Learn Work through definitions, examples and arguments, then solve unfamiliar problems. Choose a starting point and return to prerequisites whenever you need them.
The comprehensive program is being built. The units below are available now; they do not yet constitute complete school, AP or undergraduate courses. Existing introductory courses and achievements remain available.
Inspect school objectives, linked practice and remaining gaps. The dated evidence maps distinguish topic coverage from learner mastery.
Try the full-length original AP Calculus AB practice form · Try the full-length original AP Calculus BC practice form: separate calculator sections, supplied graphs and worked scoring rubrics. This is public practice, not a calibrated score prediction.
Try the full-length original AP Statistics practice form: 42 multiple-choice questions, two shared-context sets and four ten-point free responses, with worked explanations and review rubrics.
Try the full-length original AP Precalculus practice form: 42 multiple-choice questions and four free responses connecting functions, models and symbolic reasoning, with separate calculator sections and worked rubrics.
Try cumulative discrete mathematics practice D-1: twelve multipart proof and problem-solving questions, 100 explicit practice points and targeted lesson returns. Suggested time is 180 minutes; this is public practice, not a calibrated course examination.
Try cumulative real analysis practice A-1: ten multipart proof questions combining sequences, calculus and uniform limits, with 100 practice points and worked rubrics. Suggested time is 210 minutes; timing and scores are not calibrated to a university examination.
Try cumulative linear algebra practice L-1: twelve multipart questions connect systems, geometric computation and operator proofs, with 120 practice points and targeted lesson returns. The suggested 240 minutes can be split across three sessions; scores are not calibrated to a university examination.
Build a network calibration report: connect complete solution spaces, weighted least squares, sensitivity and method verification, then attempt a separate changed network.
Try cumulative abstract algebra I practice AG-1 or abstract algebra II practice AR-1. Each has ten original multipart questions, 100 practice points and worked proof criteria. These undergraduate forms are separate from school Algebra I and II; timings and scores are not calibrated to a university examination.
Build an algebra presentation manuscript: track exact changes of generators and relations, compare integer and polynomial quotients, and defend a changed-field operator case.
Try cumulative probability practice P-1 or mathematical statistics practice MS-1. Each has twelve original multipart questions, 120 practice points, worked reasoning and targeted lesson returns. Attempt the three sections separately or use the estimated four-hour rehearsal; these are public practice forms, not calibrated university examinations.
Try cumulative multivariable calculus practice MV-1: twelve multipart questions connect spatial geometry, coordinate integrals and boundary theorems, with 120 practice points, worked arguments and changed cases. The four-hour estimate can be split across three sessions; public solutions and self-review do not certify course proficiency.
Try five school cumulative portfolios and four AP repair families: original mixed cases, supplied graph, construction evidence, changed representations and criterion-level review. Public worksheets preserve the existing exam scores and completion requirements.
These selections connect existing lessons, mixed tasks and parts of the public AB/BC forms. They help you assemble evidence across a course; they are not new examinations, required checklists or calibrated university grades. Choose unfamiliar tasks, record which solutions you have seen, and ask a reviewer to check your reasoning. This guide adds no completion requirements.
The boundaries follow the dated Illinois Calculus I/II syllabi and the broader MIT single-variable sequence. Illinois places Newton’s method, shells and work in Calculus I. Some preparation lives in later-named units here; opening it requires no earlier completion. Start with the linked recap and only the selected tasks.
