a finite observing baseline can miss longer-period events. These biases mean a histogram of detections is not automatically a histogram of underlying planet occurrence.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish directional responses, activity and fitness
Organisms detect environmental information through receptors and respond through changes in physiology, growth or behavior.
Open lessonLesson · Partial differential equations and Fourier methodsDistinguish dispersive phases from diffusion
Solve the free Schrödinger equation by Fourier multipliers.
Open lessonLesson · Science · AP Chemistry preparationDistinguish dissolved totals from the ions in a solubility product
Audit historical brushite measurements, allocate a proposed free-phosphate fraction and expose what totals and source-model corrections cannot identify alone.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish dominance, codominance and multiple alleles
Dominance describes a heterozygote phenotype for a specified trait, not which allele is stronger or more common.
Open lessonLesson · Science · Physical Geology and Earth SystemsDistinguish downstream ice flow from retreat and thinning
Ice velocity describes movement of material, while thickness change describes local storage and terminus movement describes a boundary. These quantities can have different signs. A glacier can flow downslope while thinning or while its margin retreats upstream; retreat does not mean all ice reverses direction.
Open lessonLesson · Science · Biology and living systemsDistinguish drift from a selective advantage
Trace chance changes in finite populations with equal reproductive contribution.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesDistinguish edge traversal from vertex cycles
Prove the Euler tour construction.
Open lessonLesson · Science · AP Environmental ScienceDistinguish efficiency from conservation and rebound
Compute service-based energy savings while checking changed use and omitted burdens.
Open lessonLesson · Science · Physics and physical systemsDistinguish elastic and inelastic collisions
Compare separation speed, kinetic energy and internal-energy increase without losing momentum conservation.
Open lessonLesson · Science · AP Chemistry preparationDistinguish electron attachment from electron removal
Use selected experimental affinities to test signs, species and a periodic exception.
Open lessonLesson · Science · AP Chemistry preparationDistinguish energetic collisions from suitably oriented ones
Inspect a joint collision ledger, challenge independence and separate accepted fractions from event frequency.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish enzyme perturbations with controlled evidence
Compare inhibition and enzyme amount while testing environmental and measurement alternatives.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDistinguish equal potential from equal field
Use scalar superposition and vector projection as different measurements of a source.
Open lessonLesson · Science · General Chemistry IDistinguish equilibrium vapor pressure from evaporation speed
Use a supplied temperature relation while keeping phase and reference assumptions explicit.
Open lessonLesson · Science · Introductory AstronomyDistinguish expansion rate, distance and lookback time
A cosmological model relates expansion to matter, radiation, spatial curvature and other energy components. Matter dilutes as the inverse cube of scale factor, while a cosmological-constant energy density remains constant. A fitted model must be tested against observations; choosing its parameters in a workbench does not measure them.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish expression abundance from production and removal
Transcription copies information from a DNA template into RNA; translation reads an RNA message in a specified frame to build a polypeptide. A promoter helps position transcription machinery, and regulation can alter initiation without changing a coding sequence. In bacteria, transcription and translation can overlap in space and time because there is no nuclear envelope separating them. That does not mean every transcript is translated identically or that all microbes share bacterial expression architecture.
Open lessonLesson · Science · AP Chemistry preparationDistinguish fixed-volume and fixed-pressure equilibrium disturbances
Compare immediate and re-equilibrated partial pressures, including the changing volume after inert gas is added at fixed total pressure.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDistinguish four thermal-process constraints
Words such as isothermal and adiabatic describe different constraints. Isothermal means constant temperature; adiabatic means no heat transfer. One does not imply the other. Isochoric means constant volume, and isobaric means constant pressure. Always identify which condition is being imposed before selecting an energy relation.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsDistinguish free decay from a driven resonance record
A freely oscillating damped system loses mechanical energy to other forms. A periodically driven system continually receives energy from an external source. These are different experimental boundaries: a large steady driven amplitude is not evidence that damping vanished or that the oscillator created energy.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDistinguish gravitational field, mass and weight
For two point masses separated by center-to-center distance r, the gravitational attraction has magnitude GMm/r². A spherically symmetric body produces the same external field as if its mass were concentrated at its center. The inverse-square expression here applies outside that body; it is not a general formula for the field inside an arbitrary distribution.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceDistinguish grazing refraction from total internal reflection
Refraction changes direction when a wave crosses a stationary boundary between different propagation speeds. For light, index n is the vacuum speed divided by the medium phase speed. Snell law is n1 sinθ1=n2 sinθ2, with both angles measured from the normal. The boundary preserves frequency; changing speed therefore changes wavelength rather than requiring the source to change its oscillation rate.
Open lessonLesson · Science · Environmental science and sustainabilityDistinguish habitat amount from arrangement
Compare equal-area habitat maps and audit a published forest-configuration indicator.
Open lessonLesson · Science · Physics and physical systemsDistinguish heat-transfer rate from stored energy
Use a temperature profile to calculate steady conduction and accumulated transfer.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish horizontal routes using bounded mechanism evidence
Vertical inheritance follows descent, whereas horizontal transfer introduces genetic material across cell lineages. Transformation involves uptake of extracellular DNA; transduction involves a viral carrier; conjugation involves direct cell-associated transfer machinery. These are conceptual routes, not complete descriptions of every biological case. Transfer need not imply stable maintenance, expression or a particular phenotype.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDistinguish hydraulic residence from a pulse-removal time
A mean residence time describes a distribution of passage times, not a promise that every parcel exits at that time. A completely mixed reservoir contains material of different ages. Continuous outflow carries a fraction of the current tracer inventory, so an instantaneous pulse declines gradually rather than leaving as one intact packet.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingDistinguish impulse from a force-pulse maximum
Construct equal-impulse histories that make different force demands.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish incidence from prevalence in a closed record
Epidemiological summaries describe a defined population and interval. Point prevalence counts the current state at one time; incidence concerns new events among those initially at risk. They need not move together because duration, resolution, migration and competing events can change how many existing states remain at the end.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDistinguish internal recoil from an external impulse
The change in total momentum of a chosen system equals the net external impulse on that system. Internal interaction impulses occur in equal and opposite pairs and cancel when the members’ momentum changes are summed. They do not cancel on each individual object, whose momentum can change substantially.
Open lessonLesson · Science · AP Environmental ScienceDistinguish irrigation efficiency from basin water savings
Compare irrigation methods, return flows, salinity and waterlogging.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish linked regulatory DNA from shared cellular factors
Eukaryotic gene expression depends on promoter recognition, transcription factors, enhancer interactions and chromatin accessibility. A cis-regulatory element is part of the DNA context linked to the regulated sequence. A trans-acting factor is a diffusible product that can act at compatible sites on more than one DNA molecule. Their effects can be separated by comparing reporters in a shared cellular environment.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDistinguish local richness from compositional differences
Two sites can have equal richness while containing very different species. Local richness describes the number within a site; pooled richness describes the combined set. Comparing these levels makes spatial composition explicit instead of reducing biodiversity to one local count.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDistinguish mirror image signs and the focal limit
A concave spherical mirror is converging in the paraxial approximation; a convex mirror is diverging. With the real-is-positive convention, concave focal length is positive and convex focal length negative. The signed radius of curvature is twice the focal length for this approximation. Rays far from the principal axis require attention to spherical aberration, which is not modeled here.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsDistinguish no solution, one solution and a family
Read pivots, contradictions and free variables without counting equations alone.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish occupied binding sites from total transported oxygen
Most oxygen carried in blood is associated with hemoglobin in red cells, while a smaller amount is physically dissolved. Partial pressure relates to the dissolved gas and its exchange tendency, whereas saturation describes the fraction of available binding sites occupied. Content measures carried oxygen per blood volume and depends on binding capacity as well as saturation.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceDistinguish one segment force from whole-loop torque
A current-carrying segment in a uniform magnetic field experiences F=I L_vector cross B, where L_vector follows conventional current. Here the segment lies in the xy plane, has length .2 m and makes angle θ from +x toward +y. The magnetic field is .05 T along +x. Only the y component of the segment contributes, giving Fz=−I(.2)(.05)sinθ.
Open lessonLesson · Science · Physical Geology and Earth SystemsDistinguish ore grade, dilution and recovered metal
Mineral deposits concentrate material through geological processes: segregation during crystallization, transport and precipitation by hot fluids, or sedimentary concentration can produce different geometries and mineral associations. A useful resource interpretation connects those processes to observed host rocks, alteration and structure rather than assuming that every metal-bearing rock is economic ore.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish passage through a checkpoint from successful repair
Cell-cycle progression depends on regulated protein activity, not an automatic clock alone. Cyclins and cyclin-dependent kinases coordinate transitions, while inhibitors and checkpoint pathways can restrain progression when replication, DNA integrity or spindle attachment is inappropriate. Cyclin abundance, kinase modification and protein degradation contribute different kinds of control.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish passive driving forces from transport rates
Combine chemical and electrical contributions and separate net drive, route availability and measured movement.
Open lessonLesson · Science · AP Environmental ScienceDistinguish persistence, bioaccumulation and biomagnification
Track environmental residue and trophic concentrations as separate quantities.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish persistent recruitment from slow response resolution
Innate responses combine tissue sensing, soluble mediators and cellular action. Neutrophils and macrophages can engulf material, complement can mark targets and participate in other defensive effects, and inflammatory mediators change vascular recruitment and local conditions. Natural killer cells respond to altered cellular signals without using the same antigen-receptor selection process as adaptive T cells.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish pH from a buffer’s finite reserve
Water polarity organizes interactions among ions and polar groups. Hydrogen bonds are intermolecular attractions, not the covalent O–H bonds within a water molecule. The concentration of hydrogen ions influences protonation and can therefore change molecular interactions; pH is a logarithmic description of that chemical environment, not a measure of total acid in every form.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDistinguish photon energy, count rate and missing stopping evidence
In the photoelectric effect, an absorbed photon can transfer energy to an electron. The maximum emitted-electron kinetic energy is Kmax=hf−φ when the photon exceeds the surface work function φ. A retarding potential of magnitude Vs stops the fastest electrons when eVs=Kmax. Work function is a material/surface property, not energy accumulated indefinitely by one electron from a dim classical wave.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceDistinguish pollination, double fertilization and seed-resource limits
Pollination transfers pollen to a receptive reproductive structure; it is not itself fertilization. In angiosperms, pollen carries a reduced male gametophyte, and sperm delivery can lead to two fusion events. The female gametophyte and maternal sporophyte tissues contribute different parts of the developing seed.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDistinguish potential cancellation from field cancellation
Electric potential V is energy per unit charge, measured in joules per coulomb or volts. With zero potential chosen at infinity, a point source contributes V=kQ/r. Add the signed scalar contributions from sources. Unlike field arrows, potential contributions do not require direction components. A negative potential is meaningful relative to the chosen reference, not an indication of negative distance.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsDistinguish primality proofs from probable-prime tests
Prove both directions of Wilson’s criterion.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish protein organization from backbone cleavage
Use sequence, interactions and structural levels without mistaking a schematic for a predicted fold.
Open lessonLesson · Science · University Physics I with calculusDistinguish radial laws that share angular conservation
Any central force has zero torque about its center, so angular-momentum conservation alone cannot identify an inverse-square law. The supplied generator has inward acceleration μ/r² + q/r³, while the candidate model assumes only μfit/r². This is a synthetic model-comparison exercise, not a claim that a measured gravitational field has this correction.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceDistinguish radiative forcing from transient and equilibrium response
Radiative forcing is an imposed change in the energy budget; temperature response modifies outgoing energy. A continuing imbalance changes stored heat. Current warming rate, current imbalance and eventual equilibrium response are therefore related but are not interchangeable measurements.
Open lessonLesson · Science · University Physics II with calculusDistinguish radius-ratio invariance from field scale
For a very long coaxial pair, a Gaussian cylinder gives E=λ/(2πε0r)=2Kλ/r in the gap. Integrate radially from inner radius a to outer radius b to obtain ΔV=2Kλln(b/a). The logarithm requires a dimensionless radius ratio; a lone logarithm of a measured length would hide a reference.
Open lessonLesson · Science · Introductory AstronomyDistinguish reaching a radius from joining its circular orbit
A circular orbit requires a particular tangential speed at a given radius. Reaching that radius on an eccentric orbit does not supply the same velocity. The original comparison uses a central μ=1, an initial circular radius 1 and a larger target circular radius R. It isolates two ideal instantaneous prograde impulses in one plane; these are mathematical boundaries, not instructions for operating a real vehicle.
Open lessonLesson · Science · Biology and living systemsDistinguish recycling from ecological recovery
Compare exported matter, redistribution and claims about recovery.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsDistinguish redistribution from a change of rotational axis
Rotational inertia measures how mass distribution affects angular acceleration about a chosen axis. A point contributes mr², using perpendicular distance to that axis. For several points, add those contributions. Two arrangements with equal total mass can have different inertia because squared distances, not mass alone, set the weighting.
Open lessonLesson · Algebra II and precalculus: functions and structureDistinguish repeated addition from repeated multiplication
Use explicit and recursive sequence rules and derive finite sums with clear indexing.
Open lessonLesson · Science · University Physics II with calculusDistinguish required electric sources from a forbidden magnetic gradient
Maxwell equations can be used to audit a proposed field instead of merely generating a familiar wave. In a local Cartesian region take static E=(a x,0,0), where a has units V/m². Compare a transverse magnetic gradient B=(0,gx,0) with a longitudinal one B=(gx,0,0); the numerical coefficient g is a times 10⁻⁶ T/m under the explicitly linked teaching controls.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish resistance, tolerance and a heterogeneous endpoint
Resistance, tolerance and persistence are distinct evidence concepts. Resistance concerns an increased ability to grow under an inhibitory agent relative to a susceptible comparison. Tolerance concerns slower killing without necessarily shifting the growth-inhibition threshold; persistence is a heterogeneous tolerant subpopulation rather than simply any infection that lasts a long time. These phenotype definitions do not by themselves identify one molecular mechanism.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceDistinguish resolving ability from image enlargement
Magnification describes how large an object appears. Resolution describes the ability to distinguish nearby structures as separate under a specified criterion. Enlarging a blurred record can make the blur larger without restoring spatial information that the instrument did not capture.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish respiratory disruptions with matched evidence
Compare ATP synthesis, oxygen consumption and proton-gradient evidence across controlled synthetic interventions.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDistinguish scalar pressure from a normal force
Pressure describes perpendicular force per area, P=Fnormal/A, measured in pascals: 1 Pa=1 N/m². It is scalar, not a force vector. The same local pressure produces forces in different directions on differently oriented surfaces because each force acts normal to its own surface. In a static ideal fluid, there is no sustained shear stress in the model.
Open lessonLesson · Science · Engineering design and scientific computingDistinguish scenario coverage from field reliability
Interpret a fixed public suite without inventing probabilities or independent replications.
Open lessonLesson · Science · Earth and space scienceDistinguish seasonal geometry from changing solar output
Use controlled comparisons to challenge a causal explanation and design independent observational tests.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish selection by people from inheritance and trade-offs
Artificial selection occurs when people preferentially breed individuals with chosen traits.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish sensory contrast from absolute stimulus intensity
Sensory receptors transduce physical or chemical changes into cellular signals. Different receptor and pathway properties encode stimulus location, intensity, timing and quality. A receptive field specifies the stimulus region influencing a neuron’s response; it is not the same as the whole organ or the person’s entire perceptual field.
Open lessonLesson · Science · AP Environmental ScienceDistinguish sewage treatment targets and residual streams
Separate settling, biological processing, nutrient removal and disinfection.
Open lessonLesson · Social studiesDistinguish shared authority from the disappearance of states
Does joining a supranational institution give it authority over every decision?
Open lessonLesson · Science · Chemistry and matterDistinguish solution amount from concentration
Use final solution volume to connect a solute inventory with concentration.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsDistinguish source motion from force on a recipient
A positive source charge q moves along x with speed v; the observation point is at displacement r = (0,y,1 m) from its instantaneous source position.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish stabilizing feedback from reinforcement
Negative feedback makes a response oppose a deviation in a controlled variable.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsDistinguish stable feedback from perfect restoration
Homeostasis maintains regulated conditions within a workable range through interacting sensors, integrating mechanisms and effectors. A regulated variable, its sensed representation and the effector output are different quantities. Negative feedback opposes a deviation; positive feedback reinforces a change during a bounded process and requires a terminating mechanism. Neither phrase by itself specifies response speed, stability or the final value.
Open lessonLesson · Science · Physics and physical systemsDistinguish static and sliding friction
Choose the contact regime before calculating friction, net force and acceleration.
Open lessonLesson · Science · Physics and physical systemsDistinguish stopping from reversing
Find a zero-velocity instant and decide whether motion actually remains stopped.
Open lessonLesson · Orders, graph representations and certificatesDistinguish strict and non-strict partial orders
Check order axioms, separate extreme-element notions and identify where finiteness matters.
Open lessonLesson · Science · AP Chemistry preparationDistinguish successive bond breaking from a mean bond value
Build a gas-phase enthalpy ledger from reference values, cancel shared intermediates, and test a bond-energy claim.
Open lessonLesson · Science · Introductory MicrobiologyDistinguish taxon abundance from a measured community function
Community membership and community function are related but not identical. Different populations can contribute to a similar transformation, while a population can change its activity across contexts. Functional redundancy therefore refers to a specified process under specified conditions, not a guarantee that all taxa are interchangeable or that diversity never matters.
Open lessonLesson · Science · Physics and physical systemsDistinguish temperature, heat and heat capacity
Compare intensive temperature with energy transfer and sample size.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityDistinguish the data from the statistic
Construct a small exact sampling distribution and identify bias and variability.
Open lessonLesson · Social studiesDistinguish the wage from marginal hiring cost
Why can hiring one more worker raise the wage bill by more than their wage?
Open lessonLesson · Science · Chemistry and matterDistinguish titration equivalence, detection and sampling
Track reaction amounts and total volume while comparing true equivalence with a sampled endpoint.
Open lessonLesson · Science · AP Chemistry preparationDistinguish titration landmarks from indicator endpoints
Label curve landmarks and evaluate complete indicator transitions against a stated volume tolerance across matched strong and weak systems.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceDistinguish transcriptional control from downstream effects
Cells regulate expression at transcription, RNA processing/stability, translation and protein modification/degradation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDistinguish transferred charge from polarization
Begin with a system boundary, not a picture of plus and minus symbols. If electrons leave object A and enter object B, A loses negative charge and becomes positive. B gains negative charge. The combined A+B system retains its original net charge. Friction or contact can transfer charge; neither process creates a new supply of net electric charge.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsDistinguish uncertainty from omitted thermal pathways
Inferring a material property requires a model of where energy went. The synthetic hot sample cools into water and a cup. Its true specific heat is supplied so that you can test the inference, but the inference equation deliberately ignores an adjustable heat loss. A second control introduces a final-temperature sensor offset.
Open lessonLesson · Science · Earth and space scienceDistinguish unrest, eruption and a revised report
Reconcile preliminary and retrospective notices without mixing event, report and measurement dates.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceDistinguish useful conduction, leakage and a changed response law
A synthetic heater maintains a hot body while a cold calorimeter and ambient stay at 300 K. The first hot temperature is 400 K. In a linear model the useful path has conductance K and the leak has L, both in W/K. Heater power is 100(K+L); independent cold-side calorimetry reads 100K. The two measurements must have genuinely independent calibration premises.
Open lessonLesson · Science · AP Chemistry preparationDistinguish weak corrections from cancelling gas effects
Use a controlled nonideal model to test competing attraction and finite-size explanations.
Open lessonLesson · Science · Earth and space scienceDistinguish weathering from movement
Compare fictional fragmentation, alteration, transport and ambiguous-color evidence.
Open lessonLesson · Science · University Physics II with calculusDistinguish zero net force from stable orientation
A closed loop in a uniform magnetic field can have zero net force and a nonzero torque. For constant B, the integral of I dl crossed with B vanishes because the closed displacement integral is zero. Opposite forces acting at separated parts of the loop can nevertheless form a couple.
Open lessonConcept · GlossaryDistributed representation
A representation in which combinations of coordinates carry useful distinctions, rather than assigning each concept to one isolated neuron.
Open conceptCourse · Mathematics program unitDistribution calculations: events, tails and cutoffs
Use bounded local calculators to evaluate binomial events and normal regions, recover cutoffs and independently check inputs, assumptions and numerical results. Public practice is separate from demonstrated calculator proficiency.
Open courseCourse · Mathematics program unitDistribution diagnostics and calibration
Inspect distribution discrepancies, derive valid null references, refit estimated models and defend categorical model revisions.
Open courseConcept · GlossaryDistribution shift
A change in the conditions or patterns of the data, which can make previous performance estimates unreliable.
Open conceptLesson · Geometric constructions, circle proofs and coordinate lociDivide a directed segment in a chosen ratio
Connect internal ratios to affine coordinates and distinguish extrapolation from division inside a segment.
Open lessonLesson · Algebra II and precalculus: functions and structureDivide polynomials and interpret the remainder
Use the division identity to connect evaluation, remainders and factors.
Open lessonLesson · Social studiesDivided Europe, integration and post-1989 change
Compare integration, sovereignty and social change across postwar Europe.
Open lessonLesson · Science · University Physics I with calculusDo not mistake a balanced vector ledger for a passive collision
A planar momentum equation determines a missing vector when all masses and the other vector are supplied. It does not by itself prove that the event satisfies an additional energy restriction. Here a 2 kg particle enters with velocity (3, 0) m/s and a 1 kg particle starts at rest, so total momentum is (6, 0) kg m/s and initial kinetic energy is 9 J.
Open lessonExperiment · Scientific reasoning and experimental designDoes the treatment work, or did the groups differ?
Can an apparent benefit reverse while the treatment itself stays unchanged?
Open experimentExperiment · Applied mathematicsDoes twice the speed mean half the journey?
How much time can faster travel actually save?
Open experimentLesson · Single-variable calculus foundationsDomains, composition and radian measure
Check whether an expression is defined before composing functions or measuring a changing angle.
Open lessonCourse · Mathematics program unitDomains, invariants and structural counting
Find the assumptions that make a composition, accumulated reward, reachability argument or structural count valid, and repair proofs when those assumptions change.
Open courseLesson · Multivariable calculus: geometry, change and accumulationDomains, level sets and multivariable limits
Read surfaces through slices and distinguish necessary path tests from a genuine limit argument.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationDouble integrals, accumulation and density
Construct area accumulation and distinguish signed integrals from physical mass.
Open lessonLesson · Science · Environmental science and sustainabilityDraw a water-system boundary
Distinguish stores, actual flows and unmet requests in a daily water ledger.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesDraw complex arithmetic and midpoints
Construct complex sums, differences and conjugates on equal-scale coordinate axes.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsDraw forces on one object without double counting
A free-body diagram isolates one object or an explicitly stated system. Each arrow names an interaction acting on that object: Earth on cart, track on cart, or string on cart. Forces the cart exerts on something else belong on a different diagram. Velocity and acceleration can be shown nearby but are not additional force arrows.
Open lessonLesson · Science · Physics and physical systemsDraw forces on one selected object
Choose the cart as a system and separate its forces from forces it exerts on other bodies.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryDuality and cross-ratio
Find invariants of projective incidence and one-dimensional maps.
Open lessonCourse · ScienceEarth and space science
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseLesson · Social studiesEast Asian institutions: learning, office and dynastic change
Explain how education and official recruitment connected cultural authority to government while distinguishing ideals from access.
Open lessonCourse · Social studiesEconomics and public choices
Hands-on social studies course and project
Open courseLesson · Research, editing and publishingEdit with reasons and a visible decision trail
Use an editorial brief, version comparison and response memo to accept, adapt or decline changes responsibly.
Open lessonLesson · Social studiesEducation: distinguish learning from credential screening
Functionalist, conflict and interactionist perspectives ask different questions about education: preparation and socialization, unequal access, and everyday expectations. Credential screening and acquired skill are distinct possible mechanisms.
Open lessonConcept · GlossaryEffective sample size (particle weights)
The reciprocal of the sum of squared normalized particle weights. Measures weight concentration; it is not a count of unique states, independent observations or correct hypotheses.
Open conceptLesson · Linear algebra over finite fieldsEigenvalues change with the field
Keep the scalar field visible while solving eigenvector equations, constructing extension fields and testing diagonalization.
Open lessonCourse · Mathematics program unitElementary number theory: divisibility, reciprocity and Diophantine equations
A proof-based undergraduate sequence in integer arithmetic, congruences, arithmetic functions, reciprocity and Diophantine methods. Exact calculations support proofs; implementation and independent assessment remain separate evidence.
Open courseLesson · Linear algebra over finite fieldsEliminate over finite fields and count solution fibers
Use reversible row operations to classify modular systems, then prove how rank counts both attainable outputs and solutions above each output.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryElliptic geometry and antipodal identification
Compare the sphere with its projective metric quotient.
Open lessonConcept · GlossaryEmbedding
An ordered list of numbers representing something such as a word, image, or game state.
Open conceptLesson · Social studiesEmotion: arousal, appraisal and expression
Compare emotion accounts using component-specific evidence without inferring emotion from one physiological signal.
Open lessonLesson · Social studiesEmpires, authority and ethical traditions
Compare imperial authority and belief without treating official ideals as uniform practice.
Open lessonLesson · Distribution diagnostics and calibrationEmpirical CDFs and QQ evidence
Compute both sides of tied empirical jumps and explain QQ plotting positions.
Open lessonLesson · Science · University Physics II with calculusEnclose a nonuniform current and its return path
Ampère’s law becomes a local-field formula only after the symmetry is justified. The ideal source here is infinitely long and coaxial: an inner conductor of radius 1 m carries +4 A with J=J0[1+α(r/R)²], with R=1 m, while a uniform annulus from radius 2 to 3 m carries −4 A. Vacuum permeability and no end effects are stipulated.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesEncode a labeled tree reversibly
Construct and decode Prüfer sequences.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsEncode abelian groups with integer relation matrices
Prove reversible presentation changes and finite generation of integer subgroups.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceEncode and verify a matrix solution
Keep variable order, inverse existence and original equations visible.
Open lessonCourse · Mathematics program unitEncodings, diagonal arguments and limits of algorithms
Encode finite descriptions unambiguously, construct diagonal counterexamples with the correct domain, and prove why no total algorithm decides every halting question. Finite demonstrations and idealized computation have explicit limits.
Open courseLesson · Science · College Physics II — algebra-based fields and evidenceEnd an electric trajectory at its first boundary event
A uniform electric field gives a passive charge constant force qE. In this fictional model the bead has q=1 nC and mass 10⁻⁸ kg, enters halfway between plates with horizontal velocity and zero vertical velocity, and experiences no gravity. The vertical acceleration is a_y=qE/m=0.1E in SI units. Horizontal velocity stays constant because the ideal field is vertical.
Open lessonLesson · Partial differential equations and Fourier methodsEnforce boundary conditions through modes and reflection
Solve fixed-end waves and explain reflections.
Open lessonLesson · Course 2 · Geometry and control foundationsEnforce operating limits
Keep learned proposals separate from action checks, watchdogs, collision prevention and latched braking.
Open lessonLesson · Social studiesEngel and Yoder: two religion clauses
The Establishment Clause limits government establishment of religion; the Free Exercise Clause protects religious exercise.
Open lessonCourse · ScienceEngineering design and scientific computing
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseSubject · English language artsEnglish language arts
Hands-on reading, writing and revision for high-school and college learners.
Open subjectExperiment · Systems thinking and operationsEnough average capacity, yet a queue
Why can a queue form even when average demand equals capacity?
Open experimentExperiment · Environment and sustainabilityEnough energy, at the wrong time
Why can equal daily generation and demand still require grid energy?
Open experimentLesson · Social studiesEnslavement, family and cultural survival
Analyze slavery as a coercive system while attending to agency and community.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceEnumerate allowed labels before making a degeneracy claim
For a one-electron central Coulomb problem, a principal quantum number n is a positive integer. Orbital angular label l takes values 0 through n−1. For each l, magnetic projection label m takes integer values −l through +l. A proposed label outside this set is rejected rather than rounded or clipped into a different state.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesEnumerate permutations by cycle structure
Count permutations with a specified cycle type.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceEnumerate the evidence resolution of four randomized pairs
Randomization supplies a reference distribution through the assignment mechanism, not through an assumption that every observed difference is normally distributed. Under a sharp no-intervention-effect null and independent within-pair assignment, swapping the treatment/comparison labels changes each pair contrast’s sign. Four pairs produce 16 equally likely assignments.
Open lessonSubject · Experiment collectionEnvironment and sustainability
Balance flows over time and inspect the trade-offs hidden by totals.
Open subjectCourse · ScienceEnvironmental science and sustainability
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceEnvironmental Science: ecological systems and evidence
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseConcept · GlossaryEpisodic memory
Records of particular events or experiences, with their context, actions, and outcomes.
Open conceptLesson · Joint information and adaptive strategiesEquality events and joint information
Prove what a uniform independent comparison reveals, then distinguish full-pair independence from separate marginal checks.
Open lessonLesson · Compactness of function familiesEquicontinuity and common control
Distinguish local family-wide continuity from uniform control, and turn compactness and pointwise bounds into rigorous finite-grid certificates.
Open lessonLesson · Science · Physical Geology and Earth SystemsEstablish stratigraphic facing before applying superposition
A vertical sequence is not automatically a chronological sequence. Superposition orders sedimentary layers at deposition, but later overturning can place an older bed physically above a younger one. Original horizontality helps reconstruct deformation; it does not promise that present layers remain horizontal or upright.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsEstimate a difference without pooling it away
Use separate group variance estimates for an interval.
Open lessonLesson · Science · Earth and space scienceEstimate a lifting condensation level
Find where simplified parcel temperature and dew-point paths meet, then apply a separate saturated cooling rule.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsEstimate a mean when population spread is unknown
Use a t reference and investigate shape, units and independence.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsEstimate a population proportion
Construct and critique the introductory normal interval.
Open lessonLesson · Science · Physics and physical systemsEstimate a source model from measurements
Infer emf and internal resistance, then challenge the estimate with an unused reading.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceEstimate abundance with sampling assumptions and uncertainty
Population size counts individuals within a defined boundary; density divides abundance by area or volume.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsEstimate an integral and justify an error bound
Compare rules, refinement evidence and derivative-based guarantees.
Open lessonLesson · Course 1 · What has the network learned?Estimate depth and pose
Move paired cameras, reconstruct points and challenge pose estimates with wrong correspondences.
Open lessonLesson · Science · Physics and physical systemsEstimate focal length from repeated image measurements
Plan a screen experiment, analyze a labeled synthetic alternative and report uncertainty.
Open lessonLesson · Science · Earth and space scienceEstimate ideal groundwater transit and its limits
Convert Darcy flux to average pore velocity and diagnose zero-flow and transport-model limits.
Open lessonLesson · Growth estimates and controlled approximationEstimate products without losing the error
Turn positive products into logarithmic sums, prove factorial area bounds, and use a supplied Stirling remainder with explicit error semantics.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsEstimate sampling variability without inventing new information
Resample units correctly and separate bootstrap uncertainty from bias repair.
Open lessonLesson · Calculus through supplied graphs and measured tablesEstimate totals from unequal-time measurements
Build interval-weighted sums and identify the additional assumptions needed for exact totals or guaranteed bounds.
Open lessonCourse · Mathematics program unitEuclidean and coordinate geometry foundations
Twelve lessons in high-school Euclidean and coordinate geometry: diagrams, constructions, calculations and written arguments. This foundations unit is not the complete geometry and trigonometry curriculum; solid geometry, conics and the laws of sines and cosines follow separately.
Open courseCourse · Social studiesEuropean history
Hands-on social studies course and project
Open courseCourse · Social studiesEuropean history survey
Hands-on social studies course and project
Open courseLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceEvaluate a foraging return with search and handling included
A behavioral model needs an explicit objective and denominator. Energy per food item differs from energy gained per time, and gross gain differs from net gain after energetic costs. A decision that maximizes one measure may not maximize survival, nutrient balance or lifetime reproductive success.
Open lessonLesson · Course 2 · Geometry and control foundationsEvaluate a robotic system
Freeze policies and account for complete episodes, failures and exposure.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceEvaluate a sequence change in its reading-frame context
Translation reads mRNA codons in a defined frame. A ribosome coordinates mRNA with charged tRNAs whose anticodons recognize codons; aminoacyl-tRNA synthetases help establish the amino-acid assignments carried by those tRNAs. Ribosomal RNA contributes to peptide-bond catalysis. A stop codon recruits termination machinery rather than adding a stop amino acid.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsEvaluate an energy transfer rate and a power constraint
Average power is energy transferred or converted divided by elapsed time. A joule per second is a watt, W. Two devices can perform the same work but require different power if they take different times. Energy and power therefore need different units and answer different design questions.
Open lessonLesson · Science · Earth and space scienceEvaluate circular-orbit evidence and limits
Audit an analytic solution while distinguishing internal checks from real trajectory validation.
Open lessonLesson · Science · AP Environmental ScienceEvaluate climate trends and counterfactual explanations
Compare synthetic observed/model packets and distinguish mitigation from adaptation.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceEvaluate compartment function and endosymbiotic origins
Compare cellular organization and weigh sourced ancestry evidence against bounded alternatives.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceEvaluate disturbance and recovery with a reference comparison
Disturbance can change species composition, resource availability and ecosystem processes. Primary succession begins where important biological legacies such as soil are absent or greatly reduced; secondary succession retains more of those legacies. Recovery need not follow one fixed sequence toward a single inevitable endpoint, because history, dispersal and altered conditions matter.
Open lessonLesson · Science · Earth and space scienceEvaluate eclipse claims against model evidence
Distinguish alignment diagnostics from finite-size eclipse calculations and dated observations.
Open lessonLesson · Science · AP Environmental ScienceEvaluate ecosystem services without double counting
Compare a partial service ledger while identifying omitted and distributed benefits.
Open lessonLesson · Science · AP Environmental ScienceEvaluate endangered-species recovery beyond a single probability
Distinguish local persistence, species status and evidence for recovery.
Open lessonLesson · Science · AP Environmental ScienceEvaluate endocrine-response evidence without a dose shortcut
Connect signaling disruption, timing and controlled response comparisons.
Open lessonLesson · Science · AP Environmental ScienceEvaluate ENSO evidence without a deterministic local forecast
Connect equatorial winds, upwelling and ocean anomalies while preserving uncertainty.
Open lessonLesson · Science · Earth and space scienceEvaluate greenhouse evidence and model limits
Separate conservation checks, model consequences and independent observations needed for attribution or prediction.
Open lessonLesson · Science · AP Environmental ScienceEvaluate invasive-species prevention and conditional control
Separate non-native status, ecological harm and intervention arithmetic.
Open lessonLesson · Course 1 · How does AI process language?Evaluate language tasks with evidence
Design a rubric and reserved task suite, inspect actual model failures, and distinguish token loss, source support and task success.
Open lessonLesson · Course 2 · Learning through actionsEvaluate learning fairly
Compare complete seed cohorts, data and fitting budgets, frozen checkpoints and task success under a misleading reward bonus.
Open lessonLesson · Social studiesEvaluate policy timing and uncertain effects
Will a delayed response arrive after conditions change?
Open lessonLesson · Science · University Physics I with calculusEvaluate power on the same trajectory whose work you claim
A unit-mass point starts from rest at x(0) = 0 with net force F=k t. Integrating Newton's law gives v=k t²/2 and x=k t³/6. The force depends on time, so work can be evaluated as integral F(t)v(t)dt along this particular trajectory rather than pretending the endpoint force acts through the entire displacement.
Open lessonLesson · Social studiesEvaluate security beyond a single total
Security can refer to state protection and to people's access to safety, food and livelihoods.
Open lessonLesson · Social studiesEvaluate social change through mechanisms
Collective action coordinates people around shared concerns.
Open lessonLesson · Science · AP Environmental ScienceEvaluate urban runoff and ecological-footprint claims
Connect land cover, green infrastructure and demand boundaries.
Open lessonConcept · GlossaryEvent graph
A representation with events as nodes and relations such as before or after as edges, often retaining sources and timestamps.
Open conceptStudio · Optional P07 implementation studioEvidence assistant implementation studio
Dependency-free Python retrieval, exact tool receipts, corpus provenance and end-to-end public workflow tests.
Open studioLesson · Partial differential equations and Fourier methodsEvolve a profile by decaying spatial modes
Solve homogeneous boundary heat problems by separation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceEvolve interference while retaining fixed energy probabilities
Prepare equal coherent amplitudes of the first two stationary infinite-well states. After removing an irrelevant overall phase, psi=[phi1+exp(i theta)phi2]/sqrt(2), where theta=theta0−(E2−E1)t/hbar. Because E2=4E1, the relative angular frequency is 3E1/hbar, not E1/hbar.
Open lessonCourse · Mathematics program unitExact factorization and repeated solves
Prove factorization conditions, verify row-permutation receipts and reuse exact factors without confusing singularity, free variables or arithmetic cost with numerical stability.
Open courseLesson · Course 1 · Data audits and training diagnosisExamples are measurements
Inspect missing values, copied requests, unit mismatches and conflicting labels before treating a table as evidence.
Open lessonAssessment · Exact factorization and repeated solvesExceptional pivots, late swaps and reusable receipts
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesExchange and adaptation: when did horses spread?
Trace selected exchanges and test horse-dispersal claims using date ranges, research evidence and Indigenous perspectives.
Open lessonLesson · Social studiesExchange networks and states, c. 1200–1450
Compare land and sea networks and trace more than goods through them.
Open lessonLesson · Statistics: displays, designs and differencesExecute a systematic sample and a two-factor experiment
Specify random starts, treatment combinations and allocation steps that another learner can reproduce.
Open lessonConcept · GlossaryExecution runtime
The component that enforces allowed actions, validates arguments, runs operations, and returns observations independently of a model’s proposal.
Open conceptLesson · Social studiesExecutive checks: a winning coalition may not override a veto
The veto changes coalition requirements and bargaining incentives. Distinguish the president's objection, Congress's ability to override and the separate constitutional paths through which a bill can become law.
Open lessonLesson · Social studiesExecutive Order 9066: distinguish text, implementation and retrospective account
A public record proves particular wording and formal action. Implementation records and historical evidence are needed to explain whom that authority affected. Reading the literal text alone can miss discriminatory application; replacing the text with a later summary can misstate what the document actually says.
Open lessonLesson · Social studiesExpansion, markets and contested belonging
Explain how territorial and market expansion created unequal opportunities and coercion.
Open lessonConcept · GlossaryExpectimax
Choose the action with greatest expected utility using supplied probabilities for uncertain outcomes or opponent responses. Incorrect probabilities can produce poor decisions even with exact search.
Open conceptLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain a conductor’s zero field without erasing its potential
A conductor in electrostatic equilibrium cannot sustain an internal electric field: mobile charges would continue moving if a field remained. For a solid conductor with no cavity, excess charge resides on its surface. At the surface the external field is perpendicular; a tangential component would drive further surface charge motion. These statements concern equilibrium, not a conductor carrying a steady current.
Open lessonProject · Probability, uncertainty and expected outcomesExplain a decision under uncertainty
Build a fictional decision model with an explicit outcome space, conditional probabilities and numerical consequences. Compare alternatives using expected outcomes while explaining a downside that the average hides.
Open projectLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceExplain a phenotype ratio with a biochemical interaction
Dominance describes how alleles at one locus relate to a defined phenotype; it does not mean that an allele is common, stronger in every sense or always beneficial. With incomplete dominance, the heterozygote can have an intermediate score; with codominance, distinguishable contributions of both alleles can be observed. Different measurement levels can reveal different dominance relationships.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsExplain a pressure decrease with kinetic, height and loss terms
Bernoulli’s ideal relation is an energy account along a streamline for steady incompressible flow with negligible dissipation. Each term p, ρv²/2 and ρgz has units J/m³, equivalent to Pa. Pressure work can become kinetic or gravitational energy; pressure itself is not a complete energy budget.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsExplain a toy RSA round trip
Implement repeated squaring with an exact modular invariant.
Open lessonLesson · Science · Physics and physical systemsExplain acceleration at constant speed
Compare tangent velocities around a circle and distinguish speed from a changing velocity vector.
Open lessonLesson · Science · AP Environmental ScienceExplain adaptation through inherited variation
Model frequency change across generations without claiming individuals adapt by need.
Open lessonProject · Calculus: local rates and accumulated changeExplain and optimize a changing system
Choose a geometric growth, motion or bounded design model. Connect a local rate to finite change, calculate accumulated change where appropriate, and justify an optimum only within a stated feasible domain. Use captures from at least two calculus mechanisms and include a counterexample to a tempting shortcut.
Open projectLesson · Science · Physics and physical systemsExplain attraction and repulsion between currents
Calculate signed force per length and reconcile unequal fields with equal interaction forces.
Open lessonLesson · Social studiesExplain behavior without reducing people to labels
Social psychology examines how situations and other people influence behavior; personality describes patterns across contexts.
Open lessonLesson · Science · Chemistry and matterExplain boiling-point evidence with intermolecular attractions
Compare same-formula molecules using reference boiling points and structural evidence.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain carbohydrate differences through linkages and branches
Compare equal glucose inventories under prescribed linkage-specific cleavage rules.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain catalysis and choose an initial-rate window
Connect active-site chemistry and activation barriers with a blank-corrected rate estimate.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain cell-cycle checkpoints and loss of regulation
Cyclins and cyclin-dependent kinases help control transitions between cell-cycle states.
Open lessonLesson · Science · Engineering design and scientific computingExplain changing temperature with an energy balance
Connect an exact interval calculation to stored energy and ambient heat transfer.
Open lessonLesson · Sections, solid measures and geometric designExplain circle formulas by polygon bounds
Use nested polygons to justify circumference and area limits.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain crossing over and assortment without mixing mechanisms
Crossing over exchanges corresponding DNA between nonsister chromatids of paired homologs during prophase I.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain current without consuming charge
Current is the rate at which charge crosses a chosen section: I=ΔQ/Δt, measured in amperes, or coulombs per second. It has a direction convention but is not a spatial vector added by geometric components. Conventional current follows the direction positive charge would move. In a metal the mobile electrons drift in the opposite direction.
Open lessonLesson · Science · Introductory MicrobiologyExplain descendant clusters without need-directed mutation
A mutation can occur before a later condition makes its consequences visible. The future usefulness of a variant does not imply that a cell generated it because it anticipated that need. Descendants share ancestry, so an early event can produce a cluster of related carriers rather than independent mutation events in every carrier.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain differentiated cells with shared genetic information
Different cell types in a multicellular organism can retain largely the same genome while expressing different subsets of genes.
Open lessonLesson · Science · Biology and living systemsExplain diffusion through opposing flows
Follow both directions of particle exchange and check conservation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain Doppler observations before using a numerical extension
Doppler reasoning concerns the rate at which an observer encounters repeated wavefronts. In the required qualitative treatment, approaching motion raises observed pitch and receding motion lowers it for the stipulated sound geometry. A source and observer moving together with the same velocity have no relative approach and receive the original repetition rate. Source frequency and observed frequency are distinct quantities.
Open lessonLesson · Science · Chemistry and matterExplain equilibrium with two continuing reactions
Compare forward and reverse rates without confusing equal rates with equal concentrations.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain focusing changes without a clinical-eye claim
A fixed screen places a stronger constraint than merely asking whether a lens forms some image. For a given object distance and chosen image-plane distance, the required focal length follows 1/f=1/o+1/i. Moving the object while keeping the screen fixed generally requires changing focal length or repositioning optics. The lesson studies that geometric condition, not an individual eye or a medical prescription.
Open lessonLesson · Science · Chemistry and matterExplain gas pressure with a controlled volume comparison
Keep amount and temperature fixed while testing the inverse pressure-volume relationship.
Open lessonLesson · Science · AP Environmental ScienceExplain global winds beyond a deflection formula
Connect unequal heating, pressure and rotation to large-scale circulation.
Open lessonLesson · Science · Earth and space scienceExplain gravity and acceleration in orbit
Distinguish circular free fall from zero gravity and use center-based radius in an inverse-square calculation.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain how cell compartments coordinate a protein’s route
Connect organelle structures and functions while separating an RNA message from protein cargo.
Open lessonLesson · Science · Biology and living systemsExplain how light supports carbon fixation
Separate the source of carbon from the source of energy in a photosynthetic organism.
Open lessonLesson · Science · Physics and physical systemsExplain how source motion changes wavefront spacing
Reconstruct emitted fronts and distinguish emitted frequency from received frequency.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsExplain hydraulic force gain with displacement and energy
A pressure change imposed on an enclosed static ideal fluid is transmitted through it. At equal heights, two ideal pistons exposed to the same external atmosphere have equal fluid gauge pressure, so Fsmall/Asmall=Flarge/Alarge. This relates the applied forces above ambient; piston weights, friction and different outside pressures would need separate terms.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain inversion without confusing amplitude and transmitted energy
A pulse approaching a join can produce both reflected and transmitted disturbances. For two ideal strings under the same tension, the denser string has lower wave speed. Reflection from the slower side is inverted; reflection from a faster side is not. Equal strings have no reflected pulse in this idealization. Fixed-end and free-end reflection are useful limiting cases, not evidence that every boundary has the same sign.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain magnetic alignment without inventing monopoles
A magnetic field is a vector description of how moving charges, currents and magnetic materials interact. A bar magnet has north and south polarity, but its field lines do not begin on isolated magnetic charges. Outside the bar they run from north toward south and continue through the interior to close the loops. Cutting a bar into pieces creates smaller dipoles, not a free north pole and a free south pole.
Open lessonLesson · Science · Physics and physical systemsExplain magnetic turning without an energy gain
Connect perpendicular force to constant speed, circular radius and the zero-field limit.
Open lessonLesson · Science · Earth and space scienceExplain melting without a global molten layer
Distinguish partial melting, magma movement, intrusion and eruption in their geological settings.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain membrane structure and evaluate passage evidence
Repair membrane regions and distinguish routes from measured transport.
Open lessonLesson · Science · General Chemistry IExplain metallic transport without overidentifying its cause
Use a transport product to distinguish carrier availability from motion through a material.
Open lessonLesson · Science · AP Environmental ScienceExplain nuclear power and bounded radioactive decay
Connect controlled fission, energy conversion and waste evidence without inventing a safety cutoff.
Open lessonLesson · Science · AP Environmental ScienceExplain ocean acidification through carbonate chemistry
Separate a pH trend from acidity labels and full seawater equilibrium.
Open lessonLesson · Science · Earth and space scienceExplain opposite seasons with axial tilt
Compare hemispheres and a zero-tilt counterfactual while keeping solar input and orbital distance fixed.
Open lessonLesson · Science · AP Environmental ScienceExplain ozone depletion and evaluate substitution tradeoffs
Track catalytic recycling while separating ozone protection from climate impacts.
Open lessonLesson · Science · General Chemistry IExplain periodic size through competing model parameters
Separate attraction and shell extent before using a periodic trend.
Open lessonLesson · Science · AP Chemistry preparationExplain potential changes with the reaction quotient
Reason from composition to signed cell potential and distinguish direction from magnitude.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain pressure through collision momentum
Begin with the gas as a collection of moving particles, not as a substance containing a material called heat. A particle approaching a rigid wall with normal velocity vx reverses that component in an elastic collision. Its momentum change is −2 m vx; the wall receives the opposite impulse. Tangential motion does not contribute to that normal impulse. Many such events per second produce an average normal force and therefore pressure.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceExplain pumping with storage and connected-discharge evidence
Groundwater occupies connected pore and fracture spaces, but not all stored water drains freely. In an unconfined aquifer, specific yield relates a water-table change to the volume released or stored under a defined drainage assumption. It differs from total porosity and from the storage mechanisms dominating a confined aquifer.
Open lessonLesson · Science · Chemistry and matterExplain reaction enthalpy using a bond ledger
Calculate signed reaction enthalpy without confusing bond energies with activation energy.
Open lessonLesson · Science · Earth and space scienceExplain relative humidity without adding water
Compare vapor pressure with temperature-dependent saturation pressure and track the conserved mixing ratio.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain reproductive barriers and divergence
Speciation involves evolutionary divergence that establishes separately evolving lineages.
Open lessonLesson · Science · Physical Geology and Earth SystemsExplain residual melt enrichment with a closed material budget
Cooling magma can contain crystals and liquid simultaneously. In Bowen’s reaction framework, early minerals can react with remaining liquid as conditions change; plagioclase composition can also evolve. A sequence is not a promise that every magma produces every mineral: initial composition, pressure, water and reaction time matter.
Open lessonLesson · Science · AP Environmental ScienceExplain richness turnover and island constraints
Use an immigration-extinction model without treating equilibrium as no ecological change.
Open lessonLesson · Science · Biology and living systemsExplain selection without giving evolution a goal
Follow how differential reproduction changes an existing heritable variant.
Open lessonLesson · Science · Physics and physical systemsExplain self-inductance and voltage reference signs
Distinguish passive voltage drop from induced emf while current changes.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain signaling distance and receptor specificity
A signal can act through direct cell contact, diffusion to nearby cells or circulation over longer distances.
Open lessonLesson · Science · Physics and physical systemsExplain spring turning points and fastest motion
Track energy exchange through equilibrium and both turning points without confusing speed with acceleration.
Open lessonLesson · Science · Chemistry and matterExplain temperature effects using activation barriers
Compare temperature-dependent forward and reverse rate constants using an explicit energy profile.
Open lessonLesson · Science · Physics and physical systemsExplain the broad central band from a single slit
Compare slit width, wavelength and the dark minima of a diffraction pattern.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesExplain the extra obstruction in general matching
Prove the odd-component necessity in Tutte’s criterion.
Open lessonLesson · Science · Physics and physical systemsExplain the odd harmonics of a closed-open tube
Combine a rigid closed end with an open end and distinguish allowed mode index from harmonic number.
Open lessonLesson · Science · Physics and physical systemsExplain voltage sag inside a source
Separate emf, terminal voltage, load power and internal dissipation.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceExplain water’s thermal buffering without confusing heat and temperature
Compare mass and supplied energy, then distinguish warming, evaporation and freezing mechanisms.
Open lessonLesson · College compositionExplain what evidence licenses you to say
Build a paragraph that distinguishes observation, inference and value judgment, and tests the warrant connecting them.
Open lessonLesson · Course 1 · Data audits and training diagnosisExplain what your model is for
Turn results into bounded claims, identify missing evidence and draft a model card linked to your actual experiment.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain when a measurement changes the circuit
An ideal ammeter is placed in series with the current to be measured and has zero resistance. An ideal voltmeter spans the two nodes of interest and has infinite resistance, so it draws no current. These are modeling limits. A real instrument can add resistance or create an additional path, changing the circuit before it reports a value.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsExplain why dielectric insertion needs a boundary condition
A dielectric is an insulating material that polarizes in an electric field. Bound positive and negative charge distributions shift slightly relative to one another, producing an opposing field. A linear dielectric that fills an ideal capacitor increases its capacitance by a factor κ, the relative permittivity. The material response must be distinguished from free charge supplied to conducting plates.
Open lessonLesson · Science · AP Chemistry preparationExplain why dissolution is a balance of competing effects
Combine lattice, hydration and entropy terms without claiming a universal dissolution sign.
Open lessonLesson · Algebra foundations: expressions, equations and rootsExplain why expressions are equivalent
Use distribution and structure without confusing a few matching values with an identity.
Open lessonConcept · GlossaryExploration
Choosing actions to collect information about alternatives. Exploitation uses current estimates to select a promising action.
Open conceptLesson · Mathematics investigationExplore a graph without getting trapped
How does a queue find shortest unweighted paths despite cycles?
Open lessonLesson · Course 2 · Learning through actionsExplore before you decide
Choose between uncertain rewards and discover why trying a new option can improve later decisions.
Open lessonPath · Software engineers · action and planningExplore learned decisions
Compare rewards, search and predictive planning, then evaluate a puzzle agent.
Open pathLesson · Science · Earth and space scienceExplore pressure decrease and atmospheric scale height
Use the exponential pressure profile to compare equal altitude increments and pressure-level heights.
Open lessonAssessment · Precalculus: exponential and logarithmic evidenceExponential and logarithmic evidence: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Differential calculus: techniques and applicationsExponential and logarithmic rates
Connect multiplicative growth, inverse functions and logarithmic differentiation.
Open lessonLesson · Calculus II: series, curves and applicationsExpose cancellation before taking a limit
Track finite endpoints in geometric and telescoping sums.
Open lessonLesson · Mathematical modeling and senior investigationExpose the ratios controlling a model
Scale equations and reconstruct physical predictions.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeExpress laws and independence through measures
Construct probability laws as pushforwards.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeExtend consistent interval assignments
Construct an outer measure from a premeasure.
Open lessonLesson · Partial differential equations and Fourier methodsExtend energy arguments to coupled fields
Diagonalize a small system and preserve Maxwell constraints.
Open lessonExperiment · Systems thinking and operationsExtra copies cannot fix every failure
When does redundancy improve an entire service?
Open experimentLesson · Compactness of function familiesExtract a uniform limit from a family of functions
Build one subsequence that converges at a countable dense set, then use equicontinuity and finite nets to control every input simultaneously.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceExtract spectral information from repeated actions
Analyze power and shifted inverse iteration, residuals and spectral gaps.
Open lessonLesson · Abstract algebra I: groups and symmetryExtract structure from divisibility of orbit sizes
Prove nontrivial center and classify order p².
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusExtreme and intermediate values
Prove existence statements from compactness and completeness without inventing algorithms.
Open lessonLesson · Course 2 · Planning and reasoningFactor a probabilistic model
Build an eight-row joint distribution from local tables, test conditional independence, and separate observing a variable from intervening on it.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsFactor a ring map through its kernel
Track identity preservation and quotient isomorphisms.
Open lessonLesson · Algebra foundations: expressions, equations and rootsFactor structure and use zero products
Extract common factors, identify identities and distinguish factored expressions from equations.
Open lessonConcept · GlossaryFair learning evaluation
A declared comparison reports data, supervision, interactions, optimizer work, checkpoint selection and every seed or failure. Frozen evaluation keeps task success separate from a potentially misleading reward proxy.
Open conceptConcept · GlossaryFalse positive
A positive prediction or flag for a case whose reference label is negative. In anomaly review this is a false alarm on a confirmed normal case.
Open conceptLesson · Social studiesFamily change: map care beyond one household
Family, marriage, household residence and care relationships overlap but are not identical. Institutional definitions affect which relationships become visible in records.
Open lessonExperiment · Design and product developmentFast for whom?
Can a faster average conceal a design failure?
Open experimentConcept · GlossaryFeature
A pattern in inputs or activations that a computation responds to. A feature may be distributed across many neurons.
Open conceptLesson · Social studiesFederalist 10 and Brutus I disagree about scale
Madison and Brutus both worry about republican government but predict different consequences from scale.
Open lessonLesson · Social studiesFederalist 39 and 51: two ways to divide power
Federalism divides authority across national and state governments.
Open lessonLesson · Social studiesFederalist 70 and 78: executive energy and judicial independence
Hamilton connects a single executive with decisiveness and traceable responsibility.
Open lessonConcept · GlossaryFeedback selection bias
Recorded feedback reflects what was exposed, chosen and logged. It need not represent preferences over all available options, and recommendations can reinforce that selection.
Open conceptConcept · GlossaryFiltering
Estimating the current hidden-state distribution using only observations up to the current time. Prediction advances the state model; correction incorporates the current reading. Exact filtering does not guarantee the model matches reality.
Open conceptLesson · Science · AP Chemistry preparationFind a boiling condition without extrapolating a fit
Invert a bounded empirical vapor-pressure relation and separate missing support from physical impossibility.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceFind a bound state by matching a finite boundary and its tail
Take a symmetric one-dimensional well with zero potential inside absolute x less than a0 and height V0 outside. A bound state has energy strictly between zero and V0 in this well-bottom reference. For the lowest even state, the interior amplitude is A cos(kx), with k=sqrt(2mE)/hbar. Outside it decays as A cos(ka0)exp[−kappa(|x|−a0)], where kappa=sqrt[2m(V0−E)]/hbar.
Open lessonLesson · Mathematics investigationFind a direction a matrix preserves
When does applying a matrix only scale a selected nonzero vector?
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesFind a matching or a Hall obstruction
Prove Hall’s theorem with alternating reachability.
Open lessonLesson · Literary inquiry and interpretationFind a question the whole story complicates
Develop an interpretive question from a tension across the complete work rather than a theme label or an isolated quotation.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsFind a region before integrating its height
Use intersections, orientation and splitting to construct nonnegative area.
Open lessonLesson · Optimization: models, certificates and algorithmsFind a starting basis and avoid false progress claims
Use artificial variables and understand zero-length pivots.
Open lessonExperiment · Programming and software engineeringFind an item without checking everything
When does dividing the search space save work?
Open experimentLesson · Trigonometry, polar geometry and precalculus representationsFind complete interval and general solutions
Combine factoring, inverse branches and periodicity without discarding roots.
Open lessonLesson · Functions in graphs, measurements and growth decisionsFind delayed exponential crossovers
Compare exponential and polynomial values beyond a misleading initial window.
Open lessonLesson · Science · Chemistry and matterFind density from the volume a sample displaces
Combine a mass reading with a change in liquid volume, keeping the measurement boundary explicit.
Open lessonLesson · Science · Earth and space scienceFind dew point as an inverse saturation calculation
Use the vapor-pressure curve to find a saturation threshold while distinguishing it from a weather prediction.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsFind directions preserved by a linear map
Compute eigenspaces and distinguish zero eigenvalues from forbidden zero eigenvectors.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsFind every integer solution, not just one
Use gcd divisibility to solve linear integer equations and parameterize all solutions.
Open lessonLesson · School trigonometry: proofs, inverse branches and periodic decisionsFind every threshold time in a stated interval
Use inverse-trig technology, all-solution families and inequality intervals in context.
Open lessonLesson · Algebra II and precalculus: functions and structureFind intersections and classify systems
Connect simultaneous constraints with substitution, elimination and intersection geometry.
Open lessonLesson · Abstract algebra I: groups and symmetryFind invariant complements by averaging
Turn finite-group actions into invariant geometry and prove complete reducibility.
Open lessonLesson · Course 2 · Testing whether improvement is realFind out whether improvement generalizes
Select a lucky benchmark winner, remove components, and compare step accuracy with whole-workflow reliability.
Open lessonLesson · Calculus synthesis: regions, motion and approximationFind polar tangents through Cartesian derivatives
Differentiate the Cartesian parameterization and test horizontal, vertical and singular cases.
Open lessonLesson · Science · Physics and physical systemsFind pressure and displacement modes in an open tube
Apply ideal open-end boundary conditions and compare the two descriptions of a standing sound wave.
Open lessonLesson · Course 2 · Memory and shared working stateFind relevant information
Compare keyword and learned-vector retrieval, change passage boundaries, and inspect what a ranking score can and cannot tell you.
Open lessonLesson · Abstract algebra I: groups and symmetryFind rotation axes and constrain finite spatial groups
Derive axis-angle rotations and the necessary finite-group pole data.
Open lessonLesson · Linear Algebra II: operators and spectral structureFind spectral information that survives coordinates
Relate characteristic polynomials, multiplicities and triangular forms to the scalar field.
Open lessonLesson · Course 1 · Simple models and useful baselinesFind structure without labels
Alternate k-means assignments and center updates, then project onto a principal axis and inspect distinctions that vanish.
Open lessonLesson · Mathematical statistics: derivation and model criticismFind the assumptions that drive a reported standard error
Use leverage and residual covariance to diagnose model departures.
