Modeling: defend a model, its evidence and its limits
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 300 minutes, as an unvalidated planning estimate. This collection contains 10 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
These ten public synthetic investigations are sample mixed assessment tasks, not an independently reviewed capstone. Worked solutions are available; prior exposure must be disclosed in any subsequent review.
State assumptions, units, admissible parameter ranges and the evidence behind each conclusion. Submit calculations together with model-design and criticism arguments. A saved response or a long draft does not establish correctness, research novelty or professional validity.
Use the suggested duration only for planning. Calculators or reproducible code may support arithmetic, but include enough intermediate work for another reader to check the result. Written arguments and investigation evidence require human review.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
A synthetic storage system starts with 12 m³ of water. Constant inflow is 5 m³/h and commanded outflow is 3 m³/h for two hours. (a) Specify a decision contract for avoiding overflow of a 17 m³ tank, including an observable and decision time. Derive the balance and check every unit. (b) A second proposal assumes an additional unmeasured leak of 0.5 m³/h. Compute both final volumes and explain which observations could distinguish the models. (c) A precaution costs 7 units and prevents a 50-unit overflow loss. Derive the probability threshold for using it. Explain why the two deterministic trajectories alone do not supply that probability. Identify an omitted mechanism that could invalidate either balance.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For a synthetic well-mixed compartment, C′=q/V(Cin−C)−kC, with q=3 L/min, V=12 L, Cin=8 mg/L, k=1/8 min⁻¹ and C(0)=0, derive the exact solution. Set τ=qt/V and c=C/Cin; identify the dimensionless parameter and equilibrium. Now treat ε=kV/q as small and expand the dimensionless equilibrium through first order. Compute its error at the stated ε. Explain why the same dimensionless trajectory can describe another apparatus and why dimensional agreement does not establish equal mixing or valid transfer. State a meaningful positivity check.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
An inventory model is x(n+1)=0.6x(n)+10 with x(0)=10. Find its equilibrium, closed form and x(2); prove convergence. Separately, a two-state machine switches from working to failed with probability 1/5 per day and from failed to working with probability 3/10 per day, conditional only on its current state. Let p(n) be its working probability. Derive the probability recurrence, its stationary value and p(2) if initially working. Explain why these similar affine recurrences describe different objects, and construct one dependence or nonstationarity mechanism that would invalidate the Markov model. What additional evidence would be needed to compare an actual inventory or machine with these predictions?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A directed network has edges s→a capacity 4, s→b capacity 3, a→b capacity 2, a→t capacity 2 and b→t capacity 5; no other edges exist. Find a maximum flow and a cut certificate. A proposed extra unit of capacity can be installed on exactly one edge: s→a or a→t. Determine its effect in each case and explain why inspecting only the chosen edge gives the wrong answer. Then evaluate whether the maximum-throughput objective is adequate if traffic classes have different delay penalties; specify one revised decision contract without pretending the given capacity data determine delays.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A synthetic signal is y(t)=A exp(−kt)+b, with A>0, k>0 and unknown offset b. Exact equally spaced readings at t=0,1,2 are 13,7,4. Recover A,k,b and explain the algebraic identifiability mechanism. Contrast this with observing only t=0 and t=1: exhibit two admissible parameter triples producing 13 and 7. Describe why readings that are almost equal can make the three-reading recovery unstable even though it is structurally identifiable. Design a measurement protocol that checks the offset and separates structural identifiability from practical precision. Explain what happens when A=0 is allowed.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Fit y=ax through the origin to observations (x,y)=(1,2),(2,3),(3,7) by unweighted least squares. Derive the estimator, residuals and residual sum of squares. Compute how the predicted response at x=4 changes when only the third measured y is increased by δ. Under independent zero-mean measurement errors of common variance σ² and an exactly correct through-origin model, derive the variance of the fitted mean at x=4. Distinguish this from predicting one new noisy observation. Finally explain why extrapolation and a missing intercept are model uncertainties not covered by that variance formula.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Two frozen candidate models A and B are tested on a common held-out set. In context I (eight cases), their mean squared errors are 1 and 2 respectively; in context II (two cases), they are 9 and 4. Compute their pooled test errors. If deployment has equal context proportions, recompute the target-weighted errors and choose under that criterion. Explain the assumptions behind reweighting. A team chooses the winner after this comparison and tunes it using all ten test outcomes; state what this does to the evidential status of subsequent scores on the same cases. Propose a finite validation plan with a baseline, subgroup criterion and a genuinely new evaluation, and identify one uncertainty that these ten aggregate summaries cannot resolve.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the recurrence x(n+1)=r x(n)(1−x(n)), a numerical report conjectures that every starting x(0) in [0,1] remains in [0,1] whenever r is positive. Prove a sharp parameter range for this interval-invariance claim and give a counterexample outside it. Compute the first three iterates for r=2, x(0)=1/4 using exact fractions. Design checks that distinguish an implementation error from a false conjecture and from floating-point sensitivity. Specify what must be archived for another person to reproduce an experimental sweep, and explain why checking a million starting values is not a proof of the universal statement.