| Capability | Prepare or return | Attempt and explain |
|---|---|---|
| Limits and derivative reasoning | Limits and the limits of sampling; Implicit differentiation | Derivative-definition limit and implicit geometry and related rates. Show the domain and dependent-variable reasoning. |
| Global and local conclusions | Mean value theorem hypotheses; Extrema and boundary candidates | Constrained optimization and theorem quantifiers. Explain why the selected candidate or theorem applies. |
| Newton’s method | Derive tangent steps and inspect a cycle. Review tangent-line algebra; numerical-analysis course completion is unnecessary. | Attempt the lesson’s step and cycle tasks. For further evidence, use the Newton recurrence portion of multiple-root and safeguard practice; its secant/safeguard extension is optional here. |
| Accumulation and representation | FTC and accumulation; Unequal-time measured rates | Signed graph and FTC reasoning and moving-endpoint verification. Separate data assumptions from exact model calculations. |
| Geometric volume | Shells and slicing direction | Compare shells with disks. Explain bounds and why two descriptions agree without counting a shell twice. |
| Physical work | Build work from force and displacement. This topic is hosted in the series unit but does not require series. | Pumping work and physical units: begin with the pumping part. State weight density, lift distance and ignored losses; the hydrostatic-force part provides additional application practice. |
| Capability | Prepare or return | Attempt and explain |
|---|---|---|
| Choose integration methods | Substitution branches and radicals; Complete the square | Trig branches and methods, a shifted quadratic denominator, and parts and partial fractions. Verify a primitive by differentiation. |
| Improper and numerical integrals | Separate singular contributions; Error bounds versus refinement | An internal pole and choose a justified interval count. Explain what the hypotheses establish. |
| Surface and physical applications | Circumference times slant length; Work and pressure | Generating length and surface area plus force and pumping. Identify excluded caps, units and model assumptions. |
| Choose a series test | Root and ratio limits; Geometric tails | A root-test family with no ratio limit, including its changed denominator. Derive a tail guarantee and distinguish it from the exact sum and minimum required term count. |
| Taylor and endpoint evidence | Test both endpoints; Derivative data and error assumptions | Power series and two remainder arguments or a nonzero Taylor center. Explain why point data cannot establish an interval derivative bound. |
| Parametric and polar synthesis | Recover position and length; Choose the overlap region | Planar motion and a polar overlap with changing boundary. Keep displacement, distance, tracing and region selection separate. |
Keep a first attempt, a hypothesis/error log and a repaired argument. A reviewer can select a changed case; record whether it was genuinely unfamiliar. Reusing a public AB/BC question does not make it an unseen university assessment. Workload calibration, broader source coverage and independent learner evidence remain open.
Source boundaries: Illinois MATH 221 (2011/2017 syllabus), Illinois MATH 231 (2017 syllabus), and MIT 18.01SC (Fall 2010).
Construct real rotation planes, distinguish real and complex spectral statements, and prove when an inner product is compatible with multiplication by an imaginary unit.
2 lessons · 12 practice tasks
Open unitProve finite transform identities, diagonalize circular filters and derive a bounded radix-two implementation with explicit normalization and arithmetic costs.
3 lessons · 18 practice tasks
Open unitProve real symmetric positivity criteria, construct Cholesky factors and use exact energy identities to separate unique solutions from singular compatibility and unbounded objectives.
2 lessons · 12 practice tasks
Open unitProve factorization conditions, verify row-permutation receipts and reuse exact factors without confusing singularity, free variables or arithmetic cost with numerical stability.
2 lessons · 12 practice tasks
Open unitProve determinant identities over fields, derive cofactor certificates and test exact and numerical interpretations. Computation supports the proofs rather than replacing their hypotheses.
3 lessons · 18 practice tasks
Open unitSolve 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.
3 lessons · 18 practice tasks
Open unitUse oriented incidence matrices to connect conservation, cycles, potentials and orthogonal certificates. Signed real flows are allowed; capacities and nonnegative-flow constraints require additional models.
2 lessons · 14 practice tasks
Open unitProve infinite-sequence norm inequalities, distinguish coordinate and norm convergence, and construct complete spaces with explicit approximation boundaries.
2 lessons · 12 practice tasks
Open unitBuild an ordered complete field from rational Cauchy sequences, proving arithmetic, order and completeness without assuming real limits in advance.
3 lessons · 18 practice tasks
Open unitConstruct local differential-equation solutions in a complete function space, certify integral iterates, and prove the boundaries of uniqueness and continuation.
3 lessons · 18 practice tasks
Open unitConstruct Riemann–Stieltjes sums, prove existence under explicit hypotheses, and distinguish refinement convergence from arbitrary fine-mesh convergence.
4 lessons · 24 practice tasks
Open unitTurn common continuity control into uniformly convergent subsequences, distinguish a family from its closure, and justify optimization over functions.