Open lessonLesson · Rhetoric, media and public voiceFind the bridge between evidence and a proposal
Separate a claim, its evidence and its unstated assumption, then qualify the claim when the bridge is weak.
Open lessonLesson · Course 2 · Testing whether improvement is realFind the cause of an improvement
Compare all four factorial cells and distinguish interaction from extra compute.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsFind the complete symmetry set
Distinguish actual self-maps from plausible-looking axes.
Open lessonLesson · Science · Physics and physical systemsFind the energy source during a change of radius
Separate angular-momentum conservation from kinetic-energy conservation.
Open lessonLesson · Mathematics investigationFind the exponent that reaches a target
How do you solve a growth equation when time is unknown?
Open lessonLesson · Linear Algebra II: operators and spectral structureFind the input and output axes of any linear map
Derive SVD from the spectral theorem for A*A and interpret singular values geometrically.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsFind the landing event from the actual initial height
The familiar same-height projectile range formula has a condition: launch and landing heights must match. The supplied virtual point starts 2 m above the ground. Resolve its initial velocity into horizontal vx and vertical vy, with upward positive, then share one elapsed time between both component equations. Horizontal velocity remains constant only because air resistance and other horizontal forces are excluded.
Open lessonLesson · Science · Biology and living systemsFind the limiting factor in an enzyme reaction
Compare substrate changes with enzyme changes using an initial-rate model.
Open lessonLesson · Calculus II: series, curves and applicationsFind the radius, then test each endpoint
Separate open-interval absolute convergence from boundary behavior.
Open lessonExperiment · Design and product developmentFind the regression hidden in a revision
Does an improvement for one group hide harm to another?
Open experimentLesson · Linear Algebra II: operators and spectral structureFind the shortest polynomial relation an operator obeys
Construct annihilating polynomials and connect square-free splitting to diagonalizability.
Open lessonLesson · Science · Physics and physical systemsFind the threshold for total internal reflection
Test when Snell’s law has no propagating transmitted solution.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsFind the true period of a unit
Prove order divisibility and compute periods of powers.
Open lessonLesson · Science · Physics and physical systemsFind thermal equilibrium with an energy balance
Account for the samples, cup and exported energy before predicting a common temperature.
Open lessonLesson · Course 1 · Simple models and useful baselinesFind unusual examples
Fit a reference-distance detector, inspect unseen normal behavior and count false alerts when confirmed problems are rare.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsFind what a straight sampled track does not reveal
A tracking table provides positions at particular times. The difference between neighboring positions divided by elapsed time is an interval-average velocity. A straight line through the recorded points is consistent with constant velocity, but it does not prove that every unrecorded instant followed that line. The information between samples is an inference, not another measurement.
Open lessonLesson · Science · AP Chemistry preparationFind when enthalpy and entropy favor change
Convert entropy units, calculate ΔG°, and justify temperature-dependent favorability.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsFind where unique factorization fails
Use norms and a nonprincipal ideal to test integer-like claims in a quadratic ring.
Open lessonLesson · Mathematical modeling and senior investigationFind which changes matter and where
Compare derivatives, elasticities and bounded parameter sweeps.
Open lessonConcept · GlossaryFine-tuning
Further training some or all existing model parameters for a chosen task or dataset, with separate evaluation of gains and retained behavior.
Open conceptLesson · Course 1 · Simple models and useful baselinesFinish a tabular experiment
Turn a real-data comparison into a frozen, reproducible report with error examples and carefully bounded claims.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryFinite affine and projective incidence
Prove incidence properties in small algebraic models.
Open lessonLesson · Encodings, diagonal arguments and limits of algorithmsFinite descriptions and encodings
Construct reversible codes and distinguish a countable syntax from the mathematical behavior it can describe.
Open lessonCourse · Mathematics program unitFinite Fourier bases, convolution and FFT
Prove finite transform identities, diagonalize circular filters and derive a bounded radix-two implementation with explicit normalization and arithmetic costs.
Open courseLesson · Combinatorics and graph theory: enumeration, structure and certificatesFinite matroids: preserve a partial choice while exchanging
Derive basis size and the extension lemma from finite independence axioms.
Open lessonLesson · Point-set topology: spaces, constructions and separationFinite products and projections
Construct the product topology from open rectangles.
Open lessonCourse · Mathematics program unitFinite sampling guarantees and dependence
Derive a finite probability guarantee, plan a sample size and audit whether the data-collection mechanism supports the proof.
Open courseLesson · Real Analysis II: uniform limits, metric spaces and local calculusFinite-dimensional norms and operator norms
Measure linear maps uniformly over directions and justify finite-dimensional norm equivalence.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceFit a boost from one record and test unused event differences
An inverse problem needs a training record, an independent prediction and explicit uncertainty. The original training clock is at rest in S. A measured ratio of its event differences in S′ is Δx′/Δt′=−0.6 with hard bound ±e, so the frame boost lies in β=0.6±e. Only this ratio is supplied; no absolute training time increment is available to impose an additional hidden constraint.
Open lessonLesson · Course 1 · Simple models and useful baselinesFit a line and explain its errors
Move a line over licensed flower measurements, inspect residuals and compare a fitted model with a mean baseline and shuffled labels.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityFit a line and inspect its failures
Connect a least-squares model to residuals, influence and prediction limits.
Open lessonLesson · Mathematics investigationFit a line and inspect what it misses
How much can one observation change a fitted trend?
Open lessonLesson · Mathematics investigationFit a line when equations disagree
Which line minimizes the squared residuals of three observations?
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceFit a molecular assay and challenge its prediction range
A calibration relates a known reference quantity to the measurement response. Write signal y=b+mc for a linear model, with intercept b, slope m and concentration c. The inverse c=(y−b)/m is meaningful only within a validated range and under compatible measurement conditions. A high-looking fit alone is not evidence that an unknown sample follows the same model.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsFit a periodic model and test its assumptions
Translate extrema and timing into a sinusoid and distinguish fitting from evidence.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsFit an inconsistent system without hiding the error
Derive normal equations, examine uniqueness and distinguish fit from explanation.
Open lessonLesson · Science · University Physics II with calculusFit an offset and reserve a fourth record
A constant measurement offset bends a logarithm of the raw readings. The synthetic response is y=Ae^(−t/τ)+B, with amplitude, time constant and offset all unknown to the analyst. Three readings at independently exact times 0,1,2 s calibrate those three parameters; a fourth at 3 s must remain fresh.
Open lessonLesson · Fitted regression: coefficients, uncertainty and validationFit coefficients and audit residuals
Construct a numeric design and check a rank-aware least-squares fit against its defining equations.
Open lessonLesson · Precalculus: periodic and reciprocal graph evidenceFit competing periods and test new predictions
Execute conditional periodic regression and expose ambiguity with unused inputs.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceFit exact data with a unique polynomial
Construct cardinal polynomials and prove uniqueness and a smooth-function interpolation remainder.
Open lessonLesson · Functions in graphs, measurements and growth decisionsFit growth models and change time units
Recover exponential parameters from supplied graph landmarks.
Open lessonLesson · Mathematical modeling and senior investigationFit parameters and inspect what the fit misses
Derive least squares and use residual patterns.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceFit residuals without unnecessarily squaring conditioning
Derive QR least squares and distinguish residual minimization from accurate coefficient recovery.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsFit rotational inertia without hiding resistance or readout bias
The drive torque on a rotating body need not equal its net torque. In this fixed-axis model, an opposing torque q acts while the body rotates positively, so τdrive−q=Iα. For unchanged I and constant q, plotting drive torque against independently known angular acceleration gives slope I and intercept q.
Open lessonLesson · Science · University Physics II with calculusFit signed frequency evidence without consuming validation
Complex impedance measurement supplies both magnitude and phase; its imaginary part can therefore retain the sign of reactance. Here independently known resistance is 2 Ω, while inductance L and reciprocal capacitance K=1/C are unknown. Exact calibrated angular frequencies 1 and 2 rad/s give X1=L−K and X2=2L−K/2 in SI numerical units.
Open lessonLesson · Science · University Physics II with calculusFit two hypotheses before revealing a third location
Two calibrated models can fit the same two records perfectly and still make different predictions elsewhere. At exact perpendicular distances y=1,2,3 m, compare a known-half-length 1 m straight-segment shape f(y)=1/[y√(L²+y²)] with L=1 m with an infinite-line shape g(y)=1/y. Each has its own unknown amplitude and constant indicated baseline. The finite-shape amplitude has units µT m², while the infinite-line amplitude has units µT m; these coefficients are not interchangeable.
Open lessonLesson · Precalculus: constructing and validating polynomial modelsFit, predict and challenge polynomial models
Separate matching a calibration table from predicting new observations with a stated domain and error criterion.
Open lessonLesson · Distribution diagnostics and calibrationFitted nulls and parametric bootstrap
Derive the exponential rate fit and scale-invariant discrepancy.
Open lessonAssessment · Fitted regression: coefficients, uncertainty and validationFitted regression: calculation, assumptions and frozen transfer
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitFitted regression: coefficients, uncertainty and validation
Fit actual multivariable data, derive conditional uncertainty and retain an honest diagnostic and validation record. Synthetic examples and public rubrics do not establish a completed empirical investigation.
Open courseLesson · Science · Environmental Science: ecological systems and evidenceFix the delivered service before comparing lifecycle burdens
A product is not its service. A heater comparison should specify useful heat delivered at the required temperature, reliability and duration, rather than compare one appliance or one unit of electricity. This functional unit makes alternatives answer the same need. Their material, manufacturing, operating and end-of-life boundaries must then be made comparable.
Open lessonConcept · GlossaryFLOPs
A count of floating-point arithmetic operations. FLOPs per second is a processing rate; neither quantity is memory capacity.
Open conceptLesson · Differential calculus: techniques and applicationsFollow a change through nested functions
Track every intermediate rate and avoid treating derivative notation as unrestricted cancellation.
Open lessonLesson · Mathematics for AI · M05Follow a gradient through a computation
Connect a derivative, chain factors, partial gradients and a numerical check.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceFollow a molecular switch after its input ends
A G-protein-coupled receptor links extracellular recognition to an intracellular nucleotide switch. An activated receptor promotes exchange of GDP for GTP on a G-protein subunit. The GTP-bound state can regulate an effector; GTP hydrolysis returns it toward an inactive GDP-bound state. Turning the switch off does not require destroying every protein molecule.
Open lessonLesson · Course 1 · How does AI process language?Follow a token through a Transformer
Inspect a tiny numerical decoder from token IDs to the next-token distribution.
Open lessonLesson · Science · Earth and space scienceFollow core and shell changes
Locate nuclear reactions and separate production from later release.
Open lessonLesson · Science · Physics and physical systemsFollow current and voltage through a series path
Use a voltage divider and a loop balance to explain two series resistors.
Open lessonExperiment · Applied mathematicsFollow growth one period at a time
Why do repeated percentages differ from repeated fixed additions?
Open experimentLesson · Science · Physics and physical systemsFollow longitudinal motion in a sound wave
Distinguish horizontal parcel motion from the traveling pattern and from a plotted waveform.
Open lessonLesson · Proof foundations: from claims to argumentsFollow membership through a function
Compare inverse images, which preserve intersections, with images, which can merge distinct inputs.
Open lessonLesson · Partial differential equations and Fourier methodsFollow moving data along characteristic curves
Reduce a first-order PDE to an ODE on a path.
Open lessonLesson · Science · Environmental science and sustainabilityFollow nitrogen across a plot boundary
Track elemental nitrogen across four pools and external accounts.
Open lessonLesson · Social studiesFollow quota rents to their recipients
Hold an import limit fixed and test who receives its scarcity value.
Open lessonLesson · Science · Earth and space scienceFollow stored energy after forcing removal
Separate continuous state variables from a discontinuous energy input and net flux.
Open lessonLesson · Calculus II: series, curves and applicationsFollow terms toward a limit
Separate a sequence, its continuous extension and its numerical evidence.
Open lessonLesson · Partial differential equations and Fourier methodsFollow the direction of information on a grid
Test upwinding, conservation and CFL limits.
Open lessonLesson · Course 1 · How does a network learn?Follow the slope of a mistake
Turn errors into a loss curve, then find out why the size of a learning step matters.
Open lessonLesson · Science · Earth and space scienceFollow water and energy without a conveyor shortcut
Combine mechanisms and observations into bounded claims and specify missing transport evidence.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsForce a collision without finding it
Choose the right boxes and prove sharp bounds.
Open lessonLesson · Triangle-free coloring certificatesForce an extra color without creating a triangle
Prove the exact chromatic increase of the Mycielski construction and distinguish a coloring certificate from a lower-bound proof.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesForce structure in every coloring
Prove a finite Ramsey recurrence.
Open lessonLesson · Precalculus extensions: motion, matrices and transitionsForecast state proportions and test a past-state reconstruction
Construct conditional transition rows, inspect exact multiple-step forecasts and distinguish feasible, impossible and nonunique backcasts.
Open lessonExperiment · Applied mathematicsForecast without seeing the future
How do smoothing and trend assumptions change a forecast?
Open experimentConcept · GlossaryForward kinematics
Calculating body or tool position and orientation from a robot’s joint configuration and geometry. It does not by itself solve for joints, test collisions or establish physically feasible motion.
Open conceptStudio · Optional P05 implementation studioFramework debugging studio
PyTorch CPU autodiff, data loaders, dropout modes, Adam state and reproducible checkpoint resume.
Open studioLesson · Real Analysis II: uniform limits, metric spaces and local calculusFréchet differentiation as a uniform linear approximation
Require a remainder small relative to displacement in every direction.
Open lessonLesson · Social studiesFreedom movements and contemporary debates
Evaluate movement strategies and contemporary claims through multiple disciplines.
Open lessonLesson · Social studiesFrench Revolution, Napoleon and restoration
Trace revolutionary principles through changing regimes and contested inclusion.
Open lessonLesson · Social studiesFrench rights: who authorizes power?
Turn the 1789 declaration into a test of authority and institutional evidence.
Open lessonLesson · Differential geometry of curves and surfacesFrenet frames, curvature and torsion
Derive the moving frame of a nonstraight space curve.
Open lessonLesson · Social studiesFrom requisitions to national powers
A constitution makes coordination possible by assigning powers and procedures.
Open lessonLesson · Single-variable calculus foundationsFrom secant rates to a derivative
Derive an instantaneous rate from a difference quotient, with a nonzero increment before taking its limit.
Open lessonLesson · Social studiesFrom stimulus energy to a perceptual judgment
Distinguish physical stimulus properties, sensory transduction and interpretation using controlled cue comparisons.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusFunction limits and continuity
Build epsilon–delta proofs and separate a punctured limit from the function value.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusFunction series and the Weierstrass M-test
Dominate every pointwise tail by one convergent numerical series.
Open lessonCourse · MathematicsFunctions and graphs
Five interactive mathematics lessons and a separate transfer assessment.
Open courseAssessment · Algebra II and precalculus: functions and structureFunctions and structure: mixed assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitFunctions in graphs, measurements and growth decisions
A school-entry practice unit connecting quantities, measurement accuracy, unfamiliar graphs and tables, transformations, inverses, sequences and growth decisions. Graph descriptions and sketches require rubric review; this closes specific representation gaps rather than claiming complete AP preparation.
Open courseRefresher · Math refreshersFunctions, slopes, and derivatives
Practice the mathematics, then return to your lesson.
Open refresherLesson · Course 2 · Geometry and control foundationsFuse noisy measurements
Predict motion, correct with sensors and inspect the uncertainty assumptions of a linear Kalman filter.
Open lessonLesson · Differential equations: models, methods and dynamicsGain, phase and practical resonance
Derive the frequency response and distinguish three frequencies with different meanings.
Open lessonLesson · Social studiesGender and sexual orientation: constructs, development and respectful evidence
Distinguish gender identity, expression, social roles and sexual orientation, and identify what a measure does and does not establish.
Open lessonLesson · Social studiesGender: test how rules and expectations allocate opportunities
Gender socialization concerns learned expectations; institutions may reinforce or challenge them. An analysis should specify how norms, evaluation or scheduling alter opportunities, without assuming a group's members share one preference or life course.
Open lessonCourse · ScienceGeneral Biology I: cellular mechanisms and quantitative evidence
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceGeneral Biology II: diversity, organismal mechanisms and population evidence
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceGeneral Chemistry I
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceGeneral Chemistry II
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseConcept · GlossaryGeneralization
How well a model performs on examples or conditions it did not use for training.
Open conceptCourse · Mathematics program unitGeneralized linear models: fitting and evidence
Specify grouped-binomial and Poisson responses, derive vector fitting steps, distinguish finite existence from bounded computation, and audit conditional uncertainty and frozen checks.
Open courseLesson · Science · Physics and physical systemsGenerate an emf by changing loop area
Connect rod motion, changing magnetic flux and signed circuit current.
Open lessonLesson · Course 1 · Generative models beyond textGenerate by learning a flow
Train a velocity field and compare Euler and Heun generation on the diffusion lesson’s four-mode target.
Open lessonLesson · Course 1 · Generative models beyond textGenerate by removing noise
Train a small 2D noise predictor, inspect corruption arithmetic and manually step a frozen DDPM sampler.
Open lessonLesson · Course 2 · Geometry and control foundationsGenerate more than one valid action plan
Train conditional action denoising and independently verify sampled routes.
Open lessonLesson · Course 1 · Generative models beyond textGenerate with a variational autoencoder
Train a tiny probabilistic encoder and decoder, trace a noisy latent sample and compare reconstruction with KL regularization.
Open lessonLesson · Structural induction: constructions, recursion and parsingGenerated words, balance and unique parsing
Prove that a recursive grammar describes exactly the balanced parenthesis words, then use unique first-return parsing to justify recursive definitions.
Open lessonStudio · Optional Project 6 implementation practice; human review pendingGenerative implementation studio
Dependency-free Python flow training, gradient checks, Adam resume, Euler/Heun sampling, learner exercise and browser-compatible evidence.
Open studioModule · Course 1Generative models beyond text
Learn reconstruction, latent representations and the limits of sampling from small trained models.
Open moduleConcept · GlossaryGenerator
A component that proposes examples, answers, or action sequences. Its proposal is not independent evidence of correctness.
Open conceptLesson · Differential geometry of curves and surfacesGeodesic curvature and boundary orientation
Split acceleration into surface-normal and tangent-normal parts.
Open lessonLesson · Differential geometry of curves and surfacesGeodesics and their initial-value problem
Separate intrinsic straightness from ambient straightness.
Open lessonLesson · Differential geometry of curves and surfacesGeodesics as critical points of energy
Prove the first-variation formula with fixed endpoints.
Open lessonCourse · Mathematics program unitGeometric constructions, circle proofs and coordinate loci
School-entry constructions and coordinate proofs with explicit tools, reversible locus arguments and general-triangle measurement. Public written practice requires review; completion is not an independent mastery claim.
Open courseCourse · Mathematics program unitGeometric definitions, motion witnesses and theorem proofs
A school-entry reasoning unit deriving congruence and similarity criteria from explicit Euclidean motion and incidence facts, then proving parallel-line and triangle/quadrilateral results. Written proofs and constructions require review; public completion is not a mastery certification.
Open courseModule · Course 2Geometry and control foundations
Describe positions and orientations with named frames, then build toward verified simulated motion.
Open moduleCourse · MathematicsGeometry and trigonometry
Use coordinates, similarity, area, volume and angle ratios to construct and check a geometric design with explicit units and domains.
Open courseAssessment · Euclidean and coordinate geometry foundationsGeometry foundations: calculation and proof assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Axiomatic Euclidean and non-Euclidean geometryGeometry: axioms, transformations and models
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesGideon and McDonald: applying rights to states
Selective incorporation concerns applying Bill of Rights protections against states through the Fourteenth Amendment.
Open lessonLesson · Rhetoric, media and public voiceGive a message an audience and a decision
Adapt a proposal to an audience’s authority, needs and constraints without changing the underlying evidence.
Open lessonLesson · Poetry, drama and performanceGive each character something to do
Infer contrasting character objectives and connect a shift in tactic to a dramatic beat.
Open lessonLesson · Course 1 · How does AI process language?Give words a place in space
Explore vectors, similarity, and why a word’s representation can change with context.
Open lessonAssessment · Generalized linear models: fitting and evidenceGLM specification, computation and evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Generalized linear models: fitting and evidenceGLM uncertainty and model checks
GLM uncertainty and model checks. Derive, calculate and qualify a fitted response model.
Open lessonLesson · Social studiesGlobal circulation: follow cultural meaning as well as revenue
Global circulation can expand audiences while changing control over attribution, income and meaning. Neither complete homogenization nor automatic enrichment follows merely from crossing borders; trace who selects, translates, distributes and benefits.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceGlobal error budgets and reference checks
Prove and accumulate an exact global error bound over accepted half steps.
Open lessonLesson · Differential geometry of curves and surfacesGlobal Gauss–Bonnet by triangulation
Sum local curvature identities into a topological invariant.
Open lessonLesson · Social studiesGlobal inequality: trace production and retained value
Modernization and dependency approaches emphasize different processes in global inequality: domestic transformation and international relationships. Treat these as competing or complementary explanations to test, not rankings of cultures or inevitable national destinies.
Open lessonLesson · Social studiesGlobal production: value, labor and cultural exchange
Trace cross-border production without double-counting value or assuming gains are equally shared.
Open lessonLesson · Differential geometry of curves and surfacesGlobal shape, vector-field index and limits
Combine curvature, topology and extrinsic constraints.
Open lessonLesson · Social studiesGlobal war, depression and mass violence
Explain escalation and mass mobilization without reducing war to one trigger.
Open lessonLesson · Social studiesGlobalization, technology and recent history
Evaluate recent change with comparable measures and explicit uncertainty.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsGoursat’s theorem and Cauchy’s theorem
Prove triangular contour cancellation without assuming a continuous derivative.
Open lessonLesson · Science · AP Environmental ScienceGovern a shared resource without assuming inevitable collapse
Explain incentives, regeneration and enforceable sustainability rules.
Open lessonConcept · GlossaryGradient
The local sensitivity of loss to each parameter. Gradient descent takes a small step in the opposite direction.
Open conceptConcept · GlossaryGradient boosting
An additive model fitted in sequence, with each new learner approximating a direction that reduces the current loss. For squared-error regression, the next tree fits observed-minus-predicted residuals.
Open conceptAssessment · Calculus through supplied graphs and measured tablesGraph and table synthesis: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Functions in graphs, measurements and growth decisionsGraph ceiling steps and discrete decisions
Translate whole-unit charging rules into ceiling functions.
Open lessonLesson · Homotopy, fundamental groups and covering spacesGraph loops and free groups
Reduce graph loops using a maximal tree.
Open lessonConcept · GlossaryGraph neural network
A neural model that processes nodes and their relationships, often by passing and aggregating messages along graph edges before learned node updates.
Open conceptLesson · Functions in graphs, measurements and growth decisionsGraph transformed exponentials and logarithms
Construct shifted exponential and logarithmic graphs with asymptotes.
Open lessonRefresher · Math refreshersGraphs and coordinates
Practice the mathematics, then return to your lesson.
Open refresherLesson · Differential geometry of curves and surfacesGraphs and the minimal-surface equation
Derive mean curvature and stationarity of area.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryGreat circles and spherical triangles
Derive spherical metric and trigonometry from unit vectors.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationGreen's theorem and oriented planar boundaries
Convert circulation to a double integral while checking orientation and holes.
Open lessonLesson · Social studiesGroup influence: public agreement and private judgment
Distinguish normative and informational influence using public responses, private judgments and uncertainty.
Open lessonLesson · Proof foundations: from claims to argumentsGroup objects by a proved equivalence
Establish an equivalence relation, build its classes and assemble a reviewable proof portfolio.
Open lessonAssessment · Abstract algebra I: groups and symmetryGroup theory: compute, construct and prove structure
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitGrowth estimates and controlled approximation
Prove eventual comparisons, bound harmonic sums and estimate products while tracking the meaning of the error.
Open courseLesson · Social studiesHaitian independence: liberty, leadership and evidence
Compare freedom from enslavement, national independence and internal authority.
Open lessonLesson · Encodings, diagonal arguments and limits of algorithmsHalting decisions and effective reductions
Prove the impossibility of a universal halting decider in an explicit program model, then transfer that impossibility by an effective program construction.
Open lessonLesson · Science · Earth and space scienceHandle ambiguous and conflicting mineral evidence
Distinguish multiple matches, one conditional match and no compatible candidate.
Open lessonLesson · Optimization: models, certificates and algorithmsHandle constraints and nonsmooth penalties explicitly
Derive projection and proximal optimality relations.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsHarmonic functions and harmonic conjugates
Derive harmonicity of real and imaginary parts of a holomorphic function.
Open lessonLesson · Social studiesHealth institutions: trace access and exposure without diagnosing
Social determinants concern environmental and social conditions affecting opportunities and risks, including housing, transport, education and work. Population mechanisms are different from diagnosing an individual.
Open lessonLesson · Social studiesHealth, environment and unequal global connections
Explain why technical advances need institutions and why aggregate improvements can hide unequal access.
Open lessonLesson · Course 2 · Connecting tools and specialistsHelp specialists understand each other
Compare prose, structured records, and vectors; diagnose lost information and mismatched coordinate meanings.
Open lessonLesson · Social studiesHelping: noticing, interpreting and taking responsibility
Locate where a helping process may break down and distinguish situational opportunity from a global moral judgment.
Open lessonConcept · GlossaryHeuristic
An estimate or rule that helps choose promising options. A useful clue is not a proof of correctness.
Open conceptConcept · GlossaryHidden Markov model (HMM)
A model with hidden states, state-transition probabilities and observation likelihoods. Its assumptions specify how the next state and current observation depend on the state, with an initial belief distribution.
Open conceptAssessment · Real skew-adjoint geometry and complex structuresHidden planes, compatible metrics and canonical normalization
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Social studiesHistorical inquiry and local history
Hands-on social studies course and project
Open courseCourse · Social studiesHistorical methods and public history
Hands-on social studies course and project
Open courseExperiment · Physics and engineeringHit the target without hitting the ceiling
Can two trajectories reach similar distances but have different feasibility?
Open experimentLesson · Complex analysis: holomorphic functions, contours and mappingsHolomorphic functions and the Cauchy–Riemann equations
Distinguish differentiability at a point from holomorphy on an open set.
Open lessonLesson · Point-set topology: spaces, constructions and separationHomeomorphisms and topological invariants
Distinguish a continuous bijection from a homeomorphism.
Open lessonLesson · Homotopy, fundamental groups and covering spacesHomotopies and fixed endpoints
Distinguish free homotopy from homotopy relative to a subspace.
Open lessonAssessment · Homotopy, fundamental groups and covering spacesHomotopy and coverings: constructions, obstructions and classification
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitHomotopy, fundamental groups and covering spaces
A proof-based undergraduate bridge from point-set topology to algebraic topology: explicit loop and lifting arguments, circle degree, gluing, graph and surface groups, and covering classification. Homology, cohomology and higher homotopy are further courses; written solutions require mathematical review.
Open courseLesson · Social studiesHousehold work: connect time diaries to institutions
A division of labor can connect household arrangements, paid employment and public services. Describing a group difference is the first step; explaining its mechanism requires evidence about opportunities, constraints and decisions.
Open lessonExperiment · Systems thinking and operationsHow a short surge becomes repeated swings
Can correcting a stock gap too aggressively amplify orders?
Open experimentExperiment · Scientific reasoning and experimental designHow averaging changes noisy evidence
How often does an average miss the true value by more than your tolerance?
Open experimentLesson · Social studiesHow boundaries change the map
Can two accurate maps tell different stories?
Open lessonLesson · Read like a literary detectiveHow do words change the temperature?
Revise diction and syntax to create two defensible tones while preserving the scene’s events.
Open lessonModule · Course 1How does a network learn?
Start with a single neuron. Discover how small numerical steps become learning.
Open moduleModule · Course 1How does AI process language?
Follow a word from a numerical representation to a next-token prediction.
Open moduleAssessment · Growth estimates and controlled approximationHow much error does the approximation allow?
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Course 1 · Data audits and training diagnosisHow much should a small result convince you?
Draw repeated samples, inspect uncertainty and discover why more biased data can strengthen the wrong claim.
Open lessonExperiment · Data literacy and statisticsHow precise is a sample percentage?
How much more data is needed to halve random uncertainty?
Open experimentCourse · ScienceHuman Anatomy and Physiology Foundations
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · Social studiesHuman geography and world regions
Hands-on social studies course and project
Open courseLesson · Axiomatic Euclidean and non-Euclidean geometryHyperbolic triangle identities and angle defect
Use the Lorentz model to derive a non-Euclidean cosine law.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryHyperboloid and Klein coordinates
Compare straight geodesics with angle-preserving pictures.
Open lessonConcept · GlossaryIdempotency
Repeating an operation has the same relevant effect as performing it once. Stable operation identities and saved receipts can prevent duplicate effects.
Open conceptLesson · Point-set topology: spaces, constructions and separationIdentifications, saturation and gluing
Compute saturated sets as unions of equivalence classes.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsIdentify a finite return capacity without a diagnostic threshold
Tubular reabsorption uses membrane pathways with finite transport capacity, and the filtered amount depends on both filtration rate and plasma concentration. A transporter limit is an amount per time, not automatically a concentration. Converting one into a threshold requires the filtration rate and an explicit model of nephron behavior.
Open lessonLesson · Science · Earth and space scienceIdentify a galaxy from complementary evidence
Connect galaxy structure, projection and physical size with independent distance evidence.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceIdentify a nutrient constraint without turning a model into fertilizer advice
Plants require mineral nutrients as well as carbon, water and energy. Nitrogen contributes to proteins and nucleic acids; phosphorus contributes to nucleic acids, phospholipids and energy-transfer chemistry. A mineral nutrient is not interchangeable with photosynthetic carbon or with the energy used to acquire it.
Open lessonLesson · Science · Earth and space scienceIdentify a pattern with multiple lines
Retain one-feature ambiguity and test a single-species hypothesis using additional distinct transitions.
Open lessonLesson · Precalculus: exponential and logarithmic evidenceIdentify a shifted exponential from evidence
Use equal-step differences to recover a baseline, then check a prediction in original units.
Open lessonLesson · Stable matching and preference guaranteesIdentify blocking pairs and model assumptions
Distinguish a preference-stable outcome from a matching that merely pairs everyone.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeIdentify exactly which functions are integrals of derivatives
Prove indefinite integrals are absolutely continuous.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceIdentify experimental units and use paired contrasts
The unit independently assigned to a condition determines the replication available for a treatment claim. A hundred detector scans of one preparation are a hundred technical readings, not a hundred independent biological preparations. They can reveal instrument repeatability while leaving preparation-to-preparation variation almost unmeasured.
Open lessonLesson · Science · Physical Geology and Earth SystemsIdentify minerals by intersecting independent observations
A mineral is a naturally occurring solid with an ordered internal structure and a characteristic composition or compositional range. A rock can combine several minerals, glass or other constituents. Identification therefore begins with an individual material and a documented observation, not a rock color or a familiar-looking photograph.
Open lessonLesson · Science · Earth and space scienceIdentify minerals from several properties
Connect scratch resistance and cleavage to a conditional mineral identification.
Open lessonLesson · Abstract algebra I: groups and symmetryIdentify polyhedral rotations as permutation groups
Prove faithful actions and recover A4, S4 and A5 without identifying groups by order alone.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceIdentify the charge and voltage constraints of capacitor networks
Capacitor combination rules follow different common quantities from resistor rules. Across parallel capacitors, the potential difference is the same and plate charges add: Qtotal=(C1+C2)V. In a series pair with an initially neutral isolated intermediate conductor, the two capacitor charge magnitudes are equal. Their voltage drops add, so V=Q/C1+Q/C2 and 1/Ceq=1/C1+1/C2.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsIdentify the direction of friction from contact motion
A rotating body can slide at its contact while its center moves forward. The no-slip relation vCM=Rω is a constraint to test, not a universal identity. To reason about friction, compare the instantaneous contact motion relative to the surface, not just the direction of the center’s motion.
Open lessonLesson · Partial differential equations and Fourier methodsIdentify the equation and the data it needs
Separate PDE type, linearity and well-posedness.
Open lessonLesson · Science · Physics and physical systemsIdentify the force that bends a path
Separate required inward net force from the interaction supplying it, and compare fixed-speed and fixed-angular-rate changes.
Open lessonLesson · Abstract algebra I: groups and symmetryIdentify the operation before checking the axioms
Test closure, associativity, identity and inverses without assuming commutativity.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsIdentify the real inward force in circular motion
Uniform circular motion has constant speed but changing velocity direction. Over a small angular step, the velocity-vector triangle is similar to the position-radius triangle: Δv/v≈Δs/r. Divide by elapsed time and use Δs/Δt≈v to obtain inward acceleration aᵣ=v²/r. The direction changes even though the speed does not.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIdentify torsional stiffness with a known inertia increment
Use a second controlled state to break a parameter ambiguity and audit a biased period.
Open lessonLesson · Science · Earth and space scienceIdentify what stellar evidence cannot determine
Construct equal-flux and equal-spectrum examples and choose independent evidence that can distinguish them.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsIdentify which classes are squares
Count square classes, define the Legendre symbol and use primitive-root parity.
Open lessonLesson · Science · General Chemistry IIIdentify which salt ion actually transfers a proton
A salt can be acidic, basic or approximately neutral depending on its ions.
Open lessonLesson · Social studiesImperial rules, African sovereignty and resistance
Distinguish imperial claims from consent and compare responses to expansion.
Open lessonProject · Build, test and operate a small data toolImplement and test a fictional order-processing tool
Write a small program that joins customer IDs to orders, calculates a contract-defined price and reports unmatched inputs. Add independent tests and a local retry simulation that never double-applies the same operation. Work only with invented records.
Open projectLesson · Axiomatic Euclidean and non-Euclidean geometryImpossibility by field degree
Prove selected construction problems impossible under stated tools.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusImproper integrals and limits of integration
Treat infinite intervals and singular endpoints as separate limits, not formal substitutions.
Open lessonLesson · Course 2 · Planning and reasoningImprove a policy using imagined experience
Fit dynamics, optimize actions inside that model, then test in the external simulator.
Open lessonLesson · Mathematical statistics: derivation and model criticismImprove an estimator by conditioning
Derive risk reduction and show where sufficiency enters.
Open lessonConcept · GlossaryIn-context learning
Changing behavior in response to examples in the current input context, ordinarily without updating stored model parameters.
Open conceptLesson · Science · College Physics II — algebra-based fields and evidenceInclude measuring instruments in the circuit model
An ideal voltmeter draws no current, but a real finite input resistance is another parallel branch. An ideal ammeter adds no voltage drop, but a finite burden resistance changes the series path. This virtual packet makes both deviations explicit rather than calling a meter reading the untouched original quantity.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceInclude reflection phase in the optical path ledger
Thin-film interference combines a propagation difference with possible phase changes on reflection. At normal incidence, the beam that travels into and back out of a film accumulates optical path2nt, where t is physical thickness and n is the film index. Using vacuum wavelength λ0, the propagation phase is4πnt/λ0. Physical distance alone is not the optical path inside a material.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsInclude the meter in the circuit being inferred
An ideal voltmeter samples the difference between two nodes with infinite input resistance, drawing no current.
Open lessonLesson · Science · University Physics II with calculusInclude the return arc in a closed source geometry
A closed annular-sector loop makes the return-path issue explicit. The inner arc has radius 0.5 m and carries 2 A counterclockwise; the outer arc carries the same current clockwise, connected by radial segments. Both arcs subtend the same positive sector angle. Their current directions are opposite because they are parts of one closed path.
Open lessonLesson · Science · University Physics I with calculusInclude the ring area when deriving disk inertia
A disk with radial density cannot be treated as a line of equally weighted radii. An annulus has area 2πr dr, so its mass is that area times local surface density. Omitting the area factor changes both total mass and rotational inertia.
Open lessonLesson · Science · General Chemistry IIIncrease dissolved total while decreasing free metal
Complexed and free dissolved metal play different roles in a solubility product.
Open lessonLesson · Course 2 · Learning through actionsIncrease the probability of useful actions
Learn a stochastic policy from sampled returns and inspect the variance of actual updates.
Open lessonAssessment · Distribution calculations: events, tails and cutoffsIndependent event setup and numerical checks
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Differential calculus: techniques and applicationsIndeterminate forms and justified derivative ratios
Check the form and theorem hypotheses before replacing a ratio by a ratio of derivatives.
Open lessonLesson · Social studiesIndigenous societies and an Atlantic colonial world
Compare Native societies and colonial systems without treating either as uniform.
Open lessonConcept · GlossaryInductive bias
A learning process’s preferences among possible patterns. Architecture, optimization, regularization, and data all contribute.
Open conceptLesson · Social studiesIndustrial power, migration and empire
Connect industrial growth to labor organization, exclusion and overseas power.
Open lessonLesson · Social studiesIndustrialization: energy, government and working lives
Compare mechanisms of industrial change and test whether production gains improved workers' lives.
Open lessonLesson · Social studiesIndustry, nation-building and imperial expansion
Connect industrial change to class, nationalism, reform and empire.
Open lessonLesson · Science · General Chemistry IIInfer a concentration-cell polarity without changing the metal
Two identical electrode materials can differ in reduction potential because their ionic activities differ.
Open lessonLesson · Science · AP Chemistry preparationInfer a constant from an actual electrometric titration
Reconstruct apparent pK from historical NBS observations, distinguish derived columns from measurements, and inspect the authors’ endpoint exclusion.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceInfer a decay time separately from a voltage offset
A discharge trace can appear to approach a nonzero voltage because of a fixed instrument offset. The supplied true signal is 8exp(−t/2) V, but the indicated signal is Aexp(−t/τ)+o with A equal to instrument gain times true initial voltage. We deliberately keep the source generator visible for teaching while asking which parameters the observations alone identify.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsInfer a decay without confusing the meter and leakage
A capacitor discharging through leakage resistance Rl and parallel meter resistance Rm has equivalent resistance Req = RlRm/(Rl + Rm).
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingInfer a field cancellation from signed source contributions
Separate gravitational field from test-object force and preserve vector directions in superposition.
Open lessonLesson · Science · University Physics I with calculusInfer a force sum without inventing its agents
Newton's second law for a constant-mass point in an inertial frame is the vector equation Fnet=m a. A trajectory can determine the required net force after differentiation, but not uniquely identify which interactions produced that sum. First choose the recipient, then list forces acting on it; a reaction force on another recipient is not part of the same free-body diagram.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsInfer a load location while testing the support model
A uniform4 m beam has weight 12 N at its center x=2 m. A separate24 N point load is at x. Upward-only supports are at 1 and 3 m. The load and both support forces belong on the beam’s recipient diagram. Equal-and-opposite forces exerted by the beam on its supports act on different recipients and do not cancel the support forces here.
Open lessonLesson · Science · General Chemistry IInfer a mass-loss formula without confusing assumptions with observations
Combine before/after mass with a chemical hypothesis, then test whether the inferred ratio is identifiable.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer a parameter and challenge it with unused evidence
Parameter fitting is a claim about a model, not just a way to draw a line. For a release from rest at a known zero position with constant acceleration, x=0.5at². Plotting x against t² makes a straight-line model with slope k=a/2. A nonzero initial velocity or an incorrect origin would require extra terms rather than silently altering acceleration.
Open lessonLesson · Science · General Chemistry IInfer a partial-charge model from bond dipole evidence
Keep a measured dipole, an effective charge and a chemical bookkeeping label distinct.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer a photon from an energy-level difference, not a level label
A single-electron energy diagram assigns an energy to each stationary state. A transition emits a photon when the final state has lower energy, and absorbs a photon when it has higher energy. Conservation gives photon energy |Ef−Ei|=hf=hc/λ. The magnitude of one level relative to zero is not generally the energy of a transition between two bound states.
Open lessonLesson · Science · General Chemistry IInfer a shell boundary from successive ionization evidence
Keep individual ion-removal energies separate from their cumulative cost.
Open lessonLesson · Science · General Chemistry IInfer a signed lattice term from a formation cycle
Use a reaction-defined energy cycle rather than a memorized sign for lattice energy.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceInfer a source location without assuming an instrument zero
An inverse problem asks which source parameters could have produced a set of observations. Here the stipulated family is a single positive point charge at position s left of x=1 m, with indicated potential R(x)=K/(x−s)+o and K=kQ. Three readings at x=1,2,3 m contain information about source position, strength and a common offset. Assuming o=0 without evidence can produce a misleading source estimate.
Open lessonLesson · Science · Physics and physical systemsInfer a spectral line from an energy-level difference
Use differences between stipulated levels to predict line energies and challenge identification from one line.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer a spring constant and retain the restoring sign
An ideal spring near its equilibrium length exerts a restoring force Fₛ=−kx, where x is signed displacement from the equilibrium configuration and k is positive. Extension to the right produces a leftward force; compression to the left produces a rightward force. The minus sign describes direction, not a negative stiffness.
Open lessonLesson · Science · Introductory AstronomyInfer a surface history without equating craters with age
An impact crater record reflects production and preservation. Volcanic resurfacing, erosion, burial and later impacts can remove or obscure older craters. Comparing surfaces therefore requires common counting criteria, area, size range and a calibrated production history; a larger raw count is not automatically an older surface.
Open lessonLesson · Science · Introductory AstronomyInfer a total mass from the correct orbit and bounded measurements
Newton’s version of Kepler’s third law connects the relative orbit of two gravitating bodies to their total mass. In the conventional AU, year and solar-mass normalization used here, M₁+M₂=a³/P². The semimajor axis is half the largest diameter of the relative separation ellipse, not the instantaneous separation or one star’s smaller orbit around the center of mass.
Open lessonLesson · Science · General Chemistry IIInfer a weak-acid state from mass, charge and water together
Weak acid strength, total concentration and fractional ionization are distinct quantities.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsInfer acceleration while auditing the clock
For constant acceleration, position changes quadratically with time: x=x0+v0t+at²/2. This is an algebraic model with a stated interval, not an assertion that every motion is uniformly accelerated. Equal-time position samples supply a useful test: neighboring displacement increments change by a constant amount.
Open lessonLesson · Science · General Chemistry IInfer an amount from a gas slope rather than a biased reading
Separate absolute state variables from a common instrument offset.
Open lessonLesson · Science · General Chemistry IInfer an empirical ratio without assigning oxidant oxygen to the sample
Combine carbon and hydrogen capture with a conditional oxygen-by-difference account.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer an ensemble half-life without predicting an individual decay
For independent nuclei with a constant decay probability per unit time, expected surviving population follows N=N0(1/2)^(t/T), where T is half-life. Equal time intervals remove an equal fraction on average, not an equal number. After two half-lives one-quarter remains and three-quarters have decayed. The law describes an ensemble; it does not schedule the next decay of one nucleus.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsInfer an interface temperature from a steady heat ledger
Conduction transfers energy through microscopic interactions without requiring bulk material motion. For a uniform layer under steady one-dimensional conditions, heat rate P=kAΔT/L. Define thermal resistance Rth=L/(kA), measured in K/W, so P=ΔT/Rth. This is a rate law, not the heat capacity that sets energy needed to change temperature.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer an RC time scale without confusing energy and charge
A resistor-capacitor circuit changes with time because the capacitor voltage depends on accumulated separated charge. In a charging series circuit, an initially uncharged capacitor has zero voltage, so the resistor initially takes the source voltage and current is largest. As capacitor voltage rises, less voltage remains across the resistor and current decreases. At the ideal long-time limit, current approaches zero and capacitor voltage approaches the source voltage.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer an unknown charge from spatial force data
A single force reading can fit many explanations: a larger source charge, a shorter separation or a sensor offset. To distinguish them, keep the probe charge fixed and compare several known separations. For point charges in vacuum, the magnitude is F=k|Qq|/r², where r is center-to-center separation and k=8.99×10⁹ N m²/C². Like signs repel and opposite signs attract along their line of separation.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer angular change from net torque and inertia
For a rigid body rotating about a fixed inertial axis with constant inertia, the angular form of Newton’s second law is τnet=Iα. It relates the net external turning effect to angular acceleration, not directly to angular velocity. Use the same axis when computing inertia and every torque; mismatched axes describe different problems.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceInfer community metabolism from an oxygen assay with explicit assumptions
An environmental assay measures an observable response, not a process without assumptions. In a closed illuminated water sample, photosynthetic oxygen production and community respiration both affect dissolved oxygen. A dark sample suppresses photosynthesis but still contains respiring producers and heterotrophs. Their difference can support an inference only if other conditions are comparable.
Open lessonLesson · Science · General Chemistry IInfer concentration from blank-corrected equivalents
Separate endpoint volume, stoichiometric equivalents and acid formula-unit amount .
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceInfer decay while preserving background and unused records
Independent unstable nuclei have a statistical survival law. An exponential expectation does not specify which individual nucleus decays next or promise that an actual sample contains exactly half its nuclei after one half-life. Activity is λN in decays per time, whereas a detector rate additionally depends on efficiency, branching and geometry. The synthetic source contribution here begins at 640 counts/min; that rate alone does not specify an initial number of nuclei.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer density without confusing sample mass and tare
Solids, liquids and gases differ because of particle arrangements, motion and interactions. A solid resists sustained shape change; a liquid flows while retaining approximately fixed volume; a gas flows and can change volume substantially. Liquids and gases are fluids. An ideal fluid in this course is incompressible and has no viscosity, an approximation rather than a claim that real materials lack internal interactions.
Open lessonLesson · Science · AP Chemistry preparationInfer deposited material from charge and stoichiometry
Convert current and time to electron amount, deposited mass and conditional efficiency.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceInfer differential growth without assigning one universal hormone effect
Plant signals coordinate development through concentration, transport, receptor sensitivity and tissue context. Auxin can influence differential elongation, apical dominance and organ development; its effect is not a universal command to grow faster in every tissue. Gravity and light responses depend on redistribution and downstream response as well as signal presence.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceInfer direct and indirect interactions from controlled contrasts
Competition reduces performance when organisms share a limiting resource; predation and herbivory transfer material through consumption.
Open lessonLesson · Science · General Chemistry IIInfer dissolved gas from its own partial-pressure constraint
Solubility is an equilibrium composition under specified conditions.
Open lessonLesson · Science · Earth and space scienceInfer distance from seismic arrival times
Use a P–S arrival separation to calculate a conditional source distance.
Open lessonLesson · Science · AP Chemistry preparationInfer dry gas from water levels and a wet pressure
Repair an amount calculation with two signed pressure corrections and detect inconsistent observations.
Open lessonLesson · Science · AP Chemistry preparationInfer equivalence without mistaking an endpoint for proof
Convert buret differences into blank- and calibration-corrected amounts; withhold target inference when calibration or selectivity is missing.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceInfer expression mechanisms without equating abundance and synthesis
An RNA abundance measurement reflects molecules produced and molecules removed. More RNA can result from increased transcription, slower degradation or both. At steady state, production and loss balance while individual molecules continue to turn over; a stable amount does not imply that transcription or degradation has stopped.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer field direction and cancellation from two currents
A long straight current produces a magnetic field that circles the wire rather than pointing along it. For a thin wire in vacuum, B=μ₀|I|/(2πr), where r is perpendicular distance from the wire axis. The approximation neglects finite ends and the internal field inside a real wire. The direction follows curled fingers around the conventional current direction, not electron drift.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsInfer flow only from a complete oxygen balance
A steady organ oxygen balance connects blood flow Q, arterial content Ca, venous content Cv and oxygen uptake U through U=Q(Ca−Cv). This is a conservation relation, not a statement that partial-pressure difference equals content difference. If the organ stores oxygen during the observation interval or another source contributes, the simple steady equation needs additional terms.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer fluid head from an ideal jet without mixing two heights
For a large open tank with a small opening, compare the free surface and the exiting stream. Both are at atmospheric pressure, and the large surface area makes the surface speed negligible over a short interval. Bernoulli’s equation then gives exit speedv=√(2gh), whereh is the free-surface height above the opening. This is Torricelli’s result under stated ideal assumptions, not a universal discharge rate.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer gas amount and challenge a straight-line calibration
An equation of state becomes an experimental inference only after its controlled quantities are specified. At fixed amount and volume, P=nRT/V predicts a pressure-versus-Kelvin-temperature line through the origin. A plot against degrees Celsius has a nonzero intercept because T = θ + 273.15. Absolute pressure, not gauge pressure, belongs in this equation.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingInfer inertia from parallel-axis evidence without hiding bias
Use the linear relation in squared offset and test its conditioning.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsInfer inertial mass while retaining a force-reading fault
A force-versus-acceleration plot can estimate inertial mass only under explicit force and measurement assumptions. Here independent acceleration labels are stipulated exact. The synthetic true mass is 2 kg and force readings follow ma+o, where o is a common additive readout offset. A second selected fault affects only the high-acceleration reading.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer internal resistance and audit source power
A source’s emf is energy supplied per unit charge, measured in volts despite the historical word “force.” A simple nonideal source model puts an ideal emf in series with internal resistance r. During discharge into an external load, terminal voltage is Vterminal=emf−Ir. It equals emf only at zero current in this model, not at every operating point.
Open lessonLesson · Science · AP Chemistry preparationInfer isotope composition from a mass spectrum
Normalize a declared single-element signal and distinguish reference masses, sample composition and uncertainty.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsInfer liquid density without confusing it with a balance offset
A fluid changes shape under sustained shear rather than supporting a fixed shear deformation like an ideal elastic solid. Liquids and gases both qualify, but an incompressible-liquid approximation treats density as unchanged across the selected pressure range. Density is mass per volume, not weight per volume. A sample can have larger mass without being a denser material.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer mass from a force–acceleration relationship
Newton’s second law relates the net force on a fixed-mass object to its acceleration: Fnet=ma. The word net matters. A pulling-force reading alone is not net force when friction or another horizontal interaction also acts. Before fitting mass, specify which forces the sensor includes and which must be subtracted.
Open lessonLesson · Science · General Chemistry IInfer mass through an explicit calibration model
A reading is not the measurand: derive an inverse calibration and test what the standards can establish.
Open lessonLesson · Science · Physical Geology and Earth SystemsInfer path speed from summed slowness contributions
A travel time measures a sum of path length divided by speed along the ray. Even in two simple vertical layers, the effective speed is not generally the arithmetic mean of layer speeds. Each segment contributes time, and the total path divided by total time gives the corresponding effective speed.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceInfer quartet scores from an explicit original character matrix
A character matrix records taxa as rows and comparable characters as columns. Missing data, correlated characters and coding choices must remain visible. A tree score is interpretable only when a learner can trace it back to this matrix and the stated model of character change.
Open lessonLesson · Science · AP Chemistry preparationInfer radioactive half-life from corrected count evidence
Correct background, compare equal counting windows and withhold a half-life when signal or detector conditions fail.
Open lessonLesson · Science · General Chemistry IIInfer rate exponents through separate concentration interventions
Write r=k[A]^m[B]^n as a hypothesis at fixed temperature and controlled conditions.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer rotational inertia from a coupled motion model
A hanging mass and a fixed-axis pulley require separate system diagrams. For the descending mass, weight mg acts down and tension T up, giving mg−T=ma. For the pulley, string tension acts tangentially at radius R, giving TR=Iα when bearing friction is neglected. The axle forces do not contribute torque about its own fixed axis.
Open lessonLesson · Science · Physics and physical systemsInfer slit separation from a fringe pattern
Use controlled comparisons and a small-angle estimate to infer an optical length.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceInfer stage durations cautiously from cell samples
A microscope snapshot samples cells at different stages.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer standing-wave modes from displacement and pressure boundaries
Standing waves can form from oppositely traveling waves in a constrained region. Their nodes remain at fixed positions, while antinodes are locations of maximum oscillation amplitude. An antinode is not permanently at maximum displacement: it passes through equilibrium during each cycle. The allowed wavelengths follow the end conditions, not an arbitrary list of frequencies chosen by the observer.
Open lessonLesson · Course 2 · Planning and reasoningInfer state across time
Track a hidden moving object by alternating motion predictions and noisy sensor updates, while keeping a distribution over possible positions.
Open lessonLesson · Science · Physical Geology and Earth SystemsInfer stream exchange from a shared hydraulic-head datum
A stream may gain water from an aquifer or lose water into it. The direction depends on hydraulic heads and hydraulic connection, not on comparing water depths measured from unrelated ground surfaces. Elevation and pressure must be referred to a common datum.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer temperature drops across a two-layer barrier
At steady state, energy does not accumulate at the interface between the two layers. The same thermal power must therefore cross each layer. Equal power does not imply equal temperature drops or equal temperature gradients. The material parameters and geometry determine how the total drop is divided.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceInfer topology and fragment identity without overreading a gel pattern
A restriction-map hypothesis predicts fragment lengths from cut positions and molecule topology. A linear molecule has physical ends; a circular molecule has a wraparound segment. Two complete cuts therefore make three fragments from one linear molecule but two from a circular molecule, provided the cuts are distinct and no extra fragmentation occurs.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsInfer velocity from a position graph with uncertainty
An interval-average velocity is the slope joining two points on a position–time graph: (x₂−x₁)/(t₂−t₁). Instantaneous velocity is the local slope. A graph can sit above zero yet slope downward: positive position and negative velocity are perfectly compatible. Screen angle alone is not a velocity; read axis scales and units.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceInfer water movement and an unknown solution with limits
Combine solute and pressure potentials, then analyze a separate synthetic mass-change inquiry.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsInfer wave parameters without confusing propagation with material motion
A traveling wave transports a disturbance and energy; the medium’s particles need not travel with the wave pattern. Transverse displacement is perpendicular to propagation, while longitudinal displacement is parallel. A spatial snapshot gives wavelength λ, and a time record at one position gives period T or frequency f=1/T. Phase speed is v=fλ; neither record alone determines both frequency and wavelength.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInfer wavelength while testing geometry and calibration assumptions
A fringe pattern can support an inference only after its coordinate references and geometry are stated. The synthetic packet uses signed orders −3 through +3, slit separation 0.25 mm and screen distance 1.20 m. Exact generated positions use y=L tan(arcsin(mλ/d)), then add a common coordinate offset and a small declared reading error. The source wavelength is available for testing the model, not because a real instrument has been calibrated.
Open lessonLesson · Course 1 · Generative models beyond textInfer what cannot be directly observed
Fit a two-component mixture in separate EM steps and compare exact latent inference with a restricted variational approximation.
Open lessonCourse · Mathematics program unitInference workflows: calculate, check and conclude
Complete proportion, mean and categorical analyses using local numerical reference distributions. Choose methods from design, inspect assumptions and interpret uncertainty; tool output does not certify the study or award mastery.
Open courseAssessment · Statistical inference: estimates, tests and defensible conclusionsInference: choose methods and defend conclusions
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitInfinite cardinality through explicit constructions
Control digit ambiguity, build a bijection from two injections, and absorb countably many elements with explicit domain and selection assumptions.
Open courseLesson · Point-set topology: spaces, constructions and separationInfinite products and the box topology
Describe product neighborhoods with finitely many restricted coordinates.
Open lessonRefresher · Math refreshersInformation and compression (optional)
Practice the mathematics, then return to your lesson.
Open refresherLesson · Course 1 · How does a network learn?Initialize and normalize
Inspect activation and gradient scales, break hidden-unit symmetry, and compare batch and layer normalization in training and evaluation modes.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociInscribe a square, hexagon and equilateral triangle
Construct three regular figures from a given circle and prove their equal sides and angles.
Open lessonPath · Software engineers · model foundationsInspect and evaluate models
Trace gradients, compare representations and test a trained classifier.
Open pathExperiment · Cybersecurity and digital literacyInspect before you follow
Which action prevents exposure without discarding a legitimate notice?
Open experimentLesson · Optimization: models, certificates and algorithmsInspect curvature in feasible critical directions
Use the Lagrangian Hessian rather than the objective Hessian alone.
Open lessonLesson · Social studiesInspect microscopy as a collective practice
How does a private observation become public knowledge?
Open lessonLesson · Social studiesInspect strategic interaction and entry
Can individually attractive choices lower both firms’ profits?
Open lessonLesson · Science · Scientific inquiry and measurementInspect what a model misses
Use residuals to identify a systematic model mismatch.
Open lessonLesson · Science · Scientific inquiry and measurementInspect who entered the sample
Connect selection rules to a sample's representativeness.
Open lessonExperiment · Physics and engineeringInsulation and thermal mass solve different problems
What changes warm-up time, and what changes eventual temperature?
Open experimentLesson · Real Analysis I: completeness, limits and rigorous calculusIntegrability through oscillation control
Prove two distinct sufficient conditions and inspect bounded discontinuous examples.
Open lessonAssessment · Integral calculus: accumulation, methods and applicationsIntegral calculus: accumulation and method selection assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitIntegral calculus: accumulation, methods and applications
Sixteen substantive lessons linking Riemann sums, the fundamental theorem, geometric and contextual models, integration techniques and approximation. This unit is not complete AP AB/BC or Calculus II coverage. Symbolic arguments and modeling decisions need review against written rubrics.
Open courseLesson · Real Analysis I: completeness, limits and rigorous calculusIntegral properties and the fundamental theorem
Distinguish the derivative of accumulation from evaluation using a primitive.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate a circular-aperture response over detector samples
A uniformly illuminated circular aperture has a different transform from a rectangular slit. In the scalar far field, the normalized amplitude is 2J1(v)/v, with v=πDθ/λ in the small-angle approximation. J1 is the first-order Bessel function. Its limiting amplitude at v=0 is one, and the intensity response is the squared amplitude.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate a coupled system without canceling the wrong forces
Combine system dynamics with recipient equations to recover an internal connector force.
Open lessonLesson · Science · University Physics I with calculusIntegrate a density profile before evaluating pressure and buoyancy
Hydrostatic force balance on a thin fluid layer gives dp/dz = ρg when depth z increases downward. The familiar pressure increment ρgh assumes constant density. Here the stipulated stable profile is ρ(z) = 1000(1 + βz) kg/m³ with nonnegative β.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate a finite current segment before using an infinite limit
A straight segment runs from x = −a to +a with current I toward +x, and the observer is (0,y,0), y positive.
Open lessonLesson · Science · University Physics I with calculusIntegrate a force along two genuinely different paths
Work is the line integral of F·dr, not the integral of a force magnitude with every sign discarded. In two dimensions its component form is integral(Fx dx+Fy dy). Parameterize each path segment, substitute the coordinates along that segment and include only its actual displacement components.
Open lessonLesson · Science · University Physics I with calculusIntegrate a force ramp with the original initial conditions
A force law gives acceleration, but an acceleration equation still needs initial conditions. The original2 kg point has x0=0 and v0=1 m/s, with net force F(t)=F0+k t. The coefficient k has units N/s; it is a slope in time, not a spring constant.
Open lessonLesson · Science · University Physics II with calculusIntegrate a nonuniform current density
A measured total current is an integral across an oriented surface, not a point value of current density. Consider an imposed axial profile J(r)=J0[1+α(r/R)²] in a circular wire of radius R=1 mm, carrying total conventional current 2 A in the positive axial direction. The coefficient α changes the shape while total current stays fixed.
Open lessonLesson · Science · University Physics II with calculusIntegrate a nonuniform field through an oriented patch
Flux is an oriented surface integral, not a count of actual particles and not the magnitude of the electric field. For each small patch, use E dot n times its area. A tangent field can be large while contributing no flux. For an open patch, reversing the chosen normal reverses flux; enclosed-charge language applies only after specifying a closed surface.
