Courses / Mathematics / Mathematical modeling and senior investigation Mathematics · unit version 1 Mathematical 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 any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
What must you check after proposing a differential-equation solution? A formula must satisfy both the dynamics and the stated scenario.
Compare reasoning and find a refresher Differentiate it, substitute into the equation, and check initial/boundary conditions and admissible ranges. Agreement at one time or with one observation is insufficient.
Review solution verification
Why can a fitted curve fail to support a causal claim? Model evaluation depends on how observations and interventions were obtained.
Compare reasoning and find a refresher Association can result from confounding or selection. Random assignment can support causal comparisons under its assumptions; a small residual by itself cannot identify a causal mechanism.
Review study design
Why record the assumptions used in a mathematical argument? A defended investigation must show exactly where its conclusion applies.
Compare reasoning and find a refresher A proof establishes a conclusion under its hypotheses. Changing a domain, regularity condition or independence assumption can invalidate a step; explicit dependencies let a reviewer test the boundary of the claim.
Review quantified assumptions
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Modeling: defend a model, its evidence and its limits
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Senior investigation: distinguish mechanisms behind the same observations
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Define an output, scope and decision rule before choosing equations.
Specify a measurable model output. Separate prediction from decision loss. Write a falsifiable scope statement. 6 practice tasks.
Lesson details and objectives Choose a boundary and account for every transfer.
Construct a conservation equation. Distinguish internal transfers from losses. Check positivity and units. 6 practice tasks.
Lesson details and objectives Construct dimensionless groups and recognize their limits.
Check dimensional homogeneity. Solve a dimensional exponent system. Separate units from empirical constants. 6 practice tasks.
Lesson details and objectives Scale equations and reconstruct physical predictions.
Choose characteristic scales. Derive a dimensionless equation. Interpret small and large parameter regimes. 6 practice tasks.
Lesson details and objectives Analyze a recurrence and its stability.
Derive a stock recurrence. Solve its fixed point and transient. Compare a genuine discrete model with a discretized ODE. 6 practice tasks.
Lesson details and objectives Use positivity and total balances before simulation.
Derive a two-compartment system. Check an invariant region. Relate equilibrium to observed outputs. 6 practice tasks.
Lesson details and objectives Build a finite Markov model with inspectable probabilities.
Check a transition matrix. Compute one-step and stationary probabilities. State assumptions behind the Markov property. 6 practice tasks.
Lesson details and objectives Separate flow feasibility from shortest routes.
Write node balances and capacity constraints. Certify a maximum flow with a cut. Explain which network abstractions discard time. 6 practice tasks.
Lesson details and objectives Turn a model into a justified optimization problem.
Define objective and feasible set. Construct an optimality certificate. Distinguish robust and average-case decisions. 6 practice tasks.
Lesson details and objectives Derive a regular expansion and detect nonuniform limits.
Compute a first-order perturbation. Find a boundary-layer scale. Derive a diffusion similarity exponent. 6 practice tasks.
Lesson details and objectives Distinguish latent state, sensor output and sampling.
Write an observation equation. Identify aggregation and measurement errors. Design data collection that matches the question. 6 practice tasks.
Lesson details and objectives Separate structural ambiguity from noisy estimation.
Construct indistinguishable parameter pairs. Use observation rank to test a linear model. Design measurements that break an ambiguity. 6 practice tasks.
Lesson details and objectives Derive least squares and use residual patterns.
Compute a simple parameter estimate. Explain loss and noise assumptions. Diagnose residual structure without overclaiming. 6 practice tasks.
Lesson details and objectives Compare derivatives, elasticities and bounded parameter sweeps.
Compute a local sensitivity. Normalize by scale without dividing by zero. Detect interactions missed by one-at-a-time changes. 6 practice tasks.
Lesson details and objectives Compare interval, variance and simulation statements.
Compute an exact interval bound. Propagate variance with covariance. Distinguish parameter, process and model uncertainty. 6 practice tasks.
Lesson details and objectives Match evaluation evidence to the intended population.
Separate calibration and validation. Compute an evaluation error. Diagnose distribution shift and extrapolation. 6 practice tasks.
Lesson details and objectives Compare predictive, mechanistic and complexity evidence.
Calculate a held-out score. Explain the flexibility–fit tradeoff. Select a model using a prespecified purpose. 6 practice tasks.
Lesson details and objectives Use rank, precision and intervention limits.
Compare informative and redundant designs. Calculate a simple information determinant. Separate experimental information from causal assumptions. 6 practice tasks.
Lesson details and objectives Build an auditable numerical evidence package.
Separate verification from validation. Use exact cases and refinement checks. Record enough information for rerunning. 6 practice tasks.
Lesson details and objectives Critique sources without confusing summary with novelty.
State a bounded research question. Trace a claim to its primary source. Compare assumptions before transferring a theorem. 6 practice tasks.
Lesson details and objectives Design adversarial tests and prove corrected statements.
State quantifiers and boundary cases. Construct a counterexample. Distinguish finite enumeration from general proof. 6 practice tasks.
Lesson details and objectives Make computational evidence logically sufficient for its claim.
Prove an invariant or reduction. Specify a finite exhaustive search. Separate floating-point evidence from exact certificates. 6 practice tasks.
Lesson details and objectives Connect definitions, argument and evidence in clear prose.
State a complete result with hypotheses. Explain one calculation with defined notation. Write an abstract whose claims match the evidence. 6 practice tasks.
Lesson details and objectives Separate correctness, clarity and scope issues.
Classify a reviewer concern. Repair a flawed argument. Record revisions without overwriting prior evidence. 6 practice tasks.
Lesson details and objectives Specify evidence before collecting a favorable result.
Choose a coherent investigation route. Define milestones and independent checks. Plan a fallback that still yields mathematical substance. 6 practice tasks.
Lesson details and objectives Assemble a report, reproducible artifacts and a response to critique.
Support each conclusion with traceable evidence. Answer an unfamiliar challenge. Distinguish completed work from independently reviewed achievement. 6 practice tasks.
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