3 lessons · 18 practice tasks
Open unitProve how one-sided limits, oscillation and small interval covers determine integrability, then test the theorem on dense and uncountable exceptional sets.
4 lessons · 24 practice tasks
Open unitConstruct upper and lower limits from entire tails, characterize subsequential behavior, and prove convergence tests without assuming ordinary root limits exist.
2 lessons · 12 practice tasks
Open unitProve which information a probability calculation may use, and bound every admissible strategy in a finite reveal process.
2 lessons · 12 practice tasks
Open unitBuild the observation event, distinguish fresh restarts from retained information, and compare exact expected inspection work under explicit sampling assumptions.
3 lessons · 18 practice tasks
Open unitConstruct 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.
4 lessons · 24 practice tasks
Open unitProve stability under unknown comparisons, count independent families of stable outcomes, and adapt proposal invariants to unequal populations.
3 lessons · 18 practice tasks
Open unitConstruct a graph from explicit adjacency rules, verify an upper coloring bound, and prove a matching lower bound by recoloring a hypothetical solution.
2 lessons · 12 practice tasks
Open unitProve halving and subtraction rules, preserve integer Bezout coefficients, and distinguish a plausible answer from an independently checkable gcd certificate.
2 lessons · 12 practice tasks
Open unitProve which operations preserve relation axioms, repair failed closure claims, and derive short directed paths from an extremal argument.
2 lessons · 12 practice tasks
Open unitControl digit ambiguity, build a bijection from two injections, and absorb countably many elements with explicit domain and selection assumptions.
3 lessons · 18 practice tasks
Open unitFind the assumptions that make a composition, accumulated reward, reachability argument or structural count valid, and repair proofs when those assumptions change.
4 lessons · 24 practice tasks
Open unitTurn informal requirements into quantified membership rules, construct languages, and prove the limits of a precisely stated expression grammar.
2 lessons · 12 practice tasks
Open unitSpecify graph transition policies, separate stationary balance from convergence, and prove what a direct restart guarantees.
3 lessons · 18 practice tasks
Open unitDerive a finite probability guarantee, plan a sample size and audit whether the data-collection mechanism supports the proof.
1 lessons · 6 practice tasks
Open unitBuild probability estimates from the sampling mechanism, distinguish events from counts, and turn average waiting times into carefully justified deadline bounds.
2 lessons · 12 practice tasks
Open unitProve eventual comparisons, bound harmonic sums and estimate products while tracking the meaning of the error.
3 lessons · 22 practice tasks
Open unitTranslate relations into orders, construct faithful representations, count directed walks and certify graph equivalence or difference.
4 lessons · 26 practice tasks
Open unitFind mutually preferred deviations, trace deferred acceptance, and prove exactly what its stability and optimality guarantees require.
3 lessons · 18 practice tasks
Open unitTurn 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.
3 lessons · 20 practice tasks
Open unitConnect precedence proofs, combinatorial sample spaces and dependent indicators. Use targeted returns to matching, orders, planarity and modular arithmetic without requiring whole advanced courses.
2 lessons · 14 practice tasks
Open unitEncode 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.
3 lessons · 18 practice tasks
Open unitSpecify 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.
3 lessons · 20 practice tasks
Open unitBuild 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.
3 lessons · 20 practice tasks
Open unitConnect component graphs to paths, reconstruct matrix actions, and distinguish possible from impossible transition histories. These college-preparation extensions are outside AP Precalculus examination weighting.
3 lessons · 21 practice tasks
Open unitConstruct complete noncircle polar curves, restrict their traces, and distinguish signed radius from distance and interpolation evidence without calculus.
2 lessons · 12 practice tasks
Open unitConstruct 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.
4 lessons · 24 practice tasks
Open unitConstruct and read logarithmic vertical axes, recover exponential models, and compare baseline assumptions using original-scale errors and unused predictions.
1 lessons · 8 practice tasks
Open unitIdentify 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.
3 lessons · 19 practice tasks
Open unitBuild 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.
2 lessons · 12 practice tasks
Open unitCompare 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.