Open lessonLesson · Science · University Physics II with calculusIntegrate a nonuniform permittivity through the gap
A spatially varying dielectric does not generally act like its arithmetic-average permittivity. In this one-dimensional ideal gap, ε(z)=ε0(1+αz/d), there is no free volume charge and the normal displacement D is constant. The electric field is E(z)=D/ε(z), so it decreases where the permittivity is larger.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate a physical pendulum and test the best pivot distance
Combine a mass integral, parallel-axis theorem and a small-angle rotational differential equation.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate a radial density before invoking Gauss symmetry
Distinguish insulating volume charge from a conducting surface distribution.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate a rotational torque law with its initial condition
Extend fixed-inertia torque dynamics to a bounded velocity-dependent resistance model.
Open lessonLesson · Science · University Physics II with calculusIntegrate a signed current-segment contribution
Biot–Savart is a vector source integral. For a straight segment from x=−1 to +1 m carrying 3 A along positive x, an observer at (0,y,0) with positive y sees a contribution along positive z: the sign follows from dl crossed with the source-to-observer vector. We use the explicitly rounded teaching value μ0=4π×10⁻⁷, not a claim that this is an exact modern SI constant.
Open lessonLesson · Science · University Physics I with calculusIntegrate a smooth force without confusing impulse and work
For a fixed-mass particle, net force is the time derivative of momentum. Integrating a signed force over time gives momentum change, not energy change. Here a 2 kg particle initially moves at −1 m/s under the supplied net force F(t) = A sin(πt/T), from t = 0 to T. No measured impact waveform or apparatus is implied.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate a spectrum to predict a fringe envelope
Two delayed copies of one source can retain coherence at each frequency while different frequency components average incoherently in the detector. The relevant calculation averages the phase-dependent cross term over the power spectrum; it does not first add unrelated spectral fields as though all frequencies had a fixed common phase.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate a spherical capacitor from two energy accounts
For concentric conductors of radii a and b carrying ±Q, the vacuum gap field is kQ/r².
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate a temperature-dependent state-energy change
For a fixed-volume system with no nonmechanical work or phase transition, dU=n Cv(T)dT. If heat capacity varies with temperature, the finite energy change is an integral, not one arbitrarily chosen endpoint capacity times deltaT. The packet has n=2 mol, Ti=300 K and molar Cv(T)=A+BT, with A=20 J/(mol K).
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate a vector force and differentiate the resulting momentum
Connect force-time area, momentum-time slope and a vector initial condition.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate a velocity distribution into directional pressure
A zero mean velocity does not imply zero pressure. Consider independent uniform velocity components on intervals −ux..ux, −uy..uy and −uz..uz, with ux=100a, uy=100b and uz=100 m/s. Mass density is 0.6 kg/m³. This original anisotropic packet is not asserted to be thermal equilibrium: collisions would normally relax unequal directional moments unless the distribution were maintained.
Open lessonLesson · Science · University Physics I with calculusIntegrate acceleration twice and locate a reversal
Acceleration alone does not specify one unique trajectory.
Open lessonLesson · Science · Physical Geology and Earth SystemsIntegrate an actual discharge archive without inventing an instantaneous peak
A stream hydrograph records flow through time, but its interpretation depends on how each value was formed. An instantaneous measurement, daily mean and daily maximum are not interchangeable. A daily mean preserves the volume integral over its day when multiplied by that day’s duration; it cannot recover the exact timing or height of an unrecorded subdaily peak.
Open lessonLesson · Science · University Physics II with calculusIntegrate an arc while total charge stays fixed
Parameterizing a curved source requires its length element. A circular arc of radius R at angle φ has ds=R dφ when φ is in radians. For a uniform arc from −θ to +θ carrying fixed charge Q, λ=Q/(2θR). Changing the angle while holding Q fixed therefore changes its line density.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate an arc with signed component geometry
Retain angular limits, density normalization and two-component directions.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate an oriented magnetic flux before differentiating
Magnetic flux is a signed surface account, not the amount of magnetic field at one point.
Open lessonLesson · Science · University Physics II with calculusIntegrate changing flux before inferring an induced field
A changing magnetic field creates an electric circulation even if no wire occupies the contour. Choose a stationary circle with positive z normal and counterclockwise positive traversal. In the local region let Bz=(B0+βt)[1+(r/ell)²], with B0=2 T and ell=1 m. The ramp β has units T/s, while the radial factor is dimensionless.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate charging work instead of using a final-state rectangle
An initially uncharged ideal capacitor has voltage V(q) = q/C at an intermediate charge q.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate current in space and charge in time
Current is dq/dt through a selected cross-section.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate dissipation only along an accessible branch
Combine position-dependent friction with kinetic-energy feasibility and an enlarged thermal inventory.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate distance across a velocity reversal
Use zero crossings to distinguish signed displacement from accumulated distance.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate distinct work paths between identical end states
Fix the same initial and final equilibrium states of a monatomic ideal gas: initial volume V0=1 litre and pressure p0, final volume rV0 and pressure p0/r. Both endpoints have the same pV and hence the same temperature. This does not force every intervening path to be isothermal.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate elapsed clock time over a complete worldline
Proper time is the time accumulated by a clock along its actual trajectory. In flat spacetime, dτ=dt sqrt(1−v(t)²) with c=1. A constant-speed segment gives Δτ=Δt/γ, but a changing velocity requires integration or a sum over segments; one maximum speed does not specify the whole clock reading.
Open lessonLesson · Science · University Physics II with calculusIntegrate enclosed charge in a nonuniform sphere
A spherical Gaussian surface simplifies the field integral only when the complete charge distribution and boundary conditions have spherical symmetry. In this original insulating sphere, radial density is ρ0(1+αr²/R²), radius R=1 m and total charge Q=3 nC. No external field or conducting rearrangement is included.
Open lessonLesson · Science · Physical Geology and Earth SystemsIntegrate glacier mass balance over an elevation-area distribution
A glacier gains mass through accumulation and loses it through ablation. The equilibrium line separates locally positive and negative annual surface balance for a stated interval, but glacier-wide balance also depends on how much area occupies each elevation band. The line’s height alone is not a complete mass budget.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate gravitational work with a defensible reference
Recover inverse-square potential and distinguish an exact radial transfer from a local constant-field approximation.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceIntegrate local approximations with an error model
Derive composite rules, polynomial exactness and derivative-based error bounds.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate magnetic motion and recover a Hall sign
For B along +z and signed omega = (q/m)B, the Lorentz equations are dvx/dt = omega vy, dvy/dt = −omega vx and dvz/dt = 0.
Open lessonLesson · Science · University Physics II with calculusIntegrate magnetic rotation and parallel translation
Uniform magnetic force is perpendicular to velocity, so it can turn motion without changing kinetic energy. Use a fictional nonrelativistic particle with mass 1 kg, charge +1 or −1 C, field 1 T along positive z and initial velocity (vperp,0,2) m/s. These parameters define a mathematical experiment, not a real particle or apparatus.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate material resistance without confusing graph slopes
Resistivity describes a specified material state; resistance also depends on geometry.
Open lessonLesson · Science · University Physics I with calculusIntegrate mechanical loss once within a declared system
A2 kg point in a stationary medium experiences linear resistance with coefficient1 kg/s and no drive. From initial velocity v0, the solution is v=v0 exp(−t/2). Its kinetic energy is v0² exp(−t) joules because m/2=1 in this packet.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate momentum and energy over a moving distribution
Distinguish the velocity of the center from velocities distributed through an extended system.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate power without confusing one interaction with net work
Compare an instantaneous dot product, interval-average power and cumulative work.
Open lessonLesson · Science · University Physics II with calculusIntegrate resistance rather than averaging a cross section
A long, gently tapered conductor can be represented by thin resistive slices when section-average current is a justified approximation. Here length is 1 m, entrance area A0=1 mm² and radius grows as r0(1+αx/L). Resistivity is fixed at 2×10⁻⁸ Ω m. This is a stipulated one-dimensional element, not an exact three-dimensional boundary solution.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsIntegrate ring projections before taking limits
Use axial symmetry without replacing the ring by a point at every distance.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate rotational energy without double counting
Normalize a mass distribution before integrating local kinetic energy.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate rotational inertia of a nonuniform rod
Weight each differential mass by squared perpendicular distance before applying a torque law.
Open lessonLesson · Science · University Physics I with calculusIntegrate rotational relaxation and its work budget
For a rigid body rotating about a fixed axis with constant inertia, torque balance is I dω/dt = τnet. A constant applied torque does not imply constant net torque if resistance changes with angular speed. The supplied body has I = 2 kg m², drive torque 4 N m and resisting torque −bω, initially at rest.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceIntegrate shell probability rather than reading a spatial-density peak
A normalized hydrogenic 1s density per physical volume is rho(r)=exp(−2r/a)/(pi a³). It is largest at r=0. That does not mean a thin shell at the origin contains the most probability: a spherical shell has volume 4pi r²dr. Multiplying gives radial probability density P(r)=4r²exp(−2r/a)/a³, with units of inverse length.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate signed torque across a distributed load
Use a cross product and an origin declaration before replacing a load by a resultant.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate sliding motion to its first no-slip event
Evolve translation and rotation separately while kinetic friction acts.
Open lessonLesson · Science · Introductory AstronomyIntegrate stellar support rather than equating force with pressure
Hydrostatic equilibrium balances a pressure gradient against inward gravity: dP/dr=−GM(r)ρ/r². Pressure itself is not a net force; differences across a layer provide the force per volume. A stable average structure can still contain microscopic thermal motion, radiation transport and convection.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingIntegrate torque over an accessible angular path
Distinguish torque-angle work, time impulse and the turning point of an energy branch.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeInterchange nonnegative iterated integrals
Prove Tonelli by indicators and monotone approximation.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusInterchanging limits and Riemann integrals
Control integral errors by uniform error and exhibit concentration that defeats pointwise passage.
Open lessonLesson · Social studiesInterest groups: trace access, information and policy effects
Interest groups may supply expertise, mobilize members, litigate or seek access. Evidence of participation is not evidence of policy control. Trace whether an argument entered the decision record and how the final rationale treated it.
Open lessonConcept · GlossaryInterface
The agreed form and interpretation of information passed between components, including fields, shapes, units, and meaning.
Open conceptLesson · Social studiesInterference: which learning disrupts which memory?
Distinguish proactive from retroactive interference and test similarity separately from elapsed time.
Open lessonCourse · Social studiesInternational relations — High school
Hands-on social studies course and project
Open courseCourse · Social studiesInternational relations — Introductory college
Hands-on social studies course and project
Open courseLesson · Science · AP Environmental ScienceInterpret a changing capacity and instantaneous growth
Calculate density feedback without confusing an instantaneous rate with a future count.
Open lessonLesson · Science · Introductory MicrobiologyInterpret a coding change without inventing its phenotype
A coding sequence is read in triplets within a defined frame. A substitution can preserve the encoded amino acid, replace it, or create a stop signal; an insertion or deletion not divisible by three can shift later triplet boundaries. These labels describe consequences under a code and frame, not a guaranteed effect on growth, resistance or disease.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceInterpret a cooperation condition without treating it as a universal rule
Cooperative behavior can arise through several mechanisms. A direct mutual benefit differs from a cost to an actor that benefits a relative; repeated interactions and reciprocal effects add further possibilities. Merely observing a benefit to another organism does not identify the evolutionary mechanism.
Open lessonLesson · Science · Introductory MicrobiologyInterpret a curved retention record without inventing one mechanism
A retained or surviving fraction can change nonlinearly across a recorded series. Different subpopulations, shielding, recovery limitations or changing measurement can generate a tail. A curve alone therefore does not identify a unique resistance mechanism or demonstrate a universal process rate.
Open lessonLesson · Science · General Chemistry IInterpret a mass-to-charge coordinate before naming a mass
Resolve the ambiguity between charge state and particle mass using a second spectral feature.
Open lessonLesson · Science · Introductory MicrobiologyInterpret a perturbation without combining unlike ecosystem outcomes
A community perturbation can change carbon retention, nutrient export and other processes in different directions. Those outcomes use different currencies and do not automatically collapse into one score. A carbon benefit in one boundary may coexist with an energy demand or nitrogen tradeoff outside it; an explicit decision criterion is needed before ranking scenarios.
Open lessonLesson · Science · General Chemistry IInterpret a radial probability distribution without inventing an orbit
Separate probability per volume, radial shell density and probability over a finite region.
Open lessonLesson · Science · Introductory MicrobiologyInterpret a section without mistaking it for the whole object
A two-dimensional microscopy record can represent a section, an optical slice, a surface view or a projection through depth. These are different sampling operations. Multiple structures along a line of sight can overlap in projection, while a thin section can miss the widest part of an object or omit a structure located outside its plane.
Open lessonLesson · Science · Introductory MicrobiologyInterpret a supplied growth record without fitting one rate to every phase
A microbial population record reflects births, losses, changing resources and the measurement method. A typical closed-system description distinguishes an initial adjustment interval, a period of approximately exponential increase, a roughly stationary count and eventual decline, but not every actual record follows that sequence. A flat count does not imply that every cell is inactive or that division has stopped.
Open lessonLesson · Science · Earth and space scienceInterpret a temperature-only stability comparison
Compare lapse rates and identify where the parcel temperature crosses the environment without claiming a full buoyancy calculation.
Open lessonLesson · Science · Earth and space scienceInterpret a two-layer density comparison
Relate density ordering to a bounded buoyancy tendency while distinguishing stability from absence of motion.
Open lessonLesson · Science · Biology and living systemsInterpret a zero gas signal
Test whether a zero net signal distinguishes inactivity from balanced processes.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsInterpret an interval as a procedure
Separate a fixed parameter, a random interval and coverage assumptions.
Open lessonLesson · Science · Earth and space scienceInterpret assessed human and natural contributions
Distinguish combined human influence from warming and cooling components over a shared historical period.
Open lessonLesson · Science · Earth and space scienceInterpret boundary motion and its limits
Separate across-boundary and along-boundary motion before inferring a tectonic setting.
Open lessonLesson · Science · Physics and physical systemsInterpret capacitor discharge and threshold times
Track voltage decay, signed current and the difference between voltage and energy half-times.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceInterpret chromosome-linked and cytoplasmic inheritance
Genes on sex chromosomes can produce inheritance patterns different from autosomal loci.
Open lessonLesson · Science · Earth and space scienceInterpret declining storage and infeasible budgets
Distinguish release from storage from an arithmetic deficit that violates the assumptions.
Open lessonLesson · Science · AP Environmental ScienceInterpret dose-response and median-effect claims
Separate endpoint, exposure and population variation from a synthetic curve.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceInterpret gel and expression evidence with controls
Electrophoresis separates molecules according to properties that affect movement through a medium.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceInterpret localization as a distribution, not a trajectory
A prepared quantum state can produce a distribution of positions even if an ideal instrument has no additive bias. The supplied Gaussian density is exp[−(x−μ)²/(2σx²)]/[σx sqrt(2π)]. Its horizontal variable is position and its vertical unit is inverse length. A density larger than one per nanometre is not a probability larger than one: probabilities refer to areas over intervals. The whole-line integral is one, while the finite plotted range omits small tails.
Open lessonLesson · Science · Physics and physical systemsInterpret magnetic flux and coil orientation
Distinguish field, flux per turn and flux linkage using the coil normal.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsInterpret negative loop current and source absorption
Choose a loop traversal and an assumed positive current direction before writing equations. Traversing an ideal source from negative to positive terminal gives a potential rise; reversing the traversal gives a drop. Across a resistor, moving with the assumed current gives −IR. Consistent signs let the calculation reveal the actual direction rather than requiring a guess to be correct.
Open lessonLesson · Science · AP Environmental ScienceInterpret noise through logarithmic levels and exposure context
Add independent source intensities and distinguish level, duration and biological effects.
Open lessonLesson · Science · General Chemistry IInterpret nonideal gas evidence without a single-cause shortcut
Separate two model corrections and test the domain before interpreting compressibility.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceInterpret organelle enrichment using independent marker controls
Eukaryotic compartments organize processes while remaining connected by transport and signaling. The nucleus encloses much of the genetic material; ribosomes synthesize proteins; endoplasmic reticulum and Golgi participate in processing and traffic; mitochondria and chloroplasts have distinct energy-transforming roles. A compartment name is a hypothesis about location and function, not a substitute for evidence.
Open lessonLesson · Science · AP Environmental ScienceInterpret photochemical smog without a universal precursor rule
Explain sunlight-driven secondary ozone and compare conditional precursor sensitivities.
Open lessonLesson · Science · Introductory MicrobiologyInterpret recoverable units without silently equating them to cells
A colony-forming unit is an operational recoverable unit under a specified assay context, not a universal synonym for one cell. An aggregate can contain several cells yet contribute one recoverable unit, and some viable cells may fail to produce a counted result under the recorded conditions. Conversely, nonviable material can still contribute to direct counts or optical signals.
Open lessonLesson · Science · Physical Geology and Earth SystemsInterpret recurrence as a conditional probability model
A mean recurrence interval summarizes a statistical model or record; it is not a schedule. Under a stationary homogeneous Poisson model with rate λ, the expected number of events in a window T is λT and the probability of at least one is 1−exp(−λT). Expected count and probability are different quantities.
Open lessonLesson · Differential equations: models, methods and dynamicsInterpret roots through energy and response
Connect damping regimes, natural frequency and forced behavior.
Open lessonLesson · Science · Earth and space scienceInterpret saturation limits without hiding the result
Retain a supersaturated proposal and distinguish state arithmetic from condensation physics.
Open lessonLesson · Science · Earth and space scienceInterpret temperature anomalies and reference periods
Rebase recorded annual anomalies without changing the underlying differences or trends.
Open lessonLesson · Science · General Chemistry IIInterpret temperature sensitivity without confusing equilibrium
Reactants must encounter one another in suitable configurations with sufficient energy to access a reaction pathway.
Open lessonLesson · Science · AP Environmental ScienceInterpret tolerance and a realized ecological niche
Distinguish a response curve from a complete species distribution or adaptation claim.
Open lessonLesson · Science · Physical Geology and Earth SystemsInterpret volcanic unrest without reversing a conditional probability
Volcanic monitoring combines seismic activity, deformation, gas chemistry and other evidence with a volcano’s history. Unrest can arise without a subsequent eruption, and the meaning of a signal depends on background behavior and the event definition. A statement about how often a signal precedes an event is not automatically the probability of an event after observing that signal.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceInterpret yield response without confusing ET with irrigation
Crop output depends on water availability and its timing, but also nutrients, temperature, salinity, pests and management. Evapotranspiration includes plant transpiration and evaporation; it can be supplied by rainfall, irrigation or stored soil water. Irrigation withdrawal additionally includes conveyance and return-flow pathways.
Open lessonConcept · GlossaryIntervention
Deliberately changing a factor to investigate its effects, with explicit assumptions about what else stays fixed.
Open conceptLesson · Social studiesInterwar crisis: recovery policies and political rights
Compare economic interventions by mechanism and political constraints, rather than equating every expanding state.
Open lessonLesson · Differential geometry of curves and surfacesIntrinsic metrics and local isometries
Determine which geometric properties survive bending.
Open lessonCourse · Social studiesIntroduction to African American studies
Hands-on social studies course and project
Open courseCourse · Social studiesIntroduction to human geography
Hands-on social studies course and project
Open courseCourse · ScienceIntroductory Astronomy
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceIntroductory Microbiology
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseLesson · Science · University Physics III — thermal, optical and modern evidenceInvert a nonlinear count response before subtracting an independent background
A supplied nonparalyzable detector model has steady mean indicated rate m=r/(1+tau r), where r is the true interaction rate and tau is an independently calibrated dead time. Each recorded event excludes a fixed interval from further recording; incoming events in that interval do not extend it. The inverse is r=m/(1−tau m), valid only when tau m is below one.
Open lessonLesson · Science · Introductory AstronomyInvert parallax without inventing symmetric distance errors
Annual stellar parallax is a geometric angular displacement caused by the observer’s changing position around the Sun. The parallax angle convention uses a one-AU baseline, not the full peak-to-peak displacement across opposite orbital positions. With parallax p in arcseconds, distance in parsecs is 1/p; with milliarcseconds it is 1000/p.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesInvert sums on a finite partial order
Construct the incidence Möbius function recursively.
Open lessonLesson · Mathematical statistics: derivation and model criticismInvert tests into coverage statements
Connect acceptance sets, confidence sets and prospective power.
Open lessonExperiment · Media literacy and civic reasoningInvestigate apparent disagreement
Are two reports measuring the same thing?
Open experimentLesson · Science · Biology and living systemsInvestigate measured variation in penguins
Inspect measured morphology, missing observations and within-species variation.
Open lessonLesson · Social studiesIran: trace a bill through constitutional review
An elected legislature can be only one gate in a lawmaking process. Identify appointment chains and review standards separately; a twelve-member body need not use one identical voting rule for every question.
Open lessonLesson · Differential geometry of curves and surfacesJacobi fields and geodesic separation
Derive the linearized equation for neighboring geodesics.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationJacobians and general changes of variables
Transform both the region and its local volume scale, checking multiplicity.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceJoin local cubics with global smoothness
Derive natural cubic spline equations and explain boundary conditions and uniqueness.
Open lessonCourse · Mathematics program unitJoint information and adaptive strategies
Prove which information a probability calculation may use, and bound every admissible strategy in a finite reveal process.
Open courseConcept · GlossaryJoint training
Optimizing multiple connected components with a shared objective. Gradients can coordinate their representations while also introducing coupling and regressions.
Open conceptLesson · Course 1 · Generative models beyond textJudge a generated distribution
Compare fidelity, mode coverage, copying flags, task gates and licensed source receipts instead of trusting one score.
Open lessonLesson · Social studiesJudicial process: allegations, evidence and remedy
Judicial review operates through cases and procedures. A complaint is a party's allegation, a court's procedural ruling may not decide the merits, and a remedy addresses a specified legal wrong. These distinctions prevent overstating a judicial event.
Open lessonLesson · Weighted integration and jump conventionsJumps, tags and endpoint conventions
Separate refinement integrability from all-small-mesh tagged convergence and account for the exact mass carried by interior and endpoint jumps.
Open lessonLesson · School trigonometry: proofs, inverse branches and periodic decisionsJustify a principal inverse before using it
Explain one-to-one restrictions and distinguish a principal answer from all solutions.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesJustify greedy selection by exchange
Prove the safe-edge exchange argument.
Open lessonConcept · GlossaryK-means
A clustering algorithm alternating nearest-center assignments with moves to group means under squared Euclidean distance. Initialization, scaling and k affect results; groups are not verified labels.
Open conceptLesson · Science · University Physics III — thermal, optical and modern evidenceKeep a band-gap interpretation conditional on transport assumptions
A supplied intrinsic-carrier transport family has log conductivity L(T)=A+p ln(T/300)−Theta/T, where Theta=Eg/(2kB). The logarithm is of conductivity relative to a declared reference unit. The exponent p describes a stipulated mobility or prefactor temperature dependence. A is unknown to the inverse analyst even though the teaching generator reveals A=0.
Open lessonLesson · Science · Physics and physical systemsKeep a gas at constant temperature while it does work
Compare gas pressure curves and track the heat needed to hold temperature steady.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceKeep an unknown character state distinct from observed absence
Incomplete specimens and uncertain sequence alignment can leave a character unobserved. Coding an unknown as absence invents a biological observation. Uncertainty should be retained in the matrix and propagated to the inference rather than resolved in whichever direction supports a favored tree.
Open lessonLesson · Science · Introductory AstronomyKeep aperture resolution distinct from light-gathering power
Diffraction spreads an ideal point-source image even through a perfectly made aperture. For the stated Rayleigh criterion and an unobstructed circular aperture, angular scale is 1.22λ/D radians. This is an idealized point-pair criterion, not a claim that every extended source feature at that spacing is recoverable.
Open lessonLesson · Science · Introductory AstronomyKeep band luminosity and extinction in a distance modulus
Magnitudes encode brightness logarithmically: a difference m 2−m 1 corresponds to flux ratio F 2/F 1=10^(−0.4(m 2−m 1)) in a common band and calibration. Smaller magnitude means greater flux. Apparent magnitude describes received brightness; absolute magnitude describes what the same source would have at ten parsecs without intervening attenuation.
Open lessonLesson · Science · University Physics II with calculusKeep battery work in an energy-derived force
An energy derivative gives mechanical force only after the electrical constraint is stated. The supplied lateral-overlap approximation is C(x)=C0[1+3x/ell], with C0=100 pF and ell=0.1 m. It corresponds to relative permittivity 4 occupying part of a plate area at the same gap. Initial voltage at zero overlap is 10 V.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsKeep calibration ambiguity in an energy fit
A height-to-speed experiment can compare released gravitational energy with final kinetic energy, but only if both height and speed are calibrated. In this synthetic packet, an object starts at rest and descends heights1,2,3 m under g=10. The true fraction entering final kinetic energy is η, so true v²=2ηgh.
Open lessonLesson · Science · Engineering design and scientific computingKeep calibration evidence in a fair comparison
Compare supplied offset and rounding effects while retaining original readings.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceKeep calibration, source inference and a new clock test in separate stages
An indicated frequency can contain both multiplicative gain and additive offset. Two independently known anchors at physical frequencies2 and6 give indications y2 and y6. Fit g=(y6−y2)/4 and B=y2−2g. A separately known frequency4 is unused while fitting; its indication tests the affine response. This test does not establish the same response for every source, temperature or later time.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceKeep carbon mass currencies and budget residuals distinct
An annual carbon budget compares emissions, atmospheric accumulation and land/ocean sinks over a defined interval. Independent estimates need not close exactly because they contain uncertainty and potentially omitted processes. Conservation still holds physically; the residual describes an incomplete observational accounting.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceKeep carbon source separate from energy and electron source
Metabolic labels answer different questions. Autotrophs obtain much of their biomass carbon from inorganic carbon such as CO2; heterotrophs use preformed organic carbon. Phototrophy uses light as an energy source, whereas chemotrophy uses chemical energy. These axes should not be collapsed into a single plant-like versus animal-like distinction.
Open lessonLesson · Science · University Physics II with calculusKeep carrier averaging separate from pulse integration
A plane wave carries energy through its Poynting vector. For a sinusoidal carrier, cycle averaging gives intensity epsilon0c Eamplitude²/2. Now let that carrier amplitude have a slowly varying envelope Epeak sin(πt/T) over a pulse of duration T. The carrier frequency is stipulated 1 GHz while T is in microseconds, so the envelope is slow relative to the carrier.
Open lessonLesson · Science · Introductory MicrobiologyKeep clearing-zone geometry separate from a potency claim
A region without visible growth in an inhibition image is an observation under particular conditions. It does not by itself distinguish inhibition of growth from killing, establish a molecular target or rank clinical effectiveness. Diffusion, solubility, source geometry, medium and the organism’s response can affect the apparent boundary.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceKeep cytoplasmic fusion, nuclear fusion and DNA replication distinct
Fungal sexual life cycles can separate cytoplasmic fusion from nuclear fusion. Plasmogamy brings compatible cellular contents together; karyogamy fuses nuclei. A dikaryotic state contains two distinct haploid nuclei in one cytoplasmic compartment, written n+n. That is not the same nuclear organization as one diploid 2n nucleus.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceKeep decaying modes from numerical growth
Derive stability regions and distinguish stability restrictions from accuracy requirements.
Open lessonLesson · Social studiesKeep deposits, reserves and loans distinct
Change a two-bank ledger and explain which transactions create deposits, move reserves or reduce equity.
Open lessonLesson · Course 2 · Connecting tools and specialistsKeep documents from becoming instructions
Trace document-induced proposals and enforce a task-specific runtime boundary without confusing permission with correctness.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceKeep domains through rational arithmetic
Add, subtract, multiply and divide expressions without restoring excluded inputs.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceKeep electric and magnetic forces signed at entry
Electric and magnetic forces must be added as vectors before a cancellation is claimed. The supplied entry velocity is along +x, electric field .6 N/C along +y and magnetic field .2 T along +z. A right-handed coordinate system gives v cross B along −y. Therefore the net entry force is q(.6−.2v) in the y direction, with charge in coulombs.
Open lessonLesson · Science · Biology and living systemsKeep food-web energy and nutrient accounts separate
Trace production through two feeding transfers and interpret the untransferred balance.
Open lessonLesson · Science · Physical Geology and Earth SystemsKeep geothermal gradient distinct from heat flux
Temperature, temperature gradient and heat flux are different quantities. Fourier conduction in one dimension gives flux opposite the temperature gradient. With depth positive downward and temperature increasing downward, heat flows upward. Its magnitude is conductivity multiplied by gradient in kelvin per meter.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceKeep lens power, image distance and orientation distinct
A thin lens bends paraxial rays as though the refraction occurred at one plane. In the same surrounding medium on both sides, a converging lens has positive focal length and a diverging lens negative focal length. A real object before the lens has positive object distance. Positive image distance is after the lens, unlike the mirror convention in which a positive real image lies back in front of the reflecting surface.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsKeep line charge and total charge distinct in a coaxial gap
Two long coaxial conductors have inner radius a, outer radius b and active length L.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsKeep momentum while accounting for collision energy change
An inelastic collision reduces the system’s translational kinetic-energy subtotal while converting energy to other forms. A perfectly inelastic collision is the sticking case: both objects share one final velocity. Total momentum can still be conserved when external impulse is negligible. Total energy is not destroyed.
Open lessonLesson · Social studiesKeep money and loanable-funds graphs separate
Do two interest-rate diagrams measure the same market?
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceKeep peak and duration visible beside an exposure average
Environmental exposure describes contact with an agent over time; concentration, duration and frequency are different dimensions. An average can summarize a profile but erase its peaks and timing. A concentration at a monitor also need not equal the concentration at a person’s actual location.
Open lessonLesson · Science · General Chemistry IKeep phase and reaction extent in a formation-enthalpy account
Apply products-minus-reactants with explicit coefficients, standard references and physical states.
Open lessonLesson · Science · General Chemistry IKeep photon energy, photon rate and beam power distinct
Build an energy budget that distinguishes a single quantum from a stream of quanta.
Open lessonLesson · Mathematics investigationKeep scale and precision separate
How many digits should a small or large number display?
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsKeep signed angular momentum while coupling converts energy
For fixed-axis rotation, angular momentum is Iω with the axis sign convention retained. A point contribution more generally uses the perpendicular component of momentum times its moment arm, and contributions sum about one common origin. In this coaxial packet, the first rotor has inertia 2 kg m² and angular velocity+3 rad/s.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceKeep source and observer velocities in the medium frame
Sound fronts move through a medium. In this model still air carries waves toward +x at 340 m/s, the source is left of the observer and the source frequency is 500 Hz. Positive source velocity moves toward the observer; positive observer velocity moves away along the same direction. Keeping these signs explicit is more reliable than choosing a formula sign from an unlabeled word such as approaching.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceKeep source potential separate from pair energy
Potential is a scalar source property at a point, with units joules per coulomb. The potential created by several point charges is the signed sum kQ_i/r_i. Multiplying that potential by a newly introduced passive charge gives its interaction energy with those sources. It does not automatically include energy already stored in interactions among the original sources.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesKeep the labels when operating on matrices
Encode a small contextual table as a matrix with ordered row and column labels.
Open lessonLesson · Science · University Physics I with calculusKeep the same initial conditions across three damping regimes
The model xddot + c xdot + 4x = 0 has unit mass and stiffness 4 N/m. Trying exp(rt) gives the characteristic equation r² + cr + 4 = 0. Its discriminant c² − 16 separates oscillatory complex roots, a repeated real root and two distinct real roots.
Open lessonLesson · Poetry, drama and performanceKeep the words; change the relationship
Compare two staging interpretations of the same dialogue and explain how visible action supports each.
Open lessonLesson · Science · University Physics I with calculusKeep the zero-energy escape limit distinct from finite turning
For radial outward motion in the supplied μ = 1 field, angular momentum is zero and specific energy is v²/2 − 1/r. The potential reference is zero at infinity. Escape speed at starting radius r₀ is sqrt(2/r₀), the speed that makes this energy exactly zero.
Open lessonLesson · Science · Earth and space scienceKeep unknown relationships in a geological history
Represent a partial history and choose evidence that could constrain a remote outcrop.
Open lessonLesson · Science · AP Chemistry preparationKeep voltage intensive while balancing electron transfer
Relate E°, ΔG° and K with correct electron balance and equation scaling.
Open lessonLesson · Science · General Chemistry IKeep wet totals separate from dry-gas composition
Subtract a supplied vapor contribution before assigning dry product amount and mole fractions.
Open lessonLesson · Social studiesKing's letter: legality, justice and nonviolent pressure
King responds to criticism of the Birmingham campaign by explaining why delay and failed negotiation made direct action necessary in his argument.
Open lessonLesson · Course 2 · Planning and reasoningKnow what a prediction leaves uncertain
Separate noisy observations, multiple physical outcomes and disagreement across fitted models.
Open lessonLesson · Differential equations: models, methods and dynamicsKnow what an initial condition guarantees
Use local hypotheses and separate uniqueness from global continuation.
Open lessonLesson · Course 2 · Learning through actionsKnow which policy produced the data
Compare a logged behavior policy with a target policy using trajectory weights, coverage checks and bounded replay.
Open lessonConcept · GlossaryKV cache
Stored attention keys and values from earlier positions, reused during causal generation to avoid recomputing them. It costs memory and does not update weights.
Open conceptLesson · Social studiesLandscape and national culture: reading The Oxbow
Compare a preparatory sketch and finished landscape, test competing interpretations and investigate how art reaches audiences.
Open lessonLesson · Social studiesLanguage development: rules, examples and communication
Distinguish memorized examples from productive language rules while considering interaction and task context.
Open lessonLesson · Social studiesLanguage: separate words, social action and context
A literal translation can preserve a sentence's words while losing its social action. Linguistic anthropology studies how speakers use language in situations, including authority, politeness, affiliation and exclusion.
Open lessonLesson · Single-variable calculus foundationsLarge inputs and unbounded outputs
Distinguish a finite limit at infinity from an infinite limit near a finite input, using signs and dominant terms.
Open lessonConcept · GlossaryLatent state
An internal representation used by a model, often a vector. It need not be written as words or directly match observable quantities.
Open conceptLesson · Complex analysis: holomorphic functions, contours and mappingsLaurent series on annuli
Derive positive and negative powers from two boundary kernels.
Open lessonLesson · Course 2 · Learning without losing old skillsLearn a better way to adapt
Optimize a starting weight for rapid adaptation, then evaluate adapted copies on a task excluded from meta-training.
Open lessonLesson · Course 2 · Connecting tools and specialistsLearn a shared representation contract
Train an adapter from paired vectors, expose missing information, and propagate messages through typed relations.
Open lessonLesson · Course 2 · Planning and reasoningLearn continuous dynamics
Collect cart transitions, fit a next-state predictor, and compare it with simpler baselines.
Open lessonLesson · Course 2 · Learning through actionsLearn from delayed rewards
Teach an agent a route, inspect a Q-learning update, and uncover a reward rule that rewards the wrong behavior.
Open lessonLesson · Course 1 · How does AI process language?Learn from preferences
Train a response ranker from editable pairwise labels, examine rater disagreement and expose a polished but false proxy exploiter.
Open lessonLesson · Course 1 · What has the network learned?Learn from relationships between examples
Train an encoder from paired views, diagnose collapsed features and test what a simple classifier can still recover.
Open lessonLesson · Course 2 · Choosing what to do nextLearn from routing evidence without fooling yourself
Expose selection bias, compare specialists on the same contexts, and check confidence against outcomes.
Open lessonLesson · Course 1 · How does a network learn?Learn from small batches
Inspect individual gradient arrows, average a mini-batch and compare learning under explicit example and update budgets.
Open lessonLesson · Course 2 · Learning through actionsLearn from verified self-play
Generate game experience, check it against exact rules, and use verified outcomes to change future decisions.
Open lessonLesson · Course 2 · Learning without losing old skillsLearn reusable temporally extended skills
Train two tabular option policies and logistic termination functions, then compare staged navigation with a flat learner and fixed hierarchy.
Open lessonLesson · Course 2 · Learning without losing old skillsLearn something new without losing the old
Train a shared neural network on two tasks, measure forgetting, and compare replay with a parameter-change penalty.
Open lessonLesson · Course 1 · What has the network learned?Learn the pattern, not the examples
A model can fit every training point and still fail on the next one. Find out how to tell.
Open lessonLesson · Course 1 · Generative models beyond textLearn with a generator and discriminator
Train an actual tiny GAN, inspect discriminator gradients and compare missing modes with a reproducible diffusion reference.
Open lessonLesson · Course 1 · What has the network learned?Learn with objects and relationships
Train shared graph messages, change directed relationships and test whether reordering objects changes their predictions.
Open lessonLesson · Course 2 · Learning through actionsLearn without new interactions
Fit policies from a fixed log, trace unsupported action values, and test a conservative objective against behavior cloning.
Open lessonLesson · Social studiesLearning changes with experience and context
Classical conditioning concerns learned associations between events; operant conditioning concerns how consequences alter behavior.
Open lessonModule · Course 2Learning through actions
Explore how rewards and experience shape decisions.
Open moduleModule · Course 2Learning without losing old skills
Learn new tasks while preserving useful old knowledge.
Open moduleLesson · Social studiesLegislation: identical text must pass both chambers
Support for a general proposal is different from passage of identical legislative language. Committees, floor procedures, amendments and bicameral agreement create distinct decision points and opportunities to alter a bill.
Open lessonLesson · Calculus II: series, curves and applicationsLength uses slope, not vertical change alone
Derive graph length by refining chords and distinguish existence from elementary evaluation.
Open lessonExperiment · Environment and sustainabilityLess pollution added does not mean none remains
How do accumulation and removal affect intervention outcomes?
Open experimentLesson · Poetry, drama and performanceLet a metaphor do more than one thing
Develop and limit a figurative reading by testing it against the poem’s concrete details.
Open lessonLesson · College compositionLet a question earn its thesis
Turn a broad topic into a contestable question and a provisional thesis whose scope matches available evidence.
Open lessonLesson · Rhetoric, media and public voiceLet a serious objection change your argument
Represent a strong opposing reason fairly and respond through evidence, concession or a revised proposal.
Open lessonLesson · Write with purposeLet readers trace what you borrowed
Distinguish quotation, paraphrase and original inference, then provide enough source information for a reader to trace each borrowed claim.
Open lessonLesson · Differential equations: models, methods and dynamicsLet the homogeneous coefficients become functions
Derive a general particular solution from a known fundamental pair.
Open lessonLesson · Social studiesLifespan designs: age, cohort and repeated measurement
Distinguish age differences from within-person change using cross-sectional, longitudinal and sequential designs.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsLift a root one prime-power digit at a time
Derive the first-order lifting congruence.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsLift finite-field cycles to a Galois group
Construct a residue field and prove one automorphism lifts Frobenius on all roots.
Open lessonLesson · Homotopy, fundamental groups and covering spacesLift homotopies and detect injectivity
Construct a lifted square by a finite grid of sheet inverses.
Open lessonLesson · Homotopy, fundamental groups and covering spacesLift paths one sheet at a time
Construct a lift using a finite subdivision of the interval.
Open lessonLesson · Abstract algebra I: groups and symmetryLift spatial rotations to unit quaternions
Construct SU(2), prove its rotation homomorphism and track the two possible lifts.
Open lessonConcept · GlossaryLikelihood
How probable specified evidence is under a proposed state or model. Probability of an alarm given a fault differs from probability of a fault given an alarm.
Open conceptLesson · Course 2 · Learning through actionsLimit a policy update
Compare raw and clipped probability-ratio objectives on the same frozen batch, and test the limits of clipping.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusLimit laws and inequalities
Prove arithmetic laws with boundedness and preserve only justified inequality conclusions.
Open lessonLesson · Single-variable calculus foundationsLimit laws and legitimate algebra
Use algebra to establish nearby behavior while preserving exclusions and checking quotient conditions.
Open lessonLesson · Science · Earth and space scienceLimit weather claims to the recorded evidence
Handle absent precipitation and ambiguous pressure metadata without turning a small station comparison into a climate claim.
Open lessonCourse · Mathematics program unitLinear Algebra I: systems, spaces and linear maps
Sixteen undergraduate foundation lessons connecting elimination, vector spaces, linear maps, determinants, eigenvectors, orthogonality and least squares. Written arguments require review. Linear Algebra II will extend abstract operator theory, spectral results, quadratic and canonical forms, and singular-value analysis.
Open courseCourse · Mathematics program unitLinear Algebra II: operators and spectral structure
A finite-dimensional operator sequence over R and C, with quotient and dual constructions, canonical forms, spectral proofs and singular-value analysis. Written proofs require review; publication alone does not establish semester equivalence.
Open courseCourse · Mathematics program unitLinear algebra over finite fields
Solve modular systems, count bases and subspaces, and test spectral claims with the scalar field stated explicitly. These finite-field models retain the distinction between linear algebra and arithmetic over composite residue rings.
Open courseCourse · MathematicsLinear algebra: directions, systems and approximation
Use vectors and matrices to model constraints, diagnose solution structure, fit inconsistent observations and interpret preserved directions.
Open courseAssessment · Linear Algebra I: systems, spaces and linear mapsLinear algebra: systems, maps and evidence assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentConcept · GlossaryLinear regression
A numerical predictor formed from an intercept and weighted inputs. Least squares fits its coefficients by minimizing training squared error; an observed coefficient is not automatically causal.
Open conceptCourse · English language artsLiterary inquiry and interpretation
Develop a critical interpretation through sustained reading, formal analysis, responsible context and comparison, then curate an annotated exhibit that makes the reasoning inspectable.
Open courseLesson · Point-set topology: spaces, constructions and separationLocal compactness and one-point compactification
Use compact neighborhoods with the Hausdorff hypothesis explicit.
Open lessonConcept · GlossaryLocal control loop
A feedback process that responds near the component doing the work, while higher-level planning supplies objectives and handles exceptions.
Open conceptLesson · Differential geometry of curves and surfacesLocal Gauss–Bonnet from a moving frame
Prove the curvature–boundary identity on a disk patch.
Open lessonStudio · Optional P08 implementation studioLocal model operations studio
Python loopback HTTP inference, versioned receipts, delayed feedback, measured timing and verified rollback.
Open studioLesson · Science · AP Physics C: Mechanics — calculus-based modelingLocate a center by integrating a nonuniform mass distribution
Use differential mass, normalization and spatial bounds rather than a geometric midpoint shortcut.
Open lessonLesson · Science · Earth and space scienceLocate a compatible source from several stations
Intersect distance bands and test a candidate against every station.
Open lessonLesson · Science · Engineering design and scientific computingLocate a fault using node and current constraints
Explain complementary voltage and current readings with explicit topology and polarity.
Open lessonLesson · Science · AP Environmental ScienceLocate a mineral mixture on a texture triangle
Interpret sand-silt-clay proportions while separating texture from structure and fertility.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceLocate a nondisjunction event before predicting chromosome outcomes
Nondisjunction is a failure of the expected chromosome separation. A homolog-pair error in meiosis I and a sister-pair error in one meiosis II division affect different sets of formal products. Identify the stage and the separated objects before drawing gamete chromosome counts; the word error alone does not determine the pattern.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceLocate a signaling defect with rescue controls
Interventions can distinguish competing pathway orders more strongly than a single endpoint correlation.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsLocate a structure without reversing the subject’s left
Anatomy describes structures and relationships; physiology explains processes. Anatomical position provides a shared reference: upright, facing forward, arms at the sides with palms forward. Directional words belong to the subject, not the viewer. A front-view picture places the subject’s left on the viewer’s right, but rotating the picture does not rename the subject’s left arm.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityLocate a value within a model
Use z scores, normal areas and quantiles with explicit assumptions.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsLocate a virtual image without sending light behind a mirror
A plane-mirror image can be located by tracing physical reflected rays backward as straight-line extensions. The reflected light stays on the object side, but an observer interprets the diverging rays as coming from a point behind the mirror. That apparent point is a virtual image. Dashed extensions are a geometric construction, not rays passing through the reflecting surface.
Open lessonLesson · Course 1 · What has the network learned?Locate and segment objects
Draw boxes and masks, expose merged and missed objects, and evaluate a tiny pixel classifier on real images.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsLocate equilibrium and test the restoring-force law
An equilibrium position has zero net force. Stability asks what happens after a small displacement: a force back toward that position supports oscillation, while a force away from it destabilizes the equilibrium. Zero velocity alone does not identify equilibrium; a mass at a spring turning point is momentarily at rest under a nonzero restoring force.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsLocate Newton’s third-law partners on different bodies
An interaction produces a pair: the force of A on B and the force of B on A. Their magnitudes are equal and their directions opposite at the same instant. The labels reverse agent and recipient. The pair is not “a force and the motion it causes”; acceleration is a response, not the partner force.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsLocate one meeting event in two inertial frames
Choose the coordinate origin and observer before assigning a velocity. For an observer moving uniformly at speed u along the same axis, x′=x−ut when origins coincide at t=0, and v′=v−u. A fixed coordinate-origin shift is different: it changes position labels but not velocities. These are low-speed Galilean comparisons, not relativistic transformations.
Open lessonLesson · Science · Physics and physical systemsLocate real images with a converging lens
Connect intersecting rays to signed image distance and linear magnification.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsLocate skeletal landmarks before assigning an action
The axial skeleton includes skull, vertebral column, ribs and sternum; the appendicular skeleton includes limb bones and their girdles. This division describes organization, not a ranking of importance. The pectoral girdle includes clavicle and scapula, while the pelvic girdle connects the lower limbs to the axial framework. Their mobility and load-transfer roles differ.
Open lessonLesson · Science · Physics and physical systemsLocate standing-wave nodes and antinodes
Distinguish persistent nodes from temporary zero crossings and examine how phase shifts the pattern.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceLocate the energy source of a motional emf
A rod moving through a magnetic field can separate charge because its carriers experience magnetic force. Here the rod is .3 m long, moves right through .4 T into the page and increases a rectangular loop area. Positive charges are pushed upward along the rod by v cross B. The induced circulation opposes the increasing inward flux and produces magnetic drag on the moving rod.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsLocate the force before calculating its torque
A force diagram for rotation must retain where each force acts. Replacing an extended body by a dot loses the distance and orientation needed to calculate turning effects. Mark the rotation axis, the point of application and the force’s line of action. The position vector runs outward from the chosen axis to the application point.
Open lessonLesson · Partial differential equations and Fourier methodsLocate the limit of a classical characteristic solution
Track nonlinear speeds and characteristic crossing.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsLocate thin-lens images with rays and signed distances
A converging thin lens has positive focal length under the real-is-positive convention; a diverging lens has negative focal length. In the simple surrounding medium assumed here, parallel incoming rays converge to a focal point for the converging lens or diverge as though from a focal point on the incident side for the diverging lens. A lens has focal behavior on both sides.
Open lessonLesson · Euclidean and coordinate geometry foundationsLocate triangle centers and test possible sides
Distinguish medians, bisectors and altitudes, and recognize degenerate side data.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsLogarithms and branch choices
Specify a domain and argument range for a logarithm branch.
Open lessonConcept · GlossaryLogistic regression
A model that applies a sigmoid to a weighted input sum to estimate a binary class probability. Training fits probabilities; a separate threshold chooses class decisions.
Open conceptConcept · GlossaryLogit
A raw score before conversion to probabilities. Vocabulary logits score possible next tokens.
Open conceptConcept · GlossaryLoss
A number measuring prediction error. Our lab averages the squared errors over four examples.
Open conceptLesson · Social studiesLouisiana cession: territory, promises and payment
Audit a treaty claim, reconstruct the financial commitments and distinguish interstate agreement from the consent of affected communities.
Open lessonLesson · Social studiesMagna Carta: trace consent without assuming equality
Map a charter’s consent procedure, exceptions and beneficiaries, then separate written restraints from enforcement.
Open lessonProject · Money and business decisions from explicit modelsMake a defensible fictional business decision
Plan a small fictional service or shop. Compare two price/stock choices, prepare a cash reserve, and evaluate a proposed equipment investment. Include a downside demand case and distinguish worker incentives from the business objective.
Open projectExperiment · Programming and software engineeringMake a passing test suite reveal a bug
Can all your tests pass while the rule is wrong?
Open experimentLesson · Course 1 · How does a network learn?Make a problem easier to see
Build XOR with two hidden neurons and discover what nonlinear activations add.
Open lessonLesson · Write with purposeMake a reader able to act
Adapt the same information for two audiences without changing facts or hiding a limitation.
Open lessonLesson · Science · Engineering design and scientific computingMake a result reproducible from its evidence
Reconstruct a run from settings, units, event order and retained calculations.
Open lessonLesson · Rhetoric, media and public voiceMake a visual claim inspectable
Design a text-and-visual message whose hierarchy, scale and caption preserve the limits of its evidence.
Open lessonLesson · Social studiesMake an analysis reproducible without exposing people
Reproducibility means another reader can follow the data, definitions and decisions to obtain the reported analysis.
Open lessonLesson · Stories and creative writingMake an ending grow from an earlier choice
Design an ending that changes the meaning of an earlier detail while preserving causal coherence.
Open lessonLesson · Poetry, drama and performanceMake an image audible—or visible on the page
Connect a sensory image and a sound pattern to an interpretive performance choice.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceMake an iteration matrix explain convergence
Derive Jacobi and Gauss–Seidel and prove convergence under spectral or norm conditions.
Open lessonExperiment · Programming and software engineeringMake automation safe to retry
What if a charge succeeds but its response is lost?
Open experimentLesson · Science · Environmental science and sustainabilityMake conservation priorities and access visible
Compare equal total benefits with different distributions and habitat evidence needs.
Open lessonPath · Students, engineers and managers · accessible entryMake decisions from data
Frame a prediction task, audit measurements, compare errors and write claims supported by evidence.
Open pathLesson · Sections, solid measures and geometric designMake density and design decisions from geometry
Choose denominators, compare feasible designs and report model limits.
Open lessonLesson · Write with purposeMake evidence do a job
Integrate an accurately attributed quotation or number and explain the inference it supports.
Open lessonPath · Managers · hands-on decisionsMake evidence-based AI decisions
Challenge a model claim, compare costs and examine a release decision using local experiments.
Open pathLesson · Geometric definitions, motion witnesses and theorem proofsMake geometric definitions testable
Use precise definitions and counterexamples before trusting a picture.
Open lessonLesson · Linear Algebra II: operators and spectral structureMake indistinguishable vectors into one vector
Construct quotient spaces and prove the first isomorphism theorem for linear maps.
Open lessonLesson · Abstract algebra I: groups and symmetryMake multiplication independent of representatives
Connect conjugation, normality and quotient operations.
Open lessonLesson · Course 1 · How does a network learn?Make numerical computation trustworthy
Break naive softmax, repair its log-domain loss and check gradients with explicit perturbation sizes and tolerances.
Open lessonLesson · Abstract algebra I: groups and symmetryMake parity independent of a chosen factorization
Define sign and use its kernel.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceMake repeated substitution converge for a reason
Use contraction estimates to prove convergence and computable iteration errors.
Open lessonLesson · College compositionMake sources speak to the same question
Synthesize sources around a shared issue while preserving differences in what each source can establish.
Open lessonLesson · Proof foundations: from claims to argumentsMake the next case depend on proved cases
Distinguish a tested pattern from an induction argument with adequate base cases.
Open lessonLesson · Read like a literary detectiveMake the strongest case against your reading
Compare two interpretations using supporting and complicating evidence, then qualify a claim.
Open lessonLesson · Social studiesMali and the Inka: wealth, roads and the evidence of power
Compare mechanisms of political authority while respecting different chronologies and types of surviving evidence.
Open lessonLesson · Science · General Chemistry IIMap an indicator transition onto an asymmetric volume interval
A measured-looking color endpoint and stoichiometric equivalence are different definitions.
Open lessonLesson · Social studiesMap modern amendments: timing, succession and participation
Read locally packaged constitutional provisions, annotate their mechanisms and test a claim against connected clauses.
Open lessonLesson · Statistical investigations, simulation and probability decisionsMap outcomes to a probability distribution
Map elementary outcomes to a numerical variable and graph its probability law.
Open lessonLesson · Social studiesMap relationships without assuming one family form
Kinship systems organize relatedness, care and obligations in varied ways.
Open lessonLesson · Social studiesMap the limits of a religious peace
Whose freedom does a settlement protect?
Open lessonLesson · Social studiesMarbury: a right, a remedy and jurisdiction
Judicial review concerns constitutional limits on government action.
Open lessonLesson · Social studiesMask and veil: read cultural expression as evidence
Compare a complete Dunbar poem with Du Bois’s argument about seeing oneself through others’ judgments.
Open lessonExperiment · Media literacy and civic reasoningMatch a claim to its original record
Does the cited document establish the claim’s scope and time?
Open experimentLesson · Science · AP Physics C: Mechanics — calculus-based modelingMatch a friction threshold to a moving-stage integral
Use a static inequality before release and a different signed kinetic model after motion starts.
Open lessonLesson · Science · University Physics I with calculusMatch a wave at a boundary before interpreting transmitted power
At a massless join between two strings with the same tension, transverse displacement is continuous and transverse forces balance. For a sinusoidal incident wave, let r and t be signed reflected and transmitted displacement amplitudes divided by incident amplitude. The boundary conditions give 1 + r = t and Z1(1 − r) = Z2t.
Open lessonLesson · Science · Earth and space scienceMatch attribution intervals to their meaning
Distinguish ensemble spread, assessed ranges and compatible periods before comparing evidence.
Open lessonLesson · Calculus II: series, curves and applicationsMatch derivatives with a polynomial
Construct Taylor coefficients and distinguish formal matching from convergence to a function.
Open lessonLesson · Science · AP Chemistry preparationMatch photon energy, transition direction and preparation
Connect wavelength to energy units and separate energy compatibility from allowed transitions and population.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingMatch piecewise motion without inventing jumps
Carry boundary values into each integration interval and stop a model when its constraint changes.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsMatch potential branches without a false infinity reference
Integrate a Gaussian field while preserving a valid finite reference.
Open lessonLesson · Science · Environmental science and sustainabilityMatch power, energy and service timing
Compare equal available totals against a time-specific request.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceMatch pressure and displacement at tube boundaries
A tube boundary constrains pressure and displacement differently. At an ideal open end the pressure variation is approximately zero while displacement can have an antinode. At a rigid closed end displacement is zero while pressure has an antinode. Confusing these two fields reverses the interpretation of an otherwise familiar standing-wave sketch.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceMatch renewable energy to the time of service
An energy total does not identify when energy is available. A system can generate enough electricity over a day yet fail to serve an evening need. Storage moves availability through time but introduces losses and finite power, capacity and operational constraints. Demand shifting is a different intervention that changes the timing of service.
Open lessonLesson · Science · University Physics II with calculusMatch the state instead of restarting the transient
A finite resistor limits current even when an ideal source changes abruptly. The same RC=1 s element starts at zero voltage, sees 6 V until time ts, and then sees −2 V. The resistor remains connected. Capacitor voltage is continuous because a finite current cannot transfer finite charge in zero time.
Open lessonLesson · Science · University Physics II with calculusMatch three damping regimes without a singular formula
The free circuit keeps L=1 H and C=0.25 F but adds a nonnegative series resistance R. Starting from q=1 C and I=0, the equation is qsecond+R qprime+4q=0. Its roots are −R/2 plus or minus the square root of (R/2)²−4, so the critical resistance is 4 Ω.
Open lessonLesson · Euclidean and coordinate geometry foundationsMatch triangles with sufficient evidence
Use correspondence and congruence criteria without relying on scale or orientation.
Open lessonCourse · Mathematics program unitMatching completions, families and unequal sides
Prove stability under unknown comparisons, count independent families of stable outcomes, and adapt proposal invariants to unequal populations.
Open courseLesson · Social studiesMaterial culture: separate custody, ownership and interpretation
An object's material history and its institutional history answer different questions. Manufacture, use, removal, sale, accession, display and return can change its meanings and who controls access. Physical possession does not alone settle ethical authority.
Open lessonCourse · Mathematics program unitMathematical modeling and senior investigation
Twenty-six lessons develop bounded models, identifiable parameters, validated predictions and defensible mathematical investigations. Original synthetic cases support calculation, proof, reproducibility and revision. This teaching unit prepares an investigation and supplies project tasks; completion does not establish an independently reviewed capstone, research novelty or professional calibration.
Open courseCourse · Mathematics program unitMathematical statistics: derivation and model criticism
Twenty-two lessons deriving estimators, tests, intervals and regression from probability models. Exact finite-sample results, regular asymptotic approximations and model-dependent decisions are distinguished. Mixed assessment, independent analysis and proof review remain separate requirements; topic coverage alone does not establish semester equivalence.
Open courseAssessment · Mathematical statistics: derivation and model criticismMathematical statistics: derivation, decisions and model checks
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · MathematicsMathematics for AI
Seventeen contextual mathematical support lessons across all fifteen AI courses.
Open courseGuide · Mathematics programMathematics objective evidence
Inspect dated school algebra, geometry and statistics objectives, actual practice links and remaining gaps.
Open guideLesson · Precalculus extensions: motion, matrices and transitionsMatrix actions and reconstruction
Build point-set images, recover basis actions and distinguish an inverse from lost information.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesMaximize edges under clique restrictions
Prove Mantel’s bound.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsMaximum modulus and open mapping
Prove interior maximum rigidity using the Cauchy mean formula.
Open lessonLesson · Social studiesMcCulloch and Lopez: federal capacity has boundaries
McCulloch upheld a national bank as an appropriate means to legitimate federal ends and rejected Maryland's tax on it.
Open lessonConcept · GlossaryMean absolute error (MAE)
The average absolute size of prediction residuals, in the original output units. It does not allow positive and negative errors to cancel.
Open conceptConcept · GlossaryMean squared error (MSE)
The average of squared prediction residuals. Its units are squared output units, larger errors contribute disproportionately, and an empty-sample mean is undefined.
Open conceptLesson · Science · Physics and physical systemsMeasure a circuit and diagnose a mismatch
Plan meter placement, estimate resistance from readings and distinguish competing circuit faults.
Open lessonLesson · Science · Physics and physical systemsMeasure a discharge time constant and test the model
Estimate RC from readings, reserve a prediction check, and investigate uncertainty and meter loading.
Open lessonLesson · Calculus II: series, curves and applicationsMeasure a region by small sectors
Interpret signed radii and choose a nonrepeating trace interval before integrating area.
Open lessonLesson · Science · Biology and living systemsMeasure abundance without overlooking diversity
Compare edge and spread samples across a synthetic habitat gradient.
Open lessonLesson · Course 2 · Learning without losing old skillsMeasure adaptation cost
Compare zero and meta-trained initializations across linear and shifted curved task families using fixed support/update budgets.
Open lessonLesson · Science · Introductory MicrobiologyMeasure an image with a calibration ratio and reading bounds
An image becomes a length measurement only when its pixels or scale marks are connected to a known distance under the same imaging transformation. An objective label or enlarged display does not by itself supply micrometers per saved pixel. Camera sampling, cropping and resizing can change the relation between a screen distance and the specimen.
Open lessonAssessment · Measure, Lebesgue integration and the probability bridgeMeasure and integration: construction, convergence and probability
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Course 1 · Why does AI need computation?Measure and optimize inference
Profile actual local inference, separate cold and warm traffic, and inspect precision error before choosing an optimization.
Open lessonLesson · Science · Biology and living systemsMeasure cells without confusing size and magnification
Calibrate a drawing and a recorded onion micrograph, with explicit endpoint choices.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociMeasure coordinate polygons with justified lengths
Choose useful bases and perpendicular heights for tilted polygons instead of reading vertical coordinate gaps as heights.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsMeasure dispersion and recover moments from tails
Connect squared deviations, moments and tail integrals.
Open lessonLesson · Calculus II: series, curves and applicationsMeasure geometric decay in a complicated term
Use ratio and root limits and recognize their boundary failure.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceMeasure how much the mathematical answer can change
Distinguish absolute and relative sensitivity from errors made by an algorithm.
Open lessonLesson · Mathematical statistics: derivation and model criticismMeasure local likelihood sensitivity
Derive an information bound and inspect regularity failures.
Open lessonLesson · Science · Physics and physical systemsMeasure pendulum periods and test a prediction
Plan and carry out repeated timing measurements, then compare observations with a clearly labeled model.
Open lessonLesson · Science · Earth and space scienceMeasure plate motion in a shared frame
Subtract two velocity vectors and interpret the relative motion across a local boundary.
Open lessonLesson · Science · Chemistry and matterMeasure reaction rates over a chosen interval
Separate concentration change, average rate and accumulated product.
Open lessonLesson · Science · Physics and physical systemsMeasure reflection angles from the normal
Trace a ray at a smooth boundary and separate direction from reflected power.
Open lessonLesson · Social studiesMeasure responsiveness with a named convention
Can a price rise lower total revenue?
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsMeasure rotation and derive exact values
Connect arc length, signed rotation and special-triangle coordinates.
Open lessonLesson · Science · Environmental science and sustainabilityMeasure the biodiversity claim
Compare abundance, detected richness and sampling design before evaluating conservation.
Open lessonLesson · Course 2 · Testing whether improvement is realMeasure the whole workflow
Run an assistant through a fixed case suite and diagnose retrieval, evidence, tool and runtime failures separately.
Open lessonLesson · Social studiesMeasure urban development beyond an average
Does a rising city average mean everyone gained?
Open lessonLesson · Course 2 · Testing whether improvement is realMeasure variation rather than hiding it
Separate initialization variation, task clusters and paired differences.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsMeasure vector alignment and projection
Separate vectors from points and connect the dot product to angles and work.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesMeasure when a random structure appears
Compute subgraph expectations with explicit dependence.
Open lessonCourse · Mathematics program unitMeasure, Lebesgue integration and the probability bridge
A proof-based advanced undergraduate sequence constructing measures and integrals, proving convergence and Lp theorems, and connecting density and conditional-expectation arguments to probability.
Open courseLesson · Social studiesMeasurement: test the interpretation before trusting the score
Reliability concerns consistency; validity concerns evidence supporting an intended interpretation and use. A question can produce stable answers while consistently measuring the wrong thing. Validation examines comprehension, recall, judgment and response options.
Open lessonCourse · Social studiesMedia and civic inquiry — High school
Hands-on social studies course and project
Open courseCourse · Social studiesMedia and civic inquiry — Introductory college
Hands-on social studies course and project
Open courseSubject · Experiment collectionMedia literacy and civic reasoning
Investigate survey headlines and trace repeated claims back to their observations.
Open subjectConcept · GlossaryMemory
Information retained for later use, in current working state, explicit records, reusable procedures, or learned parameters. Retrieval and correctness require separate checks.
Open conceptModule · Course 2Memory and shared working state
Decide what to remember, retrieve, update, or forget.
Open moduleConcept · GlossaryMemory bandwidth
The rate at which bytes move to or from memory. Moving weights and intermediate results can limit computation speed.
Open conceptLesson · Social studiesMemory systems: facts, episodes and skills
Distinguish episodic, semantic and procedural performance and interpret a selective impairment without locating all memory in one structure.
Open lessonLesson · Social studiesMental health claims require careful interpretation
Mental and physical wellbeing involve interacting biological, psychological and social factors.
Open lessonLesson · Social studiesMental-health classification: purpose, context and limits
Distinguish a symptom report, screening result, professional assessment and explanatory model.
Open lessonConcept · GlossaryMeta-learning
Learning an initialization, representation, or strategy that helps adaptation across tasks, evaluated after a limited adaptation budget.
Open conceptConcept · GlossaryMetareasoning
Deciding which computation or evidence-gathering action to perform, considering its expected benefit, cost, and stopping conditions.
Open conceptLesson · Real Analysis II: uniform limits, metric spaces and local calculusMetric sequences, closed sets and continuity
Transport epsilon arguments to general metric spaces and characterize continuity by inverse images.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusMetric spaces and neighborhoods
Test distance axioms and build open balls without assuming Euclidean coordinates.
Open lessonLesson · Social studiesMexico: read a mixed electoral allocation
Separate an institution's stated rule, its application to an election, and a speaker's defense of that application. A source may be valuable evidence while arguing for a contested interpretation.
Open lessonLesson · Social studiesMigration, labor contracts and exclusion, 1800–1914
Explain how transport, labor demand, contracts and exclusion shaped migration without treating all movement as equally voluntary.
Open lessonLesson · Social studiesMigration, war and cultural innovation
Connect migration and cultural production without reducing art to a population statistic.
Open lessonConcept · GlossaryMinimax
Choose the action with the best worst-case opponent response in a zero-sum game. Chance outcomes still use their probabilities; guarantees depend on the model and utility.
Open conceptLesson · Social studiesMissouri Compromise: boundaries, power and admission
Test the geographic scope of a slavery restriction, explain the stakes of state admission and repair a misleading chronology.
Open lessonLesson · Social studiesMixed methods: make numbers and accounts answer each other
A joint display aligns evidence around a shared question. Agreement, complementarity and disagreement are all informative. Interview accounts can suggest mechanisms but do not automatically estimate their population prevalence.
Open lessonAssessment · Weighted integration and jump conventionsMixed weights, jump values and existence boundaries
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentConcept · GlossaryMixture-of-experts
An architecture routing inputs through selected expert blocks within a network. This differs from choosing independent full models or tools for subtasks.
Open conceptLesson · Complex analysis: holomorphic functions, contours and mappingsMöbius transformations and generalized circles
Track poles and infinity in fractional-linear maps.
Open lessonConcept · GlossaryModel
A representation or computation used to describe or predict something. A learned model fits parameters from examples; a programmed rule specifies its behavior directly.
Open conceptLesson · Differential equations: models, methods and dynamicsModel amount before modeling concentration
Use inflow-minus-outflow with changing volume and consistent units.
Open lessonProject · Linear algebra: directions, systems and approximationModel and audit a small linear system
Choose a resource, measurement or transformation scenario. Submit a matrix with units, an exact-system diagnosis, a least-squares comparison and an interpretation of a projection or eigen-direction. Keep exact equations, approximate fits and decisions separate.
Open projectConcept · GlossaryModel card
A short account of a model’s purpose, data, evaluation, limitations, and appropriate use. It connects performance claims to evidence.
Open conceptLesson · Mathematical modeling and senior investigationModel connectivity and bottlenecks with conservation
Separate flow feasibility from shortest routes.
Open lessonLesson · Mathematical modeling and senior investigationModel repeated updates without confusing them with derivatives
Analyze a recurrence and its stability.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceModel rounded arithmetic before trusting a calculation
Separate representable numbers, rounding error and exceptional range behavior.
Open lessonLesson · Mathematical modeling and senior investigationModel what is measured, not only what exists
Distinguish latent state, sensor output and sampling.
Open lessonConcept · GlossaryModel-predictive control
Planning ahead, executing the first action, observing the actual result, and planning again from the updated state.
Open conceptAssessment · Mathematical modeling and senior investigationModeling: defend a model, its evidence and its limits
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentSubject · Experiment collectionMoney, economics and business
Explore pricing, costs and uncertain demand in fictional markets.
Open subjectLesson · Course 1 · Why does AI need computation?Monitor what changed
Replay changing traffic, request delayed labels and allocate limited alert reviews.
Open lessonLesson · Homotopy, fundamental groups and covering spacesMonodromy and deck transformations
Construct the action of loop classes on a covering fiber.
Open lessonLesson · Discontinuities and Riemann integrationMonotone limits and countable jumps
Prove one-sided limits and countability of discontinuities from order, with precise endpoint conventions.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusMonotone sequences and Bolzano–Weierstrass
Use completeness to extract limits without assuming the original sequence converges.
Open lessonLesson · Social studiesMontreal Protocol: separate agreement, implementation and recovery
International environmental cooperation needs an implementation chain: scientific observation, negotiated obligations, domestic action, monitoring and adjustment. Signing an agreement and recovering an environmental system are different outcomes on different timescales.
Open lessonExperiment · Physics and engineeringMore current, less useful power?
Can drawing more current make a power source less effective?
Open experimentAssessment · Precalculus extensions: motion, matrices and transitionsMotion, transformation and transition evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesMotivation: incentives, interest and changing goals
Distinguish immediate task output from interest and evaluate whether an incentive changes each.
Open lessonLesson · Euclidean and coordinate geometry foundationsMove a figure without changing its shape
Track translations, reflections and rotations using coordinates and distance preservation.
Open lessonLesson · Optimization: models, certificates and algorithmsMove along an improving feasible edge
Derive reduced costs and the minimum-ratio step.
Open lessonLesson · Linear Algebra II: operators and spectral structureMove an operator across an inner product
Construct adjoints and relate their kernels and ranges to orthogonal complements.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsMove factorization between a UFD and its fraction field
Prove primitive products and track content.
Open lessonLesson · Science · Physics and physical systemsMove heat from cold to warm with work
Close the energy and entropy accounts of a proposed refrigerator.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsMove the axis without moving the mass distribution
A body has no single rotational inertia independent of axis. Moving a parallel axis away from its center of mass generally increases the squared distances of its particles. The parallel-axis theorem states I=Icm+Md², where d is the perpendicular separation and Icm is the inertia about the parallel center-of-mass axis. Both the parallel condition and the center-of-mass reference matter.
Open lessonLesson · Social studiesMughal expansion and revenue: conquest, collection and proof
Compare Mughal conquest and consolidation, examine competing battle explanations and trace revenue through distinct records.
Open lessonConcept · GlossaryMultilayer perceptron
A feed-forward neural network with learned affine transformations and nonlinear hidden layers. Its inputs and outputs depend on the task representation.
Open conceptLesson · Calculus-based probability: laws, dependence and limitsMultinomial counts: derive allocation and diagnose dependence
Connect ordered records, count vectors, covariance and changed collection protocols.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsMultiply complex numbers by scaling and rotating
Use modulus, argument, De Moivre and complete root families.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsMultiply dimensions with a proved basis
Use degree divisibility without assuming independence.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsMultiply matrices in the order the maps act
Track shapes, entries and noncommuting products.
Open lessonAssessment · Multivariable calculus: geometry, change and accumulationMultivariable calculus: geometry and theorem choice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitMultivariable calculus: geometry, change and accumulation
Twenty substantial lessons in three-dimensional geometry, multivariable differentiation, integration and vector-calculus theorems. Exact exercises accompany reviewed setup and proof tasks. This authored practice sequence does not by itself establish assessed Calculus III preparation; cumulative assessment, source audit and learner evidence remain distinct.
Open courseLesson · Social studiesName what a choice gives up
What is the cost of using a free room?
Open lessonLesson · Science · Introductory MicrobiologyNarrow microbial groups without treating unresolved features as absent
Microbiology includes cellular organisms across Bacteria, Archaea and Eukarya, as well as acellular infectious entities such as viruses. These are not merely size classes. Bacteria and archaea lack a membrane-bound nucleus but differ in membrane chemistry, surface structures and molecular machinery. Microbial eukaryotes include fungi, protists and microscopic algae; viruses depend on host machinery and do not contain their own ribosomes.
Open lessonLesson · Social studiesNative advertising: follow payment, presentation and audience inference
An attention-funded medium may reward clicks while audiences rely on cues of independence. Payment does not prove a claim false, but hidden sponsorship can distort how readers assess it. Trace economic incentives separately from claim verification.
Open lessonLesson · Mathematics investigationNearby behavior is not the value at the point
Can a function approach a number where its own value is undefined?
Open lessonConcept · GlossaryNearest neighbors
A method that predicts from nearby training examples under a defined distance. Classification may use a majority vote; scale, neighborhood size and tie rules affect the result.
Open conceptLesson · Proof foundations: from claims to argumentsNegate the whole claim
Turn failed universal claims into precise witnesses without overclaiming.
Open lessonLesson · Point-set topology: spaces, constructions and separationNets and general topological convergence
Define directed sets and eventual membership for nets.
Open lessonCourse · Mathematics program unitNetwork incidence and the four spaces
Use oriented incidence matrices to connect conservation, cycles, potentials and orthogonal certificates. Signed real flows are allowed; capacities and nonnegative-flow constraints require additional models.
Open courseConcept · GlossaryNeuron
A calculation that multiplies inputs by weights, adds a bias, and may apply an activation function.
Open conceptLab · ExperimentsNeuron playground
Change inputs, weights, and bias. Train a neuron.
Open labLesson · Axiomatic Euclidean and non-Euclidean geometryNeutral geometry and the exterior angle
Prove consequences without using a parallel axiom.
Open lessonLesson · Social studiesNew York Times and Citizens United: identify the regulated act
New York Times concerns an attempt to stop publication before it occurred; the government failed to justify the restraint.
Open lessonLesson · Social studiesNigeria: locate the level of legislative authority
Federalism requires asking which institution has authority over a subject before treating a conflict as a simple contest of popularity. A federal label alone does not reveal the degree of practical decentralization.
Open lessonLesson · Homotopy, fundamental groups and covering spacesNo retraction and the disk fixed-point theorem
Use induced groups to rule out a disk retraction.
Open lessonLesson · Differential geometry of curves and surfacesNormal acceleration and the second fundamental form
Measure bending relative to a chosen normal.
Open lessonLesson · Differential geometry of curves and surfacesNormal curvature in every direction
Separate curve curvature from surface bending.
Open lessonConcept · GlossaryNormalization
Rescaling numerical values using a selected rule. Layer normalization centers and rescales coordinates within each token representation.
Open conceptLesson · Science · University Physics I with calculusNormalize a density before integrating its center
For an extended object, center of mass is a normalized first moment: xcm = integral(x dm)/M. A geometric midpoint works only when the mass distribution has the corresponding symmetry. Here a 2 m rod has total mass 3 kg and density λ(x) = λ₀(1 + αx/L), with α nonnegative.
Open lessonLesson · Differential equations: models, methods and dynamicsNormalize a fundamental matrix and change coordinates
Construct a full solution operator, verify its initial normalization and carry it through a basis change.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsNormalize a joint law over its actual support
Integrate out coordinates without replacing a triangle by a rectangle.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsNormalize a line density before integrating its field
Compare an analytic field with bounded numerical sums at fixed total charge.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsNormalize a line potential and recover its field
Integrate scalar source contributions, then differentiate with respect to the observer.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsNormalize a radial current profile before using circulation
An infinite cylindrical conductor of radius R carries total current I with Jz(r) = J0(r/R)^n.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingNormalize a radial density before integrating disk inertia
Distinguish circumference, area density and rotational distance in a continuous inertia calculation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceNormalize both pieces of a continuous asymmetric amplitude
Let u=x/L on a finite support 0..L. A real triangular amplitude rises as N u/q for 0≤u≤q and falls as N(1−u)/(1−q) for q≤u≤1, with zero outside. Here q strictly between zero and one is a peak-position fraction, not a probability. The two pieces agree at the peak and vanish at the endpoints.
Open lessonLesson · Science · University Physics II with calculusNormalize charge before comparing density moments
A density coefficient is not automatically the total charge. For a rod of length L with λ(s)=λ0(1+αs/L), the total is the integral of the density, not the value at one end times length. If Q is fixed while α changes, λ0 must change too. Otherwise an apparent shape effect is mixed with a changed amount of charge.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsNormalize densities and integrate events
Treat density as a rate per unit, not a point probability.
Open lessonLesson · Science · General Chemistry IINormalize species changes before comparing reaction rates
Different species can change at different numerical rates even while describing one chemical transformation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceNormalize speed evidence without hiding the omitted tail
The Maxwell speed density describes nonnegative speeds, not signed velocity components. Write u=v/s with s=sqrt(2kT/m). Then f_v(v)dv=(4/sqrt(pi))u² exp(−u²)du. The factor u² counts velocity-space directions at a given speed; omitting it would produce a different probability distribution.
Open lessonLesson · Science · General Chemistry IINormalize stepwise proton populations without inventing the pH
A conditional species distribution is not automatically a self-consistent solution composition.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceNormalize viability selection before counting adult alleles
Natural selection requires differences in contribution to subsequent generations associated with heritable variation. Survival is one component; mating success and fertility can also matter. Relative fitness compares contributions under a defined environment and life cycle, not an organism’s strength or moral worth.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationNormals, flux and orientation
Integrate normal flow through a surface and distinguish signed flux from scalar surface mass.
Open lessonLesson · Social studiesNorms are learned and negotiated
Culture includes shared meanings and practices; norms specify expected conduct.
Open lessonLesson · Social studiesNorthwest Ordinance: statehood, land and freedom
Read the conditions behind territorial government and examine the limits of broad claims about settlement and rights.
Open lessonLesson · Mathematics for AI · M11Notice the winner selected by chance
Inspect repeated comparisons, weak signals and predeclared decision rules.
Open lessonAssessment · Real numbers, complex coordinates, vectors and matricesNumber structures: a mixed construction and reasoning portfolio
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Elementary number theory: divisibility, reciprocity and Diophantine equationsNumber theory: exact arithmetic, structure and proof
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitNumerical analysis: algorithms, errors and evidence
An undergraduate numerical sequence connecting arithmetic, approximation, solvers and ODE methods with explicit convergence and stability hypotheses. Includes an executable exact-arithmetic adaptive ODE investigation and a sustained project; independent implementation artifacts and proofs require review.
Open courseAssessment · Numerical analysis: algorithms, errors and evidenceNumerical analysis: justify the computation and its limits
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentProject · Numerical analysis: algorithms, errors and evidenceNumerical investigation: accepted work and defensible error
Build one reproducible investigation of fixed and adaptive Euler step doubling for a manufactured differential equation. Connect a predicted failure, an exact arithmetic oracle, an accepted/rejected work ledger and a proved global discretization bound. Retain revisions and a defense. A noncoding proof-and-audit route is available; implementation proficiency additionally requires your own retained executable artifact and tests. Public model solutions and text submission do not certify either route.
Open projectCourse · MathematicsNumerical methods and dynamical systems
Approximate roots and integrals with explicit error checks, analyze discrete updates, and distinguish numerical instability from modeled continuous behavior.
Open courseStudio · Optional P04 implementation studioNumerical network studio
NumPy forward/backward arrays, finite differences, simultaneous updates and bounded CPU training.
Open studioLesson · Social studiesObservational learning: knowing versus performing
Distinguish acquisition of an observed action from its immediate performance and test alternative explanations.
Open lessonExperiment · Design and product developmentObserve the task before proposing a feature
How do recruitment and question choice change the evidence?
Open experimentLesson · Science · University Physics III — thermal, optical and modern evidenceObtain a discrete spectrum from two spatial boundaries
Inside a one-dimensional infinite well from x=0 to L, the stationary Schrödinger equation is −hbar² psiDoublePrime/(2m)=E psi. For positive E, write k²=2mE/hbar² and solve psi=A sin(kx)+B cos(kx). The left boundary psi(0)=0 forces B=0.
Open lessonLesson · Social studiesOceanic connections and land empires
Connect expanding exchange to imperial authority, coerced labor and ecological change.
Open lessonConcept · GlossaryOff-policy learning and evaluation
Using data collected by a behavior policy to learn about or improve a different target policy. Adequate coverage, the estimator’s assumptions and the actual logged collection probabilities matter.
Open conceptConcept · GlossaryOffline reinforcement learning
Learning an action policy from a fixed log without collecting additional experience during fitting. Repeated updates do not fill missing state/action coverage, and conservative estimates do not create unobserved outcomes.
Open conceptProject · Abstract algebra II: rings, fields and Galois foundationsOne presentation, three scalar settings: an algebra manuscript
Develop one sustained manuscript about M(t)=[[t,1],[0,t]]: certify a polynomial presentation change, specialize to integer abelian groups, and interpret the field-polynomial quotient as an operator. Explain exactly which conclusions survive changing scalars. Preserve predictions, independent verification routes and a revision and defense record. This public supported project does not automatically establish mastery, independent authorship or an unseen assessment result.
Open projectExperiment · Systems thinking and operationsOne shipment, competing service goals
Does maximizing a weighted score produce fair service?
Open experimentLesson · Real Analysis I: completeness, limits and rigorous calculusOpen sets, closed sets and limit points on R
Use neighborhoods and sequence witnesses instead of endpoint pictures alone.
Open lessonPath · Students, software engineers and managersOperate a model with evidence
Package and inspect a model, test and stage changes, compare monitoring policies and justify incident recovery.
Open pathLesson · Algebra reasoning, constraints and solution evidenceOperate on polynomials and prove identities
Use coefficient arithmetic and identities with numerical consequences.
Open lessonAssessment · Linear Algebra II: operators and spectral structureOperators: structure, geometry and justified decompositions
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentProject · Optimization: models, certificates and algorithmsOptimization investigation: build, break and defend a certificate
Produce one coherent decision-and-certificate dossier for a synthetic two-product resource model. Derive candidates independently, submit your own exact primal/dual vectors, preserve failed checks, analyze a degenerate budget, diagnose an infeasible demand change, and defend an integer transfer. The supplied checker verifies entered certificates and never generates an optimum. A proof-and-audit route is available; implementation proficiency additionally requires your own runnable enumerator, tests and work log. Submitted text and a passing arithmetic check do not establish independent mastery.
Open projectAssessment · Optimization: models, certificates and algorithmsOptimization: certify decisions and audit algorithms
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · MathematicsOptimization: feasible choices and competing goals
Formulate objectives and constraints, solve a small linear program, trace constrained gradient updates and qualify sensitivity and preference claims.
Open courseCourse · Mathematics program unitOptimization: models, certificates and algorithms
Twenty-seven lessons derive linear and nonlinear optimization methods, their guarantees and their failure modes. Exact certificates are separated from numerical stopping evidence, and continuous relaxations from integer decisions. Independent modeling, mixed assessment and proof review remain separate requirements; publication does not establish universal course equivalence.
Open courseLesson · Mathematical statistics: derivation and model criticismOptimize a test against a specified alternative
Prove the likelihood-ratio ordering and handle randomization.
Open lessonLesson · Optimization: models, certificates and algorithmsOptimize against a declared uncertainty set
Derive robust counterparts and distinguish guarantees from calibration.
Open lessonLesson · Mathematical statistics: derivation and model criticismOptimize the whole parameter space
Handle support constraints, boundary maxima and nonexistence.
Open lessonLesson · Optimization: models, certificates and algorithmsOptimize with positive-semidefinite matrix constraints
Use trace geometry and matrix-value d primal–dual certificates.
Open lessonLesson · Mathematics investigationOptimize within a feasible polygon
Why can checking a few vertices solve this linear production problem?
Open lessonConcept · GlossaryOption / reusable skill
A temporally extended action defined by its initiation conditions, internal policy, and termination rule. It can be programmed or learned.
Open conceptConcept · GlossaryOrchestrator
A controller that selects and coordinates tools, models, and actions over time.
Open conceptExperiment · Money, economics and businessOrder before demand is known
How do overstock and missed sales pull a decision in opposite directions?
Open experimentLesson · Course 1 · What has the network learned?Order changes meaning
Train a small recurrent predictor, follow its hidden state and expose what a bag of events forgets.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsOrder dependencies without inventing an order for cycles
Construct topological orders and prove why the algorithm detects cycles.
Open lessonLesson · Science · Earth and space scienceOrder rock layers and intrusions
Use superposition and cross-cutting evidence to reconstruct a conditional sequence.
Open lessonLesson · Constructing the real numbersOrder the completion and locate its rational points
Construct an ordered field from rational Cauchy classes without borrowing real completeness, then prove rational approximation and density.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusOrder, fields and completeness
Separate algebraic order laws from the least-upper-bound property.
Open lessonCourse · Mathematics program unitOrders, graph representations and certificates
Translate relations into orders, construct faithful representations, count directed walks and certify graph equivalence or difference.
Open courseLesson · Social studiesOttoman authority: service, insignia and patronage
Investigate recruitment, inspect an imperial insignia and test how patronage could present authority without proving consent.
Open lessonConcept · GlossaryOut-of-bag evaluation
Evaluation of each training row using only trees whose bootstrap samples omitted that row. Report coverage and treat repeated tuning as development.
Open conceptConcept · GlossaryOverfitting
Fitting training-specific noise or details that do not carry over to new examples.
Open conceptLesson · Social studiesOversight: findings, recommendations and implementation differ
Oversight compares authorized goals with actual administration. A hearing can reveal a problem, a recommendation can specify a remedy, and follow-up can test implementation; none is interchangeable with the others.
Open lessonLesson · Course 1 · Why does AI need computation?Package a model as a component
Export and reload a real local model bundle; check named inputs, fitted preprocessing, dependency hashes and output meanings.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsPackage a sequence as coefficients
Use formal power series to encode counting and recurrences.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesPair an optimal matching with a vertex cover
Prove Kőnig’s equality using alternating paths.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryParallel postulates and their logical role
Compare existence, uniqueness and angle consequences of parallelism.
Open lessonLesson · Differential geometry of curves and surfacesParallel transport and holonomy
Move tangent vectors without intrinsic turning.
Open lessonConcept · GlossaryParameter
A stored number adjusted during learning, such as a weight or bias. Parameter count alone does not measure model quality.
Open conceptAssessment · Determinants from permutations to certificatesParameter, rank and block proof synthesis
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · AP Physics C: Mechanics — calculus-based modelingParameterize work along competing planar paths
Use the line integral to distinguish a selected interaction, a net force and a conservative field.
Open lessonLesson · Precalculus extensions: motion, matrices and transitionsParametric components and average motion
Coordinate a spatial path with two time graphs, interpret average displacement and prove the reached branch.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationParametric surfaces and scalar surface integrals
Construct surface patches, area factors and density integrals without confusing projection with area.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsParametrize a conic from one rational point
Derive a rational circle parametrization by intersecting a line.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationPartial derivatives and total differentiability
Use partials to build tangent planes while checking the stronger differentiability requirement.
Open lessonCourse · Mathematics program unitPartial differential equations and Fourier methods
Thirty lessons connect characteristics, Fourier expansions, heat, wave and elliptic boundary problems with energy, maximum principles, transforms and numerical stability. Breadth is informed by MIT 18.152 Fall 2011 and Stanford Math 220 Autumn 2018; this is an original sequence, not a claim of equivalent assessment or complete graduate PDE theory. General existence, regularity and spectral theorem dependencies are labeled.
Open courseLesson · Domains, invariants and structural countingPartial functions: keep the domain in the proof
Track defined inputs through composition and repair injectivity, surjectivity and inverse arguments when an operation is partial.
Open lessonConcept · GlossaryPartial observability
An agent receives observations that may omit or obscure environment state. Different states can share an observation; history or a justified belief can help, without guaranteeing exact state recovery.
Open conceptLesson · Social studiesParticipation: read the 2024 turnout denominator
Turnout rates require a defined eligible or reference population and a measurement method. Survey estimates of participation are not the same object as administrative ballot counts. A group difference alone does not reveal whether rules, resources, interest or mobilization caused it.
Open lessonConcept · GlossaryParticle filter
An approximation of a hidden-state belief using samples that move through a transition model, receive observation likelihood weights and may be resampled. Finite-sample error and lost diversity are separate from model error.
Open conceptLesson · Social studiesParties, dissent and the transition of 1800
Trace speech restrictions, an electoral tie and an appeal to minority rights without confusing parties, votes and institutions.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsPartition evidence before reversing a conditional
Compute joint branches and normalize only after adding all sources.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusPartitions, Darboux sums and integrals
Define integration through upper and lower sums rather than antiderivatives.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgePass an increasing limit through the integral
Prove the monotone convergence theorem from simple functions.
Open lessonLesson · Point-set topology: spaces, constructions and separationPaths, local connectedness and their differences
Prove path connectedness implies connectedness.
Open lessonAssessment · Statistical PCA: coordinates, reconstruction and scalingPCA evidence: reconstruct, rescale and validate
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Partial differential equations and Fourier methodsPDE: data, hypotheses and solution evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Precalculus: periodic and reciprocal graph evidencePeriodic graph evidence: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesPersonality explanations: compare predictions, not labels
Distinguish psychodynamic, humanistic, trait and social-cognitive explanations and identify testable predictions.
Open lessonLesson · Social studiesPersuasion: argument quality, expertise and involvement
Test whether persuasion depends differently on argument quality and source cues across involvement conditions.
Open lessonLesson · Social studiesPhysical development: timing, variation and measurement
Distinguish developmental timing, rate and domain-specific change without treating group averages as individual deadlines.
Open lessonCourse · SciencePhysical Geology and Earth Systems
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseSubject · Experiment collectionPhysics and engineering
Investigate nonlinear motion and power trade-offs by inspecting quantities with units.
Open subjectCourse · SciencePhysics and physical systems
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · Mathematics program unitPicard construction and uniqueness
Construct local differential-equation solutions in a complete function space, certify integral iterates, and prove the boundaries of uniqueness and continuation.
Open courseLesson · Picard construction and uniquenessPicard iterates and error certificates
Construct exact polynomial iterates, prove uniform error bounds and distinguish a failed certificate from a failed differential equation.
Open lessonLesson · Picard construction and uniquenessPicard’s operator and invariant function balls
Construct a nonlinear initial-value solution by choosing a complete invariant function ball and proving an integral operator is a contraction.
Open lessonLesson · Exact factorization and repeated solvesPivoted factors and repeated solves
Track permutations and stored multipliers through late row exchanges, then reuse a verified factorization for new right-hand sides.
Open lessonLesson · Course 2 · Geometry and control foundationsPlan a collision-free trajectory
Search joint-space waypoints, try shortcuts and independently verify the full motion.
Open lessonLesson · Mathematical modeling and senior investigationPlan a cumulative investigation with a feasible fallback
Specify evidence before collecting a favorable result.
Open lessonLesson · Course 2 · Planning and reasoningPlan briefly then observe again
Use a fitted model to choose actions, execute one, then replan from the observed result.
Open lessonLesson · Science · Engineering design and scientific computingPlan evidence for an unfamiliar failure
Use a new paper condition to identify model limits before retuning a candidate.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsPlan precision before collecting data
Use a conservative planning value and account for rounding and nonresponse.
Open lessonLesson · Science · Environmental Science: ecological systems and evidencePlan precision using independent units rather than repeated readings
Monitoring design should connect a decision-relevant precision target to independent sampling units. Many readings from one sensor or one site can improve its measurement but do not automatically create new independent sites. Temporal autocorrelation and clustering can make the nominal record count overstate information.
Open lessonLesson · Science · AP Chemistry preparationPlan reaction extent when reagent supplies have bounds
Combine purity, mass and solution concentration; test a proposed extent and distinguish a nominal limiter from an interval-supported conclusion.
Open lessonProject · Mathematical models for practical choicesPlan resources for a changing school or workplace service
Build a fictional resource plan with a budget and at least two products or activities. Use a matrix to calculate needs, test one growth forecast, and compare feasible alternatives. Explain where uncertainty or a nonlinear relationship changes your decision.
Open projectLesson · Statistical inference: estimates, tests and defensible conclusionsPlan the consequences of a decision
Distinguish Type I error, Type II error and power at a specified alternative.
Open lessonStudio · Optional Project 7 implementation practice; human review pendingPlanner implementation studio
Dependency-free Python A* and HMM exercises with separate public enumeration checks and supported browser evidence exports.
Open studioModule · Course 2Planning and reasoning
Compare search, world models, and repeated internal computation.
Open moduleLesson · Social studiesPlot demand separately from marginal revenue
Why is marginal revenue below price for a single-price seller?
Open lessonLesson · Social studiesPlymouth and Pokanoket: reading an alliance
Trace reciprocal and conditional obligations, compare two accounts, and distinguish an alliance from an English land patent.
Open lessonCourse · English language artsPoetry, drama and performance
Interpret poems and dramatic scenes by making, comparing and defending choices about language, line, voice, pause and action.
Open courseLesson · Discontinuities and Riemann integrationPoint oscillation and small interval covers
Measure local variation by all nearby values, locate closed discontinuity thresholds, and budget countable interval covers.
Open lessonCourse · Mathematics program unitPoint-set topology: spaces, constructions and separation
Twenty-eight proof-centered lessons on topological constructions, convergence, separation, compactness, connectedness, extension, compactification, category and introductory dimension. Choice principles are explicit in the ultrafilter and product arguments. Metrization and Stone–Čech are developed through stated construction lemmas; the compact-metric finite-dimensional embedding theorem is proved through general position and Baire; broader noncompact forms remain separate. Written work requires mathematical review; this is not independently assessed course equivalence.
Open courseLesson · Real Analysis II: uniform limits, metric spaces and local calculusPointwise and uniform convergence
Distinguish a cutoff chosen after a point from one controlling an entire domain.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsPoisson laws and rare-event approximations
Separate a count distribution from stronger process assumptions.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsPoisson’s formula and boundary values
Derive the disk Poisson kernel and its normalization.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationPolar coordinates and area factors
Use radius and angle to integrate disk and sector regions with the correct area element.
Open lessonAssessment · Precalculus: polar curves and signed-radius evidencePolar curve evidence: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentConcept · GlossaryPolicy
A rule or learned function for choosing an action, sometimes by assigning probabilities.
Open conceptConcept · GlossaryPolicy gradient
A derivative of expected return with respect to policy parameters. A sampled estimator can weight log-probability derivatives by returns minus an action-independent baseline; individual updates can be noisy.
Open conceptConcept · GlossaryPolicy-induced selection bias
A controller’s choices change which tasks and outcomes appear in each specialist’s records, making raw comparisons potentially misleading.
Open conceptLesson · Social studiesPolitical parties: coordination does not erase disagreement
Parties connect elections, policy agendas and governing coalitions. Party leaders can coordinate information and bargaining, but members may face different constituency interests and policy commitments. Do not treat party labels as identical preferences.
Open lessonAssessment · Precalculus: constructing and validating polynomial modelsPolynomial model evidence: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Statistical inference: estimates, tests and defensible conclusionsPool only under the equality null
Build a common-rate null model and distinguish it from estimation.
Open lessonLesson · Differential calculus: techniques and applicationsPosition, velocity, acceleration and higher rates
Use signs and units to distinguish moving, speeding up and changing direction.
Open lessonCourse · Mathematics program unitPositive matrices, factors and energy
Prove real symmetric positivity criteria, construct Cholesky factors and use exact energy identities to separate unique solutions from singular compatibility and unbounded objectives.
Open courseLesson · Social studiesPositive psychology: wellbeing, persistence and evidence
Distinguish wellbeing constructs and evaluate whether an intervention finding supports a durable, generalizable claim.
Open lessonConcept · GlossaryPosterior probability
A probability conditioned on observed evidence. For finite events, conditioning requires a positive probability of that evidence under the model.
Open conceptLesson · Multivariable calculus: geometry, change and accumulationPotential functions, curl and domain holes
Use potentials to evaluate work and distinguish local curl tests from global path independence.
Open lessonLesson · Social studiesPoverty thresholds: a classification is a policy choice
An operational definition turns a social condition into a repeatable classification. It enables comparison while drawing boundaries that can omit lived constraints. Analyze those choices without confusing an official measure with an exhaustive account of deprivation.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsPower series and analyticity
Derive Taylor coefficients from Cauchy’s formula.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusPower series and local uniform convergence
Prove interior uniform control and justify termwise calculus inside the convergence radius.
Open lessonConcept · GlossaryPPO-style objective clipping
A policy surrogate uses the smaller of a probability-ratio-weighted advantage and its clipped-ratio version. It removes some incentives for large changes, but does not enforce a hard probability or KL limit.
Open conceptCourse · Mathematics program unitPrecalculus extensions: motion, matrices and transitions
Connect component graphs to paths, reconstruct matrix actions, and distinguish possible from impossible transition histories. These college-preparation extensions are outside AP Precalculus examination weighting.
Open courseCourse · Mathematics program unitPrecalculus: constructing and validating polynomial models
Build constrained cubic and quartic models, then distinguish interpolation, regression scores and new predictive evidence. This focused school-level sequence requires algebra and graph reading, not calculus.
Open courseCourse · Mathematics program unitPrecalculus: exponential and logarithmic evidence
Identify exponential baselines, construct and fit logarithmic responses, and use transformed inverses with explicit domains. School algebra and graph reading suffice; this is a focused extension, not complete AP preparation.
Open courseCourse · Mathematics program unitPrecalculus: local rates and polynomial evidence
Compare nearby secants without calculus, derive quadratic rate patterns, and distinguish a polynomial model from what finitely many observations establish. This focused sequence extends existing average-rate lessons; it is not a complete AP course or exam-readiness certification.
Open courseCourse · Mathematics program unitPrecalculus: periodic and reciprocal graph evidence
Construct complete periodic graphs, compare finite rates and fit competing sinusoidal periods using new evidence. School algebra, unit-circle coordinates and graph reading suffice; calculus is not required.
Open courseCourse · Mathematics program unitPrecalculus: polar curves and signed-radius evidence
Construct complete noncircle polar curves, restrict their traces, and distinguish signed radius from distance and interpolation evidence without calculus.
Open courseCourse · Mathematics program unitPrecalculus: semilog scales and model evidence
Construct and read logarithmic vertical axes, recover exponential models, and compare baseline assumptions using original-scale errors and unused predictions.
Open courseConcept · GlossaryPrecision
True positives divided by all positive predictions: TP / (TP + FP). It is undefined when no positives are predicted, and depends on the population mix.
Open conceptLesson · Science · Environmental Science: ecological systems and evidencePredeclare review triggers across repeated observations
A monitoring plan should specify what observation triggers review, what confirms the signal and who is authorized to revise action. Looking repeatedly for an unusual result creates more opportunities for a false alarm. That statistical issue is distinct from the ecological consequences of missing a true problem.
Open lessonLesson · Course 1 · Simple models and useful baselinesPredict a class probability
Train a two-feature probability model, inspect its gradients and compare estimates with outcomes before choosing an action threshold.
Open lessonLesson · Science · Physics and physical systemsPredict acceleration from net force and mass
Connect a force diagram to acceleration and the subsequent motion without confusing force with velocity.
Open lessonLesson · Course 2 · Geometry and control foundationsPredict actions from observations
Train an imitation policy and diagnose closed-loop distribution shift.
Open lessonLesson · Science · Physics and physical systemsPredict an ensemble without assigning a lifetime to one nucleus
Use exponential survival and distinguish an expected ensemble count from a deterministic decay schedule.
Open lessonLesson · Science · Physics and physical systemsPredict attraction and repulsion between charges
Use charge signs and inverse-square distance scaling to interpret a force pair.
Open lessonLesson · Social studiesPredict behavior from consequences and schedules
Classify added and removed consequences by their observed effect, then trace a reinforcement schedule.
Open lessonLesson · Science · Engineering design and scientific computingPredict circuit readings from competing mechanisms
Build a finite hypothesis table before choosing a virtual measurement.
Open lessonLesson · Statistics: displays, designs and differencesPredict differences between sample means
Connect known population spreads to repeated mean differences and distinguish normal populations, large-sample approximations and paired designs.
Open lessonLesson · Statistics: displays, designs and differencesPredict differences between sample proportions
Start from known population probabilities to derive the center, variability and approximate shape of a repeated-sampling difference.
Open lessonLesson · Science · Physics and physical systemsPredict floating fraction and neutral buoyancy
Use the required displaced volume to distinguish floating, neutral suspension and sinking.
Open lessonLesson · Science · Physics and physical systemsPredict how a capacitor charges through resistance
Connect the shrinking resistor voltage to charging current and the RC time constant.
Open lessonLesson · Science · Physics and physical systemsPredict how mass changes the period
Compare cycle timing on fixed time axes and distinguish frequency from angular frequency.
Open lessonLesson · Course 2 · Planning and reasoningPredict in a compact representation
Learn a small visual code and test whether predicting that code retains useful action information.
Open lessonLesson · Science · Physics and physical systemsPredict induced voltage from flux change
Read graph slope to calculate emf, conditional current and resistor heat.
Open lessonLesson · Science · Physics and physical systemsPredict magnetic force direction and charge reversals
Use vector components to distinguish velocity, field and force.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidencePredict matter-wave directions from accelerating energy
The de Broglie relation λ=h/p associates wavelength with momentum, not with the physical diameter of a particle. An electron accelerated from effectively zero kinetic energy through potential V gains kinetic energy eV. Within the nonrelativistic approximation p²/(2m)=eV, so λ is proportional to 1/sqrt(V). The supplied rounded electron coefficient is λ=1.226/sqrt(V) nm when V is in volts. This is not the photon relation E=pc.
Open lessonLesson · Science · Earth and space sciencePredict orbital changes with radius ratios
Use inverse-square and square-root scaling to compare circular orbits without confusing radius with altitude.
Open lessonLesson · Science · Chemistry and matterPredict product mass without rounding away reactants
Convert masses to moles, apply the reaction ratio, and retain excess material.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsPredict reflection from the local surface normal
A ray represents a direction of light propagation, perpendicular to a wavefront in the simple isotropic media used here. It is not a narrow physical object whose width determines light intensity. Geometric optics follows rays when wavelength-scale interference and diffraction can be neglected; those wave effects require a different model later.
Open lessonLesson · Science · Physics and physical systemsPredict resistor–inductor current transients
Use L/R, voltage balance and distinct initial conditions to compare buildup and decay.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidencePredict root water movement from the complete potential
Water movement depends on total water potential, not solute concentration alone. Solute potential lowers the reference potential, while positive pressure can raise it; gravity and interactions with surfaces can also matter. Compare the same reference, units and boundary on both sides of a proposed flow.
Open lessonLesson · Course 2 · Geometry and control foundationsPredict several actions at once
Compare predicted action chunks with fresh-feedback execution.
Open lessonLesson · Science · Physics and physical systemsPredict the magnetic field produced by a current
Track field circulation and distinguish source current from a receiving current.
Open lessonLesson · Course 1 · How does AI process language?Predict the next token
Shift inputs and targets, inspect causal visibility, calculate cross-entropy and compare reference versus generated prefixes.
Open lessonLesson · Science · AP Biology — mechanisms and evidencePredict transcripts and distinguish processing from translation
RNA polymerase reads a DNA template and synthesizes complementary RNA 5′→3′.
Open lessonLesson · Science · Chemistry and matterPredict voltage from unequal ion concentrations
Use a same-metal cell to distinguish standard potential, actual composition and equilibrium.
Open lessonLesson · Sections, solid measures and geometric designPredict what a rotating region fills
Track distance from a declared axis instead of naming a silhouette.
Open lessonLesson · Course 2 · Planning and reasoningPredict what an action will change
Learn transitions from observations, compare imagined and actual routes, and test when replanning helps.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsPredict what opening one branch changes elsewhere
Series and parallel describe electrical connections, not how components happen to look on a page. Elements are in series only when the same charge has no alternative path between them. Parallel branches connect the same pair of nodes and therefore share a voltage difference. A shared upstream resistor carries the sum of branch currents, not the current of either branch alone.
Open lessonConcept · GlossaryPrediction
An estimated output for an input, such as a category or a number. It is not automatically an action, a verified fact or a guarantee.
Open conceptConcept · GlossaryPrediction residual
Observed value minus predicted value in these regression lessons. A signed prediction error, distinct from a neural-network residual connection.
Open conceptLesson · Mathematical modeling and senior investigationPrefer a model for a reason that can be checked
Compare predictive, mechanistic and complexity evidence.
Open lessonLesson · Social studiesPreferences meet election rules
Election procedures aggregate preferences in different ways.
Open lessonLesson · Research, editing and publishingPrepare a release you can explain and correct
Perform a claim audit, prepare an accessible publication package and write a specific correction procedure.
Open lessonLesson · Course 1 · How does AI process language?Prepare a text corpus
Inspect text provenance, repair duplicate leakage, compare filters and control training exposure.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsPresent an operator using tI−A
Prove the evaluation quotient and connect polynomial equivalence to constant similarity.
Open lessonLesson · Science · Engineering design and scientific computingPreserve a failed design and justify a revision
Turn matched captures into a reproducible failure log and a bounded design claim.
Open lessonLesson · Science · Earth and space sciencePreserve ambiguity and diagnose conflicting evidence
Distinguish limited coverage, incompatible single-species patterns and a continuum without discrete features.
Open lessonLesson · Science · University Physics II with calculusPreserve both initial states of an ideal LC oscillation
An ideal LC oscillator needs both initial charge and initial current. Here L=1 H and C=0.25 F, giving angular frequency 2 rad/s. Signed current is dq/dt. Neither initial state is forced to vanish, and the zero-state case is included rather than being assigned a fictitious oscillation amplitude.
Open lessonLesson · Linear Algebra II: operators and spectral structurePreserve geometry and diagonalize compatible operators
Characterize unitary maps, prove simultaneous spectral decompositions and derive polar decomposition.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsPreserve inductor current when the closed path changes
The first virtual topology is a series loop containing a 6 V source, 2 ohm resistor and inductor L, initially with zero current.
Open lessonLesson · Partial differential equations and Fourier methodsPreserve integral balance across a jump
Derive weak form and Rankine–Hugoniot.
Open lessonLesson · Research, editing and publishingPreserve meaning when you borrow words or ideas
Quote accurately, paraphrase independently and attribute both without altering conditions or uncertainty.
Open lessonLesson · Science · General Chemistry IIPreserve reaction direction and electron equivalents in a cell potential
A cell potential belongs to a specified redox direction and activity state.
Open lessonLesson · Science · University Physics II with calculusPreserve the energy missing from a final-state-only calculation
Final electrostatic states do not automatically explain the energy transferred while reaching them. Here two capacitors have a common bottom reference conductor and isolated top-node charges, initially at V1=6 V and signed V2. Connecting the top nodes through a positive finite resistor conserves their combined charge but allows resistive energy conversion.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsPreserve two momentum components through a point collision
Momentum is a vector. In a two-dimensional collision with negligible external impulse, conserve its east and north components separately before calculating a magnitude or direction. Adding the incoming speed magnitudes would discard the directions that the conservation statement needs.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgePrevent mass from concentrating in tails
Verify uniform integrability using tail bounds.
Open lessonExperiment · Money, economics and businessPrice a fictional pop-up shop
Does the price with highest revenue also give highest profit?
Open experimentLesson · Social studiesPrice a shared network and account for its funding
Compare three pricing rules for a fictional network, reconcile costs and transfers, and defend a qualified recommendation.
Open lessonLesson · Mathematics investigationPrice one more unit of capacity
When does a local marginal value stop predicting a larger change?
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsPrimitives and path independence
Prove endpoint evaluation for a holomorphic primitive.
Open lessonConcept · GlossaryPrincipal component analysis (PCA)
A linear representation using directions of large centered variance. Keeping fewer directions can discard task-relevant distinctions even when retained variance is high.
Open conceptLesson · Differential geometry of curves and surfacesPrincipal curvatures as a self-adjoint problem
Diagonalize bending in the tangent metric.
Open lessonCourse · Social studiesPrinciples of macroeconomics
Hands-on social studies course and project
Open courseCourse · Social studiesPrinciples of microeconomics
Hands-on social studies course and project
Open courseConcept · GlossaryPrior probability
A probability assigned before the evidence currently being considered. It should describe the relevant population or assumptions, not a convenient unrelated group.
Open conceptLesson · Calculus-based probability: laws, dependence and limitsProbability axioms and limits of events
Use countable additivity rather than intuition about infinite lists.
Open lessonProject · Calculus-based probability: laws, dependence and limitsProbability investigation: arrival dependence, route termination and incomplete evidence
Build one reproducible process-audit report with two explicitly separate experiments: counts in paired time windows and a finite routing chain. Predict exact quantities, compare bounded seeded trials with independent calculations, retain overflow and unresolved records, and defend a changed dependence/exit mechanism. These are original synthetic models, not measured service data, a queue simulation or a secure examination.
Open projectCourse · Mathematics program unitProbability protocols and expected work
Build the observation event, distinguish fresh restarts from retained information, and compare exact expected inspection work under explicit sampling assumptions.
Open courseCourse · MathematicsProbability, uncertainty and expected outcomes
Count relevant outcomes, distinguish conditional and independent events, update beliefs from base rates and compare random outcomes by expectation and variation.
Open courseAssessment · Calculus-based probability: laws, dependence and limitsProbability: laws, dependence and justified limits
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentConcept · GlossaryProcedural memory
Reusable ways to carry out tasks, represented as stored procedures, controllers, or learned skills.
Open conceptProject · Statistics: from samples to qualified conclusionsProduce a reproducible statistical evidence brief
Investigate a fictional service or study claim. Audit its sample, describe the distribution, state an uncertainty or testing procedure before choosing a conclusion, and inspect a relationship for confounding and influential observations.
Open projectProject · Quantitative reasoning and numerical senseProduce an auditable quantitative estimate
Prepare a fictional school-event or workplace-material estimate. Scale a reference quantity, convert measurements consistently, calculate whole-unit purchases and distinguish a rough estimate from a precise-looking model result.
Open projectAssessment · Statistical investigations, simulation and probability decisionsProduce and defend a statistical investigation
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Differential calculus: techniques and applicationsProducts and quotients change in two ways
Derive and apply rules that account for changes in both factors.
Open lessonSubject · Experiment collectionProgramming and software engineering
Inspect algorithms and repair failures with explicit contracts and counterexamples.
Open subjectLesson · Mathematics for AI · M03Project a cloud and inspect what is lost
Connect unit directions, reconstruction, covariance, eigenvectors and low-rank intuition.
Open lessonLesson · Partial differential equations and Fourier methodsProject a profile onto orthogonal modes
Derive coefficients and normalization before solving PDEs.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsProject loop elements and account for an arc’s return path
For a circular loop of radius R in the xy plane carrying current counterclockwise viewed from +z, transverse Biot–Savart components cancel on the axis.
Open lessonLesson · Science · Earth and space scienceProject plate displacement with explicit assumptions
Convert measured-rate units and distinguish conditional displacement from a geographic forecast.
Open lessonLesson · Science · Earth and space scienceProject sunlight onto a horizontal area
Relate solar altitude to direct top-of-atmosphere irradiance and distinguish flux from daily energy.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryProjective completion and points at infinity
Build the real projective plane from vector subspaces.
Open lessonLesson · Social studiesPromises need monitoring and response rules
Institutions can establish shared information, repeated interaction and dispute procedures.
Open lessonAssessment · Proof foundations: from claims to argumentsProof foundations: argument portfolio assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitProof foundations: from claims to arguments
A twelve-lesson proof foundations unit shared by analysis, algebra and discrete mathematics. Exact checks support practice; authored arguments require review. This unit is not the complete undergraduate discrete mathematics course.
Open courseProject · Proof foundations: from claims to argumentsProof portfolio: from a remainder procedure to a theorem
Build one coherent mathematical manuscript over several sessions. Investigate which integers can be written as ax+by, prove a terminating procedure for finding their positive generator, and use the result to classify every solution of a linear integer equation. Examples motivate claims; they do not prove universal statements. Keep conjectures, rejected arguments, revisions and reviewer feedback. Public guidance is available, so this is a supported portfolio rather than an unseen examination.
Open projectLesson · Science · Introductory AstronomyPropagate a calibrated source through orbital and radiative inferences
The original cumulative packet astro-integrated-source@1 combines measured-looking quantities only as explicitly supplied fictional evidence. It is not a real discovery, a self-consistent stellar-structure calculation or proof of a dynamically stable system. The purpose is to propagate common assumptions and identify which apparent agreements are algebraic rather than independent tests.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidencePropagate a calibration interval without inventing a fossil date
Molecular differences can help estimate divergence time only through a model connecting change, rate and time. Under a strict equal-rate clock, two lineages diverging for time t accumulate pairwise expected divergence d=2rt. The factor two represents both lineages, not an arbitrary correction.
Open lessonLesson · Science · Introductory AstronomyPropagate a correlated calibration through a distance ladder
A distance ladder links geometric distances to calibrated stellar indicators and then to more distant sources. Pulsating variable stars can supply a relation between period and luminosity, but the relation depends on class, band, composition and calibration. Not every variable star belongs to one universal standard-candle law.
Open lessonLesson · Science · General Chemistry IPropagate a difference without double-counting a shared offset
A gross-minus-tare measurement can cancel a common error while retaining other uncertainty.
Open lessonLesson · Mathematical statistics: derivation and model criticismPropagate asymptotic uncertainty through a transformation
Use first- and second-order expansions where appropriate.
Open lessonLesson · Science · University Physics I with calculusPropagate orbital observations without hiding a clock gain
For a circular test-particle orbit, gravitational acceleration μ/r² equals v²/r and v = 2πr/T. Combining them gives μ = 4π²r³/T². An inferred parameter therefore depends strongly on both radius and the clock used for period.
Open lessonLesson · Mathematical modeling and senior investigationPropagate uncertainty without hiding dependence
Compare interval, variance and simulation statements.
Open lessonExperiment · Cybersecurity and digital literacyProtect access without losing recovery
Which policy rejects a stolen password while supporting a legitimate recovery?
Open experimentExperiment · Design and product developmentPrototype and repair a navigation flow
Which control actually reaches the learner’s saved work?
Open experimentLesson · Boolean specifications, certificates and circuitsProve a circuit correct and separate work from depth
Construct a bounded-fan-in alarm circuit, prove its specification for every input, and audit padding, parallel depth and shared gates.
Open lessonLesson · Set specifications and language-expression limitsProve a limit of a language-expression grammar
Use finite and cofinite length sets as a structural invariant of a small log-filter language, then distinguish expressibility from decidability.
Open lessonLesson · Mathematics investigationProve a recurrence pattern
What makes a step from k to k+1 work for every k?
Open lessonLesson · Arrival processes, absorption and honest simulationProve a route ends before trusting its equations
Connect first-step equations to a finite tail bound and inspect what fails for a trapped state.
Open lessonProject · Discrete mathematics: logic, proof and algorithmsProve and test a small discrete method
Specify a finite problem, count its outcomes, provide one counterexample and one complete proof, and trace an algorithm on a graph or recurrence. A matching number does not replace the authored argument.
Open projectLesson · Geometric constructions, circle proofs and coordinate lociProve and use coordinate line criteria
Derive parallel and perpendicular line rules without losing vertical cases.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsProve angle relations with parallel lines
Separate vertical-angle logic, Euclidean parallel-line facts and their converses.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociProve circle similarity and recover a circle equation
Derive a circle locus in both directions and classify completed-square degeneracies.
Open lessonLesson · Constructing the real numbersProve completeness of the constructed field
Build limits from rational approximations, derive least upper bounds, and construct an irrational root without assuming it already exists.
Open lessonLesson · Determinants from permutations to certificatesProve determinant products and their limits
Derive multiplicativity from an alternating multilinear form, then distinguish invertibility, block elimination and numerical conditioning.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsProve every positive integer is a sum of four squares
Verify the four-square multiplication identity.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsProve existence and uniqueness separately
Use ideal chains and prime divisibility in a PID.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesProve existence by averaging and alteration
Use indicator expectations without independence.
Open lessonLesson · Weighted integration and jump conventionsProve existence while keeping the definitions separate
Control weighted gaps and every sufficiently fine tagged sum, including finite discontinuities away from integrator jumps.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsProve finite generation survives polynomial extension
Connect ascending chains, finite ideal generators and Hilbert’s basis theorem.
Open lessonLesson · Finite sampling guarantees and dependenceProve finite sampling guarantees with pairwise independence
Derive a finite error guarantee from squared deviations, then audit endpoints, dependent sampling designs and simultaneous claims.
Open lessonLesson · Partial differential equations and Fourier methodsProve finite-speed uniqueness by an energy balance
Track flux through fixed and moving boundaries.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsProve ideal norms multiply
Connect invertible-ideal quotients to exact index formulas and prime powers.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesProve identities by counting the same objects
Construct reversible counting correspondences.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsProve identities on their shared domains
Use definitions and algebra to justify all-input equalities rather than isolated coincidences.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsProve midpoint and median structure
Prove a midsegment theorem and the common 2:1 point of all medians.
Open lessonLesson · Algorithm reasoning: invariants, termination and costProve progress with lexicographic descent and an exact work count
Separate safety from termination, prove a two-counter process stops, and derive a rank that counts its remaining transitions exactly.
Open lessonLesson · Stable matching and preference guaranteesProve proposer-optimality and interpret its limits
Use the first rejected stable pair to prove optimality among stable outcomes, without claiming universal fairness.
Open lessonLesson · Relation closure and directed reachabilityProve short reachability from an extremal vertex
Use complete pairwise orientation to prove a two-step reachability theorem, then distinguish its sufficient condition from a characterization.
Open lessonLesson · School trigonometry: proofs, inverse branches and periodic decisionsProve symmetry by moving a unit-circle point
Use general coordinates to explain half-turns, reflection, parity and periodicity.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsProve that prime factors cannot hide inside a product
Establish Euclid’s lemma, factorization uniqueness and elementary infinitude arguments.
Open lessonLesson · Discontinuities and Riemann integrationProve the discontinuity criterion for Riemann integration
Connect compact oscillation thresholds to short interval covers and small Darboux gaps, proving both directions of the bounded-interval criterion.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsProve the sign that appears when primes are exchanged
Derive quadratic reciprocity through Gauss parity and a lattice-point count.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeProve the triangle inequality in Lp
Derive Minkowski using Hölder.
Open lessonLesson · Proof foundations: from claims to argumentsProve two sets contain the same objects
Translate membership into logic and use double inclusion.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsProve unique factorization of nonzero ideals
Derive invertibility and prime-ideal factorization from the ring hypotheses.
Open lessonLesson · Partial differential equations and Fourier methodsProve uniqueness and bounds without summing the solution
Use integration by parts and a parabolic maximum argument.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeProve weak and strong averaging results
Derive the weak law for finite-variance independent samples.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsProve what a loop preserves and why it stops
Combine initialization, maintenance, exit reasoning and a decreasing measure.
Open lessonLesson · Joint information and adaptive strategiesProve what adaptive stopping can and cannot change
Compare stopping and revealing through a full-state induction, then identify the information and cost changes that break equal strategy values.
Open lessonLesson · Arrival processes, absorption and honest simulationProve what can be separated into independent counts
Use a joint count law and independent marks instead of inferring independence from a histogram.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsProve when one residue generates all units
Establish primitive roots for primes and their permitted composite extensions.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsProve why convergents are close and reduced
Derive numerator and denominator recurrences.
Open lessonConcept · GlossaryProvenance
Where evidence came from: its source, request, version, and relevant timing. Useful for deciding whether an observation applies to the current question.
Open conceptCourse · Social studiesPsychology — selected AP-topic practice
Hands-on social studies course and project
Open courseCourse · Social studiesPsychology and human behavior — High school
Hands-on social studies course and project
Open courseCourse · Social studiesPsychology and human behavior — Introductory college
Hands-on social studies course and project
Open courseLesson · Social studiesPublic credit: who owes, who receives?
Investigate debt assumption, creditor transfers and the evidence needed to judge public credit.
Open lessonLesson · Social studiesPublic opinion depends on measurement
Political beliefs develop through experience, socialization and information.
Open lessonLesson · Social studiesPullman in 1894: wages, housing and collective action
Connect company housing, a wage dispute, union networks and federal intervention through a dated evidence map.
Open lessonLesson · Stories and creative writingPut a character under a small pressure
Reveal a character through a consequential choice rather than relying only on trait labels.
Open lessonExperiment · Systems thinking and operationsPut capacity where the demand arrives
Do equal staffing budgets produce equal service?
Open experimentLesson · Read like a literary detectivePut two texts into conversation
Build a comparative argument about how two texts treat a shared question through different formal choices.
Open lessonStudio · Optional P02 implementation studioPython data-loading studio
Named CSV columns, schema and unit validation, target ID joins, training-only preprocessing and runnable Python starter.