3 lessons · 21 practice tasks
Open unitInspect distribution discrepancies, derive valid null references, refit estimated models and defend categorical model revisions.
4 lessons · 24 practice tasks
Open unitThree 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.
3 lessons · 18 practice tasks
Open unitFit 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.
3 lessons · 18 practice tasks
Open unitSpecify grouped-binomial and Poisson responses, derive vector fitting steps, distinguish finite existence from bounded computation, and audit conditional uncertainty and frozen checks.
4 lessons · 24 practice tasks
Open unitComplete 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.
3 lessons · 18 practice tasks
Open unitUse 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.
2 lessons · 12 practice tasks
Open unitConstruct 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.
6 lessons · 36 practice tasks
Open unitSix 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.
6 lessons · 36 practice tasks
Open unitInterpret 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.
4 lessons · 24 practice tasks
Open unitSchool-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.
12 lessons · 72 practice tasks
Open unitSchool-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.
14 lessons · 84 practice tasks
Open unitA 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.
12 lessons · 72 practice tasks
Open unitSchool-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.
8 lessons · 48 practice tasks
Open unitA 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.
11 lessons · 66 practice tasks
Open unitA 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.
12 lessons · 72 practice tasks
Open unitA 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.
16 lessons · 97 practice tasks
Open unitFour 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.
4 lessons · 25 practice tasks
Open unitTen 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.
10 lessons · 50 practice tasks
Open unitTwelve 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.
12 lessons · 76 practice tasks
Open unitTwelve 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.
12 lessons · 72 practice tasks
Open unitSixteen 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.
16 lessons · 96 practice tasks
Open unitA 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.
12 lessons · 52 practice tasks
Open unitSixteen 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.
16 lessons · 96 practice tasks
Open unitTen 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.
10 lessons · 51 practice tasks
Open unitTwelve 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.
12 lessons · 72 practice tasks
Open unitSixteen 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.
16 lessons · 96 practice tasks
Open unitEighteen 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.
18 lessons · 108 practice tasks
Open unitEighteen 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.
18 lessons · 108 practice tasks
Open unitEighteen 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.
18 lessons · 116 practice tasks
Open unitEighteen 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.
18 lessons · 108 practice tasks
Open unitTwenty 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.
20 lessons · 120 practice tasks
Open unitTwenty-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.
25 lessons · 150 practice tasks
Open unitLessons 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.
22 lessons · 138 practice tasks
Open unitTwenty 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.
20 lessons · 138 practice tasks
Open unitA 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.
33 lessons · 198 practice tasks
Open unitA 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.
20 lessons · 138 practice tasks
Open unitTwenty 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.
20 lessons · 128 practice tasks
Open unitProof-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.
46 lessons · 287 practice tasks
Open unitAn 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.
24 lessons · 146 practice tasks
Open unitTwenty-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.
22 lessons · 132 practice tasks
Open unitProof-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.
26 lessons · 156 practice tasks
Open unitA 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.
30 lessons · 194 practice tasks
Open unitTwenty-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.
27 lessons · 164 practice tasks
Open unitTwenty-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.
28 lessons · 172 practice tasks
Open unitA 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.
16 lessons · 96 practice tasks
Open unitA 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.
26 lessons · 158 practice tasks
Open unitA 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.
34 lessons · 212 practice tasks
Open unitThirty 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.
30 lessons · 183 practice tasks
Open unitTwenty-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.
26 lessons · 156 practice tasks
Open unitThirty 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.
30 lessons · 182 practice tasks
Open unitA proof-based advanced undergraduate sequence constructing measures and integrals, proving convergence and Lp theorems, and connecting density and conditional-expectation arguments to probability.
32 lessons · 196 practice tasks
Open unitExplore the introductory mathematics courses · Your saved work and backups
The planned scope includes high-school mathematics, the AP mathematics courses and a full undergraduate major. A linked foundations unit provides part of a course's preparation; it does not mean that the whole course is available or that an examination requirement has been met. Introductory investigations remain available separately.
Expressions, equations, inequalities, systems, functions, polynomial and rational structure, modeling.