Open studioConcept · GlossaryQ-learning
An action-value learning method that moves an estimate toward observed reward plus discounted maximum next-state value; terminal outcomes have no future contribution.
Open conceptLesson · Social studiesQualitative sampling: seek information and disconfirming cases
Purposeful sampling selects cases for a question, such as variation, unusual experience or a proposed mechanism. A run of repeated themes in a narrow recruitment network may reveal similarity in that network rather than adequate coverage of the question.
Open lessonLesson · Science · University Physics II with calculusQuantify finite-line end corrections
A source may look like a point from far away and like a long line from close to its midpoint, but those approximations have different domains. The uniform rod extends from −L to L and carries total charge Q. Its perpendicular-bisector observer is at y>0. Pairing elements at opposite source coordinates cancels horizontal components before evaluating the remaining integral.
Open lessonLesson · Optimization: models, certificates and algorithmsQuantify progress under smoothness and curvature
Derive descent and convergence bounds rather than trusting a trace.
Open lessonCourse · MathematicsQuantitative reasoning and numerical sense
Translate fractions, percentages, ratios and units into useful calculations; estimate magnitude and communicate precision without overstating evidence.
Open courseConcept · GlossaryQueries, keys, and values
An attention query compares with keys to determine mixture weights. Those weights combine value vectors into an output.
Open conceptConcept · GlossaryQuery set
Examples used to evaluate an adapted model. During meta-training, their loss can train shared meta-parameters through the outer objective.
Open conceptLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceQuestion the age distribution behind a stage-duration estimate
A snapshot gives a fraction of cells in a recognizable stage, not a direct stopwatch measurement of each cell. Multiplying that fraction by a known cycle duration is a useful classroom approximation only under an appropriate sampling model. A stable asynchronous population need not have uniformly distributed cell ages.
Open lessonLesson · Point-set topology: spaces, constructions and separationQuotient maps and final topology
Define quotient openness through full inverse images.
Open lessonLesson · Social studiesRace and ethnicity: examine classification and institutional treatment
Sociological analysis examines how categories, identities and unequal treatment are produced and maintained. Race and ethnicity must not be treated as fixed explanations for a person's abilities, behavior or worth.
Open lessonConcept · GlossaryRandom forest
An ensemble of trees trained on resampled data with random subsets of features considered at each split.
Open conceptAssessment · Precalculus: local rates and polynomial evidenceRate and polynomial evidence: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Differential calculus: techniques and applicationsRates for inverse functions and inverse trigonometry
Use a local inverse carefully, with nonzero slope and selected branches.
Open lessonLesson · Social studiesRatification and rights: compare safeguards
Reconstruct arguments for different safeguards and compare them with later amendment wording.
Open lessonLesson · Constructing the real numbersRational Cauchy classes and arithmetic
Build field operations on equivalence classes of rational Cauchy sequences using rational estimates, including a justified reciprocal construction.
Open lessonLesson · Abstract algebra I: groups and symmetryRead a classification theorem without overextending it
Use primary cyclic factors and uniqueness invariants.
Open lessonLesson · Science · Physics and physical systemsRead a complete oscillation in three motion graphs
Connect displacement, velocity and acceleration at the same instant.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceRead a compound expression as a model
Separate meaningful parts before expanding or calculating.
Open lessonLesson · Mathematics for AI · M04Read a density as probability per unit
Connect continuous probability, area, Gaussian scale and numerical integration.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsRead a distribution through its CDF
Connect events, point masses and interval probabilities.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsRead a field from a potential landscape
Use spatial derivatives and equipotential geometry rather than a potential value alone.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsRead a force-versus-speed curve without calling it a time history
Choose downward as positive for a falling point in still fluid. The stipulated drag magnitude is kv² upward for nonnegative downward speed v, and gravity is 10 m/s². The parameter k has units kg/m, so kv² has force units. This is one assigned resistance law, not a universal description of all fluids, shapes or speeds.
Open lessonLesson · Social studiesRead a frontier, an inefficient point and a technology shift
Does producing less of both goods prove resources are scarce?
Open lessonLesson · Science · Physical Geology and Earth SystemsRead a geologic contact map before drawing a section
A geological map records where geological surfaces meet the land surface, while a cross-section shows an interpretation through depth. Strike is the horizontal direction within a plane; true dip is its steepest downward inclination, perpendicular to strike. A section oblique to that direction usually shows a smaller apparent dip.
Open lessonLesson · Science · Earth and space scienceRead a geological isotope clock
Connect half-life, isotope inventories and the event a mineral can date.
Open lessonLesson · Mathematics · Functions and graphsRead a graph with its units
Connect a table, a coordinate pair and a point without being misled by axis scales.
Open lessonLesson · Optimization: models, certificates and algorithmsRead a linear objective on a polyhedron
Identify feasible faces, finite optima and improving rays.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceRead a longitudinal evolution study without inventing observations
Evolution can be investigated in records spanning generations, not only inferred from distant fossils.
Open lessonLesson · Course 2 · Testing whether improvement is realRead a paper as a set of claims
Separate a paper’s proposed mechanism, measured result and broader hypothesis.
Open lessonLesson · Science · General Chemistry IRead a phase map as coexistence evidence rather than a heating trace
Keep pressure, temperature, coexistence and endpoint meanings separate.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceRead a probability distribution of drift rather than one lucky trajectory
Genetic drift is variation in inherited frequencies due to finite sampling. It can occur without a selective advantage. Different replicate populations started from the same frequency can move in opposite directions, so one realized increase does not establish that an allele was favored.
Open lessonLesson · Mathematics investigationRead a quadratic from its shape
When does a curved model reach zero?
Open lessonLesson · Social studiesRead a real urban trend through a changed definition
Can an urban share fall while the urban population grows?
Open lessonLesson · Science · Earth and space scienceRead a recorded ocean profile
Audit eight selected actual Argo levels and compare properties without inventing movement.
Open lessonLesson · Linear Algebra II: operators and spectral structureRead a self-adjoint operator through real energy values
Prove real spectra and orthogonal diagonalization, then derive Rayleigh bounds and extremizers.
Open lessonLesson · Sections, solid measures and geometric designRead a solid through its sections
Distinguish a planar slice, a projection and a three-dimensional object.
Open lessonLesson · Science · Introductory AstronomyRead a stellar population without treating it as one trajectory
A coeval population can constrain age because stars with short central-hydrogen lifetimes leave the main sequence while lower-mass members remain. The turnoff is a population feature, not the instant when every member disappears. Real comparisons require membership, distance, extinction, composition and selection, and can be complicated by binaries or non-coeval formation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsRead a wave snapshot without mistaking it for particle travel
A wave transports a disturbance and energy through space. In the small-amplitude string model, a marked piece of string moves transversely about its equilibrium position while the pattern travels horizontally. It does not ride along with a crest. Longitudinal sound instead involves local motion parallel to propagation and alternating pressure compressions and rarefactions; the pressure profile is not a picture of the air itself moving across the entire room.
Open lessonLesson · Science · Environmental science and sustainabilityRead actual daily streamflow evidence
Audit USGS dates, discharge units, daily means and source qualifiers.
Open lessonLesson · Science · AP Environmental ScienceRead age-specific survival without changing denominators
Compare cumulative survivorship with conditional mortality and curve patterns.
Open lessonLesson · Network incidence and the four spacesRead all four spaces of a network incidence matrix
Construct bases in vertex and edge coordinates, then prove how connected components and cycles determine the four spaces.
Open lessonLesson · Social studiesRead all ten amendments in the Bill of Rights
Read locally packaged constitutional provisions, annotate their mechanisms and test a claim against connected clauses.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsRead an angular encoder without losing a revolution or reversal
Angular position requires a reference and a positive rotation direction. This packet starts at an unwrapped350° with positive counterclockwise rotation. An encoder reports the same direction as an angle between 0° and 360°, but that wrapped coordinate discards the number of full turns. A jump from 350° to 20° can be a coordinate wrap rather than a 330° clockwise rotation.
Open lessonLesson · Social studiesRead an exchange-rate quotation and external accounts
What does a currency appreciation make cheaper?
Open lessonLesson · Science · Scientific inquiry and measurementRead an instrument with units
Distinguish a physical quantity from its recorded digits.
Open lessonLesson · Linear Algebra II: operators and spectral structureRead an operator through preserved subspaces
Connect invariant subspaces, quotient maps and block-triangular matrices.
Open lessonLesson · Science · Earth and space scienceRead and audit an earthquake catalog snapshot
Trace real catalog estimates to their event identifiers, timestamps and magnitude methods.
Open lessonLesson · Algebra foundations: expressions, equations and rootsRead and evaluate an expression
Translate quantities into expressions and preserve signs, grouping and units.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsRead angular histories through a reversal
A rigid system keeps its shape while its points follow different paths. A rotating disk cannot be reduced to one point if the question concerns its spin: the center might not move at all while the rim travels. Choose a fixed axis and viewing side, call counterclockwise positive, and measure unwrapped angular position. One radian is the angle subtended by an arc equal in length to the radius; a complete turn is 2π radians.
Open lessonLesson · Science · AP Environmental ScienceRead biome evidence beyond annual averages
Infer climatic constraints without treating one index as a biome label.
Open lessonLesson · Science · AP Chemistry preparationRead bond length and separation energy from a curve
Compare controlled potential wells without confusing energy, force, distance or finite separation.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceRead bound energy levels and ionization thresholds
In the simple hydrogen approximation, bound energies are En=−13.6/n² eV relative to an electron and proton separated at rest. The negative sign encodes binding relative to this chosen zero; it is not a negative photon energy. A transition from a higher level to a lower level releases the positive difference Eupper−Elower. Reverse absorption requires the same gap when that transition is allowed. Ionization from level n instead requires reaching the continuum zero, an energy 13.6/n² eV.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceRead branching relationships rather than tip order
A rooted phylogenetic tree represents hypotheses about common ancestry.
Open lessonLesson · Statistics: displays, designs and differencesRead categorical displays with their denominators
Construct bars and pie sectors, then distinguish conditional percentages from mosaic areas when group sizes differ.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsRead coefficients as choices
Derive the binomial expansion and connect algebraic and combinatorial identities.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceRead common ancestry without using tip order as a ladder
A rooted tree supplies a direction from an ancestral lineage toward descendant lineages. An unrooted tree represents relationships without specifying that direction. An appropriate outgroup and other evidence can help infer a root, but an outgroup is not automatically the ancestor of the ingroup.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsRead conics from equations and geometric definitions
Complete squares and distinguish ellipses, hyperbolas and parabolas.
Open lessonLesson · Differential equations: models, methods and dynamicsRead corners and jumps as generalized derivatives
Translate a piecewise graph into slopes, signed impulses and an action on test functions.
Open lessonLesson · Social studiesRead Cyprus ceasefire lines without inventing a settled border
What does an operational boundary map establish about sovereignty?
Open lessonLesson · Abstract algebra I: groups and symmetryRead decompositions from characters
Prove orthogonality, multiplicities and completeness using averaged Hom spaces.
Open lessonLesson · Science · AP Environmental ScienceRead demographic age structure without a deterministic forecast
Calculate age composition and distinguish momentum from an inevitable prediction.
Open lessonLesson · Algebra II and precalculus: functions and structureRead domains and ranges across representations
Distinguish allowed inputs, attained outputs and the information a table cannot supply.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsRead epithelial architecture before measuring thickness
Epithelial tissue forms closely connected sheets at surfaces and glands. The apical domain faces a lumen or exterior, lateral surfaces contact neighboring cells, and the basal domain attaches to a basement membrane. Underlying connective tissue supplies support and exchange; an epithelial sheet itself lacks blood vessels. A lumen can be continuous with the exterior even when it lies deep inside the body, so “inside the organ” and “inside the body’s internal fluid compartment” are different boundaries.
Open lessonLesson · Science · Earth and space scienceRead erosion surfaces and missing time
Distinguish an observed erosional contact from an unmeasured duration.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsRead factor degrees as Frobenius orbits
Prove how irreducible factors partition finite-field roots into permutation cycles.
Open lessonLesson · Precalculus: local rates and polynomial evidenceRead finite differences without overclaiming a model
Test equally spaced patterns, repair unequal-spacing comparisons and construct distinct functions that fit the same finite observations.
Open lessonLesson · Science · Chemistry and matterRead formulas without changing the substance
Distinguish a molecular formula from a coefficient and count atoms across several species.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsRead geometry and stored energy from a capacitor model
A capacitor separates equal amounts of positive and negative charge. The combined device may have zero net charge while maintaining a substantial field and stored energy between its plates. The quantity Q in Q=CV means the magnitude on one plate, not the net charge of the two-plate system. Capacitance C measures charge separated per volt.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsRead group invariants and integer solutions from a certificate
Prove uniqueness and classification, then solve integer systems completely.
Open lessonLesson · Differential calculus: techniques and applicationsRead increase and curvature from derivative signs
Combine sign charts with domain information rather than relying on a plotted appearance.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsRead induced polarity from changes in a flux record
Faraday’s law relates average induced emf to flux change: emf=−NΔΦ/Δt for N turns with the same per-turn flux. The minus sign encodes Lenz’s law: induced current, if a closed path permits it, produces a field that opposes the change in flux. It does not always oppose the original field; when the original field is decreasing, the response can support it.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceRead intersections as simultaneous solutions
Use supplied graphs to find candidates and equations to verify them.
Open lessonLesson · Functions in graphs, measurements and growth decisionsRead inverses without a formula
Read inverse values from unseen tables and supplied graphs.
Open lessonLesson · Social studiesRead language landscapes without inferring identity
What can a map of public signs reveal about cultural change?
Open lessonCourse · English language artsRead like a literary detective
Build, challenge and revise interpretations using precise textual evidence, then compare how two texts create meaning.
Open courseLesson · Calculus through supplied graphs and measured tablesRead limits and derivatives from supplied graphs
Separate marked function values from nearby behavior and use a derivative graph without assuming it is the original function.
Open lessonLesson · Differential equations: models, methods and dynamicsRead local directions before solving
Construct a field and distinguish its slopes from solution values.
Open lessonLesson · Science · Earth and space scienceRead lunar phases as changing viewing geometry
Use neighboring positions to distinguish waxing and waning while keeping angles separate from dates.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceRead nucleic-acid structure with chemical identity and direction
Distinguish nucleotide components, backbone bonds and antiparallel complementary sequences.
Open lessonLesson · Social studiesRead population structure and transition models
Can birth rates fall while population still grows?
Open lessonLesson · Science · Physics and physical systemsRead position, displacement and distance
Reconstruct a journey from a position-time graph and distinguish signed change from total travel.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsRead quotient structure from an ideal
Prove the domain and field characterizations.
Open lessonLesson · Functions in graphs, measurements and growth decisionsRead rates from uneven tables and graph scales
Calculate change per input unit on unequal intervals.
Open lessonLesson · Algebra II and precalculus: functions and structureRead roots, multiplicity and end behavior
Use factors and leading terms to predict a graph without assuming its entire shape.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsRead rotational work from an angle-dependent torque
A tangential force acting through a small arc transfers work FₜΔs. With Δs=rΔθ and τ=rFₜ, the same transfer is τΔθ. For a constant signed torque about a fixed axis, W=τΔθ, where radians are used. Torque and angular displacement with opposite signs transfer negative work.
Open lessonLesson · Precalculus: semilog scales and model evidenceRead semilog scales and test exponential evidence
Make the axis transformation visible, recover a response rule and evaluate what a straight-looking fit actually supports.
Open lessonLesson · Science · Earth and space scienceRead stellar positions and pathways
Compare qualitative surface categories while retaining competing interior histories.
Open lessonLesson · Science · Introductory AstronomyRead temperature and luminosity as distinct stellar constraints
An H–R diagram compares luminosity with effective temperature or a calibrated spectral proxy. Its conventional temperature axis decreases toward the right. Main-sequence stars, giants and white dwarfs occupy different regions because temperature alone does not determine total power or radius. A population diagram is a collection of stars, not automatically the path followed by one star.
Open lessonLesson · Euclidean and coordinate geometry foundationsRead the arc before using the angle
Relate central and inscribed angles, chords and tangent lengths with explicit circle hypotheses.
Open lessonLesson · Social studiesRead the constitutional structure: Preamble and Articles I–VII
Read locally packaged constitutional provisions, annotate their mechanisms and test a claim against connected clauses.
Open lessonLesson · Geometric constructions, circle proofs and coordinate lociRead the correct arcs in circle angle proofs
Connect inscribed, cyclic and tangent configurations using named arcs and full angle arguments.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsRead turning points and force directions from potential energy
For a conservative one-particle model with total mechanical energy E, kinetic energy at position x is K=E−U(x). A position is accessible only when E≥U there. Where E=U, speed is zero; such positions can be turning points. Where E<U, a negative number is an energy deficit, not a physical negative kinetic energy.
Open lessonLesson · Science · Earth and space scienceRead what each deformation instrument measures
Compare tilt, displacement and interpreted uplift while preserving their geometry and qualifiers.
Open lessonLesson · Science · Earth and space scienceRead what spectral features represent
Distinguish continuum, emission and absorption while keeping wavelength evidence separate from intensity claims.
Open lessonCourse · Mathematics program unitReal Analysis I: completeness, limits and rigorous calculus
Twenty proof-centered lessons developing the real-number completeness property through sequences, topology of the real line, continuity, differentiation and Darboux integration. Written proofs require mathematical review. Function sequences, uniform convergence, metric-space theory and rigorous multivariable analysis belong to the subsequent Analysis II sequence.
Open courseAssessment · Real Analysis I: completeness, limits and rigorous calculusReal Analysis I: construct proofs and test hypotheses
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitReal Analysis II: uniform limits, metric spaces and local calculus
Twenty proof-centered lessons on function convergence, metric structure and rigorous finite-dimensional differentiation. Written arguments and counterexamples require mathematical review. Measure theory, Lebesgue integration, infinite-dimensional calculus and full functional analysis are later subjects; this sequence does not claim independently assessed course equivalence.
Open courseLesson · Complex analysis: holomorphic functions, contours and mappingsReal integrals and contour estimates
Choose contours whose auxiliary pieces have justified limits.
Open lessonCourse · Mathematics program unitReal numbers, complex coordinates, vectors and matrices
A school-entry bridge from fraction and coordinate arithmetic to justified rational powers, complex constructions, directed quantities and labeled matrix operations. Written proofs and coordinate drawings require human review; this unit preserves the existing algebra, geometry and linear algebra courses.
Open courseCourse · Mathematics program unitReal skew-adjoint geometry and complex structures
Construct real rotation planes, distinguish real and complex spectral statements, and prove when an inner product is compatible with multiplication by an imaginary unit.
Open courseStudio · Optional P06 baseline; independent implementation review pendingReal-image perception studio
Train and replay a Python pixel segmentation baseline with licensed images, source checks and failure analysis.
Open studioLesson · Combinatorics and graph theory: enumeration, structure and certificatesRealize or refute a degree sequence
Prove the Havel–Hakimi reduction by switches.
Open lessonLesson · Algebra foundations: expressions, equations and rootsRearrange formulas with conditions
Solve for a chosen symbol while keeping units and zero-coefficient cases visible.
Open lessonPath · Software engineers · systemsReason about AI applications
Inspect tool contracts, retrieval evidence, resource decisions and release checks.
Open pathLesson · Course 2 · Planning and reasoningReason about an opponent
Play a small stochastic game, reveal its search tree and compare adversarial planning, probabilistic planning and a policy learned from sampled returns.
Open lessonLesson · Course 2 · Geometry and control foundationsReason about forces and contact
Compare force and position commands, friction limits and the forces needed to hold a load.
Open lessonLesson · Science · AP Chemistry preparationReason from particle access to entropy change
Predict the sign of a system entropy change while distinguishing it from the total entropy criterion.
Open lessonLesson · Course 2 · Planning and reasoningReason with procedures and search
Compare a local shortcut with a procedure that explores alternatives and checks its result.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsRecalculate friction when the pull changes the normal force
A 2 kg object is initially sliding to the right on a horizontal surface. A force of magnitude F pulls at angle θ above horizontal. Gravity is 10 m/s² and the kinetic coefficient is 0.3. This lesson evaluates the instantaneous state while rightward slipping is stipulated; it does not assume that the same direction of slip persists forever.
Open lessonLesson · Science · General Chemistry IIRecalculate partial pressures after a gas-volume perturbation
An immediate pressure change and a later compositional relaxation are distinct states.
Open lessonConcept · GlossaryRecall
True positives divided by all actual positives: TP / (TP + FN). Also called the true-positive or detection rate; undefined if no positive reference cases are available.
Open conceptLesson · Mathematical statistics: derivation and model criticismRecognize a common likelihood architecture
Derive moments from the log normalizer and respect the natural parameter space.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsRecognize angular momentum in straight-line motion
Angular momentum is specified relative to a reference point or axis. A rigid body about a fixed principal rotation axis has planar angular momentum L=Iω. A moving point particle has magnitude mvr sinφ, where r runs from the reference point to the particle and φ is the angle between that radius and its velocity. The same expression is momentum times perpendicular distance to the line of motion.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsRecognize minimal connected structure
Distinguish walks and paths and prove the characteristic properties of trees.
Open lessonLesson · Mathematics · Functions and graphsRecognize when the rate changes
Compare equal input steps on a line and a curve without assuming a constant rate.
Open lessonLesson · Course 1 · Simple models and useful baselinesRecommend from incomplete feedback
Complete a small ratings picture with predictions, compare simple and learned recommenders, and inspect cold start and exposure bias.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsReconcile charged-particle deflection with an energy ledger
A uniform electric field exerts constant force F=qE on a particle whose charge remains fixed. Dividing by its mass gives constant acceleration. This resembles a projectile under gravity, except reversing the charge reverses electric acceleration. The horizontal velocity remains unchanged here because the stipulated field is entirely vertical.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsReconcile coaxial volume charge with piecewise field
Use enclosed area, not radial thickness, in a cylindrical charge account.
Open lessonLesson · Science · Introductory MicrobiologyReconcile community, genetic and functional evidence before telling a causal story
A complete microbial inference must connect quantities without erasing how each was obtained. An image object, recoverable unit, sequence read, marker carrier, expression proxy and carbon flux are not automatically the same unit. The final comparison combines community composition, observation bias, genetic state and function, while keeping actual source microscopy separate from the original numerical records.
Open lessonLesson · Science · University Physics II with calculusReconcile current surfaces without inventing gap conduction
A charging capacitor exposes why an electric-flux derivative belongs in Ampère’s law. A spanning surface through the lead sees conduction current, while another surface through the vacuum gap sees no material charge crossing it. Both surfaces have the same boundary loop and must predict the same magnetic circulation.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceReconcile ecosystem budgets across nested boundaries
An ecosystem includes interacting organisms and their physical environment, but the analytical boundary is chosen for a question. A soil patch, wetland and watershed can each be legitimate systems. Changing the boundary changes which movements count as external fluxes without changing the underlying physical transfer. State the element or material tracked, spatial extent and time interval before adding quantities.
Open lessonLesson · Social studiesReconcile expenditure, income and the business cycle
Why are imports subtracted from an expenditure total?
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsReconcile falling-mass rotation with a separate stopping receiver
This cumulative mathematical packet joins motion, force, rotation, work and thermal accounting. A 2 kg mass descends from rest while unwinding a massless no-slip cord from a drum of radius 0.1 m and inertia 0.02 kg m². Gravity is 10 m/s² and a constant axle torque 0.2 N m opposes rotation. These are fictional low-energy parameters, not instructions to build or load a machine.
Open lessonLesson · Science · University Physics II with calculusReconcile field pressure with source and separating work
For vacuum parallel plates in the no-fringe approximation, field energy density is ε0E²/2 and attractive pressure has the same numerical value in SI. The plates have area 0.01 m², initial gap 0.001 m and initial voltage 10 V. We change the gap slowly while declaring either fixed initial charge or maintained voltage; these are different mechanical experiments.
Open lessonLesson · Science · University Physics I with calculusReconcile generalized force with a first-branch energy integral
A final mechanics packet links a 2 kg translating mass to a rotor with inertia 0.02 kg m² at effective radius 0.1 m. The ideal rigid coupling imposes ω = v/R and can transmit forces of either sign. It is explicitly bilateral, not a cord that would fail when asked to push. No construction or real loading is requested.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsReconcile motional emf, magnetic drag and mechanical work
A moving conductor can separate charge because its mobile charges experience magnetic force. For a straight rod of length L moving perpendicular to itself and to uniform B, the emf magnitude is BLv. In a closed rail circuit this drives current; in an open rod, charges separate until the internal electric field balances magnetic deflection, without a sustained loop current.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceReconcile nutrient processing through connected reaches
Nitrogen moves through chemical forms and physical compartments. Assimilation can transfer dissolved inorganic nitrogen into biomass; mineralization can return it; nitrification changes oxidation state; denitrification can transfer nitrogen to gases. Removing nitrogen from one water-column ledger does not destroy nitrogen atoms or guarantee permanent storage in the whole ecosystem.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingReconcile orbital energy with angular momentum
Use two conserved quantities without mistaking an eccentric orbit for a circular one.
Open lessonLesson · Social studiesReconcile product differentiation with long-run entry
Can a differentiated seller have market power and zero economic profit?
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsReconcile six signed fluxes without guessing a local field
Use area orientation and surface integration to infer only what flux constrains.
Open lessonLesson · Science · General Chemistry IIReconcile solvent, solution and entity denominators
Concentration is not one interchangeable number.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceReconcile water delivery with head and electricity
Delivering water can require lifting it, overcoming pressure differences and overcoming friction. Total dynamic head expresses the defined mechanical energy per unit weight. The same volume can therefore need very different energy depending on source depth, terrain, pressure and system condition.
Open lessonLesson · Differential geometry of curves and surfacesReconstruct a curve from bending data
Use an ODE to turn curvature and torsion into a curve.
Open lessonLesson · Social studiesReconstruct a GDP release before calling it growth
Did the economy change when an estimate of the same quarter was revised?
Open lessonLesson · Science · Introductory AstronomyReconstruct a lens without mistaking images for sources
Gravity changes the paths of light, so a foreground mass can create multiple apparent images of one background source. Lensing depends on projected mass and the observer–lens–source geometry. It provides a mass-sensitive observation different from the source’s own luminosity or a tracer’s orbital motion.
Open lessonLesson · Science · Physical Geology and Earth SystemsReconstruct a site from materials, chronology, map and subsurface evidence
The original cumulative packet geo-site-defense@1 describes a fictional site with granite basement B, sandstone S containing granite clasts, an ash A within the top of the sandstone sequence, tilted strata T, an erosional surface E and horizontal gravel G. These supplied observations require B before S before A before T before E before G. A fault F cuts the ash, but its relation to tilt and erosion has not been observed. All drawings, ages and observations are invented; earlier actual USGS records retain their own locations and are not measurements of this site.
Open lessonLesson · Science · Physical Geology and Earth SystemsReconstruct a source from explicitly modeled S–P ranges
P and S waves travel at different speeds, so their arrival-time difference can constrain distance under an assumed propagation model. In a homogeneous medium with speeds Vp and Vs, S−P=r(1/Vs−1/Vp). The source origin time cancels, but velocity structure and correct phase identification do not.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceReconstruct a uniform field from a voltage survey
A local uniform-field region can be described by a linear potential V=ax+3y+C in SI units. Three probes occupy (0,0), (2,0) and (0,1) m. Voltage differences along horizontal and vertical baselines determine the components without calculus: electric work gives Ex=−[V(2,0)−V(0,0)]/(2 m) and Ey=−[V(0,1)−V(0,0)]/(1 m). The minus sign says field points toward decreasing potential.
Open lessonLesson · Science · Physical Geology and Earth SystemsReconstruct all histories consistent with a cross-cutting packet
A geological history can be represented as constraints rather than a guessed story. If feature X cuts feature Y, X is younger than Y under the stated identities. If a younger cover seals a fault, faulting predates the cover. Each relationship supplies an edge from older event to younger event.
Open lessonLesson · Science · University Physics I with calculusReconstruct an orbit from two turning radii
At an apsis of a noncircular orbit, radial velocity vanishes and speed is tangential. With specific angular momentum h, that speed is h/r. The same total energy must therefore equal h²/(2rp²) − μ/rp at pericenter and h²/(2ra²) − μ/ra at apocenter.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingReconstruct angular momentum from a torque history
Integrate signed torque in time and test its derivative against the momentum record.
Open lessonLesson · Science · Physical Geology and Earth SystemsReconstruct channel discharge from unequal flow panels
Discharge is the volume crossing a section per time. For velocity normal to a cross-section, Q is the integral of velocity over area. An approximate panel method multiplies each panel’s own area by its mean normal velocity, then adds the contributions. Averaging speeds first can be wrong when panel areas differ.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceReconstruct cohort reproduction with the correct survival denominator
A life table follows survival and reproduction by age or stage. Cohort survivorship is measured from an initial cohort, whereas conditional survival asks what fraction of those already alive at one age reach the next. Mixing these denominators changes the interpretation of mortality and reproduction.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsReconstruct current sum and difference from signed field components
Two long z-directed wires lie at x = −1 and +1 m.
Open lessonLesson · Abstract algebra I: groups and symmetryReconstruct cyclic and dihedral rotation groups
Turn a two-point orbit into a complete geometric classification.
Open lessonLesson · Science · Physical Geology and Earth SystemsReconstruct deformation without equating shape and stress
Strain describes a change in shape or size; stress describes force per area. A deformed marker can constrain a deformation history only if its original geometry and later preservation are independently justified. The same final shape does not uniquely identify the applied stress, temperature, strain rate or material response.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsReconstruct displacement and distance from velocity
A velocity–time graph answers two different questions. Its slope gives acceleration, with units (m/s)/s. Its signed area gives displacement, with units (m/s)×s. A point on the vertical axis is velocity itself, not displacement or acceleration. Before calculating, write the quantity and the units you need.
Open lessonLesson · Statistics: displays, designs and differencesReconstruct keyed stems and define percentile rank
Preserve individual values in a stem-and-leaf display and state how ties enter an empirical percentile calculation.
Open lessonLesson · Science · Physical Geology and Earth SystemsReconstruct magnetic stripes with an independent time scale
Magnetic minerals in cooling oceanic rocks can retain a record related to the geomagnetic field at formation. Alternating polarity patterns on opposite sides of ridges provided a test of seafloor spreading. A polarity sign by itself is not a unique age, because reversals repeat and later heating or alteration may modify a magnetic record.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingReconstruct motion with two integration constants
Integrate variable acceleration and apply independent position and velocity initial conditions.
Open lessonLesson · Statistical PCA: coordinates, reconstruction and scalingReconstruct observations and account for lost variation
Prove the projection loss and compare mean reconstruction with new-data performance.
Open lessonLesson · Abstract algebra I: groups and symmetryReconstruct polyhedra from rotation orbits
Use convex hulls and forced faces to finish the exceptional conjugacy cases.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsReconstruct sarcomere bands without shortening the filaments
A skeletal muscle contains fascicles, fibers, myofibrils and repeated sarcomeres. A muscle fiber is a multinucleate cell, not a bundle of independent cells. Endomysial, perimysial and epimysial connective tissues organize and transmit force across these scales. Within a myofibril, Z lines bound a sarcomere and thin filaments extend toward its center.
Open lessonLesson · Social studiesReconstruction, citizenship and institution-building
Explain Reconstruction through legal change, community action and opposition.
Open lessonLesson · Social studiesReconstructive memory: wording, source and confidence
Distinguish event information from later suggestions and evaluate false reports without assuming all memories are inaccurate.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeRecover a density from absolute continuity
Prove the finite Radon–Nikodym theorem.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsRecover a flux history without inventing its missing anchor
A fixed ten-turn coil of area 0.2 m² has normal +z and a supplied uniform field history B(t) = 3 T + a t + b t² for 0 to 2 s.
Open lessonLesson · Science · Earth and space scienceRecover a fractional shift after calibration
Compare virtual reference features with independently corrected wavelengths.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsRecover a function and evaluate a definite integral
Separate a family of primitives, an initial-value solution and a definite numerical accumulation.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsRecover a gcd and the combination that produces it
Prove Euclidean reduction, termination and the Bézout identity.
Open lessonLesson · Science · University Physics II with calculusRecover a vector field and verify path-independent work
Electric potential is a scalar whose spatial variation determines a vector field. A partial derivative changes one coordinate while holding the others fixed. For V=a(x²−y²)+bxy+7 V in the stated SI convention, Ex=−2ax−by and Ey=2ay−bx. The negative sign means a positive test charge accelerates toward decreasing potential if electric force is the only force.
Open lessonLesson · Domains, invariants and structural countingRecover accumulated rewards from endpoint potentials
Prove a local reward identity, telescope it between arbitrary configurations and separate stopping choices from terminal states.
Open lessonLesson · Calculus through supplied graphs and measured tablesRecover accumulation from a signed graph
Use exact geometric areas, initial conditions and FTC to connect an integrand graph to its accumulation function.
Open lessonLesson · Science · Introductory AstronomyRecover an elliptical position from elapsed time
Kepler’s equal-area law describes time, not uniform angular motion. An isolated two-body relative orbit is an ellipse with the attracting focus at one focus. The separation changes around an eccentric orbit, and the body moves faster near periapsis. A drawing with equally spaced angles therefore does not normally represent equally spaced observation times.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingRecover an instantaneous rate from finite secants
Derive the limit and its finite-spacing error before treating a graph slope as instantaneous velocity.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeRecover an integrable function from local averages
Prove a one-dimensional weak maximal estimate.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingRecover center motion from an internal-force audit
Compare individual trajectories with the externally driven center of a fixed material system.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsRecover coefficients from symmetric root data
Construct elementary-symmetric expressions and derive Newton identities without hidden division.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingRecover force and classify equilibrium from an energy landscape
Differentiate a potential and test nearby response rather than treating every zero-force point as stable.
Open lessonLesson · Partial differential equations and Fourier methodsRecover functions and propagate frequency energy
Use inversion and norm identities with precise classes.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingRecover harmonic timing from an energy integral
Use energy to obtain speed branches and an integrable turning-point time.
Open lessonLesson · Differential equations: models, methods and dynamicsRecover independent solutions when roots repeat or leave the real line
Use polynomial factors and sine/cosine pairs with real initial data.
Open lessonLesson · Partial differential equations and Fourier methodsRecover interior values from surrounding averages
Derive mean-value and rigidity mechanisms.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceRecover interval recombination without losing double crossovers
Genes on one chromosome can travel together more often than independently assorting markers. Homologous recombination exchanges corresponding segments between nonsister chromatids during meiosis. A testcross can reveal the gamete combinations made by a heterozygote when the partner’s recessive alleles make marker states distinguishable and scoring and viability are comparable.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceRecover invariant mass and bound a low-speed approximation
In c=1 units a massive particle has energy E=γm and momentum p=γmv. Energy is measured in reference mass times c² and momentum in reference mass times c; the numerical c=1 convention does not erase their physical dimensions.
Open lessonLesson · Science · University Physics I with calculusRecover oscillator phase from both initial conditions
Newton’s law for an ideal spring gives m xddot = −kx. For m = 2 kg and k = 8 N/m, the equation is xddot + 4x = 0. Its two independent solutions are cos(2t) and sin(2t), so one initial position alone cannot determine the subsequent motion.
Open lessonLesson · Calculus synthesis: regions, motion and approximationRecover planar position from velocity
Integrate components with an initial point and separate displacement, speed and traveled length.
Open lessonLesson · Precalculus: periodic and reciprocal graph evidenceRecover sinusoidal shape and changing rates
Construct a complete sinusoid and separate output level, motion direction and concavity.
Open lessonLesson · Sections, solid measures and geometric designRecover sphere volume from a comparison solid
Match every sphere section with a cylinder minus two cones.
Open lessonExperiment · Cybersecurity and digital literacyRecover the club register
Does faster restoration also recover more lost records?
Open experimentLesson · Functions in graphs, measurements and growth decisionsRecover transformations from supplied graphs
Infer translation, reflection and scale from plotted landmarks.
Open lessonConcept · GlossaryRecurrence
Reusing an update operation over successive internal states. Repetition can be programmed or learned and does not guarantee progress.
Open conceptLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsRedistribute charge by equal potential
Consider two widely separated conducting spheres joined by an ideal connector.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsReduce a network by nodes rather than its drawing
Series elements share a current because no junction offers an alternative path between them.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsReduce integer matrices without losing their lattice
Prove termination and divisibility in Smith reduction while tracking exact transformations.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsReduce polynomial presentations with degree descent
Extend Smith reduction to field polynomials and prove monic invariant-factor uniqueness.
Open lessonLesson · Weighted integration and jump conventionsReduce smooth weights and prove integration by parts
Use a derivative density only when justified, define signed smooth integrators, and derive boundary terms from sums.
Open lessonLesson · Course 2 · Planning and reasoningRefine an idea over several passes
Watch distance information spread through a grid and discover why stable internal states can still be wrong.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceRefine where evidence says the integral is difficult
Derive Richardson error estimates and explain adaptive allocation and failure detection.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceReflect a direction about the oriented normal
The law of reflection compares angles with a surface normal, not with the surface itself. A normal is perpendicular to the local reflecting surface. In the displayed plane, the incoming direction is u=(sinβ,−cosβ), while the outward normal is n=(sinα,cosα). The incoming arrow points toward the surface, so its outward-facing angle is measured using −u. A ray diagram that uses u directly as the outward incident arm can accidentally replace an angle by its supplement.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryReflections, rotations and glide reflections
Build every plane isometry from elementary reflections.
Open lessonLesson · Mathematics investigationRefute a claim—or justify every case
What can a finite search establish about a universal statement?
Open lessonLesson · Social studiesRegional states and connected communities
Explain how trade, belief and institutions connected regions without erasing diversity.
Open lessonAssessment · Calculus synthesis: regions, motion and approximationRegions, trajectories and local models: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Differential geometry of curves and surfacesRegular curves and arc length
Separate the geometric trace from its parametrization.
Open lessonLesson · Point-set topology: spaces, constructions and separationRegularity, normality and shrinking neighborhoods
State point–closed-set and closed-set separation with an explicit T₁ convention.
Open lessonConcept · GlossaryRegularization
A method that discourages some solutions during learning, such as penalizing large weights. It can reduce overfitting but must be evaluated rather than assumed helpful.
Open conceptLesson · Science · General Chemistry IIReject a basic prediction for an extremely dilute strong acid
Complete dissociation does not eliminate water from the charge balance.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsReject a circular path that needs a pulling surface
At the top of a convex circular support, inward points downward toward the circle center. A 2 kg point has gravity downward and a surface normal upward. The term centripetal force names the inward resultant needed for circular motion; it is not a third interaction to add to gravity and normal.
Open lessonLesson · Science · University Physics II with calculusReject a coupling matrix that permits negative field energy
Signed mutual inductance depends on the reference orientations of two circuits. For a reciprocal linear pair, flux linkages are L1I1+MI2 and MI1+L2I2. The model fixes L1=1 H and L2=4 H, while a proposed M may be positive or negative. Changing the reference direction can change the sign of M without making self-inductance negative.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsReject a floating prediction when required displacement is impossible
A freely floating object in vertical equilibrium has buoyancy equal to weight: ρfluidgVsub=mg. Therefore required submerged volume is m/ρfluid. For a homogeneous solid, the submerged fraction equals its average density divided by fluid density when that ratio is below one. For a hollow sealed object, use total mass and external displaced volume, not the density of its wall material alone.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceReject a two-reservoir claim using entropy as well as energy
A cyclic engine absorbs 1000 J from a fixed hot reservoir at Th and rejects heat only to a 300 K reservoir. A claimed work fraction determines W and Qc=1000−W. This first-law relation is necessary but insufficient: a mathematically closed energy ledger can still require a forbidden entropy decrease.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsReject an energy-balanced engine that violates the second law
A cyclic heat engine returns its working substance to its initial state. Over a cycle ΔU=0, so absorbed heat Qh equals work output W plus rejected heat Qc, using positive magnitudes. Efficiency W/Qh describes conversion to work. An energy balance is necessary but does not by itself establish a physically possible engine.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsReject an impossible cord force before reading apparent weight
Fluid pressure increases with depth, so the resultant pressure force on a submerged body is upward. Archimedes’ principle gives its magnitude as the weight of the displaced fluid. For a completely submerged rigid object that displacement is the whole object volume; for a floating object only the immersed portion displaces liquid.
Open lessonLesson · Science · University Physics II with calculusReject an incompatible scalar-potential claim
A proposed vector field must pass a consistency check before it is called electrostatic. Consider candidate E=(a y,b x) on a simply connected planar region. The coefficients may be supplied by separate fits. Even if each component fits its own data, the pair need not be the negative gradient of one scalar potential.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsReject an oxygen-demand case without inventing a feasible steady state
Integrated oxygen supply links ventilation, diffusion, blood binding, regional mixing, circulation and tissue extraction. A change at one stage need not repair a limitation at another. Delivery is blood flow times arterial oxygen content, while uptake is the portion removed by tissues. Delivery and actual metabolic use are not the same quantity.
Open lessonLesson · Mathematics for AI · M07Relate a latent approximation to its evidence
Enumerate KL and ELBO exactly, then inspect numerical integration separately.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceRelate animal body plans to exchange and internal transport
Animal diversity is not a ladder from inferior to superior organisms. Sponges organize water flow without the same true-tissue arrangement as cnidarians or bilaterians; cnidarians commonly have radial organization, whereas many bilaterians coordinate directional movement and sensory structures toward an anterior end. Adult echinoderm radial form also warns against inferring ancestry from adult symmetry alone.
Open lessonLesson · Science · Physics and physical systemsRelate capacitor geometry to charge and voltage
Change plate area and gap while distinguishing capacitance from stored charge.
Open lessonLesson · Science · Physics and physical systemsRelate ideal-gas pressure, volume and temperature
Compare equilibrium states while keeping amount, units and controlled variables explicit.
Open lessonLesson · Science · Physics and physical systemsRelate liquid pressure to depth and density
Predict pressure increments from the weight of a stationary liquid column.
Open lessonLesson · Science · Physics and physical systemsRelate mass distribution to angular acceleration
Move equal masses farther from a fixed axis and predict their response to the same torque.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsRelate orbit shape, conserved quantities and escape energy
Consider a satellite much less massive than a central body, so central-body recoil is negligible. With gravity as the only interaction, central force exerts zero torque about the central point and the satellite’s angular momentum about it is constant. The satellite–central-body system also conserves total mechanical energy. The gravitational potential reference is zero at infinite separation, giving U=−GMm/r.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceRelate plant tissue roles to a growing stem cross-section
Plant organs integrate dermal, ground and vascular tissues. Dermal tissues provide interfaces and protection; ground tissues include photosynthetic, storage and supporting cells; vascular tissues transport water/minerals and organic solutes. Roots, stems and leaves arrange these tissues differently rather than functioning as interchangeable cylinders.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceRelate replication direction to parallel-fork completion time
DNA polymerases extend a new strand by adding to its 3′ end, so synthesis proceeds 5′→3′ while the template is read in the opposite direction. A primer supplies an extendable end. Helicase separates strands, topoisomerases relieve torsional strain, and ligase joins appropriate backbone discontinuities after processing. These tasks are distinct; calling every enzyme a copying enzyme hides the mechanism.
Open lessonLesson · Differential calculus: techniques and applicationsRelate simultaneous rates before substituting
Build a geometric relationship, differentiate in time, then evaluate an instant.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceRelate variation, changing environments and extinction
Genetic diversity supplies variants on which selection may act, but diversity does not guarantee persistence.
Open lessonLesson · Science · Physics and physical systemsRelate wavelength, frequency and wave speed
Hold propagation speed fixed and test how wavelength changes the timing of local oscillation.
Open lessonCourse · Mathematics program unitRelation closure and directed reachability
Prove which operations preserve relation axioms, repair failed closure claims, and derive short directed paths from an extremal argument.
Open courseLesson · Social studiesRelative change needs an absolute baseline
A percentage change compares a difference with its starting value.
Open lessonConcept · GlossaryRelease gate
A declared requirement a candidate version must satisfy before promotion, such as a critical accuracy floor or resource ceiling.
Open conceptLesson · Social studiesReligion: measure belief, participation and institutional authority separately
Sociology studies religious beliefs, practices, organizations and social consequences without deciding theological truth. Secularization claims need a specified dimension: affiliation, participation or institutional authority can change differently.
Open lessonLesson · Social studiesReligious revival: organization, participation and autonomy
Read a camp-meeting plan, compare religious claims about agency, and trace institutional responses without equating attendance with equality.
Open lessonLesson · Course 2 · Memory and shared working stateRemember selectively and retrieve evidence
Compare similarity with exact and evidence-aware retrieval, age records, and consolidate a carefully scoped procedure.
Open lessonLesson · Course 2 · Planning and reasoningRemember what the current view hides
Train recurrent state and compare information available during an occlusion.
Open lessonLesson · Social studiesRemembering involves encoding and retrieval
Memory depends on attention, encoding, storage and retrieval.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsRemovable singularities, poles and essential singularities
Classify isolated singularities by their Laurent principal parts.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsRemove old directions before adding a new one
Construct an orthogonal basis, detect dependence and connect the result to QR.
Open lessonLesson · Social studiesRenaissance, print and Reformation
Explain how media, patronage and religious conflict changed authority.
Open lessonLesson · Course 1 · Data audits and training diagnosisRepair a broken training run
Inspect predictions, gradients and actual parameter changes. Compare a failure and repair from the same checkpoint.
Open lessonExperiment · Data literacy and statisticsRepair a messy event log
Which rows are observations and which are recording errors?
Open experimentLesson · Science · AP Chemistry preparationRepair a solution diagram using atoms, charge and concentration
Build the missing species count and distinguish formula concentration from dissolved-particle concentration.
Open lessonLesson · Science · AP Chemistry preparationRepair an enzyme inventory before claiming a steady rate
Propose free and bound enzyme, balance opposing fluxes and distinguish stationary turnover from chemical equilibrium.
Open lessonProject · Operate a system, not an isolated metricRepair an operating system under constraints
Model a service desk, workshop or small supply chain. Choose a bottleneck or shortage, compare two interventions, and stress-test the recommendation against timing, delayed delivery or shared failure. Use evidence from at least two experiment mechanisms; improved local utilization alone is not sufficient.
Open projectExperiment · Communication and evidence-based writingRepair an unclear instruction
Which revision removes consequential ambiguity?
Open experimentLesson · Science · Scientific inquiry and measurementRepair an unfair comparison
Separate treatment effects from group composition.
Open lessonLesson · Science · Chemistry and matterRepair covalent structures with two electron checks
Distinguish the molecule-wide electron budget from local duet and octet bookkeeping.
Open lessonLesson · Course 1 · Data audits and training diagnosisRepair data without leaking answers
Audit real flower measurements, separate related rows, and turn a tempting score into a defensible claim.
Open lessonExperiment · Communication and evidence-based writingRepair the survey headline
Which people does a percentage actually describe?
Open experimentAssessment · Binary gcd and integer certificatesRepair the trace and certify the result
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Abstract algebra I: groups and symmetryReplace a map by a quotient and an embedding
Prove the first theorem and use the subgroup/quotient correspondences.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsReplace a varying function by an equal-total constant
Use interval normalization and distinguish average value from endpoint averages.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsReplace field scalars with ring scalars
Build modules, quotients and operator actions with explicit scalar-ring hypotheses.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsReplace many field generators with one
Prove the finite-separable primitive-element theorem and test its assumptions.
Open lessonConcept · GlossaryReplay
Reusing old examples during new training so earlier behavior still influences parameter updates.
Open conceptLesson · Functions in graphs, measurements and growth decisionsReport what measurements support
Convert instrument resolution into a defensible measurement interval.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsRepresent a point with radius and angle
Convert coordinates while preserving quadrants and nonunique representations.
Open lessonLesson · Differential equations: models, methods and dynamicsRepresent inputs that switch or strike
Use delayed functions with correct arguments and distinguish ideal impulses from ordinary forces.
Open lessonLesson · Course 2 · Geometry and control foundationsRepresent motion in three dimensions
Compare fixed and moving axes, matrix and quaternion descriptions, and the limits of a projected 3D picture.
Open lessonLesson · Orders, graph representations and certificatesRepresent orders with lower sets and cover diagrams
Prove faithful order representations, build product orders and locate the finite boundary of cover reconstruction.
Open lessonLesson · Mathematical modeling and senior investigationRepresent random transitions and distinguish means from paths
Build a finite Markov model with inspectable probabilities.
Open lessonAssessment · Functions in graphs, measurements and growth decisionsRepresentations and decisions: mixed evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Orders, graph representations and certificatesRepresentations that preserve exactly the right information
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Constructing the real numbersRepresentatives, diagonals and a new irrational cut
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesReproduce a population-growth claim from real data
Choose an explicit comparison set, distinguish count and rate changes, and export an analysis another learner can reproduce.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceReproduce an analysis without overwriting its source records
A reproducible environmental analysis needs identifiable records, units, provenance and a declared transformation recipe. An edited spreadsheet without the original values cannot show what changed. Keeping raw data separate from corrected values allows another analyst to recover the result and challenge its assumptions.
Open lessonProject · AI cumulative project; human review pendingReproducible research study
Saved comparisons, replayed evidence, bounded reports and separate objective transfer assessment.
Open projectStudio · AI implementation practice; independent review pendingReproducible research study implementation studio
Run a local reference, implement a separate learner exercise and preserve reproducible evidence.
Open studioLesson · Science · AP Chemistry preparationRequire a mechanism before adding favorable reactions
Balance a coupled reaction and distinguish thermodynamic sufficiency from mechanistic connection.
Open lessonLesson · Science · University Physics II with calculusRequire both curl equations for a vacuum wave
Perpendicular sinusoidal arrows do not automatically form a valid electromagnetic wave. Test the candidate Ey=E0 cos(kx−omega t) and Bz=B0 cos(kx−omega t), with E0=2 V/m and k=π per metre. Controls propose phase speed a times c and signed magnetic amplitude bE0/c. These proposals include deliberately inconsistent choices.
Open lessonLesson · Science · Physics and physical systemsRequire both force and torque balance
Test whether a supported beam can remain at rest rather than checking force alone.
Open lessonLesson · Science · Introductory AstronomyRequire nodal alignment before an eclipse claim
New alignment places the Moon near the Sun’s direction, while full alignment places it near the opposite direction. Neither alone guarantees an eclipse because the lunar orbital plane is inclined to the ecliptic. Near the nodes, the two planes intersect and suitable alignment can allow disk or shadow overlap.
Open lessonResearch · Optional researchResearch notebook
Twelve-week plan, exact baselines, frozen protocols, and reports.
Open researchCourse · English language artsResearch, editing and publishing
Investigate a bounded question, evaluate and synthesize sources, collaborate through accountable edits, and produce a research feature with a source dossier and correction plan.
Open courseLesson · Science · University Physics III — thermal, optical and modern evidenceReserve fresh temperatures after equal-budget heat-capacity calibration
For a quantum oscillator with levels En=(n+1/2)hbar omega, the geometric partition sum gives mean energy U=hbar omega[1/2+1/(exp(Theta/T)−1)], where Theta=hbar omega/kB. Differentiate at fixed oscillator frequency to obtain C/kB=F(Theta/T), with F(x)=x²exp(x)/(exp(x)−1)². The temperature-independent zero-point energy contributes no heat capacity.
Open lessonConcept · GlossaryResidual connection
An addition that combines a block’s input with a transformed result, preserving a direct path for information and gradients.
Open conceptLesson · Complex analysis: holomorphic functions, contours and mappingsResidues and contour integrals
Extract the coefficient that survives closed integration.
Open lessonLesson · Social studiesResistance, abolition and free Black institutions
Compare strategies for freedom and explain how institutions sustained collective action.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceResolve a field before assigning a probe force
A field represents the source configuration at an observation point. For each point source, draw the displacement from source to observation point and use E=kQ r_vector/r³. Its magnitude is k|Q|/r², but signed components are essential when fields are not collinear. Add components from different sources before calculating the magnitude of the resultant; adding magnitudes discards cancellation.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingResolve circular acceleration and test force constraints
Differentiate angular motion and combine perpendicular acceleration components before judging feasibility.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceResolve curvature without inventing a neutral gyroperiod
A charged particle moving perpendicular to a uniform stationary magnetic field can change direction without changing speed. For mass m, charge q and field B along +z, define signed rate w=qB/m. Starting at the origin with velocity +1 m/s along x, the supplied path is x=sin(wt)/w and y=[cos(wt)−1]/w for nonzero w. Its initial curvature is toward −y for positive charge and toward +y for negative charge.
Open lessonLesson · Science · Physical Geology and Earth SystemsResolve effective normal force before interpreting slope resistance
A slope interpretation combines material strength, discontinuity orientation, water conditions and loading. Gravity has a component parallel to a potential sliding plane and a component normal to it. Pressure within connected water-filled fractures or pores can reduce the contact force available for friction, even if the assumed block weight is held fixed.
Open lessonLesson · Science · Physical Geology and Earth SystemsResolve fault displacement before interpreting map offset
A fault is a fracture or zone across which relative displacement has occurred. For an inclined plane, the hanging wall lies above the plane and the footwall below it. Normal and reverse components refer to downward and upward hanging-wall movement relative to the footwall, not simply to which present ground surface is higher.
Open lessonLesson · Science · Physics and physical systemsResolve forces on an incline
Choose axes along a slope and connect weight components to normal force and motion.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsResolve gas diffusion across connected barriers
Gas exchange connects alveolar gas, a thin epithelial barrier, intervening material and capillary endothelium to blood. A partial-pressure difference drives diffusion of a gas; total gas pressure and a particular gas’s partial pressure must not be confused. The oxygen and carbon-dioxide gradients can point in opposite directions even though both gases cross the same anatomical interface.
Open lessonLesson · Science · Physical Geology and Earth SystemsResolve hydraulic head loss across layered geology
Groundwater flow depends on hydraulic head and conductivity, not simply on which rock contains more pore space. Head combines elevation and pressure contributions. Porosity describes the amount of pore volume; permeability and hydraulic conductivity describe connected transport, with conductivity also depending on the fluid.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingResolve independent components and an event boundary
Use a shared event time while permitting different acceleration histories in perpendicular directions.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsResolve interface orientation before combining dielectric regions
This supplied piecewise-continuum extension follows the fully filled dielectric core case.
Open lessonLesson · Science · Physical Geology and Earth SystemsResolve plate motion in a common reference frame
A plate boundary is a relationship between moving bodies, not a speed attached to one plate. In a local east–north frame, subtract velocity of plate A from plate B. Then project that relative vector onto a boundary normal directed from A toward B and a perpendicular tangent. Positive normal motion increases separation; negative normal motion closes it.
Open lessonLesson · Science · Physical Geology and Earth SystemsResolve wave depth, propagation and refraction
A wave crest travels at phase speed, while the envelope and energy of a small wave group travel at group speed. Neither is automatically the speed at which all water particles move toward shore. Water motion in linear surface waves is largely oscillatory, with geometry depending on depth and wavelength.
Open lessonLesson · Science · AP Chemistry preparationRespect finite solid inventory when applying a solubility product
Repair signed dissolution or precipitation proposals using conserved ions, phase presence and a conditional solubility-product equality.
Open lessonLesson · Science · Introductory MicrobiologyRespect time order before inferring a transmission path
Contact information can constrain possible transmission routes, but contact is not proof that a transmission occurred. Host susceptibility, barriers, timing, source viability and the mode of spread all matter. A shared environmental source or missing contact can generate patterns that resemble direct spread between the recorded nodes.
Open lessonLesson · Course 1 · Why does AI need computation?Respond to an incident
Trace a release regression, contain its impact and verify recovery without rewriting the incident history.
Open lessonLesson · Generalized linear models: fitting and evidenceResponse families and link scales
Response families and link scales. Derive, calculate and qualify a fitted response model.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityRestrict the reference set before multiplying
Use conditional probabilities and independence with positive denominators.
Open lessonLesson · Science · Introductory MicrobiologyRetain a censored endpoint as a bound rather than zero
When an endpoint is below a detection or reporting limit, the observation is censored. It supplies an inequality rather than an exact count. “Not detected” and “proved absent” are different claims, particularly when recovery and sampling cover only part of a system.
Open lessonLesson · Science · Introductory AstronomyRetain electron inventory in a white-dwarf scaling
A white dwarf can be supported mainly by electron degeneracy pressure after sustained central fusion ends. The Pauli exclusion principle limits the occupation of quantum states; compression raises occupied momenta and pressure. This support is not simply ordinary thermal gas pressure, although actual white dwarfs are not literally at zero temperature.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsRetain every term when a coil and its field change
A coil has N turns only if the winding geometry actually supplies that many linked surfaces.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsRetain heat storage when a requested cooling response is infeasible
Thermal regulation links sensory information, hypothalamic integration, metabolic heat production, skin blood flow and sweat evaporation. Dry heat exchange includes conduction, convection and radiation, while evaporation removes latent heat when water actually changes phase. Sweat production without effective evaporation does not remove the same amount of heat; ambient conditions and airflow matter.
Open lessonLesson · Science · Introductory AstronomyRetain horizon-grazing cases instead of forcing a visibility label
A fixed celestial direction traces a daily circle around a celestial pole as Earth rotates. Depending on latitude and declination, that circle can cross the horizon, stay entirely above it or stay entirely below it. Circumpolar describes the observer-dependent geometry, not a special physical type of star.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceRetain information in nondetects without fabricating values
A reported nondetect does not establish that a substance is absent. It means the measurement did not quantify a value above a specified reporting limit under the method used. The limit and detection/quantification convention must be retained with the observation rather than replacing every censored result with an exact invented number.
Open lessonLesson · Science · University Physics II with calculusRetain initial current and the inductive energy cross term
An ideal linear inductor stores a current state rather than a capacitor voltage state. With a maintained 4 V source and resistance 2 Ω, the equation is L di/dt+2i=4. The independently prepared initial current i0 may oppose the source; no physical preparation or energized-switch procedure is requested.
Open lessonLesson · Science · Introductory MicrobiologyRetain maintenance and an independent nutrient cap before interpreting yield
Substrate uptake does not all become new biomass. Cells can expend resources maintaining ion gradients, repairing components and supporting turnover, while growth also requires additional elements and energy. A measured biomass-to-substrate ratio is therefore conditional on the observation interval and resource boundaries, not a universal organism constant.
Open lessonLesson · Science · Introductory AstronomyRetain spectral convention before inferring a temperature
An ideal blackbody continuum is a useful physical reference, not a universal description of every astronomical source. Its temperature controls both shape and emitted power. The Planck function Bλ=2hc²/[λ⁵(exp(hc/(λkT))−1)] describes spectral radiance per unit wavelength and solid angle. The subscript matters: Bν per frequency is a different density representation of the same radiation.
Open lessonLesson · Social studiesRetrieval practice: immediate fluency and later retention
Compare study conditions across delays and distinguish immediate performance from durable retrieval.
Open lessonConcept · GlossaryReturn
Accumulated rewards over time, often discounted so that more distant rewards contribute less.
Open conceptLesson · Science · Physics and physical systemsReturn a gas to its starting state and obtain net work
Close all four legs of a cycle and interpret its enclosed pressure-volume area.
Open lessonLesson · Course 1 · What has the network learned?Reuse a representation
Compare a new network with frozen and partly adapted features under matched target-label budgets.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsReverse a chain rule without losing the bounds
Transform the integrand, differential and endpoints consistently.
Open lessonLesson · Algebra II and precalculus: functions and structureReverse a function on a suitable domain
Recover inputs, restrict many-to-one rules and distinguish inverses from reciprocals.
Open lessonExperiment · Applied mathematicsReverse a probability question
Does being late mean it probably rained?
Open experimentLesson · Proof foundations: from claims to argumentsReverse and negate with care
Prove an implication through its contrapositive while keeping the converse separate.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsReverse the product rule strategically
Choose parts that simplify a problem and handle endpoint terms explicitly.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsReverse the relation between subgroups and intermediate fields
State finite Galois hypotheses and identify proof dependencies.
Open lessonLesson · Social studiesRevise a conclusion without hiding the change
Responsible communication distinguishes what is established, uncertain and corrected.
Open lessonLesson · Science · Engineering design and scientific computingRevise a controller while retaining its failure
Compare a declared redesign on public changed conditions without claiming a hidden test.
Open lessonLesson · Science · AP Environmental ScienceRevise a cross-system recommendation without losing its history
Retain an original plan, mixed evidence and a reviewer-driven revision with a changed boundary.
Open lessonLesson · Science · Engineering design and scientific computingRevise reasoning without erasing the first attempt
Apply seven review criteria to a bounded design argument and plan unfamiliar transfer evidence.
Open lessonLesson · College compositionRevise the argument before polishing the sentences
Use a reverse outline and reader feedback to change a draft’s reasoning, structure or scope, then document the consequences.
Open lessonLesson · Stories and creative writingRevise the story’s decisions, not just its adjectives
Use workshop evidence to revise a story’s causal, viewpoint or structural choices while preserving an intentional voice.
Open lessonLesson · Social studiesRevolution, rights and constitutional choices
Distinguish revolutionary claims from institutions and exclusions that followed.
Open lessonLesson · Social studiesRevolutions, industrialization and imperialism
Compare political and industrial revolutions and their unequal consequences.
Open lessonConcept · GlossaryReward
A numerical signal supplied after an action. It specifies what learning is encouraged to increase, which may differ from the designer’s intended task.
Open conceptConcept · GlossaryReward hacking
Obtaining a high reward by exploiting the reward rule while failing the intended task.
Open conceptStudio · Optional Project 10 practice; standard-simulator P09 and human review remain pendingReward-agent implementation studio
Implement exact transitions and neural gradients, run a Python Q-learning reference and resume browser-compatible checkpoints.
Open studioLesson · Real numbers, complex coordinates, vectors and matricesRewrite radicals without losing the domain
Rewrite products, quotients and nested rational powers on explicitly stated domains.
Open lessonCourse · English language artsRhetoric, media and public voice
Create an accountable public argument in two media, test its assumptions and revise for a skeptical audience.
Open courseLesson · Social studiesRights claims and equal access
Civil liberties constrain government interference; civil rights concern protection against unequal treatment.
Open lessonAssessment · Abstract algebra II: rings, fields and Galois foundationsRings and fields: quotient, factorization and extension arguments
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Social studiesRitual and heritage: preserve transmission without freezing change
A ritual includes patterned action and participants' meanings. Repetition does not imply that a practice never changes. Recording a performance, sustaining its teaching and protecting participants' control are different interventions.
Open lessonProject · AI cumulative project; human review pendingRobot-learning project
Saved comparisons, replayed evidence, bounded reports and separate objective transfer assessment.
Open projectStudio · AI implementation practice; independent review pendingRobot-learning project implementation studio
Run a local reference, implement a separate learner exercise and preserve reproducible evidence.
Open studioConcept · GlossaryRollback
Restoring a recorded previous version after a change. It recovers an earlier state but does not prove that state meets every current requirement.
Open conceptLesson · Differential calculus: techniques and applicationsRolle and the mean value theorem
Turn endpoint information into a justified intermediate-rate guarantee.
Open lessonLesson · Calculus II: series, curves and applicationsRotate a curve and measure its skin
Use circumference times slant length rather than volume slices.
Open lessonLesson · Science · AP Chemistry preparationRotate a dipole and test the competing charge interactions
Construct an orientation comparison, sum four electrostatic contacts and examine the point-dipole approximation.
Open lessonLesson · Science · General Chemistry IRotate a domain model without changing its molecular shape
Separate the arrangement of all electron domains from the arrangement of bonded nuclei.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsRouché’s theorem and stable root counts
Prove root-count invariance using a boundary homotopy.
Open lessonLesson · Algorithm reasoning: invariants, termination and costRounded recurrences and the operations they count
Derive a divide-and-conquer cost from a real partition, prove all-integer bounds and contrast balanced splitting with a quadratic alternative.
Open lessonLesson · Differential geometry of curves and surfacesRuled surfaces and developability
Separate straight generators from intrinsic flatness.
Open lessonLesson · Social studiesRules and informal networks shape access
Formal organizations use roles and procedures; informal networks transmit information and assistance.
Open lessonLesson · Course 2 · Testing whether improvement is realRun a frozen final study
Freeze a recipe and decision rule, expose final rows explicitly and retain every seed/task result.
Open lessonLesson · Social studiesRussia: executive appointment and a rejection contingency
A formal approval requirement must be read together with its contingency rules. Record whether an institution may act or must act; those words generate different decision trees.
Open lessonLesson · Social studiesRwanda 1994: a mandate without sufficient capacity
Collective security depends on decisions by multiple actors as well as formal authority. Separate warnings, interpretation, political decisions, resources, command and action; failure at one link can constrain the others.
Open lessonExperiment · Scientific reasoning and experimental designSame people, different order
Does comparing each participant with themselves remove every bias?
Open experimentLesson · Mathematics for AI · M06Sample a signal without hiding aliasing
Inspect waveform samples, frequency bins, windows and lost high-frequency information.
Open lessonLesson · Mathematics investigationSatisfy two rules at once
When do two linear requirements agree?
Open lessonLesson · Multivariable calculus: geometry, change and accumulationScalar line integrals and wire mass
Accumulate density along a curve using arc length rather than a coordinate differential.
Open lessonLesson · Mathematics investigationScale a fraction without losing the whole
What stays the same when a reference batch changes?
Open lessonLesson · Euclidean and coordinate geometry foundationsScale lengths, areas and arguments
Use equal angles and proportional sides to distinguish similarity from congruence.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesScale, reverse, and handle the zero vector
Construct positive, negative and zero scalar multiples with exact endpoint descriptions.
Open lessonConcept · GlossaryScaling law
An empirical relationship between resources such as data or model size and measured behavior such as loss. Its reliability is limited to the evidence and conditions supporting it.
Open conceptLesson · Mathematics investigationScaling length does not scale everything equally
What happens to area and volume when every length doubles?
Open lessonLesson · Course 1 · How does a network learn?Schedule the learning rate
Compare warmup, step decay and cosine decay, then choose a checkpoint using declared development checks and stopping rules.
Open lessonLesson · Social studiesSchenck, Tinker and changing speech doctrine
Speech decisions depend on facts and legal standards.
Open lessonCourse · Mathematics program unitSchool trigonometry: proofs, inverse branches and periodic decisions
Four lessons close specific school-level reasoning gaps through unit-circle constructions, derived angle formulas, justified inverse domains and calculator-based contextual threshold intervals. Exact checks and human-reviewed reasoning are distinct. This focused extension reuses the existing trigonometry foundations; it does not establish complete school, AP or college-course readiness.
Open courseAssessment · Positive matrices, factors and energySchur classification, factor reuse and singular energy
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · ScienceScientific inquiry and measurement
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseSubject · Experiment collectionScientific reasoning and experimental design
Separate an intervention from a misleading comparison, and precision from instrument bias.
Open subjectLesson · Research, editing and publishingScope a question you can actually investigate
Turn a broad public-interest topic into a feasible question with a source plan and clear limits.
Open lessonLesson · Generalized linear models: fitting and evidenceScore, information and IRLS
Score, information and IRLS. Derive, calculate and qualify a fitted response model.
Open lessonConcept · GlossarySearch
Exploring explicit alternatives, such as action sequences, while tracking which possibilities remain to be checked.
Open conceptExperiment · Scientific reasoning and experimental designSearch enough outcomes and find a surprise
How does an outcome search change the chance of a false alarm?
Open experimentLesson · Optimization: models, certificates and algorithmsSearch integer decisions with valid pruning
Track incumbent value s, node bounds and exhaustive coverage.
Open lessonCourse · Mathematics program unitSections, solid measures and geometric design
School-entry solid geometry develops section and rotation reasoning, justified polygon and slab limits, Cavalieri comparisons, contextual volume calculations and density-based design. Exact computations and written arguments supply reviewable evidence; this unit does not claim complete geometry mastery.
Open courseProject · Protect, verify and recoverSecure a fictional club service
Deliver an access and response plan for a club inventory service. Use captured experiments to justify permissions, message verification, input boundaries, recovery objectives and maintenance priorities. Include a tested sandbox contract or equivalent decision table; never use real credentials.
Open projectExperiment · Cybersecurity and digital literacySecure a tiny inventory handler
Does parsing a numeric prefix enforce a whole-input contract?
Open experimentConcept · GlossarySelf-play
Generating training experience by having an agent or copies of its policy play a game. Reliable outcomes and independent evaluation still matter.
Open conceptConcept · GlossarySemantic memory
Retained facts, concepts, and relationships, with applicable scope and evidence when represented as explicit records.
Open conceptAssessment · Precalculus: semilog scales and model evidenceSemilog modeling: mixed practice
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Course 1 · How does a network learn?Send the error backward
Trace a mistake through two weights, calculate both gradients, and then update them together.
Open lessonProject · Mathematical modeling and senior investigationSenior investigation: distinguish mechanisms behind the same observations
Develop, test and defend a bounded mathematical investigation over several work sessions. The default project asks which measurements distinguish two loss channels when total stock follows the same decay law. All supplied data are synthetic and all guidance is public. You may instead pursue the supported theorem project or propose a comparably scoped question with a reviewer. Saved milestones record learner-authored evidence; they do not authenticate data, assess proof quality, or establish an independently reviewed capstone. A final manuscript, reproducible artifacts and a human defense/revision review are separate evidence requirements.
Open projectLesson · Science · Human Anatomy and Physiology FoundationsSeparate a barrier from downstream immune removal
Immune defense begins at anatomical boundaries rather than only after an organism enters tissue. Keratinized skin, epithelial junctions, mucus, ciliary movement and chemical secretions contribute different barriers. Resident microbial communities can also affect colonization. Barrier defense and tissue responses interact, but their effects should not be collapsed into one undifferentiated percentage.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate a biological mechanism from its readout
A biological explanation connects an intervention to a process and then to an observable quantity. Those are separate links. A lower detector reading might mean less product, an altered detector response, or both. Before calculating a treatment effect, draw the chain intervention → biological state → measurement and identify a competing explanation for each arrow.
Open lessonLesson · Social studiesSeparate a celestial observation from a complete world system
What can an instrument change about an argument?
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate a changed input from a delayed environmental stock
Environmental stocks retain material from earlier inputs. A reduction in present input can therefore coexist with continued loss or accumulation while the system approaches a new balance. Comparing only this year’s input with this year’s stock can confuse a transient response with the absence of an intervention effect.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate a charged cavity from conductor material
A conductor in electrostatic equilibrium has no electric field inside its material: a sustained field would continue driving mobile charge. This statement does not say that every empty cavity inside a conductor is field-free. A charge placed in a cavity can produce a field there while charges on the cavity wall rearrange so that the surrounding conducting material remains at constant potential.
Open lessonLesson · Science · Introductory MicrobiologySeparate a community from its observation process
Different cells or molecules need not contribute equally to an observed community dataset. Recovery, lysis susceptibility, amplification, marker-copy differences and classification can create class-specific representation. A read fraction is therefore not automatically a cell fraction. This lesson interprets already supplied recovery factors and gives no extraction or amplification procedure.
Open lessonLesson · Science · Introductory AstronomySeparate a compact surface from a black-hole horizon
Massive stellar cores can collapse when their pressure support becomes inadequate. Nuclear interactions and degenerate matter can contribute to neutron-star support; sufficiently compact collapse can instead form a black hole. The outcome depends on core structure, mass loss, rotation, equation of state and explosion dynamics, not a single photospheric temperature.
Open lessonLesson · Social studiesSeparate a comparison from a causal claim
A causal question asks what would change under a different action, holding relevant conditions comparable.
Open lessonLesson · Mathematics for AI · M11Separate a comparison from a causal claim
Change cohort composition and diagnose an aggregate reversal.
Open lessonLesson · Science · Introductory AstronomySeparate a condensation threshold from a formation history
A collapsing star-forming cloud can produce a rotating disk through angular-momentum conservation. Solids collide and accumulate, larger bodies gravitationally influence nearby material, and sufficiently massive cores can acquire gas while disk gas remains. These processes depend on time, density and transport; the final planetary arrangement need not preserve every formation radius.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSeparate a conducting material from its cavity
Mobile carriers in an ideal conductor rearrange when a field acts on them.
Open lessonLesson · Science · Introductory AstronomySeparate a cosmic angular scale from ruler calibration
The cosmic microwave background records radiation released when the early plasma became transparent as neutral atoms formed. Its nearly thermal spectrum and small anisotropies constrain an earlier hot dense state. Acoustic patterns reflect coupled matter–radiation dynamics before that release, not sound currently traveling through empty space.
Open lessonLesson · Science · Introductory MicrobiologySeparate a crude association from matched comparison evidence
Host condition, exposure history and sampling can differ between comparison groups. A crude association can therefore mix within-stratum response differences with unequal group composition. Stratification and standardization make a comparison population explicit, but neither automatically removes every confounder or establishes a causal protective mechanism.
Open lessonLesson · Euclidean and coordinate geometry foundationsSeparate a drawing from a geometric argument
Use incidence, betweenness and definitions to justify geometric claims.
Open lessonLesson · Science · AP Chemistry preparationSeparate a favorable direction from an accessible rate
Explain kinetic persistence without changing equilibrium thermodynamics.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsSeparate a growth bound from a timing claim
Use quantified O, Ω and Θ definitions and count operations.
Open lessonLesson · Science · Introductory AstronomySeparate a local expansion estimate from velocity offsets
At sufficiently low redshift, a Hubble–Lemaître relation approximates recession velocity as H 0d. Galaxies also have peculiar motions relative to the large-scale expansion, and observer-frame corrections matter. A nearby object’s velocity divided by distance can therefore differ substantially from the underlying expansion slope.
Open lessonLesson · Science · AP Chemistry preparationSeparate a molecular solid, a layer and a connected framework
Remove selected contacts, trace retained paths and compare covalent fragments across separation directions.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsSeparate a path from the way it is traveled
Use paired coordinate functions, parameter domains and tracing direction.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsSeparate a pendulum length offset from a clock-scale error
A simple pendulum model has a point-like bob, a massless fixed-length support and small angular displacement. Its approximate period is T=2π√(L/g). Length is measured from pivot to the bob’s center, not merely the visible support segment. The approximation becomes less accurate at large amplitudes; amplitude independence is a model statement, not a universal fact about swinging systems.
Open lessonLesson · Science · University Physics II with calculusSeparate a physical time constant from a displayed one
The time coordinate itself can require calibration. Suppose displayed time s equals g times physical time t, with exact zero but unknown positive gain g. A zero-offset voltage 4e^(−t/τ) becomes 4e^[−s/(gτ)] on the displayed axis. The product gτ, not either factor separately, controls every point on that graph.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate a plate capacitance from a stray contribution
Capacitance is the proportionality C=Q/ΔV for the chosen linear device. An ideal capacitor has equal opposite charges on its plates, so the usual stored charge Q is a plate-charge magnitude, not the net charge of the entire two-plate object. For broad vacuum plates with negligible edge fields, C=ε0A/d. Convert area into square metres and separation into metres before using that equation.
Open lessonLesson · Science · Introductory AstronomySeparate a radial-velocity amplitude from an instantaneous sample
A spectroscopic binary produces changing line-of-sight velocities as its components orbit. A fitted velocity amplitude K is not the same as one instantaneous velocity sample. In an original circular orbit with systemic motion removed, the two radial signals are opposite sinusoids. Near a zero crossing, neither star has stopped moving; its instantaneous velocity is directed mostly across the line of sight.
Open lessonExperiment · Communication and evidence-based writingSeparate a result from its explanation
Can group composition explain a reported tool advantage?
Open experimentLesson · Social studiesSeparate a satellite observation from a land-use claim
Can a changed pixel tell you why a neighborhood changed?
Open lessonLesson · Science · Biology and living systemsSeparate a sequence change from a trait claim
Compare synonymous substitutions, amino-acid substitutions and gained stop signals.
Open lessonLesson · Social studiesSeparate a short-run tradeoff from changing expectations
Can policymakers hold unemployment below its assumed natural rate forever in this model?
Open lessonLesson · Science · Introductory AstronomySeparate a spectral shift from an additive calibration error
A line’s observed wavelength can differ from a reference because of motion, gravity, cosmic expansion or measurement calibration. Before assigning a physical cause, identify the transition, wavelength convention and observer reference frame. Air and vacuum wavelengths are not interchangeable reference lists; a shared convention is part of the evidence.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate a spectral threshold from an instrument intercept
A stopping-potential line is a measurement model, not an automatic work-function measurement. For above-threshold photons, Einstein energy balance gives eVs=hf−phi. Here frequency f is reported in units of 10^14 Hz and the supplied rounded slope is 0.4135667696 V per frequency unit. The instrument adds an unknown voltage offset B but has independently assumed unit gain. Its indication is y=m f−phi/e+B.
Open lessonLesson · Science · Introductory AstronomySeparate a star’s fuel inventory from its mass defect
Net hydrogen fusion produces helium with less rest mass than the initial reactants; the difference appears in released energy and escaping particles. The proton–proton chain and the CNO catalytic cycle provide different reaction routes, with different temperature dependence, to a similar net conversion. Fusion is not chemical combustion and does not consume every kilogram of a star at once.
Open lessonLesson · Mathematics · Functions and graphsSeparate a starting value from a rate
Change slope and intercept independently and calculate a rate from two points.
Open lessonLesson · Social studiesSeparate a subsidy wedge from an income-driven demand shift
Will a subsidy and stronger demand affect the seller in the same way?
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsSeparate a thermal spectrum’s peak, area and net exchange
A dense thermal emitter can produce a broad spectrum rather than isolated atomic lines. For an ideal blackbody, temperature determines the spectral shape. Wien’s displacement law λpeak T=b locates the maximum of intensity per unit wavelength. Stefan–Boltzmann emission P=AσT⁴ gives the total emitted power over all wavelengths for area A. A peak height or one narrow spectral band is not that total.
Open lessonLesson · Science · Introductory AstronomySeparate a transit radius signal from blended light
A transit reduces observed light when an opaque object crosses the projected stellar disk. For a uniform star and a fully overlapping small planet, the blocked fraction of the host’s light is k², where k is planet radius divided by stellar radius. The measured depth concerns all light in the observing aperture, not necessarily just the host.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSeparate a vector field from a signed test force
Add source vectors before applying the passive test charge.
Open lessonLesson · Linear Algebra II: operators and spectral structureSeparate a vector into measured and invisible parts
Prove orthogonal decomposition, best approximation and finite-dimensional Riesz representation.
Open lessonLesson · Science · Introductory MicrobiologySeparate a visible field composition from underlying abundance
A microscopy field is a sample filtered through detection and selection. Contrast, object size, focus, segmentation and the decision to choose one field rather than another can affect what is counted. Multiple fields from one preparation can be useful spatial subsamples but do not automatically provide independent biological replication.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsSeparate a wire’s field from the force it feels
A current-carrying segment in an external field experiences magnetic force of magnitude |I|LB sinθ, where θ is the angle between the segment’s conventional-current direction and B. The field in this formula is external to the segment; inserting a wire’s own field as though it pushed the entire wire sideways confuses source and response.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate absorptive capacity from available nutrient supply
Small-intestinal folds, villi and epithelial microvilli create a large exchange surface, but surface area alone does not determine absorption. Apical and basolateral transport proteins, local concentration, transit, digestion and perfusion jointly constrain movement. Some solutes use coupled active transport, others facilitated or passive processes; the epithelial cell is a polarized barrier rather than an empty hole.
Open lessonLesson · Science · Environmental science and sustainabilitySeparate accumulation from biological effects
Compare equal-total input histories and identify missing organism evidence.
Open lessonLesson · Science · Chemistry and matterSeparate acid strength, concentration and percent ionization
Compare a full weak-acid equilibrium calculation with a common square-root approximation.
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate active deposition rate from whole-section apparent rate
A rock column records preserved thickness, not every moment of elapsed time. Deposition can stop, material can be eroded and later compaction can alter thickness. Dividing total preserved thickness by elapsed time therefore answers a different question from the active rate while grains were accumulating.
Open lessonLesson · Science · Earth and space scienceSeparate age from evolutionary order
Compare fuel/power ratios without turning stages or EEP indices into years.
Open lessonLesson · Science · Biology and living systemsSeparate allele divergence from species formation
Track how exchange reduces a difference while examining the limits of a one-locus model.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate an angular baseline from wavelength dependence
A measured deviation angle may contain both refraction and an instrument zero error. For a thin prism at small angles, the supplied first-order relation is δ≈(n−1)A, where A is prism apex angle and n the refractive index relative to the surrounding air approximation. A common angular readout offset adds a constant. All angles must use a consistent unit in this first-order relation.
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate an antiform from an anticline using facing evidence
Fold names can encode different kinds of evidence. An antiform is arch-like in cross-section and a synform is trough-like. An anticline has older material toward its core and a syncline younger material toward its core. Shape and age classification agree in many upright examples, but cannot be treated as synonyms without facing evidence.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate an emission clock from photon arrival timing
A receding inertial source emits successive light pulses separated by proper period T0. The laboratory observer is at rest to the source’s left, so the returning photons travel opposite the source motion. This geometry is part of the problem; an approaching or transverse arrangement has a different arrival comparison.
Open lessonLesson · Science · Introductory AstronomySeparate an interior’s volume, mass and inertia
Bulk density alone does not reveal a unique planetary interior. Rock, metal, volatile material, temperature and pressure can produce overlapping mass–radius relations. Differentiation redistributes material when sufficiently mobile dense components sink and lighter components rise; a layered interior is an inference constrained by gravity, rotation, composition and thermal history, not a direct photograph of the core.
Open lessonLesson · Linear Algebra II: operators and spectral structureSeparate an operator into its primary pieces
Use Bézout identities to decompose a space and define generalized eigenspaces under splitting.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsSeparate apparent weight, mass and support contact
A scale measures contact force and may display that force divided by an assumed gravitational acceleration. It does not directly replace an object’s inertial mass when the support accelerates. Use upward as positive, with gravity10 m/s² downward and a selected object mass m.
Open lessonLesson · Science · AP Environmental ScienceSeparate atmospheric pressure from layered temperature
Explain atmospheric layers and gas roles while testing a bounded pressure model.
Open lessonLesson · Science · Introductory AstronomySeparate atmospheric thickness from an escape claim
Hydrostatic balance relates pressure decrease with height to the weight of overlying gas. For an isothermal ideal gas with constant gravity, dP/dz=−ρg and P=ρRT/μ combine to give P(z)/P(0)=exp(−z/H), where H=RT/(μg). Molar mass μ must be in kilograms per mole with the stated gas constant.
Open lessonLesson · Science · Introductory MicrobiologySeparate background from a nonlinear population proxy
Optical scattering or attenuation can provide a convenient population proxy, but it is not intrinsically a viable-cell count. Cell size, shape, aggregation, debris and instrument geometry can affect the relation between signal and abundance. A measured signal needs a documented background and a validated response range before it can be inverted into a biological quantity.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate behavioral mechanism, development and evolutionary function
A bird’s response to a seasonal cue can be explained at several levels. Sensory and endocrine pathways describe immediate mechanism; developmental experience describes how the response changes during life; reproductive consequences and evolutionary history address why a response may have spread. These explanations can complement one another rather than compete as mutually exclusive answers.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate binding energy, reaction energy and channel feasibility
An abstract reaction ledger has initial nuclei labeled (A,Z)=(12,6) and (4,2), with final labels (15,7) and (1,1). Both sides total A=16 and Z=8. These fictional labels are an inventory exercise, not evidence that a particular reaction occurs or has an appreciable cross section. Nuclear rest masses are used consistently; electron masses are not silently mixed into the sum.
Open lessonLesson · Science · General Chemistry ISeparate binding-energy evidence from detector-response assumptions
Infer shell populations from spectral areas only after specifying how the instrument responds.
Open lessonLesson · Social studiesSeparate bond pricing, money measures and an identity
Does a larger money stock by itself identify the cause of inflation?
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate burial pressure, temperature and differential stress
Metamorphism changes mineral assemblages and textures predominantly in the solid state. Parent composition, temperature, pressure, fluids and reaction time determine which transformations are possible. A limestone can recrystallize toward marble, while quartz-rich sandstone can form quartzite; neither must become foliated merely because it experienced heat.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate carbon gain, water loss and an efficiency ratio
Stomata connect internal leaf gas spaces to the atmosphere. Guard-cell osmotic and turgor changes alter pore opening, affecting CO2 entry and water-vapor escape. The same opening can therefore support carbon uptake and impose a water cost. Cuticle, boundary layers and leaf structure also influence exchange.
Open lessonLesson · Science · AP Environmental ScienceSeparate carbon storage from atmospheric exchange
Close a bounded carbon budget and avoid equating export with respiration.
Open lessonLesson · Science · Introductory MicrobiologySeparate carrier prevalence from a unique transfer history
A marker can become common because an existing carrier lineage expands, because the marker crosses lineage boundaries, because similar states arise repeatedly, or because the sampling frame changes. A pooled carrier frequency alone usually cannot distinguish these histories. Stable lineage information and time ordering add constraints but do not automatically identify a mechanism.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate catalytic temperature effects from enzyme history
Temperature can affect both how rapidly active enzyme molecules catalyze a reaction and how many active molecules remain after exposure. These are different mechanisms. A low endpoint rate after a warm incubation does not automatically imply that the intrinsic catalytic step becomes slower at the higher temperature.
Open lessonLesson · Science · University Physics I with calculusSeparate center motion from collision energy
For two masses, kinetic energy splits into center-of-mass translation plus relative motion: K = MVcm²/2 + μurel²/2, where μ = m₁m₂/M. Expanding the two individual kinetic terms verifies this identity and shows why the cross terms cancel.
Open lessonLesson · Science · University Physics I with calculusSeparate changes in speed from changes in direction
A vector derivative can be nonzero even when a scalar speed derivative is zero.
Open lessonLesson · Science · AP Chemistry preparationSeparate charge, distance and removal-energy claims
Use a controlled two-charge comparison to distinguish force from energy and explain the limits of atomic trend shortcuts.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate classical mode counts from a supplied activation law
Classical equipartition assigns mean energy kT/2 to each independent quadratic energy term. Three translational terms give molar Cv=3R/2. A diatomic molecule with two active rotational terms adds R, giving 5R/2 before vibrational and other contributions. Quantum level spacing can suppress a mode at low temperature, so a classical count is a regime statement rather than a universal constant.
Open lessonLesson · Science · Earth and space scienceSeparate climate observations from attribution claims
Write reproducible claims while distinguishing baseline effects, descriptive trends and causal evidence.
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate clock slope from initial composition using an isochron
A single daughter/parent ratio cannot generally identify both elapsed time and initial daughter. Multiple cogenetic aliquots can help when they shared an initial daughter/reference ratio but began with different parent/reference ratios. A stable nonradiogenic reference isotope supplies a common normalization.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate clonal specificity, precursor number and response timing
Adaptive immunity selects among preexisting receptor-bearing lymphocyte populations rather than creating a needed specificity by conscious demand. B-cell receptors can bind native antigen, while conventional T cells recognize peptide–MHC complexes with required signaling context. Helper T cells coordinate responses, cytotoxic T cells can act on target cells, and regulatory mechanisms limit inappropriate activation.
Open lessonLesson · Science · Earth and space scienceSeparate common calibration from scatter
Compare shared distance rescaling with wider conditional feasible bounds.
Open lessonLesson · Science · General Chemistry IISeparate common-ion suppression from a prescribed activity correction
A change in concentration needed for saturation need not mean the thermodynamic constant changed.
Open lessonLesson · Science · Introductory MicrobiologySeparate community composition from absolute abundance
A microbial community contains interacting populations, but a composition table reports fractions of whatever was counted. Relative abundance is a part divided by a total. A change in another class can therefore change a focal class’s fraction even when the focal abundance itself stays constant. Sequence fractions, cell fractions and biomass fractions are different quantities unless their measurement relationship is established.
Open lessonLesson · Statistics foundations: data, design and sampling variabilitySeparate concentration from normal approximation
Explain what repeated sampling does and does not guarantee.
Open lessonLesson · Science · University Physics II with calculusSeparate conductor equilibrium from grounding
A connected conductor in electrostatic equilibrium has zero field inside its material and one potential throughout that material. Neither statement says that a cavity contains zero field when charge lies inside it. This original concentric shell has inner radius 1 m and outer radius 2 m, with a point charge q at its center and isolated shell net charge Q.
Open lessonLesson · Social studiesSeparate crop origins from present-day production
Does the largest modern producer identify where a crop was domesticated?
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate daughter-genome allocation from cell separation
The spindle connects replicated chromosomes to a mechanical segregation system. In normal mitosis, sister kinetochores attach toward opposite poles, cohesion is released at the appropriate transition, and separated chromosomes move into daughter nuclear regions. Attachment surveillance reduces segregation errors but is not a promise that every division is error-free.
Open lessonLesson · Science · Earth and space scienceSeparate deposited energy from neutrino escape
Close the energy ledger and distinguish required rates from achieved fusion.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsSeparate detectable differences from useful changes
Interpret effect sizes and intervals against a decision threshold.
Open lessonLesson · Social studiesSeparate detecting a signal from deciding it is there
Use hits and false alarms to distinguish response policy from improved sensory discrimination.
Open lessonLesson · Science · Introductory MicrobiologySeparate detection, location and a causal disease claim
Host-associated microorganisms can be present without causing disease. Colonization concerns establishment at a site; infection and disease require further biological context, and disease involves harmful disruption rather than mere detection. A normally associated population can behave differently at another site or in a changed host context. Detection alone does not establish viability, invasion or causation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate diffraction scale degeneracy from an unused prediction
The original paraxial diffraction packet gives positions 6.4,12.4,18.4 mm at orders 1,2,3. Its forward law is x=c+m g λ/100, with wavelength λ in nm, gain g dimensionless and the conversion factor fixed by the stated geometry. Fitting a line estimates the product gλ, not wavelength and gain separately.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate digestive processing, absorption and animal production
Ingestion brings food into a feeding system; digestion breaks material into usable components; absorption crosses an interface into the organism; assimilation and later allocation determine what supports metabolism or new tissue. Egested undigested material is not identical to nitrogenous metabolic waste. A large meal is not automatically a large growth increment.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate dissolved concentration from total nutrient removal
Phosphorus can occur in dissolved forms and associated with particles. Weathering supplies mineral phosphorus, biological uptake transfers it to organic material, and adsorption or precipitation can change its distribution. Erosion transports particle-associated material; settling and resuspension move it across water-column boundaries. Unlike nitrogen, its major environmental cycle lacks a comparable dominant atmospheric gas reservoir.
Open lessonLesson · Science · AP Environmental ScienceSeparate diversity from a species count
Measure richness and abundance balance while distinguishing genetic, species and habitat diversity.
Open lessonLesson · Science · Introductory MicrobiologySeparate division events from gross daughters and net counts
Binary fission partitions a replicated cell into two daughters, so one completed division adds one to a cell count rather than two net cells. Other microbial division patterns include budding and fragmentation, and a counting method may treat attached products differently. Cell count, biomass and recoverable units therefore need distinct definitions.
Open lessonLesson · Science · General Chemistry ISeparate donor identity, acceptance and contact geometry
Apply an explicit screening rule without mistaking it for a universal measured interaction definition.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceSeparate drift, gene flow and mutation from selection
Genetic drift is random sampling of heritable variants between generations; its effects are especially pronounced in small populations.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate each ion’s equilibrium from zero total membrane current
An ion’s chemical gradient and the electrical potential difference jointly determine its electrochemical driving force. The equilibrium potential is the voltage at which those forces balance for that ion. A membrane permeable to several ions can have zero total current while individual ionic currents oppose one another. Electrical steady state is not necessarily thermodynamic equilibrium for every ion.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate echo travel time from electronic latency
A timing record can include both propagation and instrument delay. For a single identified reflected pulse, the round-trip path is twice the one-way target distance d. With constant propagation speed c and common latency ℓ, the recorded elapsed time is t=2d/c+ℓ. Treating the full recorded time as travel time overestimates range whenever the latency is positive.
Open lessonLesson · Science · Environmental science and sustainabilitySeparate efficiency from service demand
Compare matched service, changed use and a heat-transfer example.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate ejected volume, fractional ejection and external work
A pressure–volume loop connects ventricular filling, pressure rise at nearly fixed volume, ejection and pressure fall at nearly fixed volume. Stroke volume is end-diastolic minus end-systolic volume. Ejection fraction divides stroke volume by end-diastolic volume, so two ventricles can have the same fraction but different absolute ejected volumes. Neither quantity alone measures myocardial energy use.
Open lessonLesson · Science · Physics and physical systemsSeparate electric field from force on a charge
Compare positive, negative and zero charges at one source-field location.
Open lessonLesson · Science · Introductory AstronomySeparate electromagnetic frequency from photon arrival rate
Electromagnetic wavelength and frequency describe the wave, while photon energy describes a quantum of that radiation. In vacuum, frequency ν=c/λ and energy E=hν=hc/λ. Shorter wavelengths carry more energy per photon. Radio, infrared, visible, ultraviolet and higher-energy bands are not different kinds of matter; their interactions with instruments and intervening material differ because wavelength and energy differ.
Open lessonLesson · Science · AP Chemistry preparationSeparate electron geometry, molecular shape and polarity evidence
Inspect three-dimensional directions, lone-pair placement and controlled bond-vector cancellation.
Open lessonLesson · Science · Earth and space scienceSeparate emitted power from received flux
Predict inverse-square dilution while preserving the source’s temperature, luminosity and spectral shape.
Open lessonLesson · Science · Physics and physical systemsSeparate emitted radiation from net heat transfer
Account for emission, absorption and reflection before calculating net radiation.
Open lessonLesson · Science · Physics and physical systemsSeparate emitted-electron rate from stopping voltage
Compare a supplied quantum yield with photon rate while holding maximum electron energy fixed.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate endocrine steady concentration from its response time
A circulating hormone’s concentration reflects secretion, distribution, binding and removal, not secretion alone. Amount and concentration differ by distribution volume. Clearance expresses the effective volume cleared per time, while a first-order elimination constant has inverse-time units. Confusing those units can make an apparent physiological explanation dimensionally inconsistent.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate energy accounting from a body-mass prediction
Metabolism transforms absorbed molecules through catabolic and anabolic pathways. ATP couples many energy-releasing reactions to cellular work, while glycogen and lipid provide different storage forms. Assimilated energy is not identical to ingested food mass, and expenditure includes multiple processes rather than only visible movement.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate energy calibration from source identification
A measured spectrum first needs a response model. This fictional instrument indicates y=gE+b for true transition energy E, common gain g and baseline b. Independently stipulated reference energies2 and4 eV provide two equations for two response parameters. Their difference removes b, giving g=(y4−y2)/2; then b=y2−2g. A reference must be known independently of the instrument being calibrated, otherwise the algebra merely restates its own output.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate entropy production from unequal body entropy changes
Two fixed-volume bodies form an isolated system. The initially hot body has total capacity CA(T)=2+0.01T J/K, while the cold body has CB=4 J/K. No external work, heat or phase transition is allowed. Energy, rather than an arithmetic temperature average, determines their common final equilibrium temperature.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate entry oxidation from the citric-acid-cycle ledger
Pyruvate conversion and the citric acid cycle are related but distinct accounting stages. A three-carbon pyruvate supplies a two-carbon acetyl group, releasing one CO2 and reducing one NAD+ during entry oxidation. In eukaryotic cells this processing is associated with the mitochondrial matrix; the glycolytic pyruvate originated in the cytosol.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsSeparate equilibrium from being at rest
Newton’s first law describes inertial motion: with zero net external force, velocity remains constant. Rest is one possible constant velocity, not the only one. A cart gliding at 3 m/s on an ideal frictionless horizontal track requires no forward force to maintain that velocity.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsSeparate factorization from linear association
Test independence and construct uncorrelated dependent examples.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsSeparate force balance from torque balance
Add torques only after choosing one common axis. For a horizontal rigid body, an upward force to the left tends to rotate clockwise while an upward force to the right tends counterclockwise. Equal upward forces on unequal arms do not cancel their torques. Force components determine translational acceleration; force locations also matter for rotation.
Open lessonLesson · Partial differential equations and Fourier methodsSeparate forcing from homogeneous evolution
Derive Duhamel and lift inhomogeneous boundaries.
Open lessonLesson · Science · General Chemistry IISeparate formal charge, donor count and geometric possibilities
A coordination number counts bound donor atoms, not ligand names or oxidation number.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceSeparate genotype, environment and their interaction
A phenotype can depend on genotype and environment.
Open lessonLesson · Science · University Physics II with calculusSeparate geometric slope from a parallel stray capacitance
Capacitance measured at one gap cannot by itself separate geometric capacitance from an added parallel stray term. The synthetic family is C=S/d+Cs, with d in independently exact millimetres, C in picofarads and S in pF mm. Gain is stipulated calibrated. Two distinct gaps fit its two parameters; a third gap is reserved for validation.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate glycolytic investment, payoff and net cellular demand
Glycolysis transforms one six-carbon glucose into two three-carbon pyruvates while transferring some energy into ATP and reduced electron carriers. Its reaction sequence occurs in the cytosol and does not directly consume molecular oxygen. Whether it can continue depends partly on restoring the oxidized carrier NAD+.
Open lessonLesson · Social studiesSeparate growth in output from growth per person
Can total production rise while output per resident falls?
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate half spreading rates from plate separation
A dated crustal marker records displacement accumulated since formation, not necessarily the instantaneous present rate. On each side of an ideal ridge, distance from the ridge divided by marker age gives a mean ridge-frame half spreading rate. Comparing markers of the same age allows the two half rates to be added.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate handling-limited intake from predator population change
Increasing prey density can raise the number consumed by one predator, but handling and processing can limit intake. This per-consumer functional response differs from a numerical response in predator abundance. Neither one alone establishes the cause of a population cycle.
Open lessonLesson · Algebra II and precalculus: functions and structureSeparate holes, poles and end behavior
Factor a rational function while preserving its original domain and directional behavior.
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate igneous texture, mineral modes and mass composition
Igneous rocks crystallize from melt, but composition and cooling history answer different questions. Coarse interlocking crystals commonly indicate prolonged growth below the surface; fine textures indicate less growth time, while glass records inhibited crystallization. A porphyritic texture combines larger early crystals with a finer later matrix and can record changing conditions.
Open lessonLesson · Science · Introductory MicrobiologySeparate immune-memory evidence from a protection guarantee
Adaptive responses select antigen-recognizing lymphocyte populations, expand relevant clones and generate effector and memory populations. B cells can differentiate into antibody-secreting cells; antibody classes and binding properties can change with response context. Antibodies can neutralize, opsonize or recruit other mechanisms, but a binding signal alone does not measure every protective function.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate independent ecological replication from repeated assays
A measurement replicate and an independently treated ecological unit answer different questions. Reassaying one sample can reveal instrument precision, while independently sampled or assigned sites provide information about ecological variability and treatment response. Counting all technical repeats as if they were independent treatment units exaggerates the amount of causal information.
Open lessonLesson · Science · AP Environmental ScienceSeparate indoor source control from ventilation
Compare a bounded tracer balance with real pollutant-specific pathways.
Open lessonLesson · Science · Introductory MicrobiologySeparate inherited-state frequency from expression per carrier
A changed aggregate response can reflect a changed population mixture or a changed response within the same carriers. DNA marker frequency, transcript/protein expression and a functional phenotype are different measurements. Their agreement can support a mechanism, but one cannot be silently substituted for another.
Open lessonLesson · Science · Introductory MicrobiologySeparate innate recognition from response certainty
Innate recognition detects recurring molecular patterns and signs of damage through receptors and soluble systems. Phagocytes can bind and engulf material, while complement pathways can promote opsonization, inflammatory signaling and, for susceptible targets, membrane attack. These outcomes are distinct; recognizing a pattern does not guarantee that every downstream effect occurs or that harm is prevented.
Open lessonLesson · Social studiesSeparate input substitution from a shift in labor supply
Can cheaper equipment change hiring without changing labor productivity?
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate interference order from finite-width brightness
A principal interference direction is a phase condition, not a guarantee of a bright observed spot. Equal coherent slits separated by d have constructive path differences d sinθ=mλ and destructive differences (m+1/2)λ. The angle is measured from the central axis. A screen perpendicular to that axis at distance L records y=L tanθ, whereas y≈Lmλ/d adds a small-angle approximation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsSeparate interference orders from a finite-slit envelope
Coherent slit contributions have a stable phase relationship, so their path differences can produce repeatable bright and dark directions. For slit spacing d, nominal constructive orders satisfy d sinθ=mλ. Increasing wavelength moves a given noncentral order farther from the center. White light has a common central order but dispersed noncentral colors, with longer-wavelength red farther out than violet for the same order.
Open lessonLesson · Science · Introductory MicrobiologySeparate internal reaction capacity from delivery through a layer
Biofilms are structured microbial communities associated with surfaces or interfaces and extracellular material. Their matrix and spatial organization can retain cells and molecules while creating heterogeneous access to resources. Attachment, dispersed states and signaling-mediated regulation are biological processes, but a chemical profile alone does not identify which species or communication mechanism is present.
Open lessonLesson · Science · General Chemistry ISeparate internal-energy change from heat and boundary work
Choose a system and sign convention before combining transfers.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate joint rotation from the displacement of a limb endpoint
Joints connect skeletal elements. Fibrous joints join through fibrous connective tissue, cartilaginous joints through cartilage, and synovial joints have a joint cavity with articular surfaces, capsule and synovial fluid. Structural classification is not identical to a single mobility number. Ligaments connect and constrain bones; tendons transmit muscle force to attachments.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSeparate LC components with independent amplitude evidence
In an ideal LC circuit, T = 2pi sqrt(LC).
Open lessonLesson · Mathematics for AI · M04Separate likelihood from a prior
Fit a Bernoulli probability and compare maximum likelihood with explicit prior assumptions.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate locus histories from a single species-tree claim
A gene tree describes the history inferred for sampled sequence copies; a species tree concerns lineages of organisms. These need not coincide. Sequence duplication can create paralogs, horizontal transfer can move material across lineages, and ancestral polymorphism can persist through successive divergences before different variants are retained.
Open lessonLesson · Science · Introductory AstronomySeparate lunar illumination from image orientation
Ordinary lunar phases arise because the observer sees different portions of a sunlit hemisphere. They are not a monthly passage through Earth’s shadow. The Moon rotates synchronously in the broad sense that its average rotation and orbital periods match, while libration and perspective allow the visible presentation to vary.
Open lessonLesson · Science · Earth and space scienceSeparate lunar illumination from the visible disk
Explain why a sunlit hemisphere can appear as a crescent, quarter or full disk.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSeparate magnetic field geometry from material response
A magnetic field assigns a vector to each observation point and can predict forces on moving charges, currents and magnetic dipoles.
Open lessonLesson · Science · University Physics II with calculusSeparate magnetic guiding drift from electric work
A particle released from rest in crossed uniform fields is not the same experiment as a force-free velocity selector. Here a fictional mass 1 kg and charge ±1 C starts at the origin, with B=1 T along positive z and E along positive x. The electric field initially accelerates it; magnetic force then bends the growing motion.
Open lessonLesson · Science · Earth and space scienceSeparate magnitude, shaking intensity and damage
Match earthquake claims to evidence that the catalog subset actually contains.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsSeparate material modulus from specimen geometry and validity
A material can deform even when its overall translational motion is in equilibrium. For uniform axial tension, stress is force divided by cross-sectional area and strain is extension divided by original length. Stress has pressure units; strain is dimensionless. The Young modulus E is stress/strain within a stated linear regime, not the same quantity as a particular specimen’s spring stiffness.
Open lessonLesson · Science · Earth and space scienceSeparate metamorphism from melting
Contrast solid-state change with melting and solidification, including a contact-zone evidence problem.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate moved air from fresh alveolar ventilation
Conducting airways carry air through the nose or mouth, pharynx, larynx, trachea, bronchi and smaller branches. Gas exchange occurs mainly across respiratory regions including alveoli rather than throughout the entire conducting pathway. Airway epithelial defenses, smooth muscle and supporting cartilage have different distributions and roles; a volume model does not reproduce all of that anatomy.
Open lessonLesson · Optimization: models, certificates and algorithmsSeparate multiplier necessity from convex sufficiency
Use active sets, sign conventions and a qualification counterexample.
Open lessonLesson · Science · AP Environmental ScienceSeparate natural increase from total population change
Use births, deaths and migration with explicit per-thousand and percent conversions.
Open lessonLesson · Science · General Chemistry IISeparate nuclear bookkeeping, remaining parents and measured activity
Nuclear identity conservation, ensemble decay and detector response are distinct evidence layers.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeSeparate null-set completion from arbitrary measurability
Describe the completion of a measure.
Open lessonLesson · Differential equations: models, methods and dynamicsSeparate numerical instability from unstable physics
Construct Euler steps and compare their error and stability with an exact solution.
Open lessonLesson · Science · Physics and physical systemsSeparate observer motion from source motion
Compare equal closing speeds and use arrival events to explain different Doppler shifts.
Open lessonLesson · Science · AP Environmental ScienceSeparate ocean heat uptake, sea-level change and coral stress
Calculate a bounded thermal-expansion ledger without inventing a biological forecast.
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate ocean rise from vertical land motion in shoreline interpretation
Relative sea level is the ocean surface measured against the land at a location. Ocean-level change and land uplift or subsidence both contribute. With upward land motion positive, relative rate equals ocean-level rise minus land uplift. A site can experience relative fall even while the ocean rises against a fixed reference.
Open lessonLesson · Science · AP Environmental ScienceSeparate one-year survival from sustainable resource use
Balance resource renewal and consumption and test per-capita sensitivity.
Open lessonLesson · Science · General Chemistry IISeparate optical background from a transformed kinetic slope
The supplied optical signal is S(t)=100exp(−kt)+B, with a reactive contribution and a constant background.
Open lessonLesson · Optimization: models, certificates and algorithmsSeparate oracle progress from practical running time
Derive a central-cut ellipsoid update and qualify polynomial-time claims.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceSeparate origin-of-life hypotheses from ancestry evidence
Origin-of-life research asks how chemical systems capable of heredity and evolution arose.
Open lessonLesson · Social studiesSeparate output growth from price changes
Did the economy produce more or just charge more?
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate overlapping hazard events from expected consequences
Hazard describes a potentially damaging process and its occurrence or intensity; exposure identifies what lies in its path, and vulnerability concerns how those exposed things respond. A geological susceptibility map is not automatically a calibrated annual probability map, nor does identical hazard imply identical consequences for different sites.
Open lessonLesson · Science · General Chemistry IISeparate particle-factor evidence from a unique dissociation claim
A colligative particle count is not an atom count.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate penetration and generation in a steady particle balance
Indoor particles can originate outdoors or be generated inside. Penetration, air exchange, deposition, filtration and activity all affect a concentration. A lower outdoor reading alone cannot identify an indoor source contribution, and an indoor/outdoor ratio is not itself a health-risk measure.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate phenotype sorting from an inherited response
Phenotypes can vary because of inherited differences, environmental differences and their interactions. Selection can associate a phenotype with contribution to offspring, but a next-generation response also depends on what is transmitted. A changed phenotype mean under a changed environment is not by itself evolutionary genetic change.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate photon supply from an emission threshold
Light can transfer energy in individual photon events even when its propagation also shows interference. A monochromatic photon has energy hf=hc/λ. For the supplied rounded constant hc=1240 eV nm, a 3 eV photon has wavelength about 413.33 nm. Optical power is energy per time; dividing power by photon energy in joules gives a photon expectation rate. The conversion 1 eV=1.602176634×10⁻¹⁹ J is essential when combining those units.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate photon, electron and ATP budgets in photosynthesis
Photosynthetic light reactions convert absorbed photon energy into chemical and electrochemical changes. In linear electron flow, water oxidation supplies electrons to photosystem II; downstream transfer and photosystem I excitation support reduction of NADP+ to NADPH. The released oxygen comes from water, not from the carbon of CO2.
Open lessonLesson · Science · University Physics I with calculusSeparate physical parameters from a shared measurement gain
A period fit can be internally consistent while leaving important parameters unidentified. For an ideal spring carrying unknown base mass m plus known added mass M, a clock with elapsed-time gain g reports Tread² = (4π²g²/k)(m + M). The intercept-to-slope ratio identifies m, but the slope alone identifies only k/g².
Open lessonLesson · Social studiesSeparate political borders from everyday connections
Who governs a shared watershed?
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceSeparate polynomial approximation from a poor node choice
Analyze cardinal-weight amplification, Runge behavior and Chebyshev node geometry.
Open lessonLesson · Social studiesSeparate population change from migration
Did arrivals alone explain growth?
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeSeparate positive and negative mass
Prove Hahn decomposition for a finite signed measure.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate preserving a medullary gradient from generating it
Kidney loops and medullary blood vessels have countercurrent arrangements, but they do not perform identical jobs. Selective water permeability and salt transport in loop segments, together with flow and urea recycling, contribute to the medullary osmotic environment. Vascular exchange can reduce washout of that environment while still carrying reabsorbed material away.
Open lessonLesson · Science · Physical Geology and Earth SystemsSeparate pressure-storage release from water-table drainage
An aquifer’s head can fall without the same physical process occurring everywhere. In an unconfined aquifer, a falling water table can drain connected pore space. In a confined aquifer that remains saturated, pressure decline releases water through compressibility of water and deformation of the porous skeleton.
Open lessonLesson · Science · AP Environmental ScienceSeparate producer production from ecosystem carbon gain
Calculate GPP, NPP and net ecosystem production with explicit respiration boundaries.
Open lessonLesson · Science · Introductory AstronomySeparate projected rotation from a galaxy mass inference
A galaxy contains stars, gas, dust and gravitationally inferred matter whose distribution need not match its visible image. Disks can show organized rotation, while spheroidal components can have substantial random motions. A circular-tracer calculation therefore has a particular dynamical context rather than applying identically to every morphology.
Open lessonLesson · Science · Introductory AstronomySeparate propagation loss from distance and source strength
Luminosity is total emitted power, while flux is power received per area. For an isotropic source in a simple transparent geometry, flux=L/(4πd²). The inverse-square factor describes spreading over a larger sphere; it does not imply photons lose energy merely by crossing that transparent space.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate protein conformation from primary-chain cleavage
A protein’s primary structure records its amino-acid sequence and peptide connectivity. Local secondary structures and the overall three-dimensional fold depend on interactions among backbone and side-chain groups. A complex can also contain multiple associated polypeptide chains. These levels describe different features and require different evidence.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsSeparate pulse area from a force-reading baseline
Impulse is the signed area under net force versus time, equal to the change in momentum. For a triangle, area is half base times height, so no calculus is needed to compare pulse shapes in this packet. Peak force, average force and impulse are different quantities with different units or roles.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingSeparate quadrature error from an uncertain force record
Audit an approximate work integral and avoid inventing precision from additional samples.
Open lessonLesson · Science · Scientific inquiry and measurementSeparate random variation from systematic error
Explain why more readings do not solve every measurement problem.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceSeparate receptor binding from a cellular response
A signal reaches a cell only through a chain of physical and molecular events. Local paracrine signals, circulating endocrine signals, self-directed autocrine signals and direct cell contacts differ in delivery. None guarantees a response in every nearby cell: receptor expression and downstream machinery determine which cells can interpret a signal.
Open lessonLesson · Social studiesSeparate recovery, prior peaks and potential output
Use dated GDP, potential-output and payroll records to test three different recovery claims.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate recruitment from rate coding in a finite muscle model
A motor unit consists of a motor neuron and the skeletal muscle fibers it innervates. Recruiting additional units can increase total force, while changing firing frequency alters the time course and overlap of responses in already active units. These mechanisms do not require action potentials to become proportionally taller. A fiber’s all-or-none electrical event is distinct from graded force of a whole muscle.
Open lessonLesson · Science · Introductory MicrobiologySeparate recruitment, persistence and resolution
Inflammation coordinates vascular changes, signaling and cellular recruitment in response to infection or damage. Increased permeability and recruited cells can support defense and repair, but an excessive or persistent response can also contribute to tissue injury. Protective function and harmful consequences are therefore not a single quantity that increases monotonically with every response marker.
Open lessonLesson · Science · Introductory MicrobiologySeparate reduced recovery from actual remaining material
A lower recovered count can reflect fewer recoverable units in the original system, a weaker recovery process or interference carried into the measurement. These alternatives require controls with a clear purpose. A control is not merely an extra sample label: it should test an assumption that could change the interpretation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsSeparate refraction, grazing transmission and total internal reflection
Total internal reflection requires light to travel from higher refractive index to lower index. Snell’s law then predicts increasing transmitted angle as incidence increases. At the critical angle, the ideal transmitted direction is along the interface, 90° from the normal. Above that angle, no real transmitted angle satisfies the geometric relation.
Open lessonLesson · Science · Introductory MicrobiologySeparate relative descendants from inherited-state loss
Changes in the frequency of a genetic state can reflect differential reproductive contribution, state transmission, mutation, transfer or changing population composition. These mechanisms are not interchangeable. Selection changes representation of existing variation; it does not require a future benefit to direct the creation of that variation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate relative motion from the fixed-nucleus spectral approximation
A one-electron atom is a two-body problem before the heavy nucleus is approximated as fixed. With electron mass m and nuclear mass M, transform to center-of-mass and relative coordinates. The kinetic energy separates into center motion plus relative kinetic energy with reduced mass mu=mM/(m+M). The Coulomb potential depends only on the separation, so the relative-coordinate spectrum uses mu rather than m.
Open lessonLesson · Science · Introductory MicrobiologySeparate removal, inactivation and a sterilization claim
Microbial control terms describe different goals and boundaries. Cleaning can remove material; disinfection reduces or inactivates relevant microorganisms on nonliving objects without necessarily eliminating every resistant form. Antiseptic use concerns living tissue, while sterilization is a validated process intended to eliminate viable microorganisms including bacterial endospores. The words are not interchangeable, and this course supplies no method for performing any of them.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate reproductive anatomy, meiotic DNA and functional products
Seminiferous tubules in the testes support sperm production, with Sertoli cells providing local support and Leydig cells contributing androgen production. Sperm pass through epididymal maturation and ducts including the ductus deferens; accessory glands contribute fluid. Ovarian follicles support oocyte development, and an ovulated oocyte is received by a uterine tube rather than released directly into the uterine cavity. The uterus, cervix and vagina have distinct structural roles.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsSeparate resistance geometry from a power claim
Resistance R describes a particular component, whereas resistivity ρ describes a material under specified conditions. For a uniform wire, R=ρL/A. Doubling its length doubles resistance; doubling its cross-sectional area halves resistance. Doubling diameter instead quadruples area for a circular wire, so diameter and area changes must not be confused.
Open lessonLesson · Science · General Chemistry ISeparate resonance, formal charge and measured electron density
Preserve electron count and atom positions while changing a Lewis contributor.
Open lessonLesson · Science · Earth and space scienceSeparate response time from equilibrium
Compare adjustment speeds and conditional targets before and after a finite forcing interval.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceSeparate richness, evenness and sampling completeness
Biodiversity includes genetic, species and ecosystem variation.
Open lessonLesson · Science · Earth and space scienceSeparate rounding, grid resolution and model uncertainty
Interpret arrival residuals without claiming more precision than the evidence supports.
Open lessonLesson · Science · Introductory AstronomySeparate same-clock observations from star-relative rotation
A day can be referenced to the Sun or to a distant celestial direction. As Earth advances along its orbit, it must rotate slightly more than one star-relative turn before the mean Sun returns to the meridian. Consequently the mean solar day is longer than the sidereal rotation period.
Open lessonLesson · Science · General Chemistry ISeparate sample spread from accuracy and common bias
Derive mean, sample spread and mean uncertainty while keeping a shared bias outside the repeatability estimate.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate scientific forecasts from value choices in a score
Environmental decisions often compare pollution reduction, water use and cost. Science can inform predicted outcomes and uncertainty, but it does not uniquely determine how stakeholders should value them. A composite score should expose its normalization, weights and any assumption that one benefit can compensate for another harm.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate seasonal coverage from a comparable climate anomaly
Temperature and many pollution variables have seasonal cycles. Comparing raw averages from differently sampled seasons can confound environmental change with coverage. A season-specific anomaly subtracts a corresponding fixed reference value, while a target mean also requires explicit temporal and spatial weights.
Open lessonLesson · Linear Algebra II: operators and spectral structureSeparate sensitive problems from inaccurate algorithms
Derive spectral condition numbers, perturbation bounds and the costs of normal equations.
Open lessonLesson · Social studiesSeparate sensory, motor and autonomic pathways
Trace a response through central and peripheral pathways and distinguish neural from endocrine signaling.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceSeparate sequence predictions from functional outcomes
Mutations include substitutions, insertions, deletions and larger chromosome changes.
Open lessonLesson · Science · Introductory MicrobiologySeparate sequencing evidence from certainty about a variant
A sequence read is an observation, not automatically an independent cell or error-free genome. Repeated copies of one starting molecule, shared preparation artifacts or correlated calling errors can make nominally separate reads dependent. A variant call also does not establish that a biological phenotype follows from the sequence.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceSeparate service demand from engineering efficiency
Energy intensity describes energy per unit of service; total energy also depends on the amount of service. An efficient transport, cooling or water system can still use more total energy when activity expands. Whether expanded service meets unmet needs is a social question distinct from the energy arithmetic.
Open lessonLesson · Social studiesSeparate shared benefits, policy costs and income distribution
Can private willingness to pay understate a service’s total benefit?
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate shared sensory input from somatic and autonomic output chains
A reflex pathway links sensory detection, central integration and an efferent response, but local circuits can also be modulated by descending signals. Somatic motor neurons innervate skeletal muscle. Autonomic output typically uses a preganglionic neuron, a ganglion and a postganglionic neuron to reach smooth muscle, cardiac muscle or glands. These are organizational distinctions, not a claim that one system is always conscious and the other never influenced by context.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate signal propagation from calcium activation and mechanical response
At a skeletal neuromuscular junction, a motor-neuron action potential triggers transmitter release. Acetylcholine activates receptors in the end plate, producing a local depolarization that can initiate a muscle action potential. The end-plate response and propagated action potential are not the same signal. Transmitter breakdown and membrane recovery help terminate signaling.
Open lessonLesson · Precalculus: polar curves and signed-radius evidenceSeparate signed radius, distance and estimates
Prove distance intervals and use a polar table without inventing between-point accuracy.