Available foundations:
The remainder of this full course is in development.
Euclidean and coordinate proof, constructions, measurement, solid geometry, trigonometry and conics.
Available foundations:
The remainder of this full course is in development.
Polynomial, rational, exponential, logarithmic, trigonometric and polar functions, with the dated AP framework and separate nonexamined motion/matrix/transition extensions.
Available foundations:
The remainder of this full course is in development.
Complete single-variable calculus, series and parametric/polar topics, with AP-format mixed practice.
Available foundations:
The remainder of this full course is in development.
Study design, distributions, probability, sampling and statistical inference, with the dated AP framework.
Available foundations:
The remainder of this full course is in development.
Limits, differential calculus, integration and the fundamental theorem, with modeling and cumulative assessment.
Available foundations:
The remainder of this full course is in development.
Integration methods, improper integrals, sequences, series, Taylor approximation and parametric/polar calculus.
Available foundations:
The remainder of this full course is in development.
Partial derivatives, multiple integrals, coordinate changes and line/surface integrals with integral theorems.
Available foundations:
The remainder of this full course is in development.
Systems, subspaces, rank, linear maps, exact reusable factorizations, positive matrices and energy solves, finite Fourier transforms and FFT, determinant proofs, orthogonality, eigenvalues, least squares and network conservation certificates.
Available foundations:
The remainder of this full course is in development.
Quantifiers, proof methods, sets, relations, recursive structures, cardinality, counting, graphs, algorithm correctness/cost and computability limits.
Available foundations:
The remainder of this full course is in development.
First- and second-order equations, systems, existence/uniqueness, transforms and qualitative modeling.
Available foundations:
The remainder of this full course is in development.
Abstract finite-dimensional spaces, finite-field systems and counting, field-sensitive eigenvalues, operators, spectral theory, quadratic forms and decompositions.
Available foundations:
The remainder of this full course is in development.
Discrete, continuous and joint distributions, conditional expectation, inequalities and limit theorems.
Available foundations:
The remainder of this full course is in development.
Estimation, likelihood, tests, intervals, power, regression and model critique.
Available foundations:
The remainder of this full course is in development.
Real-number construction, completeness, sequences, continuity, differentiation and rigorous integration.
Available foundations:
The remainder of this full course is in development.
Uniform convergence, metric-space arguments, sequence-space completeness, function sequences, local differential-equation construction and multivariable analysis.
Available foundations:
The remainder of this full course is in development.
Groups, actions, homomorphisms, quotients and structure theorems.
Available foundations:
The remainder of this full course is in development.
Rings, ideals, polynomials, field extensions and introductory Galois theory.
Available foundations:
The remainder of this full course is in development.
Analytic functions, contour integration, series, residues and conformal mappings.
Available foundations:
The remainder of this full course is in development.
Conditioning, stability, approximation, numerical integration and differential equations.
Available foundations:
The remainder of this full course is in development.
A substantial independent investigation with mathematical writing, validation and review.
Available foundations:
The remainder of this full course is in development.
Convexity, linear programming, duality, nonlinear optimality, algorithms and sensitivity.
Available foundations:
The remainder of this full course is in development.
Topological spaces, continuity, compactness, connectedness, products and quotients; homotopy, fundamental groups and covering spaces. Homology and higher algebraic topology require further courses.
Available foundations:
The remainder of this full course is in development.
Divisibility, congruences, reciprocity, arithmetic functions and Diophantine reasoning.
Available foundations:
The remainder of this full course is in development.
Enumeration, generating functions, recurrences, extremal arguments and graph proofs.
Available foundations:
The remainder of this full course is in development.
Axiomatic systems, transformations, constructions and contrasting geometric models.
Available foundations:
The remainder of this full course is in development.
Curves, surfaces, curvature, geodesics and intrinsic versus extrinsic geometry.
Available foundations:
The remainder of this full course is in development.
Canonical equations and boundary problems, Fourier methods and solution hypotheses.
Available foundations:
The remainder of this full course is in development.
Measurable sets and functions, Lebesgue integration, convergence and probability-space foundations.
Available foundations:
The remainder of this full course is in development.