Open lessonLesson · Science · University Physics II with calculusSeparate signed reactance, phase and dissipated power
In steady sinusoidal response, derivative and integral terms appear with different phases. With a cosine convention, series impedance is Z=R+i[omega L−1/(omega C)]. This packet fixes L=1 H,C=0.25 F and source RMS voltage 6 V, while resistance stays positive. The initial transient is assumed to have decayed.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate sinusoidal peak, rms and transformed power
A rotating coil in a uniform field has sinusoidally changing flux. The supplied ideal generator has50 turns, area .02 m² and field .1 T, so peak emf is NABω=.1ω with ω=2πf. One complete cycle has zero signed mean voltage, but nonzero mean squared voltage. A resistor therefore can absorb positive mean power even though the voltage alternates sign.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate skin structure from an external heat-transfer boundary
The integumentary system includes skin and accessory structures such as hair, nails and glands. Skin proper has an epithelial epidermis and a connective dermis; the hypodermis lies beneath and connects it to deeper structures. Epidermis is avascular, whereas dermal vessels supply exchange and can change heat transfer. Calling the hypodermis a closely associated layer is more precise than treating all three regions as identical skin tissue.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate slit spreading from circular-aperture resolution
A finite aperture spreads a wave even in a perfectly manufactured optical system. For a single slit of width w, the first dark directions satisfy w sinθ=λ. On a perpendicular screen at L, the central bright region extends between the two signs, so its width is2L tanθ. Confusing this full width with its half-width gives a factor-of-two error.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate spatial attenuation, counting spread and dose claims
A spatial attenuation model and a radioactive survival law can both be exponential while describing different independent variables. Here a fixed narrow beam has source expectation800×2^(−x/H) counts/min after thicknessx, whereH is a supplied half-value thickness in centimetres. Its source strength is not decaying with elapsed time in this model. The equivalent coefficient is μ=ln2/H with inverse-length units; it is not a decay constant with inverse-time units.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate spherical spreading from background intensity
An ideal isotropic source distributes acoustic power over a sphere of area 4πr², so its source-only intensity is P/(4πr²). A separate uniform incoherent background adds intensity, giving I(r)=S/r²+B when S is the source intensity at one metre. This sum assumes the sources are uncorrelated; coherent waves must instead have their pressure amplitudes and phases combined.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceSeparate stage-structured transients from long-run growth
Individuals at different stages contribute differently to future population size. A population with many juveniles can change differently from one with many adults even when both share the same vital-rate parameters. A single total-count ratio therefore need not identify the long-run growth tendency.
Open lessonLesson · Science · General Chemistry IISeparate standard free energy from the current driving force
Composition enters reaction free energy through a dimensionless activity quotient.
Open lessonLesson · Directed random walks and stationary guaranteesSeparate stationary balance from convergence
Solve invariant probability equations and use finite counterexamples to distinguish existence, uniqueness and limiting behavior.
Open lessonLesson · Science · Earth and space scienceSeparate suspended load from catchment erosion
Trace sediment production, transport and storage without attributing an outlet total to an unmeasured source.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate synaptic current from depolarization and spiking
Synapses can change membrane conductance through ligand-gated channels or slower signaling pathways. The postsynaptic effect depends on the channel’s reversal potential and the membrane state, not only the transmitter’s name. Spatial and temporal summation combine inputs, while electrical synapses and chemical synapses have different transmission mechanisms.
Open lessonLesson · Science · Chemistry and matterSeparate target product from total charge
Partition electrode charge, compare current efficiencies and diagnose unreachable product targets.
Open lessonLesson · Course 2 · Geometry and control foundationsSeparate task planning from motor control
Use explicit skill preconditions, termination receipts and bounded recovery.
Open lessonLesson · Social studiesSeparate the next decision from money already spent
Should an unrecoverable fee determine the next unit?
Open lessonLesson · Science · AP Environmental ScienceSeparate thermal pollution from a chemical concentration
Explain heat discharge and biological consequences using a declared mixing balance.
Open lessonLesson · Social studiesSeparate time of day from time awake
Design a comparison that separates a daily rhythm from accumulated wakefulness.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsSeparate translational and rotational kinetic-energy ledgers
A spinning body can have kinetic energy even when its center of mass is at rest. Each small mass element moves at speed r|ω| relative to a fixed axis. Adding its energy gives Σ½m(rω)²=½(Σmr²)ω²=½Iω². This derivation uses the rigid body’s common angular velocity and the same axis for all particle distances.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceSeparate traveling phase from local particle motion
A wave speed describes how a disturbance advances, not how fast a marked piece of medium moves. In the supplied transverse wave, y=.01 sin[2π(t/.5−x/λ)] metres, t is time in seconds and x is position in metres. The period is .5 s, the amplitude is .01 m and the wavelength λ is controlled. A crest has a constant phase, so increasing time must be accompanied by increasing x: the pattern travels toward +x.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsSeparate urine concentration from conserved osmole excretion
Collecting ducts adjust final water handling after extensive upstream processing. ADH is synthesized in hypothalamic neurons and released from their terminals in the posterior pituitary; signaling in collecting-duct principal cells promotes aquaporin trafficking that changes water permeability. Permeability alone is insufficient: a suitable osmotic gradient must also exist.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsSeparate using every edge from visiting every vertex
Apply Euler parity conditions and recognize the limits of Hamiltonian heuristics.
Open lessonLesson · Differential equations: models, methods and dynamicsSeparate variables without deleting equilibria
Track divisions, constants and solution intervals.
Open lessonLesson · Science · AP Chemistry preparationSeparate water and solute-depletion approximations
Independently choose two simplifying assumptions and audit their species, charge and ion-concentration consequences for weak acids and bases.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSeparate width from separation using two spatial scales
Two finite slits are not two mathematical points. Each slit contributes an aperture transform, while shifting their centers adds opposite phase factors. For identical width w and center separation d, summing the two fields yields a cosine factor multiplying the single-slit sinc.
Open lessonLesson · Science · AP Environmental ScienceSeparate wild fisheries from aquaculture tradeoffs
Explain depletion, bycatch and species-specific aquaculture impacts.
Open lessonLesson · Social studiesSeparate worker averages from distribution
Did higher output improve every worker’s outcome?
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusSequence limits and epsilon–N proofs
Choose a tail index after a tolerance and distinguish convergence from selected samples.
Open lessonLesson · Sequence spaces and completenessSequence norms and inclusions
Build elementary sequence norms from finite inequalities and infinite partial sums, then prove strict inclusions and distinguish coordinatewise from norm convergence.
Open lessonCourse · Mathematics program unitSequence spaces and completeness
Prove infinite-sequence norm inequalities, distinguish coordinate and norm convergence, and construct complete spaces with explicit approximation boundaries.
Open courseLesson · Point-set topology: spaces, constructions and separationSequences and the limits of sequential criteria
Prove closure can be tested by sequences in first-countable spaces.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusSequential criteria for limits and continuity
Convert failed epsilon–delta statements into explicit sequences of witnesses.
Open lessonLesson · Real Analysis I: completeness, limits and rigorous calculusSeries, absolute convergence and rearrangements
Apply the Cauchy criterion to tails and distinguish absolute from conditional behavior.
Open lessonCourse · Mathematics program unitSet specifications and language-expression limits
Turn informal requirements into quantified membership rules, construct languages, and prove the limits of a precisely stated expression grammar.
Open courseLesson · Course 2 · Memory and shared working stateShare the right information at the right time
Control a three-slot event workspace and compare shared processing with a fast local response loop.
Open lessonConcept · GlossaryShared workspace
A shared place where selected goals, observations, and memory references are made available to multiple components.
Open conceptConcept · GlossaryShortcut learning
Using a predictive signal that works in the observed data but does not represent the intended task, such as a background instead of an animal’s shape.
Open conceptLesson · Mathematics investigationShrink an interval without dividing by zero
When does an average rate become a unique local rate?
Open lessonConcept · GlossaryShrinkage
In boosting, a learning-rate multiplier applied to each new learner before adding it to the existing prediction. Smaller corrections often need more stages; they do not guarantee better generalization.
Open conceptConcept · GlossarySigmoid
A smooth function that maps a numerical score to a value between 0 and 1. Interpreting it as a probability does not guarantee calibration.
Open conceptLesson · Integral calculus: accumulation, methods and applicationsSigned area and integral bookkeeping
Apply orientation, splitting, linearity and comparison without mistaking net change for total area.
Open lessonLesson · Differential geometry of curves and surfacesSigned curvature and turning angle
Recover local bending from a unit tangent.
Open lessonLesson · Social studiesSikh history: scripture, community and historical memory
Trace Sikh community authority and distinguish historical development from the dates and purposes of surviving representations.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometrySimilarities and the hierarchy of invariants
Determine what survives when rigid motions are enlarged.
Open lessonModule · Course 1Simple models and useful baselines
Compare neighbors, fitted trees, numerical regression and class probabilities with simple references before adding complexity.
Open moduleLesson · Mathematical modeling and senior investigationSimplify a model while tracking the error regime
Derive a regular expansion and detect nonuniform limits.
Open lessonLesson · Statistical investigations, simulation and probability decisionsSimulate treatment reassignment under a null
Build a sharp-null assignment distribution and compare a seeded approximation with exact enumeration.
Open lessonCourse · Mathematics program unitSingle-variable calculus foundations
Ten substantial lessons connecting functions, limits and derivative foundations. This published tranche is not a complete Calculus I or AP Calculus course. Exact practice checks values; written reasoning needs review against the supplied rubrics.
Open courseLesson · Course 1 · What has the network learned?Slide a detector across an image
Reuse one filter at many positions and trace how stride, borders and pooling change the result.
Open lessonLesson · Course 2 · Planning and reasoningSmall errors become different futures
Compare fresh-state predictions with a model that must live with its own earlier mistakes.
Open lessonLesson · Social studiesSmith: private exchange and public responsibilities
Smith explains how exchange can appeal to mutual advantage while also discussing public responsibilities.
Open lessonCourse · Social studiesSocial research methods — High school
Hands-on social studies course and project
Open courseCourse · Social studiesSocial research methods — Introductory college
Hands-on social studies course and project
Open courseLesson · Social studiesSocial Security in 1935: assistance, insurance and exclusions
Investigate program design, run a bounded 1937 payroll experiment, and compare the original law with a later payment case.
Open lessonLesson · Social studiesSocial-emotional development: relationships, roles and context
Explain how relationship patterns and changing social roles interact without treating a childhood observation as destiny.
Open lessonCourse · Social studiesSociology and social institutions — High school
Hands-on social studies course and project
Open courseCourse · Social studiesSociology and social institutions — Introductory college
Hands-on social studies course and project
Open courseConcept · GlossarySoft actor–critic
A continuous-action learning approach that combines a stochastic actor, replay, critics and an entropy-weighted objective. Entropy temperature trades off predicted return and action-distribution spread; it is not a safety guarantee.
Open conceptConcept · GlossarySoftmax
A transformation from numerical scores into nonnegative weights that sum to one, using exponentials and normalization.
Open conceptAssessment · Sections, solid measures and geometric designSolid geometry: sections, limits and design evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · General Chemistry ISolve a composition balance and reject impossible mixtures
Use two independent mass equations to infer a mixture fraction, then check feasibility and sensitivity.
Open lessonLesson · Course 2 · Planning and reasoningSolve a constraint problem
Build a small timetable, remove unsupported choices and compare backtracking with constraint propagation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSolve a finite-body balance with unequal capacities
Two finite bodies exchange heat inside an isolated fixed-volume boundary. Body A has total heat capacity CA(T)=2+0.01T J/K; body B has CB=4 J/K. Their initial temperatures TA and TB differ. There is no external heat, work or phase transition, so the sum of their internal-energy changes must vanish.
Open lessonLesson · Science · University Physics I with calculusSolve a nonlinear falling branch by separating variables
Choose downward positive for a1 kg point released from rest. A quadratic resisting force is−k v|v|, not always−k v² regardless of direction. On this falling branch v remains nonnegative, so dv/dt=g−k v²/m. Here g=10 and k has units kg/m.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingSolve a quadratic-drag branch with explicit sign limits
Compare a nonlinear resistance transient with a linear model without silently changing its velocity domain.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingSolve a signed linear-resistance initial-value problem
Use separation of variables and initial conditions to distinguish terminal equilibrium from an initial-speed cap.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSolve an LR circuit without erasing its initial current
An ideal source E, resistor R and constant inductor L form one series loop.
Open lessonLesson · Science · University Physics II with calculusSolve an RC initial-value problem with energy return
The common zero-charge formula is only one initial-value problem. Take a maintained 6 V source, resistance 2000 Ω and capacitance 500 µF, so RC=1 s. Initial voltage Vi can be below, equal to or above 6 V, including a negative polarity. Positive current enters the capacitor positive plate.
Open lessonLesson · Algebra foundations: expressions, equations and rootsSolve and interpret inequalities
Track order, endpoints and feasible values instead of treating an inequality as an equation.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingSolve and verify a harmonic initial-value equation
Use two initial conditions to select an oscillator solution, rather than guessing its phase.
Open lessonLesson · Science · General Chemistry IISolve association while rejecting impossible roots
A valid equilibrium solution must satisfy both a quotient and nonnegative material inventories.
Open lessonLesson · Science · University Physics I with calculusSolve both velocity components before integrating the path
With isotropic linear drag per unit mass gamma, horizontal and vertical equations are dvx/dt=−gamma vx and dvy/dt=−g−gamma vy. The virtual point starts at(0,5) m with velocity(4,6) m/s. Gravity is vertical but resistance acts on both components relative to the stationary medium.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsSolve coupled carts and test the system boundary
When objects share a motion constraint, choose whether to analyze each object or the whole system. Two carts connected by a taut inextensible massless string share an acceleration along a straight track. The leading cart is pulled by an external 20 N force; the trailing cart is pulled only by the string. Friction is absent in the stated model.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSolve coupled junctions and interpret a signed bridge current
A bridge joins the midpoint of two source-to-return arms.
Open lessonLesson · Science · University Physics II with calculusSolve coupled mechanical and inductive storage rather than assume instantaneous drag
The earlier driven-rod model neglected loop inductance. Here that approximation is deliberately removed. In a virtual uniform applied field, mass m=1 kg, constant loop inductance L=1 H and coupling B ell=1 T m are stipulated. Initial velocity is v0 and initial current is zero; no external mechanical drive acts. Resistance R is nonnegative. Geometry is idealized so the inductance remains constant as the rod moves; real rails and finite field boundaries would require a new model.
Open lessonLesson · Algebra foundations: expressions, equations and rootsSolve distance conditions with absolute value
Turn distance into two branches or an interval, including impossible negative distances.
Open lessonLesson · Algebra foundations: expressions, equations and rootsSolve equations with reversible steps
Preserve the solution set, classify exceptional cases and check the original equation.
Open lessonLesson · Science · Earth and space scienceSolve for an effective radiative equilibrium
Use Kelvin and the fourth-power emission relationship to locate the balance condition.
Open lessonLesson · Algebra reasoning, constraints and solution evidenceSolve inequalities with an unknown coefficient
Separate sign cases before dividing by a parameter.
Open lessonLesson · Linear Algebra II: operators and spectral structureSolve least squares while exposing lost directions
Construct the Moore–Penrose inverse, prove minimum-norm least squares and derive optimal truncated SVD.
Open lessonLesson · Algebra II and precalculus: functions and structureSolve logarithmic equations on their real domains
Use logarithms as inverse exponents, expand valid products and reject forbidden candidates.
Open lessonLesson · Science · University Physics II with calculusSolve net-charge constraints on coupled floating conductors
A floating conductor need not be at zero potential or have zero charge on every attached capacitor. Its conserved net charge is the sum of its plate charges. The original network has 2 nF from node 1 to ground, 3 nF from node 2 to ground and 1 nF between the nodes. Ground fixes the reference; the two node charges are specified.
Open lessonLesson · Science · University Physics II with calculusSolve node voltages and audit delivered power
Current paths must be defined before writing a matrix. A 6 V source connects to node 1 through Rs. Node 1 connects to ground through 2 Ω and to node 2 through 1 Ω. Node 2 connects to ground through R2. All resistances are positive, constant and lumped; ground is an explicitly chosen zero potential.
Open lessonLesson · Algebra foundations: expressions, equations and rootsSolve rational equations without restoring excluded inputs
Record denominator restrictions before clearing fractions, and filter candidates afterward.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesSolve recurrences as formal power series
Translate initial conditions into a rational generating function.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsSolve systems without changing their solutions
Explain elementary row operations and trace elimination in an augmented matrix.
Open lessonLesson · Functions in graphs, measurements and growth decisionsSolve target times and justify rounding
Solve contextual exponential target equations with logarithms.
Open lessonLesson · Course 2 · Learning through actionsSolve the Bellman equations
Compute values in a small looping decision process, separate policy evaluation from improvement, and compare exact expectations with sampled returns.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsSolve the RC differential equation with a signed initial condition
For a source emf E, series resistor R and capacitor C, loop balance is E − IR − V = 0.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsSolve triangles from sides or an included angle
Derive the cosine law from a coordinate projection and use it with triangle conditions.
Open lessonLesson · Social studiesSoviet planning, collectivization and the problem of reports
Trace how state targets, procurement and coercion connected industrial policy to rural outcomes.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationSpace curves, speed and curvature
Separate velocity from speed and compute arc length and curvature of regular curves.
Open lessonProject · Multivariable calculus: geometry, change and accumulationSpatial transport dossier: one vessel, three descriptions
Build and revise one dossier connecting a synthetic vessel’s geometry, mass distribution and instantaneous boundary transport. Reconcile independently derived volume and surface calculations, audit a coordinate change, retain numerical evidence and defend a changed-boundary case. This is a supported public investigation; saved text, downloaded calculations and matching an answer do not certify independent proficiency or a physical vessel design.
Open projectLesson · Distribution diagnostics and calibrationSpecified continuous-null calibration
Prove the continuous probability-integral transform.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsSpecify a frame before measuring motion
A position is a coordinate assigned relative to an origin, an axis direction and a time. A motion description is incomplete if any of those choices is hidden. Mark a point on the object rather than switching between its leading edge and center as it moves. State the clock reference and the unit used to calibrate the image or ruler.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsSpecify a test before seeing its answer
Define hypotheses, tail direction and a conditional p-value.
Open lessonLesson · Set specifications and language-expression limitsSpecify membership and construct language expressions
Translate nested-set requirements into predicates and justify language constructions through word membership and complete split cases.
Open lessonLesson · Mathematical statistics: derivation and model criticismSpecify the experiment before estimating
Separate the sample space, parameter, statistic and sampling design.
Open lessonAssessment · Set specifications and language-expression limitsSpecify the filter and prove its expressive limits
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Partial differential equations and Fourier methodsSpecify what it means for a Fourier expansion to recover data
Separate L², pointwise and uniform conclusions.
Open lessonLesson · Science · Chemistry and matterSpeed both directions without moving equilibrium
Lower an assigned barrier and distinguish faster equilibration from a changed equilibrium mixture.
Open lessonLesson · Science · General Chemistry IISpend a finite buffer reserve before using a logarithmic ratio
Buffer calculations need a proton inventory before an equilibrium relation.
Open lessonLesson · Course 1 · Why does AI need computation?Spend a limited compute budget
Balance model size and training data, distinguish active weights from stored weights, and measure a small browser computation.
Open lessonLesson · Stories and creative writingSpend story time where it matters
Use scene and summary deliberately to control pace and emphasis.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsSplit a motion interval at its actual reversal
Negative acceleration does not always mean decreasing speed. With positive initial velocity, a negative acceleration first slows the object. If the same acceleration continues after velocity reaches zero, the object moves increasingly fast in the negative direction. The model must say whether a controller stops the process at rest; this one does not.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsSplit a quotient using comaximal ideals
Construct the map and prove its kernel and surjectivity.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsSplit a rational function into integrable pieces
Use division, factor structure and coefficient identities without losing singularities.
Open lessonLesson · Linear Algebra II: operators and spectral structureSplit a space without losing its kernel
Build complements, prove rank-nullity, and distinguish an internal direct sum from an arbitrary sum.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsSplit a vector into parallel and perpendicular parts
Derive orthogonal projection and explain why it gives the nearest point.
Open lessonLesson · Partial differential equations and Fourier methodsSplit a wave into two traveling profiles
Derive the whole-line wave formula and its data dependence.
Open lessonLesson · Calculus synthesis: regions, motion and approximationSplit an overlap by its radial boundary
Locate intersections, identify which boundary is nearer, and integrate the actual shared region.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeSplit density mass from singular mass
Construct absolutely continuous and singular parts.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeSplit every test set without losing mass
Prove closure of Carathéodory measurable sets.
Open lessonLesson · Course 2 · Learning through actionsStabilize neural Q-learning
Train a small neural action-value model and compare replay sampling with delayed target networks.
Open lessonCourse · Mathematics program unitStable matching and preference guarantees
Find mutually preferred deviations, trace deferred acceptance, and prove exactly what its stability and optimality guarantees require.
Open courseAssessment · Stable matching and preference guaranteesStable outcomes, changed rules and uniqueness
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentConcept · GlossaryStandardization
Subtract a fitted coordinate mean and divide by its fitted spread. Fit statistics on the intended development/training cohort and reuse them; constant coordinates need explicit zero-spread handling.
Open conceptLesson · Science · Environmental Science: ecological systems and evidenceStandardize repeated surveys before calling an ecological trend
A repeated survey can change because conditions changed within habitats or because the sample contains a different mixture of habitats. These are different explanations. Define whether the target is a fixed-composition trend, the current landscape total or the mean of whatever sites happened to be visited.
Open lessonLesson · Course 1 · Simple models and useful baselinesStart with a simple baseline
Fit majority, nearest-neighbor and shallow-tree predictors; compare the same validation cases before adding complexity.
Open lessonLesson · Picard construction and uniquenessState exactly where uniqueness and continuation hold
Compare solutions through their integral equations, continue across controlled intervals, and separate failed hypotheses from actual nonuniqueness.
Open lessonLesson · Algorithm reasoning: invariants, termination and costState models and invariant discovery
Specify a nondeterministic process, repair a weak invariant and prove safety for every permitted transition.
Open lessonConcept · GlossaryState representation
Information used to describe a situation for prediction or decision-making. Missing relevant variables can make different situations look identical.
Open conceptLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceState the offspring denominator before calculating X-linked probabilities
In the stipulated XX/XY system, an XY parent contributes its X to XX offspring and its Y to XY offspring. An X-linked locus without a corresponding Y allele is therefore represented differently in the two classes. This is one specified inheritance system, not a universal model for sex determination across organisms or a description of every individual’s biology.
Open lessonConcept · GlossaryState value (V)
Expected return from a state when following a particular policy. In the one-step bandit, it is the policy-weighted average of action values.
Open conceptLesson · Social studiesState, regime and government are different
The state is the enduring set of territorial institutions; a regime describes rules governing access to and exercise of power; a government is the current leadership.
Open lessonLesson · Social studiesStates, science and Enlightenment
Compare claims about reason and authority with institutional limits.
Open lessonCourse · Mathematics program unitStatistical inference: estimates, tests and defensible conclusions
Eighteen lessons develop proportion, mean and categorical inference with design checks, supplied reference values and written interpretation. Examples are original synthetic data. Public practice and report completion are not independently graded mastery or complete AP Statistics certification; complex surveys, regression inference and advanced resampling need further study.
Open courseProject · Mathematical statistics: derivation and model criticismStatistical investigation: vehicle efficiency, model revision and changed targets
Build one reproducible investigation using the local UCI Auto MPG data pack. Compare a declared mean baseline and two regression candidates on the same complete-case cohort, inspect assumptions, freeze a public chronological validation record and defend a revised manuscript. Then change the response target to fuel consumption rather than merely rerunning the same fit. This supported portfolio does not establish independent mastery, causal effects, or secure unseen evaluation.
Open projectCourse · Mathematics program unitStatistical investigations, simulation and probability decisions
School-entry workflows that construct displays, inspect fitted models, develop simulation evidence and justify probability-based decisions. Original synthetic practice is labeled; public artifacts and written reports require review and do not certify mastery.
Open courseCourse · Mathematics program unitStatistical PCA: coordinates, reconstruction and scaling
Three original data-based lessons derive centered covariance, scores and reconstruction, then test scaling and degenerate directions. Public worked synthetic cases and written review tasks do not establish population structure or independently assessed statistical proficiency.
Open courseCourse · Mathematics program unitStatistics foundations: data, design and sampling variability
Eighteen lessons build statistical investigations, descriptive analysis, probability models and sampling-distribution readiness. This is partial preparation for the Fall 2026 five-unit AP Statistics framework, not complete AP coverage. Confidence intervals, hypothesis-testing procedures and cumulative inference assessment need the separate inference sequence. Original datasets here are synthetic.
Open courseAssessment · Statistics foundations: data, design and sampling variabilityStatistics: design, variation and model selection
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitStatistics: displays, designs and differences
Construct and interpret statistical evidence, plan data collection and reason about differences between independent samples. Six focused lessons extend the school statistics sequence; probability technology and independently scored examination work remain separate.
Open courseCourse · MathematicsStatistics: from samples to qualified conclusions
Audit observations, describe distributions, reason about sampling and interval estimation, test a prespecified hypothesis and inspect regression without confusing association with causation.
Open courseLesson · Multivariable calculus: geometry, change and accumulationStokes' theorem and compatible boundary direction
Relate circulation to curl flux through an oriented spanning surface.
Open lessonLesson · Course 2 · Geometry and control foundationsStop when the controller leaves its limits
Inspect runtime rejection, delayed observations and bounded fallback without equating a stopped task with success.
Open lessonLesson · Science · Physics and physical systemsStore energy in an ideal spring
Connect restoring force to elastic potential energy and compare compression with extension.
Open lessonCourse · English language artsStories and creative writing
Develop a character-driven story, test choices of viewpoint, scene, dialogue and structure, and produce a substantial revision with an adaptation.
Open courseLesson · Linear Algebra II: operators and spectral structureStrengthen triangularization with orthonormal geometry
Prove complex Schur triangularization and the orthonormal spectral theorem for normal operators.
Open lessonLesson · Science · Environmental science and sustainabilityStress-test an energy recommendation
Separate immediate service, stored reserve and future uncertainty.
Open lessonLesson · Science · Environmental science and sustainabilityStress-test water supply and timing
Inspect shortages, spill and storage across supplied dry, pulse and demand scenarios.
Open lessonLesson · Social studiesStress: demands, appraisal, resources and time
Distinguish stressors from responses and evaluate acute versus sustained patterns without diagnosing individuals.
Open lessonCourse · Mathematics program unitStructural induction: constructions, recursion and parsing
Build finite objects, choose induction hypotheses, prove recursive functions correct and distinguish a generated word from its construction. Written arguments need review; finite traces do not establish a universal theorem.
Open courseLesson · Point-set topology: spaces, constructions and separationSubbases and initial topologies
Generate a basis from finite subbasic intersections.
Open lessonLesson · Point-set topology: spaces, constructions and separationSubspaces and relative openness
Construct the topology inherited by a subset.
Open lessonLesson · Science · Introductory AstronomySubtract common acceleration before interpreting a tidal effect
Gravity accelerates the whole body toward a source, but it also varies across the body. A tidal effect concerns that spatial difference rather than the common acceleration. In a freely falling center frame, the near side is pulled toward the source more strongly than the center, while the far side falls behind it. Both inertial accelerations still point toward the source; the opposite relative signs do not require a repulsive gravitational force.
Open lessonLesson · Social studiesSultanates, coinage and cultural authority in South Asia
Use coins and buildings to distinguish a ruler's public claims from actual control and religious uniformity.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceSum interface phases and the complete reflected series
At a lossless normal-incidence interface the electric-field reflection coefficient is r_ij=(n_i−n_j)/(n_i+n_j). A negative coefficient is a π phase reversal; a positive coefficient is not. This interface sign is separate from the propagation phase accumulated while a wave traverses a film.
Open lessonLesson · Statistics foundations: data, design and sampling variabilitySummarize variation without hiding extremes
Calculate mean, median, sample variance and robust alternatives.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsSupply a collision response instead of inventing one from momentum
A short collision with negligible net external impulse preserves the total momentum of the selected pair. It does not automatically preserve kinetic energy or specify both final velocities. Here a 2 kg point moving at 4 m/s meets an initially resting target of mass M on the same line.
Open lessonConcept · GlossarySupport set
Examples supplied for task-specific adaptation. Separate query examples measure the adapted result.
Open conceptLesson · Real Analysis I: completeness, limits and rigorous calculusSuprema, infima and the Archimedean property
Use approximation from below and completeness to prove unboundedness of natural numbers.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusSupremum norm and the uniform Cauchy criterion
Prove completeness of bounded-function spaces from completeness of the scalar field.
Open lessonLesson · Differential geometry of curves and surfacesSurface area and invariant integration
Derive the area element from tangent parallelograms.
Open lessonLesson · Differential geometry of curves and surfacesSurface charts and regular values
Verify that a displayed parametrization describes a surface.
Open lessonLesson · Differential geometry of curves and surfacesSurfaces of revolution and special examples
Reduce surface calculations to a profile curve.
Open lessonSubject · Experiment collectionSystems thinking and operations
Explore queues and bottlenecks before recommending extra capacity.
Open subjectAssessment · Linear algebra over finite fieldsSystems, subspaces and changing scalars
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Point-set topology: spaces, constructions and separationT₀, T₁ and Hausdorff separation
Separate the three point-separation axioms.
Open lessonLesson · Tail limits and root testsTail extrema and subsequential limits
Define limsup and liminf using infinite tails, construct their finite or infinite subsequential limits, and separate sampled bounds from eventual claims.
Open lessonCourse · Mathematics program unitTail limits and root tests
Construct upper and lower limits from entire tails, characterize subsequential behavior, and prove convergence tests without assuming ordinary root limits exist.
Open courseLesson · Differential geometry of curves and surfacesTangent maps and orientation
Define tangent vectors independently of a chosen chart.
Open lessonConcept · GlossaryTanh
A smooth nonlinear activation between −1 and 1. Its slope at an output a is 1 − a², which is used during backpropagation.
Open conceptLesson · Social studiesTaxation and representation: repeal without surrender?
Compare colonial claims with Parliament’s repeal and continuing claim of authority.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusTaylor expansion and quadratic remainders
Reduce finite-dimensional Taylor estimates to one variable along a segment.
Open lessonLesson · Course 1 · What has the network learned?Teach useful invariances
Train on transformed images, check whether labels survive, and compare failure modes under a matched budget.
Open lessonLesson · Stories and creative writingTell the same event from another edge
Control what a narrator can know and compare the effects of narrative viewpoint and distance.
Open lessonLesson · Science · University Physics I with calculusTerminate a draining solution at its physical endpoint
For a large tank with surface area Atank and a small outlet, a quasisteady Torricelli model gives outflow Cd a sqrt(2gH). Mass continuity then gives Atank dH/dt = −Cd a sqrt(2gH). The discharge coefficient is stipulated, and surface kinetic energy is neglected under the disclosed small outlet-to-tank area ratio.
Open lessonLesson · Science · AP Chemistry preparationTest a bond-stiffness claim against isotope evidence
Use actual spectroscopic reference coefficients to separate a mass effect from a local-curvature inference.
Open lessonLesson · Science · Chemistry and matterTest a constant-fraction reaction model
Use half-life and exponential decay to predict a fresh interval and distinguish an assigned law from measured evidence.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsTest a constant-resistance claim against curved data
A resistance inferred from one voltage-current pair is not a test that resistance is constant. To test an ohmic model, collect several current and voltage pairs while controlling relevant conditions. Here the supplied graph places voltage on the vertical axis and current on the horizontal axis, so a straight-line slope has units V/A=Ω. Reversing the axes would make the slope conductance instead.
Open lessonLesson · Social studiesTest a developmental explanation against a changed task
Distinguish a proposed developmental change from language, familiarity and task demands.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceTest a developmental light signal without confusing it with energy supply
Plants use light as both an energy source and a developmental signal. Photoperiodic responses connect light/dark timing with seasonal development. A brief light signal can change a response even when it contributes little photosynthetic energy, so photon energy input and timing information are different explanatory variables.
Open lessonLesson · Science · Introductory AstronomyTest a dimming interpretation against calibration alternatives
A standardizable candle can link observed flux to luminosity distance once its source properties and calibration are controlled. Type Ia supernovae are useful examples, but their light curves and spectra require standardization and population checks; they are not all identical uncorrected bulbs.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceTest a finite-capacity carrier without equating saturation with active transport
A membrane’s hydrophobic interior creates different barriers for different substances. Small nonpolar molecules can use pathways unlike those available to charged or large polar solutes. Channels and carriers provide selective routes; the presence of a protein route does not by itself imply that movement is uphill or directly ATP-driven. Phospholipids are amphipathic: polar heads face water while nonpolar tails form the bilayer interior. Cis-unsaturated tails disrupt close packing compared with otherwise similar saturated tails, tending to maintain fluidity during cooling. Cholesterol can restrain movement in a warmer fluid membrane while disrupting tight packing at lower temperatures; it buffers temperature effects rather than always increasing fluidity. Its packing effects can also reduce passive water/small-solute permeability in model bilayers, but fluidity, permeability and carrier activity are distinct measurements.
Open lessonLesson · Science · Physics and physical systemsTest a friction-limited stopping prediction
Reject unreachable targets and stop applying sliding friction when the block stops.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsTest a historical single-electron atom model against quantization
A nucleus contains positive protons and neutral neutrons. Proton count Z identifies an element; changing neutron count gives isotopes, whereas removing or adding electrons changes charge state. A one-electron ion with nuclear charge +Ze has net charge (Z−1)e. These bookkeeping facts do not require a literal planetary orbit for the electron.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingTest a local harmonic approximation against a nonlinear force
Use a potential derivative to distinguish exact SHM from a local approximation.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceTest a magnetic source geometry after removing a probe baseline
A magnetic-field reading depends on source geometry and sensor response. For the long straight vacuum wire approximation, magnitude B=μ0I/(2πr), and the field circles the wire according to the right-hand rule for conventional current. At radii .05 and .10 m the physical fields differ by a factor of two, but a common additive sensor offset can hide that ratio.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsTest a mean using estimated standard error
Interpret a signed t statistic and supplied tail probabilities.
Open lessonLesson · Science · Scientific inquiry and measurementTest a model beyond its tuning cases
Distinguish curve fitting from transferable explanation.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTest a molecular explanation against independent evidence
Combine controlled model comparisons with an explicitly synthetic functional-evidence packet.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceTest a nonideal branch before treating it as equilibrium
The van der Waals equation supplements an ideal gas with an excluded-volume correction and an attraction term. In reduced variables t=T/Tc, v=V/Vc and p=P/Pc, the supplied model is p=8t/(3v−1)−3/v². The domain v>1/3 is essential; extending the expression through its pole is not a physical compression.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTest a phase boundary before solving a final temperature
Heat is energy transferred because of a temperature difference, not a substance stored inside a body. In a fixed phase with constant specific heat, the energy change is mcΔT. During an ideal phase change at specified pressure, mL can be transferred without changing temperature. A final-temperature equation must therefore be chosen only after checking which phases remain.
Open lessonLesson · Science · Physics and physical systemsTest a photoelectric threshold
Use photon energy and a supplied work function to distinguish no emission from positive maximum kinetic energy.
Open lessonLesson · Science · University Physics I with calculusTest a potential shape with evidence not used to fit it
Suppose the synthetic potential is U=x²+q x⁴+C. The constant C chooses an energy zero; it does not change force or energy differences. Fit a purely quadratic potential using only U(1)−U(0)=1+q J. This gives fitted stiffness2(1+q) N/m, but one energy difference cannot establish the whole spatial shape.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceTest a predeclared segregation ratio with an honest decision boundary
A genetic model predicts a distribution of offspring categories, not an exact finite count. Sampling variation can produce departures even when the model’s assumptions hold. A goodness-of-fit test compares the observed discrepancy with a reference distribution rather than deciding by whether the observed numbers equal the expected ratio exactly.
Open lessonLesson · Statistical inference: estimates, tests and defensible conclusionsTest a proposed success probability
Use null-based uncertainty and check its expected counts.
Open lessonLesson · Course 2 · Testing whether improvement is realTest a proposed system improvement
Expose a hidden regression, set release gates, and promote or restore a version with its evidence attached.
Open lessonLesson · Science · General Chemistry IITest a quantitative selectivity target against the second onset
A separation window needs a numerical removal criterion and a margin, not just two different constants.
Open lessonLesson · Science · University Physics I with calculusTest a rolling constraint before using its acceleration
A no-slip rolling relation is a constraint that friction must be able to sustain. Consider a rigid circular body released on a stationary incline with I = κmR². Translation down the slope gives mg sinθ − f = ma, while center torque and no-slip acceleration give fR = I a/R.
Open lessonLesson · Science · General Chemistry ITest a sample model against an independent quantitative constraint
Fit one measurement, predict another, and distinguish model compatibility from unique identification.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceTest a source estimate against a reacted-tracer counterexample
Endmember mixing uses distinct source compositions to infer contributions to a mixture. A two-source concentration relation is linear when volumes mix consistently, source concentrations are known and stable, and the tracer is conservative along the relevant flow paths. Those are causal assumptions, not consequences of solving the equation.
Open lessonLesson · Science · Introductory AstronomyTest a wavelength calibration with an unused spectral reference
A spectrum maps detector coordinates to wavelength before physical line interpretation. Reflection, refraction or diffraction can disperse different wavelengths to different positions, but a pixel number is not itself a wavelength. Calibration needs identified reference features and a specified model of the coordinate relation.
Open lessonLesson · Science · AP Chemistry preparationTest alloy-property claims against matched reference data
Normalize wire geometry, compare source uncertainty bounds, and diagnose confounded material claims.
Open lessonLesson · Science · General Chemistry ITest an electron budget before assigning geometry
Distinguish bonding electrons, directional domains and a feasible central allocation.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTest an energy target before taking a square root
An ideal spring stores energy kx²/2 when compressed by x from its relaxed length. The square matters: doubling compression multiplies stored energy by four. This model releases a virtual1 kg point from rest at height zero using a 40 N/m spring. After the spring relaxes it detaches, and a frictionless guide sends the point upward under g=10 m/s².
Open lessonLesson · Science · General Chemistry ITest an enthalpy network instead of averaging incompatible paths
Use reversal, scaling and bounded cycle closure as independent consistency checks.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTest an inheritance null with chi-square reasoning
An inheritance model predicts category probabilities before the offspring are counted.
Open lessonLesson · Science · General Chemistry IITest an intermediate profile without singular equal-rate arithmetic
The candidate sequence A→B→C forms B in one step and consumes it in the next.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceTest an inverse-square force after removing a baseline
Coulomb force for two stationary point charges has magnitude k|qQ|/r² and acts along the line joining them. Like signs repel and unlike signs attract. The two reciprocal forces have equal magnitude and act on different recipients; changing one mass would change its acceleration, not break the force-pair relation. This lesson studies force readings rather than asserting that an instrument directly reveals the ideal force.
Open lessonLesson · Science · AP Environmental ScienceTest biomass carbon claims across time and boundaries
Distinguish renewable feedstock, combustion, additional uptake and land-carbon effects.
Open lessonLesson · Science · AP Chemistry preparationTest concentration inference with blanks and two wavelengths
Convert transmitted signals to absorbance, test a calibration proposal and reject unsupported or confounded inferences.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsTest conservation claims with bounded observation error
A measured before-and-after difference does not have to be exactly zero to be compatible with conservation. Conversely, a small numerical difference alone is not enough: compare it with a justified measurement bound and estimate relevant external impulses. The supplied packets are original synthetic observations, not records of a performed physical experiment.
Open lessonLesson · Science · Environmental science and sustainabilityTest demand and explain resource limits
Separate population, per-person demand, constraints and distribution.
Open lessonLesson · Functions in graphs, measurements and growth decisionsTest even and odd symmetry with domains
Check domain symmetry before applying a parity identity.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceTest feedback stability without claiming a global ecosystem forecast
Feedback connects a change in a system to processes that influence later change. A restoring response can damp a disturbance, while reinforcing feedback can amplify it. The observed outcome depends on response strength, delays, interactions and the time step, not merely whether a verbal feedback label sounds negative or positive.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTest how an unchanged organ responds to a shared supply constraint
Tissue identification describes a component, but organ function depends on how components are connected. A supply pathway, exchange surface and downstream demand can jointly constrain performance. Two organs connected to a shared supply do not behave like two isolated containers with independently fixed inputs. A local change can alter another organ’s boundary conditions without changing that organ’s tissue structure.
Open lessonLesson · Science · Earth and space scienceTest how sampling timing changes a sediment estimate
Compare paired interval products with averages and single-sample extrapolations.
Open lessonLesson · Science · General Chemistry IITest ideal and nonideal vapor-composition predictions
For an ideal binary liquid, each partial pressure is its liquid mole fraction times its pure-component vapor pressure at the same temperature.
Open lessonLesson · Science · General Chemistry IITest integrated kinetic models with withheld raw concentrations
Any flexible two-parameter description can look convincing when judged only on the points used to set it.
Open lessonLesson · Science · AP Chemistry preparationTest mixture proposals with matched chromatograms
Use bounded relative migration and changed conditions to distinguish compatible hypotheses from identified substances.
Open lessonLesson · Science · University Physics II with calculusTest numerical decay against passivity and exact time
An RC discharge with normalized initial voltage one satisfies dv/du=−v, where u=t/(RC). Its exact response e^(−u) remains positive and decreases, and normalized stored energy is v². A numerical update can violate those physical features without the underlying circuit becoming active.
Open lessonLesson · Science · University Physics I with calculusTest numerical update order against an exact trajectory
A plausible plotted curve is not evidence that a time integrator is accurate.
Open lessonLesson · Science · University Physics I with calculusTest orbital invariants against an independent exact trajectory
A numerical orbit can preserve one invariant while accumulating trajectory error. Use the normalized central field μ = 1 and initial position (1, 0), velocity (0, 1). The exact orbit is the unit circle with period 2π, energy per mass −0.5 and angular momentum per mass 1.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceTest partner benefits rather than assuming every association is mutualistic
Symbiosis describes close association, not a guaranteed benefit to every participant. Mutualism, commensalism and parasitism concern measured effects in a context, and those effects can change with resources, life stage and the endpoint being measured. A short-term growth response is not automatically lifetime reproductive fitness.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceTest patch persistence without equating occupancy and abundance
A metapopulation perspective follows occupancy across patches, with local extinction and colonization linking habitat units. Occupancy is the fraction of patches containing a population; it is not the number of individuals within an occupied patch. Patch quality, connectivity and sampling detection affect how an observed occupancy pattern should be interpreted.
Open lessonLesson · Science · AP Chemistry preparationTest periodic explanations with ionization evidence
Compare evaluated reference data, explain departures from smooth trends and identify successive-removal shell changes.
Open lessonLesson · Mathematical modeling and senior investigationTest predictions outside the fitting calculation
Match evaluation evidence to the intended population.
Open lessonLesson · Science · Environmental science and sustainabilityTest resilience and explain adaptation
Stress trip availability and revise an urban recommendation using explicit consequences.
Open lessonLesson · Tail limits and root testsTest series using upper limits of roots
Turn an eventual root bound into an absolute tail estimate, detect infinitely recurring large terms, and keep the boundary and sparse-index cases honest.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingTest support feasibility after integrating a distributed load
Require both force and torque balance, then check whether the assumed contacts can supply the reactions.
Open lessonLesson · Science · University Physics II with calculusTest symmetry of the charge-weighted integrand
A symmetric support does not guarantee a symmetric charge distribution. On −1≤s≤1 m, prescribe λ(s)=1 nC/m times (1+αs/(1 m)). The gradient term integrates to zero in total charge, so Q remains 2 nC while its distribution changes. The registered |α|≤0.75 keeps every density positive.
Open lessonLesson · Social studiesTest the boundary between civil law and conscience
How can a toleration argument limit government without removing all regulation?
Open lessonLesson · Course 1 · Why does AI need computation?Test the complete system
Build a test suite and expose what unit, numerical, data, integration and behavior checks catch or miss.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsTest the energy and friction conditions for rolling
Rolling without slipping on a stationary surface requires the point of contact to have zero instantaneous velocity relative to that surface. For a circular body, vCM=Rω in magnitude and aCM=Rα for the corresponding motion. The center moves while the contact point is instantaneously at rest; neither statement means the whole body is at rest.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTest the no-slip condition before partitioning rolling energy
For a rigid body rolling without slipping on a stationary surface, center speed and angular speed obey v=Rω, with consistent signs. That relation is a constraint supported by contact forces, not a universal description of every moving, spinning object. While slipping, translation and rotation must be analyzed separately.
Open lessonLesson · Course 2 · Planning and reasoningTest the parts of a planning system
Compare direct action, search, and refinement under a shared planning budget, then test unfamiliar layouts.
Open lessonExperiment · Programming and software engineeringTest the whole input contract
Which tests expose a missing input guard?
Open experimentLesson · Social studiesTest transport costs in agricultural location
Why might a perishable crop locate nearer a market?
Open lessonLesson · Science · AP Chemistry preparationTest when a pre-equilibrium substitution is defensible
Derive an intermediate substitution, compare exact held-input relaxation and distinguish transient error from persistent consumption.
Open lessonLesson · Science · AP Chemistry preparationTest when adding substance changes vapor pressure
Combine a reference saturation curve with a finite particle inventory.
Open lessonLesson · Science · Physics and physical systemsTest where the small-angle approximation is useful
Compare a linear formula with the full sine restoring law and choose a stated error tolerance.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTest whether a requested straight crossing is feasible
A velocity stated relative to moving water is not the same as velocity relative to the bank. Add vectors: boat relative to ground equals boat relative to water plus water relative to ground. The supplied virtual channel has width 12 m northward and a uniform current eastward. This is a mathematical exercise only, not swimming, boating or rescue advice.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsTest whether agreement between methods is independent evidence
Two methods that agree can still share the same error. A freefall timing method infers gravity fromg=2h/t²; a small-angle pendulum infersg=4π²L/T². Their mechanisms differ, but if both use the same miscalibrated clock, their resulting estimates are not fully independent. The purpose of this capstone is to design evidence that could expose such a shared cause.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceTest whether migration and selection can be swapped
Gene flow moves inherited variants between populations when migrants or their gametes contribute to reproduction. Movement alone is not necessarily genetic contribution. A mixture model must define whether migrants replace residents, add to them or contribute only a fraction of the next generation.
Open lessonLesson · Course 2 · Planning and reasoningTest whether the model helps action
Freeze a model and planner before comparing prediction, control and changed-dynamics outcomes.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceTest which inference survives isolated vacuum expansion
An insulated gas expands into an initially evacuated part of a rigid enclosure. The external boundary does no work and no heat crosses it, so delta U=0. The actual expansion is not a sequence of equilibrium piston states. A pressure-volume curve drawn through intermediate equilibrium states must not be mistaken for its actual work path.
Open lessonLesson · Science · Chemistry and matterTest why buffers resist change but have finite capacity
Compare pH response at different conjugate-pair amounts and diagnose approximation failure after exhaustion.
Open lessonModule · Course 2Testing whether improvement is real
Use independent evidence to test your architectural ideas.
Open moduleLesson · Complex analysis: holomorphic functions, contours and mappingsThe argument principle
Relate logarithmic-derivative residues to zero and pole orders.
Open lessonLesson · Social studiesThe Compromise of 1850: measures, coercion and resistance
Disaggregate a legislative settlement, inspect unequal procedures and connect federal enforcement with a documented freedom-seeking case.
Open lessonLesson · Homotopy, fundamental groups and covering spacesThe covering lifting criterion
Prove necessity of the subgroup inclusion for a based lift.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryThe disk model and conformality
Transport hyperbolic geometry into a bounded picture.
Open lessonLesson · Multivariable calculus: geometry, change and accumulationThe divergence theorem and closed-boundary flux
Convert outward flux to volume accumulation and account for missing surfaces and singularities.
Open lessonLesson · Differential geometry of curves and surfacesThe exponential map and normal neighborhoods
Prove a local shortest-path result from radial orthogonality.
Open lessonLesson · Differential geometry of curves and surfacesThe Gauss equation and intrinsic curvature
Prove that Gaussian curvature is determined by the metric.
Open lessonLesson · Differential geometry of curves and surfacesThe Gauss map and oriented area change
Interpret curvature as the derivative of the normal map.
Open lessonLesson · Axiomatic Euclidean and non-Euclidean geometryThe hyperbolic upper half-plane
Derive distances and geodesics in a concrete non-Euclidean model.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusThe implicit function theorem
Solve constraints locally using an invertible derivative block and derive sensitivities.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusThe inverse function theorem
Use a local contraction to explain why an invertible derivative gives a local inverse.
Open lessonLesson · Social studiesThe Jamaica Letter: unity at what scale?
Compare regional government with cooperation among states.
Open lessonLesson · Differential geometry of curves and surfacesThe metric in surface coordinates
Measure tangent lengths and angles with a Gram matrix.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusThe multivariable chain rule and local estimates
Compose derivative maps while controlling how an inner remainder changes scale.
Open lessonLesson · Social studiesThe national bank: powers, means and evidence
Compare constitutional tests for a bank and distinguish authorization, ownership and performance.
Open lessonLesson · Social studiesThe SEC account incident: authenticate the action, not just the account
Account identity, message authorization and the underlying institutional action are separate questions. A familiar account can be compromised; several reposts may repeat the same unauthorized claim. Verification needs a dated authoritative action record and an independent route to it.
Open lessonReading · Literary inquiry and interpretationThe Story of an Hour — complete reading
Kate Chopin’s public-domain short story with 23 stable paragraph links, vocabulary and edition notes.
Open readingLesson · Differential geometry of curves and surfacesThe surface connection in coordinates
Differentiate tangent fields without leaving the surface.
Open lessonLesson · Social studiesThe Truman Doctrine: infer a mechanism from a proposal
Foreign policy combines diagnoses, instruments and predicted consequences. A presidential speech directly supports claims about what its speaker proposed and argued; evaluating the diagnosis or the result requires additional evidence.
Open lessonLesson · Social studiesThinking: search strategies, framing and base rates
Compare efficient heuristics with systematic search and check a judgment against the relevant reference group.
Open lessonLesson · Point-set topology: spaces, constructions and separationTietze extension by successive corrections
Construct a bounded extension using Urysohn separators and geometric residual control.
Open lessonLesson · Social studiesTitanic reporting: compare what could be known when
Publication time and evidence access shape what a report can establish. Later information can correct an early claim, but lateness alone does not guarantee reliability. Trace dependence: several outlets repeating one dispatch are not several independent witnesses.
Open lessonConcept · GlossaryToken and tokenizer
A token is an entry in a model’s vocabulary. A tokenizer converts text into token IDs, often splitting words into smaller pieces.
Open conceptConcept · GlossaryTool call
A structured request to an externally implemented operation. A proposal only becomes an action if the execution runtime accepts and runs it.
Open conceptLesson · Point-set topology: spaces, constructions and separationTopological spaces and open sets
Check the three topology axioms on a finite family.
Open lessonAssessment · Point-set topology: spaces, constructions and separationTopology: distinguish hypotheses, models and invariants
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Real Analysis II: uniform limits, metric spaces and local calculusTotal boundedness and finite nets
Distinguish finite approximation at every scale from ordinary boundedness.
Open lessonLesson · Social studiesTrace a claim to its origin
Provenance identifies where information came from and how it changed.
Open lessonExperiment · Money, economics and businessTrace a loan balance
When does making a payment still leave more debt?
Open experimentLesson · Science · Biology and living systemsTrace a negative-feedback response
Connect a disturbance, sensed deviation and corrective response over time.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrace a reading frame from initiation to termination
Translation reads mRNA codons through a ribosome and links amino acids into a polypeptide.
Open lessonLesson · Science · Engineering design and scientific computingTrace a sensor–controller–actuator loop
Connect measurement, requested power and true temperature without confusing information with energy.
Open lessonLesson · Social studiesTrace a signal across two neurons
Separate electrical signaling within a neuron from chemical signaling across a synapse.
Open lessonLesson · Social studiesTrace a tariff from prices to lost surplus
Change a tariff, inspect a fixed-scale market graph and reconcile effects on buyers, sellers and government.
Open lessonLesson · Social studiesTrace a tax wedge and a shared-service decision
Does the person sending tax payment necessarily bear all its burden?
Open lessonLesson · Science · Earth and space scienceTrace a temperature-dependent feedback loop
Distinguish an evolving feedback flux from an imposed parameter and stable amplification from runaway.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrace absorbed material through distinct vascular routes
Absorption does not send every nutrient directly into the same vascular path. Many monosaccharides and amino acids enter intestinal capillaries and travel through portal blood to the liver before reaching the general circulation. Enterocytes reassemble many long-chain lipids into transport particles that enter lacteals, pass through lymph and subsequently reach venous blood.
Open lessonLesson · Science · Earth and space scienceTrace and repair rock-cycle pathways
Compare alternative routes and diagnose a missing transformation without inventing a fixed cycle.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrace antiparallel replication and repair limits
DNA replication is semiconservative: each daughter duplex contains one parental strand and one newly synthesized strand.
Open lessonLesson · Science · Chemistry and matterTrace both pathways through a galvanic cell
Interrupt electronic or ionic transport and reconcile material inventories with transferred charge.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrace circulation before inferring a valve state
The heart has two atria that receive blood and two ventricles that eject it. Systemic venous return enters the right atrium, passes the tricuspid valve to the right ventricle and exits through the pulmonary valve to the lungs. Pulmonary venous return enters the left atrium, passes the mitral valve to the left ventricle and leaves through the aortic valve to systemic tissues. Arteries carry blood away from the heart and veins toward it; oxygen content is not their defining distinction.
Open lessonLesson · Social studiesTrace constitutional change through Amendments XI–XIX
Read locally packaged constitutional provisions, annotate their mechanisms and test a claim against connected clauses.
Open lessonLesson · Social studiesTrace cultural diffusion without flattening people
Can adoption reveal how an idea traveled?
Open lessonLesson · Stable matching and preference guaranteesTrace deferred acceptance and prove stability
Use provisional holds and permanent rejection history to prove termination, completeness and stability.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrace digestive transformations without losing atoms
The alimentary route passes from mouth through pharynx, esophagus, stomach, small intestine and large intestine before elimination. Mechanical processing and peristaltic or segmentation movements change particle size, mixing and location; chemical digestion breaks particular bonds. Absorption crosses the epithelial boundary, so movement through the lumen and entry into the body are distinct events.
Open lessonLesson · Course 1 · How does a network learn?Trace disappearing and exploding gradients
Follow exact derivatives through depth and time, compare identity residual paths and inspect globally clipped parameter updates.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrace distinct neural and hormonal routes through a feedback axis
The hypothalamus links neural information with endocrine regulation. Anterior pituitary secretion is controlled in part by hypothalamic signals delivered through a portal vascular system; posterior pituitary terminals release hormones synthesized by hypothalamic neurons. Anterior glandular tissue and posterior neural tissue therefore have different anatomical relationships to the hypothalamus. A concrete axis is hypothalamic TRH → anterior-pituitary TSH → thyroid T3/T4. Thyroid hormones feed back negatively on hypothalamic and pituitary signaling. If thyroid response to the same TSH falls, the immediate tendency is lower thyroid output; reduced feedback can then increase upstream drive, without guaranteeing complete restoration. In contrast, posterior-pituitary release of ADH and oxytocin comes from hormones synthesized in hypothalamic neurons and transported to their terminals, not from the same anterior-pituitary tropic relay.
Open lessonLesson · Science · Introductory MicrobiologyTrace DNA copying without assuming balanced inheritance
Semiconservative copying gives each daughter DNA duplex one template strand and one newly synthesized strand. DNA polymerases extend a new strand in the 5′ to 3′ direction by adding to its 3′ end. At a moving replication fork, antiparallel templates therefore require continuous leading-strand synthesis and discontinuous lagging-strand fragments. Priming provides an extendable end; ligation joins processed fragments. These molecular roles are not a method for replicating or engineering a microorganism.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrace energy coupling through an activated intermediate
Compare ATP-consuming pathways and audit intermediates, products and open-system limits.
Open lessonLesson · Science · AP Environmental ScienceTrace energy pathways and test a food-web explanation
Analyze trophic energy and distinguish a conditional pathway model from causal population evidence.
Open lessonLesson · Social studiesTrace exchanges from buyer and seller schedules
Which trades remain worthwhile after costs rise?
Open lessonLesson · Social studiesTrace government borrowing and a debt stock
Can public borrowing alter the private financing comparison?
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTrace lost oscillator energy through a finite thermal receiver
A mechanical oscillator that settles loses its organized mechanical energy, but that energy is not destroyed. To infer a temperature rise, first name the receiver and the rest of the environment. The chosen receiver may capture only part of the energy, and its heat capacity is independent information, not automatically equal to the oscillator’s mass.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrace one flux through multiple tissue barriers
An organ combines multiple tissues rather than one repeated cell type. An intestinal wall, for example, contains an epithelial interface, connective support with vessels, muscle layers and neural elements. Each contributes a different causal role. Changing an epithelial transporter, matrix thickness or vascular boundary can alter transfer through the whole wall even if the other structures are unchanged.
Open lessonLesson · Science · Introductory MicrobiologyTrace peptide presentation without assuming activation
Adaptive T-cell recognition depends on a peptide displayed with a compatible major histocompatibility complex context, not simply on the presence of a whole microorganism. In the canonical distinction, MHC I on most nucleated cells presents peptides to CD8-associated T-cell recognition, while MHC II on professional antigen-presenting cells supports CD4-associated recognition. Specialized exceptions and cross-presentation mean these are organizing pathways rather than exhaustive rules for every cell state.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrace photosynthetic electrons and proton coupling
Repair thylakoid topology and distinguish two causes of reduced ATP production.
Open lessonLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceTrace plant generations without treating spores as gametes
Land plants alternate a multicellular diploid sporophyte with a multicellular haploid gametophyte. These generations are connected by meiosis and fertilization, but those processes do not occur at the same transitions as in the familiar animal life cycle. Naming each stage prevents chromosome accounting from becoming a memorized list.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsTrace polar curves instead of treating radius as height
Recognize circles, signed radii and repeated tracing through coordinated tables.
Open lessonLesson · Science · Environmental science and sustainabilityTrace pollution from source to receptor
Distinguish source strength, transport, contact and biological response.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrace reception, relay and cellular response
A receptor converts a binding event into a change in intracellular activity.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrace reproductive feedback without forecasting a cycle
Ovarian follicles and the uterine lining respond to coordinated endocrine signals but are not the same tissue cycle. FSH and LH support follicular and gonadal processes; estradiol from developing follicles contributes to endometrial proliferation. After ovulation, the corpus luteum produces progesterone and other signals that support a secretory endometrial state. If luteal support declines without the relevant pregnancy signals, hormone withdrawal contributes to menstrual shedding.
Open lessonLesson · Science · Environmental science and sustainabilityTrace resources into useful service
Compare conversion chains and environmental boundaries.
Open lessonLesson · Social studiesTrace resources, assignment and achievement in Project STAR
Examine an actual class-size experiment and audit what happens when assignment, attendance and missing outcomes diverge.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrace respiration and explain fermentation’s carrier-recycling role
Audit carbon, reusable electron carriers and ATP investment across three finite glycolysis preparations.
Open lessonLesson · Science · Environmental science and sustainabilityTrace storage limits and dispatch losses
Distinguish capacity, power, internal energy and delivered output.
Open lessonLesson · Science · Physics and physical systemsTrace the upright virtual image of a diverging lens
Use a negative focal length and compare smaller virtual images across object distances.
Open lessonLesson · Science · Environmental science and sustainabilityTrace urban heat, water and materials
Compare mechanisms and controlled methods alongside a repair ledger.
Open lessonLesson · Mathematics for AI · M03Trace vectors through a matrix
Compute dot products, check shapes and compose transformations in order.
Open lessonLesson · Science · AP Environmental ScienceTrace water pollutants from source to watershed response
Distinguish source categories, transport and wetland retention without confusing load with concentration.
Open lessonLesson · Science · Physical Geology and Earth SystemsTrack a coastal sediment cell before calling redistribution a loss
Waves approaching obliquely can drive longshore transport, while rivers supply sediment and offshore pathways remove or temporarily store it. Beaches, spits and bars are parts of a connected sediment system. A change at one beach does not necessarily equal the net change of the whole coastal cell.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsTrack a dilation beyond the origin
Follow every point under a dilation and distinguish a line through the center from a parallel image.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceTrack a Gaussian phase without changing its position density
A normalized Gaussian can carry more information than its visible density. Take psi(x)=(2πsigma²)^−1/4 exp[−x²/(4sigma²)+i beta x²/(4sigma²)], centered at zero. The real parameter beta is a quadratic phase chirp. Its magnitude squared is the same Gaussian position density for every beta, with intrinsic position standard deviation sigma.
Open lessonLesson · Science · General Chemistry IITrack a protonated anion family under an imposed pH
A protonation equilibrium can change free-ion availability without changing the dissolution constant.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTrack an internal energy release in two frames
Two points of masses 1 and 3 kg initially move together at ground velocity U. An internal store separates them with relative final speed u=v1−v2, with no external impulse. Their total momentum remains 4U even though their kinetic energy can increase. The energy source is the internal store, not a violation of momentum conservation.
Open lessonLesson · Euclidean and coordinate geometry foundationsTrack angles through a transversal
Connect vertical and supplementary angles with theorems about parallel lines.
Open lessonLesson · Science · Chemistry and matterTrack atoms and end groups as polymer units join
Build an ideal ester-linked fragment and account for every atom in the product, leftover monomers and water.
Open lessonLesson · Science · Biology and living systemsTrack carbon through fixation and respiration
Reconcile plant carbon gain or loss with exchanges across a system boundary.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsTrack charge transfer before calculating recipient forces
Separate a conserved signed charge account from force on one recipient.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceTrack cofactor regeneration across fermentation branches
Glycolysis reduces NAD+ to NADH, so sustained flux needs a way to regenerate oxidized carrier. Fermentation transfers those electrons to an organic acceptor derived from metabolism. In the standard lactate route, pyruvate is reduced; in the ethanol route, carbon is released as CO2 before the remaining organic intermediate is reduced.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsTrack covariance when adding uncertain quantities
Separate linearity of means from additivity of variances.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrack DNA and chromosome number through a cell cycle
A chromosome is counted by its centromere organization, while DNA content measures the amount of DNA.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrack energy flow and distinguish production from standing biomass
Energy enters ecosystems through primary producers and moves through food webs as organisms consume organic material.
Open lessonLesson · Science · Physics and physical systemsTrack energy through a phase change
Use the heating curve to separate warming from melting.
Open lessonLesson · Science · Chemistry and matterTrack energy through warming and melting
Predict temperature and liquid fraction from a conserved energy ledger.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsTrack gauge pressure across a density interface
A small static fluid element is in force balance. The pressure on its lower face exceeds that on its upper face by the weight of the intervening fluid divided by area. In a uniform-density region this gives Δp=ρgΔh for a downward depth change. Pressure acts normally on each face; it is not a downward vector like the weight.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceTrack homologs and chromatids through meiosis
Meiosis reduces chromosome-set number through one DNA replication followed by two divisions.
Open lessonLesson · Science · Earth and space scienceTrack hydrogen through the pp-I chain
Cancel intermediates and returned protons while keeping nuclear and atomic bookkeeping separate.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsTrack ideals through a quotient
Prove ideal correspondence and preserve the hypotheses in quotient calculations.
Open lessonLesson · Science · Physics and physical systemsTrack inductor energy and sustained resistor heating
Reconcile source work, stored magnetic energy and dissipated heat in two histories.
Open lessonLesson · Science · Earth and space scienceTrack initial water and remaining capacity
Distinguish final inventory from storage change and locate filling within an hour.
Open lessonLesson · Binary gcd and integer certificatesTrack integer Bézout coefficients through binary gcd
Normalize a common power of two, repair odd coefficient rows before halving, and verify the final gcd with an independent arithmetic certificate.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesTrack inversions with a generating polynomial
Derive the inversion insertion recurrence.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceTrack magnetic storage separately from resistor heat
Self-inductance quantifies how a changing current links changing magnetic flux back to the same circuit. In the supplied ideal RL circuit, a6 V source drives a10 Ω resistor and inductance L from zero current. A finite applied voltage cannot create an instantaneous current jump in the ideal inductor. Initially current is zero, resistor voltage is zero and the full source voltage appears across the inductor in passive sign convention.
Open lessonLesson · Science · Physics and physical systemsTrack mechanical energy during a descent
Follow the exchange between gravitational potential energy and kinetic energy while keeping system boundaries explicit.
Open lessonLesson · Science · AP Environmental ScienceTrack nitrogen form without inventing nitrogen atoms
Distinguish fixation, assimilation, ammonification, nitrification and denitrification.
Open lessonLesson · Science · Introductory MicrobiologyTrack nitrogen species without losing nitrogen atoms
Microorganisms transform nitrogen among chemical forms. Nitrogen fixation reduces dinitrogen into forms that can enter organic synthesis; ammonification returns organic nitrogen to ammonium. Nitrification oxidizes ammonia/ammonium through nitrite toward nitrate, while denitrification reduces nitrate through intermediates toward gaseous products under suitable electron-donor and acceptor conditions. These mechanisms differ from simply removing nitrogen atoms from existence.
Open lessonLesson · Science · Biology and living systemsTrack nitrogen through an ecosystem
Account for uptake, litter and mineralization without creating matter.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsTrack nuclear identity through four decay types
Radioactive decay changes an unstable nuclear state to a lower-energy final state, possibly with a different nucleus and emitted particles. A gamma transition can leave proton and neutron counts unchanged while reducing excitation energy. Thus “decay always creates a different element” is false. Whether a particular isotope can undergo a channel depends on its actual energy structure; balanced symbols alone do not establish instability.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceTrack passive diffusion without an infinite reservoir assumption
Passive transport moves a permeating substance down its relevant chemical or electrochemical potential gradient. In the neutral-solute model here, a concentration difference drives net flux. Individual molecules can cross in both directions even when net movement is toward the lower-concentration compartment.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsTrack refraction without changing the source frequency
Refraction changes ray direction when light passes between media with different speeds, except at normal incidence where direction can remain unchanged. Refractive index is n=c/v, with c the vacuum speed and v the phase speed in the stipulated medium. A larger index means a lower speed, not a larger frequency supplied by the source.
Open lessonLesson · Science · Human Anatomy and Physiology FoundationsTrack renewal without equating surface output with current basal input
Epidermal renewal combines production, differentiation, movement and shedding. Basal progenitor activity supplies cells that change their molecular and structural properties as they move outward. The outer barrier therefore carries a history of earlier production and differentiation, not an instantaneous display of what basal cells are doing now. A count at one layer cannot alone establish the rate in another.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceTrack replication and separation without changing count conventions
A cycling eukaryotic cell normally passes through G1, S, G2 and M. G1 supports growth and preparation, S duplicates nuclear DNA, and G2 precedes segregation. Cells may enter a noncycling G0 state rather than continuously progressing. A stage label describes a biological program, not merely a DNA amount: a DNA measurement alone may not distinguish G2 from a mitotic stage.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceTrack return flows before claiming real water savings
Water withdrawn from a river or aquifer is not necessarily consumed. Some returns to usable downstream water or groundwater; some is evaporated, transpired, exported or degraded beyond the defined recovery boundary. Field application efficiency and basin depletion therefore answer different questions.
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsTrack tangent and reciprocal-function domains
Distinguish ratios from reciprocals and locate zeros and poles.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceTrack template direction and processed transcript length
Transcription makes RNA complementary to a DNA template read 3′→5′, with RNA extended 5′→3′. The coding strand, written in the same 5′→3′ orientation as the RNA, matches its sequence except DNA T corresponds to RNA U. A promoter helps define initiation and which strand is used for a particular gene; neither strand is the universal template for an entire chromosome.
Open lessonLesson · Science · Biology and living systemsTrack water while solute stays behind
Predict water redistribution under an explicitly ideal membrane model.
Open lessonLesson · Partial differential equations and Fourier methodsTrack waves on spheres and inside cones
Interpret the three-dimensional Kirchhoff formula.
Open lessonLesson · Optimization: models, certificates and algorithmsTrack when a shadow price remains valid
Derive value bounds and basis-preserving ranges.
Open lessonLesson · Abstract algebra I: groups and symmetryTrack which structure a map preserves
Prove kernel and image properties and distinguish injectivity from surjectivity.
Open lessonLesson · Literary inquiry and interpretationTrack who knows what, and when
Explain how narrative order, access to consciousness and compressed duration shape the reader’s interpretation.
Open lessonLesson · Course 2 · Learning through actionsTrade off reward and exploration with entropy
Train a bounded continuous actor and twin critics, and compare task reward with action-distribution entropy.
Open lessonLesson · Course 1 · How does AI process language?Train a tiny language model
Train a causal attention decoder, inspect actual gradients, restore checkpoints and compare position-level losses against a count baseline.
Open lessonProject · Project 10 reward-learning extension; five separate transfer cases; human review pendingTrain and evaluate a reward-learning agent
Save three-seed Q-learning models, freeze evaluation and inspect learned, authored and search baselines.
Open projectLesson · Course 2 · Choosing what to do nextTrain and evaluate a small neural router
Train a 6 → 8 → 5 network, inspect its action probabilities, and compare it with a strong rule under the same tool budgets.
Open lessonConcept · GlossaryTraining data
Examples used to fit model parameters or preprocessing. Evaluation evidence must be separated according to the claim, including related groups or time periods when needed.
Open conceptProject · Foundation projectTraining diagnosis project and assessment
Compare matched neural runs, report a controlled repair and solve unfamiliar update cases.
Open projectLab · Foundation lessonTraining-debugging lab
Neural training lesson playground: SGD, Adam, saturation, learning rates and blocked gradients.
Open labLesson · Course 2 · Geometry and control foundationsTransfer a policy to a richer embodiment
Compare planar velocity and 3D-engine torque contracts with explicit missing evidence.
Open lessonLesson · Differential equations: models, methods and dynamicsTransfer functions, startup and periodic input
Carry harmonic response into causal transforms and justify an infinite periodic synthesis.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeTransfer integration along a measurable map
Prove the pushforward integration formula.
Open lessonLesson · Science · Introductory AstronomyTransform a celestial direction into a local horizon frame
The celestial sphere is a directional construction, not a claim that stars share one distance. Right ascension and declination label directions relative to a specified equatorial reference. Altitude and azimuth instead use an observer’s horizon: altitude is positive above it, and the azimuth convention here runs clockwise from north through east. A coordinate value is incomplete without its frame and convention.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsTransform a law while preserving every inverse branch
Use CDF arguments and absolute derivative factors.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceTransform an event pair and preserve its causal class
An event specifies a place and a time; two events need not be successive positions of the same object. Begin by naming the event pair before assigning a speed, duration or length. The original coordinates use c=1: one reference length equals the distance light travels in one reference time. This is a unit convention, not a claim that meters and seconds are interchangeable.
Open lessonLesson · Differential equations: models, methods and dynamicsTransform an initial-value problem into algebra
Track the transform domain and initial terms rather than replacing derivatives blindly.
Open lessonAssessment · Finite Fourier bases, convolution and FFTTransform conventions, singular filters and padding
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Science · University Physics I with calculusTransform displacement and kinetic energy in the same frame
A constant net force2 N acts on a = 2 kg point for2 s. Its velocity increases by2 m/s, and with initial ground velocity v0 its displacement is2v0+2 m. Ground work is2 times that displacement and equals the ground kinetic-energy change.
Open lessonLesson · Science · General Chemistry IITransform equilibrium constants with exact reaction statements
A constant is not a property of a chemical name alone; the complete written equation matters.
Open lessonLesson · Algebra II and precalculus: functions and structureTransform inputs and outputs in the right order
Map points through shifts, reflections and stretches while tracking the transformed domain.
Open lessonLesson · Precalculus: exponential and logarithmic evidenceTransform inverses without losing domains
Recover full inverse rules and solve inequalities using the correct monotonic direction.
Open lessonLesson · Science · University Physics I with calculusTransform relative walking into two ground displacements
An internal displacement is not necessarily a ground displacement. Model a point person of mass m on a translating 80 kg base, with no external horizontal force and initially stationary whole-system center. The model specifies only a relative displacement ℓ; it supplies no forces, walking time or energy conversion.
Open lessonLesson · Science · AP Chemistry preparationTransform the equation and its equilibrium constant together
Construct a target from scaled reversible equations, inspect cancellation and verify multiplicative constants against independent activity expressions.
Open lessonConcept · GlossaryTransformer
A neural architecture combining attention with feed-forward transformations, residual connections, and normalization. Decoder-only models can predict the next token.
Open conceptLesson · Distribution calculations: events, tails and cutoffsTranslate a counting event into a calculator query
Evaluate discrete tails without losing the boundary count, and check the binomial model before interpreting a result.
Open lessonExperiment · Media literacy and civic reasoningTranslate a dramatic headline
What does a relative reduction mean in actual counts?
Open experimentLesson · Science · AP Physics 1: algebra-based mechanicsTranslate a force-time pulse into momentum change
For fixed mass, Newton’s second law can be written Fnet=Δp/Δt over an interval when Fnet is interpreted as the interval-average force. Multiplying gives impulse J=FavgΔt=Δp. For a varying force, impulse is the signed area under its force–time graph. The vector direction of impulse determines the direction of momentum change, not necessarily final velocity. Conversely, on a momentum-versus-time graph the interval slope Δp/Δt gives average net force; the local slope gives instantaneous net force. A rising, falling or horizontal momentum segment therefore indicates positive, negative or zero net force along the chosen axis, not necessarily positive, negative or zero momentum.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsTranslate a projectile between inertial observers
A projectile trajectory is a relationship between positions in a chosen frame. An observer moving horizontally at constant velocity u measures x′=x−ut if both origins coincide at launch. The vertical coordinate is unchanged because the observer has no vertical motion. Time is shared in this nonrelativistic model.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsTranslate geometric operations into quadratic extensions
Prove a degree obstruction and state its limits.
Open lessonLesson · Functions in graphs, measurements and growth decisionsTranslate recursive and explicit sequences
Convert arithmetic and geometric rules in both directions.
Open lessonLesson · Mathematics investigationTrap a root before approximating it
What can a shrinking interval prove about an approximate answer?
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsTreat a larger field as a vector space
Separate algebraic and transcendental elements.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsTreat a motion as a function on the plane
Track every point through translations, rotations and reflections.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceTreat a renewal maximum as a model property, not a harvest instruction
A renewable resource can replenish on a decision timescale, but its replenishment rate may depend on the stock and environment. A nonrenewable stock has negligible replacement over that same timescale, even if geological formation occurs over much longer periods. Classifying a resource as renewable does not mean any extraction rate is sustainable.
Open lessonLesson · Linear Algebra II: operators and spectral structureTreat coordinates as descriptions of a space
Work with polynomial and matrix spaces, then distinguish a map from its coordinate matrix.
Open lessonLesson · Social studiesTreatment literacy: compare evidence and outcomes
Evaluate an educational treatment claim using comparison groups, outcomes, follow-up and uncertainty.
Open lessonCourse · Mathematics program unitTriangle-free coloring certificates
Construct a graph from explicit adjacency rules, verify an upper coloring bound, and prove a matching lower bound by recoloring a hypothetical solution.
Open courseLesson · Differential calculus: techniques and applicationsTrigonometric derivatives need radian measure
Connect circle geometry and limiting rates; preserve domain and angle units.
Open lessonAssessment · School trigonometry: proofs, inverse branches and periodic decisionsTrigonometric reasoning: mixed coordinate, inverse and threshold evidence
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentAssessment · Trigonometry, polar geometry and precalculus representationsTrigonometry and representations: mixed assessment
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentCourse · Mathematics program unitTrigonometry, polar geometry and precalculus representations
Sixteen lessons connect exact trigonometry, periodic models, triangles, vectors, polar and parametric representations. Written and symbolic reasoning is reviewed with rubrics, not silently graded as decimal matching. This unit includes introductory conics and complex polar form; it does not alone establish full AP Precalculus or college analytic-geometry coverage.
Open courseLesson · Multivariable calculus: geometry, change and accumulationTriple integrals in Cartesian, cylindrical and spherical coordinates
Build volume elements and bounds for solids, then choose coordinates that simplify them.
Open lessonLesson · Literary inquiry and interpretationTry a lens, then test what it leaves out
Apply two explicit interpretive methods and compare the details each explains or neglects.
Open lessonLesson · Mathematical modeling and senior investigationTurn a broad concern into a testable decision
Define an output, scope and decision rule before choosing equations.
Open lessonLesson · Social studiesTurn a broad question into a measurable one
Operationalization states how a concept will be observed.
Open lessonLesson · Directed random walks and stationary guaranteesTurn a directed graph into a probability model
Specify outgoing choices and dangling behavior, prove mass conservation, and distinguish weighted path probabilities from graph counts.
Open lessonLesson · Differential equations: models, methods and dynamicsTurn a first-order equation into a product derivative
Normalize the equation and choose the integrating factor on a valid interval.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsTurn a hypothetical solution into a smaller one
Identify a positive integer that strictly decreases.
Open lessonLesson · Domains, invariants and structural countingTurn a modular obstruction into a reachability theorem
Discover an invariant on an integer grid, then solve for move counts to distinguish reversible and forward-only reachability.
Open lessonLesson · Science · Engineering design and scientific computingTurn a need into testable design requirements
Compare insulation choices against separate power and thickness constraints.
Open lessonLesson · Calculus II: series, curves and applicationsTurn a positive tail into an integral comparison
Use the term test, p-series and remainder bounds with their hypotheses.
Open lessonExperiment · Applied mathematicsTurn a production plan into resource needs
How can one table calculate several resource totals?
Open experimentLesson · Discrete structures: counting, graphs and algorithmsTurn a recursive definition into a proved formula
Solve first-order recurrences and use initial conditions in second-order models.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceTurn a sign bracket into a certified root interval
Prove bisection convergence and distinguish interval guarantees from residual heuristics.
Open lessonLesson · Course 2 · Geometry and control foundationsTurn a target into motion
Find alternative target-reaching postures and separate geometric reach, joint limits and obstacle contact.
Open lessonLesson · Optimization: models, certificates and algorithmsTurn a zero gap into coordinatewise conditions
Use primal and dual slacks to certify an optimum.
Open lessonLesson · Abstract algebra I: groups and symmetryTurn abstract multiplication into motion of a set
Prove orbit-stabilizer through a coset bijection.
Open lessonLesson · Mathematics for AI · M08Turn an angle into local motion
Connect trigonometry, rotations, Jacobians, least-squares motion and singularities.
Open lessonLesson · Mathematics investigationTurn an angle into two coordinates
Why can sine or cosine be negative when a triangle side length cannot?
Open lessonLesson · Calculus II: series, curves and applicationsTurn an approximation into a justified guarantee
Use derivative bounds and alternating bounds with correct degree and interval.
Open lessonLesson · Course 2 · Testing whether improvement is realTurn an idea into a falsifiable experiment
Specify a paired baseline, one change, metric, budget and decision rule before final evaluation.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceTurn an occupation probability into a count using the state measure
The Fermi occupation probability for one state is f(E)=1/[1+exp((E−mu)/(kBT))]. It lies between zero and one and equals one half at E=mu. Occupation probability alone is not a particle count. To count occupied states in an energy interval, multiply by the density of states D(E) and integrate over energy.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceTurn an ODE slope into a controlled first-order step
Derive explicit Euler and separate one-step defect from accumulated global error.
Open lessonLesson · Partial differential equations and Fourier methodsTurn boundary problems into coercive variational equations
Separate weak existence from classical regularity.
Open lessonLesson · Mathematical modeling and senior investigationTurn criticism into a documented mathematical improvement
Separate correctness, clarity and scope issues.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesTurn disjoint paths into a cut problem
Reduce vertex-disjoint paths to split-vertex flow.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsTurn division into a continued fraction
Compute finite continued fractions by Euclidean division.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesTurn endpoints into a directed quantity
Compute displacement components from ordered endpoints and preserve units.
Open lessonLesson · Science · Engineering design and scientific computingTurn failed requirements into a useful failure log
Separate a failing criterion, its observed symptom and a proposed cause.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceTurn local defects into a global convergence statement
Prove a one-step error bound and interpret adaptive error control with clear limitations.
Open lessonLesson · Course 2 · Geometry and control foundationsTurn observations into estimates
Distinguish noisy sensor readings, delayed information, filtered estimates and simulator truth.
Open lessonLesson · Linear Algebra II: operators and spectral structureTurn polynomial identities into operator calculations
Prove Cayley–Hamilton and reduce powers and inverses modulo a polynomial relation.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsTurn residue tests into complete solution sets
Solve prime quadratic congruences and combine component roots with CRT.
Open lessonLesson · Course 1 · What has the network learned?Turn sound into features
Connect samples, local frequency measurements and a spectrogram, then expose aliasing and window tradeoffs.
Open lessonLesson · Partial differential equations and Fourier methodsTurn spatial derivatives into frequency multipliers
Fix transform conventions before solving a PDE.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsTurn square existence into a signed count
Prove Euler’s criterion and Gauss’s lemma, including supplementary residue laws.
Open lessonLesson · Mathematics investigationTurn statements into sets and truth values
Is “A implies B” the same as “A and B”?
Open lessonLesson · Optimization: models, certificates and algorithmsTurn tangent information into global bounds
Connect convexity, subgradients and positive-semidefinite curvature.
Open lessonLesson · Mathematical statistics: derivation and model criticismTurn unbiasedness into uniqueness
Prove a complete-family example and the Lehmann–Scheffé conclusion.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsTurn unique factorization into a convergent Euler product
Derive finite-prime Euler products by geometric series.
Open lessonExperiment · Media literacy and civic reasoningTwenty headlines, one observation
How much new evidence does a repost add?
Open experimentConcept · GlossaryTyped intermediate representation
A shared structured form with explicit fields, entities, and relations that connects different processing stages.
Open conceptLesson · Social studiesUncertainty: distinguish precision from missing people
A sample estimate varies across repeated samples. A confidence procedure is evaluated by how often its intervals cover the population value in repeated use under its assumptions. More precision does not repair a biased frame or an ambiguous question.
Open lessonCourse · Artificial intelligenceUnderstand Neural Networks
Build intuition for predictions, errors and learning, then test a classifier.
Open courseLesson · Social studiesUnderstand practices in their context
Anthropology studies human life across time and space through multiple methods.
Open lessonPath · Students · guided experimentsUnderstand your first model
Explain how a network learns, spot overfitting, and evaluate a small classifier.
Open pathLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsUndo a divisor sum by controlled cancellation
Define the Möbius function and prove inversion through a finite double sum.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsUndo a square linear system when possible
Construct inverses by elimination and recognize singularity.
Open lessonLesson · Science · Earth and space scienceUnequal heating and circulation patterns
Connect broad heat transport and winds with rotation and transient weather systems.
Open lessonAssessment · Structural induction: constructions, recursion and parsingUnfamiliar constructions and recursive proofs
Mixed problems with first-response history and explicit reasoning rubrics.
Open assessmentLesson · Real Analysis I: completeness, limits and rigorous calculusUniform continuity and compact domains
Distinguish one shared input tolerance from point-dependent tolerances.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusUniform limits preserve continuity
Use a three-term estimate and distinguish limit regularity from convergence mode.
Open lessonConcept · GlossaryUnit quaternion
Four constrained numbers used to represent a 3D rotation. The component order must be declared; opposite signs represent the same orientation. A quaternion does not describe translation or establish feasible motion.
Open conceptLesson · Social studiesUnited Kingdom: confidence is a relationship
Distinguish a party's own seat total from the support needed to maintain a government. An institutional relationship can depend on conventions as well as written statutes.
Open lessonCourse · Social studiesUnited States history
Hands-on social studies course and project
Open courseCourse · Social studiesUnited States history I: through 1877
Hands-on social studies course and project
Open courseCourse · Social studiesUnited States history II: after 1877
Hands-on social studies course and project
Open courseCourse · ScienceUniversity Physics I with calculus
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceUniversity Physics II with calculus
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseCourse · ScienceUniversity Physics III — thermal, optical and modern evidence
Investigate measurements, fair tests and scientific explanations. Explore authored school, AP preparation and college-level pathways, with course-specific scope and evidence limits.
Open courseLesson · Proof foundations: from claims to argumentsUnpack definitions to build a proof
Use integer witnesses rather than treating divisibility as unexplained symbolism.
Open lessonLesson · Mathematics for AI · M04Update a belief from counts
Distinguish conditional directions, joint mass, expectation and posterior probability.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityUpdate a probability without losing the base rate
Build natural-frequency trees and interpret reverse conditionals.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceUpdate an occupancy claim after repeated nondetection
An occupied site may produce a nondetection because organisms are cryptic, inactive or outside the sampled portion. Occupancy concerns whether a site is used or contains the species under a defined period and scale; detection concerns the observation process conditional on that state. Treating every zero as absence confounds these two quantities.
Open lessonLesson · Course 2 · Planning and reasoningUpdate beliefs from evidence
Use sensor frequencies to distinguish detection from posterior probability and test how base rates and repeated evidence change a belief.
Open lessonLesson · Course 2 · Learning without losing old skillsUpdate without silently changing the contract
Train a real candidate, recompute old/new regression gates and retain both checkpoints for a replayable rollback record.
Open lessonLesson · Social studiesUrbanization: distinguish improvement from population replacement
Urban analysis connects land use, housing, employment, transport and power. A neighborhood average can change because residents' circumstances change or because different residents move in and out.
Open lessonLesson · Point-set topology: spaces, constructions and separationUrysohn separation and a metrization construction
Construct a continuous separator by nested dyadic neighborhoods.
Open lessonLesson · Science · General Biology I: cellular mechanisms and quantitative evidenceUse a bypass to test pathway order without overclaiming
Receptor tyrosine kinases can transmit information when ligand-dependent arrangements bring intracellular kinase domains into an activating configuration. Phosphorylation creates or changes interaction sites and can recruit relay proteins. A kinase transfers phosphate from ATP; a phosphatase removes phosphate. Depending on the target, phosphorylation can activate, inhibit or alter localization rather than always turning a protein on.
Open lessonLesson · Course 2 · Learning through actionsUse a critic to guide learning
Train separate policy and value heads, trace their coupled updates, and audit a biased bootstrap.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsUse a force-position graph to calculate work
When force changes with position, multiply force by displacement only after accounting for that change. Partition the path into small intervals and add force component times interval length. On a force-versus-position graph, this is signed area. Straight segments allow exact triangle and trapezoid geometry without calculus.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsUse a force’s line of action instead of its full lever length
Torque describes the tendency of a force to change rotation about a specified axis. For a lever on the positive x axis, a force at its end produces signed torque rFy. Equivalently τ=rFsinθ, where θ is the signed angle from the outward lever to the force. Counterclockwise torque is positive in this packet.
Open lessonLesson · Mathematics · Functions and graphsUse a function as an input–output rule
Substitute values, follow the order of operations and check the permitted inputs.
Open lessonLesson · Course 2 · Geometry and control foundationsUse a Jacobian
Predict small tool movements, inspect missing directions and compare inverse velocity proposals.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceUse a matching solution without confusing amplitude and flux
Consider a rectangular barrier of height V0 and width L, with identical zero-potential leads on both sides. In each constant-potential region solve the stationary Schrödinger equation; impose continuity of amplitude and derivative at both interfaces. The supplied closed matching result is expressed using e=E/V0 and w=L sqrt(2mV0)/hbar.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceUse a nonzero initial voltage in an RC transient
A capacitor need not start uncharged. This packet begins with 100 µF at 2 V, then an ideal switch connects it through R to a maintained 6 V source. With finite resistor-limited current, capacitor voltage cannot jump at the switching instant. The initial resistor drop is 6−2=4 V, so initial current is 4/R, not 6/R.
Open lessonLesson · Course 1 · How does AI process language?Use a pretrained model responsibly
Inspect token IDs, context budgets, genuine recorded completions, output schemas and source support.
Open lessonLesson · Write with purposeUse a reader’s confusion as evidence
Prioritize substantive revisions from specific feedback and explain which suggestions to accept, adapt or reject.
Open lessonLesson · Science · AP Environmental ScienceUse a reference to test disturbance effects
Interpret before-after contrasts without confusing shared environmental change with treatment effects.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceUse a signed intermediate object in a two-lens system
Two lenses cannot generally be replaced by adding their separated focal lengths or image distances. First solve what the first lens alone would do. Here lens1 at global x=0 has f1=10 cm and object x=−30 cm, so its predicted image is at x=15 cm with magnification−1/2. A second lens at x=d then acts on the rays reaching that plane, not on the original object distance thirty centimetres away.
Open lessonExperiment · Applied mathematicsUse a slope to improve a choice
When does following the downhill direction overshoot?
Open experimentLesson · Science · General Biology II: diversity, organismal mechanisms and population evidenceUse a specificity control before attributing an immune response to memory
Animal defenses combine physical interfaces, cellular responses and signaling. Innate recognition can respond to broad molecular patterns, while vertebrate adaptive responses involve antigen-specific lymphocyte selection. B-cell antibody functions and T-cell functions are not interchangeable, and an immune cell count alone does not establish elimination of a threat.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceUse a testcross to estimate recombination and retain limits
A testcross to a double recessive can make the gametes of a heterozygous parent visible in offspring categories, provided phenotype classification and viability are suitable.
Open lessonLesson · Science · University Physics II with calculusUse a withheld spatial ratio without erasing gain ambiguity
A model can fit one field reading by adjusting its charge even if its geometry is wrong. Here a uniform finite rod of half-length L is compared with a point-source hypothesis. Fit the point-source amplitude using only the first field at distance 1 m and withhold the second at 2 m to test that point hypothesis. A finite rod with both unknown length and amplitude cannot be fitted from that first reading alone. In a second stage, use both readings to calibrate the rod, then reserve a fresh third field at 3 m for validation. An unknown common positive sensor gain multiplies both readings.
Open lessonLesson · Science · Chemistry and matterUse absolute temperature and specify what is held fixed
Separate constant-volume heating from constant-pressure expansion and avoid Celsius-ratio errors.
Open lessonLesson · Science · University Physics I with calculusUse an energy plot as one diagnostic, not a full certificate
For the supplied unit-mass,unit-stiffness harmonic model, x(0) = 1 and v(0) = 0 give exact x=cos t,v=−sin t and constant energy(x²+v²)/2=.5 J. The benchmark runs to10 s with a bounded step count. This exact solution is used as an independent numerical reference, not a measured oscillator record.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsUse an independent field to challenge a potential fit
Separate an exact fit from an identifiable, independently tested source claim.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsUse an inequality before assigning a friction value
A point starts at rest on an incline. Resolve its weight into mg sinθ down the slope and mg cosθ normal to the surface. The normal reaction is mg cosθ because there is no normal acceleration or other normal applied force. These components are parts of one weight vector, not extra forces in addition to weight.
Open lessonLesson · Course 2 · Planning and reasoningUse an informed search
Compare uniform-cost search and A* on a weighted map, audit estimates and reopen a location when a cheaper route arrives.
Open lessonLesson · Science · University Physics II with calculusUse an integral law without inventing a constant field
Two infinite parallel wires need not produce a field with circular symmetry about their midpoint. Put them at x=−0.4 and +0.4 m, with signed currents +2 A and a selectable second current along z. A centered circular path of radius at least 1 m encloses both wires, but is centered on neither individual field pattern.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceUse an oriented normal in a signed flux-change account
Magnetic flux through a flat surface in a uniform field is Φ=BAcosθ, with θ measured from the oriented surface normal. Flux is signed; it is not simply field magnitude times an unlabeled area. The normal also fixes a positive boundary circulation by the right-hand rule. Reversing that convention reverses both reported flux and emf signs without changing the physical event.
Open lessonLesson · Science · University Physics I with calculusUse an unused speed to challenge a resistance law
A force observation at one speed can fit different models. Write the synthetic resistance magnitude as Fref(v/vref)^p, where vref=1 m/s and p is1 or2. The ratio is dimensionless, so Fref always has force units. If rewritten as b v or k v², the two fitted coefficients have different dimensions.
Open lessonLesson · Science · Introductory AstronomyUse angular momentum as well as energy to reconstruct a trajectory
In an isolated inverse-square point-mass field, specific orbital energy ε=v²/2−μ/r and specific angular momentum h=r v_t remain constant. Specific means per unit orbiting mass. Negative energy corresponds to a bound ellipse, zero to the parabolic limit and positive to an unbound hyperbola, but energy alone does not determine the ellipse shape or closest approach.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceUse area weights when survey effort is deliberately unequal
Stratification divides the target population into defined groups before sampling. It can ensure representation of rare habitats and improve precision when groups differ. Allocating equal effort to two habitats does not imply they occupy equal area. Distinguish why samples were allocated from the population quantity being estimated.
Open lessonLesson · Science · University Physics I with calculusUse both area integrals for a nonuniform lamina
A planar center of mass requires two first moments over the same area and the same total mass. Our rectangle spans 0 to 2 m east and 0 to 1 m north, with density σ = σ₀(1 + ax + by). The supplied a and b are nonnegative gradients in inverse metres; total mass is fixed at 4 kg.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsUse both initial conditions in an LC oscillation
Choose one capacitor plate and let q be its signed charge; define i = dq/dt as current toward that plate.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsUse center distance in gravity and orbital comparisons
For an ideal spherical source, exterior gravitational force has magnitude GMm/r², where r is center-to-center separation. Altitude above the surface is h=r−R, not r. Substituting altitude into the denominator would predict a singular field at the surface even though the source-center distance remains finite.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsUse characteristic-p powers as automorphisms
Prove Frobenius and count its fixed points.
Open lessonLesson · Science · AP Physics C: Electricity and Magnetism — calculus field modelsUse circulation when a scalar potential is insufficient
A changing magnetic flux produces an electric field even if no conducting loop is present.
Open lessonLesson · Discrete structures: counting, graphs and algorithmsUse colors to detect structural obstructions
Prove the odd-cycle characterization and separate coloring from drawing aesthetics.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceUse complementary evidence to identify inherited information
DNA stores sequence information in an ordered polymer.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceUse complex projections and test the order of optical elements
The introductory Jones extension represents a fully polarized monochromatic field by two complex components. The original normalized input is E=(1,exp(iδ))/sqrt2, so |Ex|²+|Ey|²=1. Relative phase changes the polarization state even though each component intensity remains one half.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesUse conjugates to divide and measure
Compute complex quotients by a nonzero conjugate multiplier and verify by multiplication.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsUse contact geometry to predict capillary rise or depression
Cohesion refers to attraction between molecules of the same material; adhesion refers to attraction across different materials. Surface tension γ is force per length, with units N/m. It describes the tendency of an interface to reduce area. It is not the same quantity as pressure, although a curved interface can sustain a pressure difference.
Open lessonLesson · Literary inquiry and interpretationUse context without inventing a history
Distinguish textual evidence, editorial history and contextual hypotheses, and cite the version that supports an interpretation.
Open lessonLesson · Optimization: models, certificates and algorithmsUse continuous solutions as bounds, not rounded answers
Compare integer feasibility with the relaxed polyhedron.
Open lessonLesson · Science · Earth and space scienceUse continuum shape as a conditional thermometer
Relate a blackbody wavelength peak to temperature while distinguishing normalized shape from received power.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceUse contrasting patterns without overstating an exact fit
Detection asks whether a change can be distinguished from a specified variability model; attribution asks which causes explain it under tested alternatives. A single global average discards spatial, temporal and vertical information that can help distinguish drivers. Real conclusions combine physical mechanisms, observations and model evaluation rather than one numerical fit.
Open lessonLesson · Optimization: models, certificates and algorithmsUse curvature without confusing local and global guarantees
Analyze Newton step s and safeguarded Hessian approximations.
Open lessonLesson · Course 1 · Data audits and training diagnosisUse data with a purpose
Choose justified fields, examine provenance and permission, and build a practice data card with explicit handling limits.
Open lessonLesson · Statistics foundations: data, design and sampling variabilityUse design to support a causal claim
Plan controls, random assignment, replication, blocking and paired comparisons.
Open lessonLesson · Linear Algebra I: systems, spaces and linear mapsUse determinants to detect collapse and track signed volume
Compute determinants with row-operation rules and qualify geometric interpretations.
Open lessonLesson · Mathematical modeling and senior investigationUse dimensions to reject impossible formulas
Construct dimensionless groups and recognize their limits.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsUse discriminants to constrain Galois groups
Prove the sign criterion and classify irreducible cubic examples with explicit boundaries.
Open lessonLesson · Abstract algebra II: rings, fields and Galois foundationsUse division to control ideals and gcds
Prove Euclidean domains are principal and produce Bézout coefficients.
Open lessonLesson · Social studiesUse education to test a claim about natural inequality
Can unequal training be mistaken for unequal capacity?
Open lessonLesson · Differential equations: models, methods and dynamicsUse eigenvectors to organize coupled rates
Solve diagonalizable constant systems and interpret their modes.
Open lessonLesson · Science · Chemistry and matterUse electrode potentials without multiplying the voltage
Use signed reduction potentials to distinguish reaction direction from equation scale.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsUse energy to reconstruct speed and turning points
For a horizontal ideal spring, choose spring potential zero at equilibrium. Then U=½kx² and total energy E=½mv²+½kx². At either turning point x=±A, velocity is zero, so E=½kA². In the absence of dissipation or external work, this same total applies at every other position.
Open lessonLesson · Science · Physics and physical systemsUse energy to test a stopping target
Find the turning point and reject an unreachable target instead of accepting negative kinetic energy.
Open lessonLesson · Mathematical modeling and senior investigationUse examples to refine rather than certify a claim
Design adversarial tests and prove corrected statements.
Open lessonLesson · Algebra II and precalculus: functions and structureUse exponent laws without losing the base conditions
Compare multiplicative change, equivalent exponential forms and exact equations.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesUse faces to certify planar obstructions
Prove Euler’s formula by deleting cycle edges.
Open lessonLesson · Optimization: models, certificates and algorithmsUse feasible geometry to certify nearest points
Prove convex closure rules and the projection variational inequality.
Open lessonLesson · Mathematical statistics: derivation and model criticismUse geometry to derive sampling laws
Whiten normal vectors and identify independent orthogonal quadratic forms.
Open lessonLesson · Science · Earth and space scienceUse hydraulic head to determine flow direction
Interpret a shared head datum and distinguish a driving drop from the slope of a head profile.
Open lessonLesson · Integral calculus: accumulation, methods and applicationsUse identities to expose an inner derivative
Choose odd-power substitutions or half-angle identities in radian trigonometry.
Open lessonLesson · Science · College Physics II — algebra-based fields and evidenceUse independent acoustic calibration to interpret a decay timescale
A recorded time can have both a scale error and an additive latency. Stationary echo targets at independently known distances0.34,0.68 and1.02 m have physical round-trip times2,4 and6 ms if sound speed is independently340 m/s. The echo instrument indicates g times physical travel time plus a latency. The first two records determine g and latency under those premises; the third is unused validation of the response, not another calibration input.
Open lessonLesson · Science · AP Chemistry preparationUse K and Q to decide the actual direction
Connect standard free energy to K and actual reaction free energy to Q/K.
Open lessonLesson · Science · Physics and physical systemsUse Lenz’s law with a clear viewing direction
Predict circulation by opposing flux change and account for reference signs.
Open lessonLesson · Differential equations: models, methods and dynamicsUse local linearization without claiming a global portrait
Distinguish hyperbolic conclusions, inconclusive centers and nonlinear counterexamples.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsUse microscopic multiplicity without turning likelihood into certainty
A macroscopic description leaves many microscopic details unspecified. In the counting illustration, only the number of particles on the left is reported. Every complete left/right assignment has the same probability, but different reported counts correspond to different numbers of assignments. Thus equally likely microstates do not imply equally likely macrostates.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceUse molecular groups to frame biological predictions
Compare atom inventories, connectivity and specified charge without inventing a pH model.
Open lessonLesson · Course 1 · How does a network learn?Use momentum and adaptive steps
Compare gradient descent, accumulated momentum and Adam while inspecting their state and actual parameter changes.
Open lessonLesson · Science · General Chemistry IUse orbital occupancy to compare bond order and magnetism
Fill a supplied degenerate orbital scheme and check two independent consequences.
Open lessonLesson · Science · AP Biology — mechanisms and evidenceUse per-capita evidence to identify density dependence
Density-dependent effects vary with population density, such as competition for a limiting resource.
Open lessonLesson · Science · Chemistry and matterUse periodic patterns without turning trends into laws
Compare periods and groups using outer-shell arrangements and explain limits of property predictions.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsUse permutation of units to control powers
Prove Euler’s theorem, Fermat’s theorem and the limits of converse primality claims.
Open lessonLesson · Science · AP Environmental ScienceUse pest evidence without confusing control with evolution
Explain resistance selection and integrated pest-management decisions.
Open lessonLesson · Combinatorics and graph theory: enumeration, structure and certificatesUse planar topology in coloring and dual graphs
Prove the five-color theorem with Kempe chains.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsUse polarization to distinguish transverse structure from intensity
An electromagnetic plane wave has electric and magnetic fields perpendicular to each other and to its propagation direction. Its polarization specifies the electric-field oscillation orientation. A longitudinal sound disturbance in an ordinary fluid has no comparable independent transverse direction, so the same polarizer argument does not apply to it. This wave polarization is different from separating positive and negative charges in an electrostatic material.
Open lessonLesson · Science · University Physics III — thermal, optical and modern evidenceUse probability moments rather than reading off a density peak
Consider a supplied position density per unit u=x/L: f(u)=(p+1)(p+2)u^p(1−u) for 0≤u≤1, with positive integer p. Integrating u^p−u^(p+1) gives 1/[(p+1)(p+2)], which proves normalization. Physical position density is f(x/L)/L, not f alone.
Open lessonLesson · Science · General Chemistry IIUse product selectivity to separate indistinguishable decay traces
Fictional isomer A independently forms B and C through irreversible first-order branches with constants kB and kC.
Open lessonLesson · Geometric definitions, motion witnesses and theorem proofsUse proportional cuts and altitude similarity
Prove proportional cuts in both directions and recover Pythagoras from altitude similarities.
Open lessonLesson · Differential equations: models, methods and dynamicsUse proportional rates with a reference state
Connect growth, decay and cooling to initial data and interpretation.
Open lessonLesson · Science · AP Physics 2: algebra-based thermal, electromagnetic and optical systemsUse PV area without confusing path and state
Pressure and volume locate a state on a PV diagram. A sequence of states specifies a path, and the path matters for work. For a quasistatic expansion, the moving boundary transfers energy to the surroundings. The work done by the gas is positive; the AP convention for work done on the gas is negative.
Open lessonLesson · Science · General Chemistry IUse reaction extent to explain a limiting-reactant switch
Derive a piecewise yield bound while keeping every species nonnegative.
Open lessonLesson · Science · Environmental Science: ecological systems and evidenceUse recapture overlap while recognizing weak identification
Mark–recapture methods use the overlap between samples to learn about individuals not observed. The basic idea compares the marked fraction in a later sample with the marked fraction of the population. This requires a defined closed population and a sampling process that makes the marked individuals appropriately representative.
Open lessonLesson · Abstract algebra I: groups and symmetryUse relations without assuming rotations commute with reflections
Describe polygon symmetries with normal forms.
Open lessonLesson · Science · Physics and physical systemsUse reservoir temperatures to bound engine efficiency
Derive the Carnot limit and distinguish a reversible bound from a practical power rating.
Open lessonLesson · Science · Physical Geology and Earth SystemsUse sediment and bedrock evidence to test a glacial history
Glaciers erode, transport and deposit material, while meltwater sorts and redistributes it. Abrasion by debris at the bed can polish and groove rock; plucking removes blocks under suitable bed conditions. Valley modification can produce broad U-shaped profiles, cirques and hanging tributary valleys, but interpreting one shape requires the wider geological context.
Open lessonLesson · Science · General Chemistry IUse signed electronic energy changes without literal orbital trajectories
Connect discrete bound-state energies, absorption/emission and the continuum reference.
Open lessonLesson · Partial differential equations and Fourier methodsUse signs to bound elliptic solutions
Prove comparison before trusting a computed profile.
Open lessonLesson · Science · College Physics I — algebra-based measurement and mechanicsUse system boundaries without inventing a pulling contact
Two initially resting carts touch on a frictionless line. The rear cart has mass 2 kg and the front has selected mass M. A force F acts only on the rear cart. Take rightward, toward the front cart, as positive. The contact can push the carts apart but cannot pull them together: there is no string, glue or magnetic attachment.
Open lessonLesson · Numerical analysis: algorithms, errors and evidenceUse tangent information without promising global success
Derive Newton iteration, quadratic local error and multiplicity effects.
Open lessonLesson · Science · AP Physics C: Mechanics — calculus-based modelingUse the chain rule to audit an oscillator clock
Distinguish a phase shift from a rescaled time axis in a synthetic motion record.
Open lessonLesson · Science · University Physics I with calculusUse the driven phase to close the average-power account
A continuing sinusoidal force supplies energy to a damped oscillator. Consider xddot + c xdot + 4x = 0.2 cos(Ωt) after transients have decayed. A steady response can be written A cos(Ωt − φ), with amplitude and phase determined together by substitution or a complex-amplitude calculation.
Open lessonLesson · Science · University Physics I with calculusUse the force derivative without discarding a flat minimum
A one-dimensional conservative potential defines force by F=−dU/dx. For U=−a x²/2+b x⁴/4 with positive b, force is a x−b x³. Equilibrium requires zero force, not zero potential. Additive potential constants never move these equilibria.
Open lessonLesson · Elementary number theory: divisibility, reciprocity and Diophantine equationsUse the least positive integer as a proof tool
Build divisibility and division with remainder from integer order.
Open lessonLesson · Social studiesUse the right labor-market denominator
Can unemployment fall without anyone finding a job?
Open lessonLesson · Trigonometry, polar geometry and precalculus representationsUse the sine law without losing a triangle
Compare side-angle pairs and enumerate zero, one or two SSA triangles.
Open lessonLesson · Science · General Chemistry IIUse the water constant instead of assuming neutral always means seven
Neutrality is a comparison of acid and base species under a declared activity convention.
Open lessonLesson · Social studiesUse theories to make different predictions
A theory simplifies relationships to explain patterns.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsUse transforms only on their valid domain
Differentiate or identify an MGF with explicit existence conditions.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsUse two conservation laws in an elastic collision
An elastic collision preserves the total kinetic energy of the colliding system. If external impulse is negligible, it also preserves total momentum. Individual objects can exchange both momentum and kinetic energy; neither conservation law says each object keeps its own initial value.
Open lessonLesson · Science · AP Physics 1: algebra-based mechanicsUse two equilibrium conditions for an extended load
Static equilibrium means both zero translational acceleration and zero angular acceleration, with the object initially at rest. For a planar rigid body, solve force components and signed torques about a convenient axis. A torque calculation alone cannot determine whether the body is also translating.
Open lessonExperiment · Physics and engineeringUse two measurements to isolate a fault
Does a low loaded voltage identify a unique problem?
Open experimentLesson · Science · University Physics I with calculusUse v dv/dx without crossing an inaccessible endpoint
For a monotonic piece of one-dimensional motion, the chain rule gives dv/dt=(dv/dx)(dx/dt)=v dv/dx. This can turn a position-dependent force equation into a relation between speed and position, but the selected motion branch must remain explicit.
Open lessonLesson · Science · General Chemistry IIValidate an approximation using depletion and output error
A small equilibrium constant alone does not justify neglecting a concentration change.
Open lessonLesson · Course 1 · Simple models and useful baselinesValidate without reusing the answers
Inspect cross-validation folds, repair duplicate and time leakage, and fit preprocessing inside the training boundary.
Open lessonConcept · GlossaryValidation data
Examples excluded from parameter fitting and used to choose settings. A separate test set evaluates the final frozen choice.
Open conceptConcept · GlossaryValue of information
The expected improvement in a decision made after seeing evidence. Net value subtracts the cost of obtaining that evidence.
Open conceptLesson · Homotopy, fundamental groups and covering spacesVan Kampen and gluing loops
State the open-cover and path-connected-intersection hypotheses.
Open lessonConcept · GlossaryVariance
Average squared deviation from a mean, describing numerical spread. These cohort examples divide by the number of observations; retained variance is not task accuracy or semantic importance.
Open conceptLesson · Course 2 · Geometry and control foundationsVary the world during training
Compare randomized collection with separately changed evaluation conditions.
Open lessonConcept · GlossaryVector
An ordered list of numbers. Matching coordinates are multiplied and added in a dot product; the order of coordinates matters.
Open conceptLesson · Multivariable calculus: geometry, change and accumulationVector line integrals, work and circulation
Compute tangential accumulation and track the effect of direction and parametrization.
Open lessonRefresher · Math refreshersVectors and matrices
Practice the mathematics, then return to your lesson.
Open refresherLesson · Multivariable calculus: geometry, change and accumulationVectors, cross products, lines and planes
Use three-dimensional vectors to construct lines, normals, planes and volumes.
Open lessonConcept · GlossaryVerifier
A procedure that checks a claim or result against specified criteria, such as exact game rules. Its reliability is limited by what those checks actually establish.
Open conceptLesson · Science · University Physics I with calculusVerify a traveling wave through derivatives and energy flux
The ideal small-slope string equation is ytt = (T/μ) yxx, where T is tension and μ is mass per unit length. It follows by balancing the small difference of transverse tension components on a short string segment. The approximation keeps tension constant and neglects terms beyond the small transverse slope.
Open lessonLesson · Orders, graph representations and certificatesVerify graph isomorphisms and refute incomplete invariants
Supply a vertex bijection with an edge/nonedge certificate, or prove that an invariant separates two graphs.
Open lessonConcept · GlossaryVersion check
Comparing a proposed change’s base version with the current artifact version to detect stale writes before accepting them.
Open conceptLesson · Course 1 · Why does AI need computation?Version data and experiments
Reconstruct a result from local data, code, configuration and artifact records; locate changed preprocessing.
Open lessonLesson · Measure, Lebesgue integration and the probability bridgeView conditioning as a contraction and projection
Prove conditional Jensen with supporting affine functions.
Open lessonLesson · Social studiesVotes do not automatically equal seats
District boundaries, electoral formulas and thresholds affect representation.
Open lessonLesson · Social studiesVoting expansion and exclusion in New York
Compare a constitution and amendment, inspect fictional evidence records, and explain how voting expansion could preserve racial exclusion.
Open lessonLesson · Calculus-based probability: laws, dependence and limitsWait for a success target: negative-binomial laws and censoring
Derive the stopped-trial law, its moments and the exact effect of a recording cap.
Open lessonLesson · Social studiesWar becomes global: diplomacy, production and civilians
Trace escalation through contingent decisions and distinguish military plans from realized outcomes.
Open lessonLesson · Social studiesWar with Mexico: disputed borders and treaty promises
Test a war justification, reconstruct the treaty sequence and distinguish citizenship rules from lived outcomes.
Open lessonLesson · Social studiesWar, alliance and peace: which event changed what?
Trace military reversals and compare an alliance, a capitulation and a peace treaty.
Open lessonConcept · GlossaryWeight
A learned multiplier. In our price example, it controls how much the prediction increases for each extra item.
Open conceptCourse · Mathematics program unitWeighted integration and jump conventions
Construct Riemann–Stieltjes sums, prove existence under explicit hypotheses, and distinguish refinement convergence from arbitrary fine-mesh convergence.
Open courseLesson · Discrete synthesis: schedules, sampling and certificatesWeighted precedence and scheduling
Build earliest schedules, prove their optimality and diagnose the resource assumptions behind the proof.
Open lessonLesson · Weighted integration and jump conventionsWeighted sums and refinement
Build weighted Darboux integration from nonnegative increments, prove its refinement criterion, and state exactly how arbitrary tagged sums converge.
Open lessonLesson · Single-variable calculus foundationsWhat a limit says—and what samples cannot prove
Read nearby behavior from tables and described graphs without confusing a limit with the value at a point.
Open lessonLesson · Course 2 · Learning through actionsWhat can the agent observe?
Hide a cart’s velocity, compare aliased observations and recover motion from a short exact history.
Open lessonLesson · Read like a literary detectiveWhat changes when the ending comes first?
Distinguish event chronology from narrative order and explain the effect of withholding information.
Open lessonLesson · Read like a literary detectiveWhat did the text actually say?
Separate a textual observation from an inference and give a plausible alternative explanation.
Open lessonExperiment · Money, economics and businessWhat does the reward encourage?
Can rewarding output increase defects and reduce business value?
Open experimentLesson · Course 1 · What has the network learned?What does this neuron notice?
Draw an image with 25 pixels, inspect each weight, and see what makes a detector respond.
Open lessonModule · Course 1What has the network learned?
Look inside learned features and find the shortcuts that can fool a model.
Open moduleLesson · Course 1 · Data audits and training diagnosisWhat should the machine predict?
Define inputs, outputs, users and success before choosing a model. Test how a changed target changes the labels.
Open lessonLesson · Single-variable calculus foundationsWhen a tangent slope does not exist
Compare one-sided derivatives, jumps and vertical tangents; continuity alone does not promise a finite derivative.
Open lessonLesson · Social studiesWhen an overall result reverses within groups
Compare overall and within-difficulty success rates before crediting an intervention.
Open lessonLesson · Real Analysis II: uniform limits, metric spaces and local calculusWhen derivatives pass through a limit
Use derivative convergence plus one anchor value to recover a differentiable limit.
Open lessonLesson · Mathematics investigationWhen do equations identify one point?
How can row subtraction reveal one, no or infinitely many solutions?
Open lessonExperiment · Systems thinking and operationsWhen faster work creates a bigger pile
Which stage should receive extra capacity?
Open experimentExperiment · Scientific reasoning and experimental designWhen two improvements cancel each other
Is the effect of one intervention independent of the other?
Open experimentExperiment · Design and product developmentWhere should the team improve?
Do equal percentage-point improvements produce equal results?
Open experimentExperiment · Data literacy and statisticsWhich average tells the story?
What changes when the slowest service disappears from the log?
Open experimentLesson · Real numbers, complex coordinates, vectors and matricesWhich number laws survive for matrices?
Compute dimension-compatible products and demonstrate noncommutativity with a small example.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesWhich number types survive arithmetic?
Prove closure of rational numbers under addition and multiplication using integer fractions.
Open lessonLesson · Proof foundations: from claims to argumentsWhich properties survive composition?
Trace witnesses through two functions and distinguish properties of the composite from those of its factors.
Open lessonLesson · Relation closure and directed reachabilityWhich relation properties survive an operation?
Use witnesses and complete finite counterexamples to test inverse, intersection, union and composition of relations.
Open lessonExperiment · Data literacy and statisticsWho answered the survey?
Can a large survey still give the wrong answer?
Open experimentLesson · Social studiesWho can authorize a public action?
A constitution allocates public powers and constrains their use.
Open lessonExperiment · Cybersecurity and digital literacyWho can read the payroll?
Can you remove unnecessary access without breaking legitimate work?
Open experimentLesson · Proof foundations: from claims to argumentsWho chooses first?
Track how witnesses depend on universally chosen values.
Open lessonLesson · Social studiesWho is absent from the dataset?
A sampling frame lists units eligible for selection.
Open lessonLesson · Social studiesWho made this source, and for whom?
Use provenance and intended audience to evaluate what a document can show.
Open lessonLesson · Course 1 · Data audits and training diagnosisWho receives the errors?
Trace a limited review budget through two channels and compare who is missed, reviewed unnecessarily or absent from the evidence.
Open lessonModule · Course 1Why does AI need computation?
Understand the tradeoffs between model size, data, and computation.
Open moduleLesson · Science · Earth and space scienceWhy does air move?
Separate pressure-gradient acceleration, moving-air deflection and surface friction.
Open lessonExperiment · Physics and engineeringWhy double speed needs more than double space
Which part of stopping distance grows fastest with speed?
Open experimentLesson · Real numbers, complex coordinates, vectors and matricesWhy every quadratic has two complex roots
Prove the real-coefficient quadratic case of the Fundamental Theorem of Algebra, counting multiplicity.
Open lessonLesson · Real numbers, complex coordinates, vectors and matricesWhy fractional powers mean roots
Derive positive-base rational powers by preserving integer exponent laws.
Open lessonExperiment · Scientific reasoning and experimental designWhy more measurements can still be wrong
Does repeating a measurement remove every kind of error?
Open experimentLesson · Combinatorics and graph theory: enumeration, structure and certificatesWhy weighted matroid greedy works—and when it does not
Prove optimality with a preserved-prefix invariant and audit signed objectives.
Open lessonLesson · Homotopy, fundamental groups and covering spacesWinding numbers and circle degree
Define degree through lifted arguments.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsWinding numbers and contour topology
Compute the integer winding number of oriented closed curves.
Open lessonLesson · Social studiesWork organizations: rules, discretion and goal displacement
Bureaucratic organization coordinates work through roles, procedures and authority. Rules can improve consistency while producing failures when a proxy target displaces the underlying purpose.
Open lessonConcept · GlossaryWorking memory
Task-relevant information kept available for current decisions, such as the active goal, constraints, and recent observations.
Open conceptLesson · Social studiesWorking memory: maintenance, manipulation and competing tasks
Compare dual-task costs to test resource-sharing hypotheses without treating working memory as a fixed universal number.
Open lessonCourse · Social studiesWorld history and global connections
Hands-on social studies course and project
Open courseCourse · Social studiesWorld history I: early societies–c.1450
Hands-on social studies course and project
Open courseCourse · Social studiesWorld history II: c.1450–present
Hands-on social studies course and project
Open courseConcept · GlossaryWorld model
A model that predicts how an environment changes, conditioned on its state and an action. Prediction errors can compound during planning.
Open conceptLesson · Social studiesWorld War II: connected fronts, civilian lives and persecution
Connect expansion, mobilization and civilian experiences while distinguishing military conflict from targeted persecution.
Open lessonLesson · Social studiesWorld wars, dictatorship and genocide
Explain political breakdown through choices and institutions while centering victims' evidence.
Open lessonProject · AI cumulative project; human review pendingWorld-model project
Saved comparisons, replayed evidence, bounded reports and separate objective transfer assessment.
Open projectStudio · AI implementation practice; independent review pendingWorld-model project implementation studio
Run a local reference, implement a separate learner exercise and preserve reproducible evidence.
Open studioLesson · Research, editing and publishingWrite a feature that keeps its evidence visible
Synthesize records and testimony into an accessible feature while distinguishing reporting, inference and unanswered questions.
Open lessonLesson · Science · Scientific inquiry and measurementWrite a procedure someone can repeat
Make a scientific method traceable.
Open lessonLesson · Course 2 · Testing whether improvement is realWrite a reproducible research report
Connect every claim to a frozen result, reproducibility artifact and limitation.
Open lessonLesson · Mathematical modeling and senior investigationWrite a result another reader can reconstruct
Connect definitions, argument and evidence in clear prose.
Open lessonLesson · Science · Earth and space scienceWrite a supported historical attribution claim
Connect archived comparisons with the IPCC synthesis while preserving causal and observational scope.
Open lessonLesson · Stories and creative writingWrite what characters will not say directly
Write dialogue in which speakers pursue different goals and use action beats to make tension inspectable.
Open lessonCourse · English language artsWrite with purpose
Plan, draft, test and revise an explanatory feature whose claims, structure and language serve a specific audience.
Open courseLesson · Course 1 · How does a network learn?Your first neuron
Turn inputs, a weight, and a bias into a prediction.
Open lessonLesson · Complex analysis: holomorphic functions, contours and mappingsZeros, multiplicity and the identity theorem
Factor a nonzero holomorphic germ by its zero multiplicity.
Open lesson