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A hypothetical source proves that x(n+1)=Tx(n) converges to zero for every initial vector when the spectral radius ρ(T)<1. A draft cites it as proving that every matrix with determinant below 1 has decaying iterates, and illustrates the claim with T=diag(2,1/5). Audit the claim, calculate its determinant, spectral radius and the first three iterates from (1,1), and repair the citation and conclusion. Write a short replacement paragraph that distinguishes the cited theorem from your calculation. Supply a review-response record identifying the error, the replacement evidence and a new regression check. Explain what bibliographic and proof details must be checked in the real source before citing this hypothetical theorem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prepare a bounded investigation proposal and defense for the synthetic retention process R(t)=R0 exp(−kt), with t measured in days. Exact pilot observations are R(0)=18 and R(2)=8. (a) Fit k and predict R(4). Compare a constant-rate linear alternative matching the same pilot data; calculate its R(4) and identify a physical-domain problem. (b) Design a new measurement that distinguishes these candidates, including precision, timing, replication and a predeclared comparison rule. (c) Supply a deliverable plan linking a mathematical derivation, reproducible computation, uncertainty/validation evidence, a limitation statement and reviewer questions. (d) Defend what can and cannot be concluded from a future measurement near one prediction. Explain why completing this public task does not itself establish an independently reviewed capstone or a novel research result.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A synthetic storage system starts with 12 m³ of water. Constant inflow is 5 m³/h and commanded outflow is 3 m³/h for two hours. (a) Specify a decision contract for avoiding overflow of a 17 m³ tank, including an observable and decision time. Derive the balance and check every unit. (b) A second proposal assumes an additional unmeasured leak of 0.5 m³/h. Compute both final volumes and explain which observations could distinguish the models. (c) A precaution costs 7 units and prevents a 50-unit overflow loss. Derive the probability threshold for using it. Explain why the two deterministic trajectories alone do not supply that probability. Identify an omitted mechanism that could invalidate either balance.
For a synthetic well-mixed compartment, C′=q/V(Cin−C)−kC, with q=3 L/min, V=12 L, Cin=8 mg/L, k=1/8 min⁻¹ and C(0)=0, derive the exact solution. Set τ=qt/V and c=C/Cin; identify the dimensionless parameter and equilibrium. Now treat ε=kV/q as small and expand the dimensionless equilibrium through first order. Compute its error at the stated ε. Explain why the same dimensionless trajectory can describe another apparatus and why dimensional agreement does not establish equal mixing or valid transfer. State a meaningful positivity check.
An inventory model is x(n+1)=0.6x(n)+10 with x(0)=10. Find its equilibrium, closed form and x(2); prove convergence. Separately, a two-state machine switches from working to failed with probability 1/5 per day and from failed to working with probability 3/10 per day, conditional only on its current state. Let p(n) be its working probability. Derive the probability recurrence, its stationary value and p(2) if initially working. Explain why these similar affine recurrences describe different objects, and construct one dependence or nonstationarity mechanism that would invalidate the Markov model. What additional evidence would be needed to compare an actual inventory or machine with these predictions?
A directed network has edges s→a capacity 4, s→b capacity 3, a→b capacity 2, a→t capacity 2 and b→t capacity 5; no other edges exist. Find a maximum flow and a cut certificate. A proposed extra unit of capacity can be installed on exactly one edge: s→a or a→t. Determine its effect in each case and explain why inspecting only the chosen edge gives the wrong answer. Then evaluate whether the maximum-throughput objective is adequate if traffic classes have different delay penalties; specify one revised decision contract without pretending the given capacity data determine delays.
A synthetic signal is y(t)=A exp(−kt)+b, with A>0, k>0 and unknown offset b. Exact equally spaced readings at t=0,1,2 are 13,7,4. Recover A,k,b and explain the algebraic identifiability mechanism. Contrast this with observing only t=0 and t=1: exhibit two admissible parameter triples producing 13 and 7. Describe why readings that are almost equal can make the three-reading recovery unstable even though it is structurally identifiable. Design a measurement protocol that checks the offset and separates structural identifiability from practical precision. Explain what happens when A=0 is allowed.
Fit y=ax through the origin to observations (x,y)=(1,2),(2,3),(3,7) by unweighted least squares. Derive the estimator, residuals and residual sum of squares. Compute how the predicted response at x=4 changes when only the third measured y is increased by δ. Under independent zero-mean measurement errors of common variance σ² and an exactly correct through-origin model, derive the variance of the fitted mean at x=4. Distinguish this from predicting one new noisy observation. Finally explain why extrapolation and a missing intercept are model uncertainties not covered by that variance formula.
Two frozen candidate models A and B are tested on a common held-out set. In context I (eight cases), their mean squared errors are 1 and 2 respectively; in context II (two cases), they are 9 and 4. Compute their pooled test errors. If deployment has equal context proportions, recompute the target-weighted errors and choose under that criterion. Explain the assumptions behind reweighting. A team chooses the winner after this comparison and tunes it using all ten test outcomes; state what this does to the evidential status of subsequent scores on the same cases. Propose a finite validation plan with a baseline, subgroup criterion and a genuinely new evaluation, and identify one uncertainty that these ten aggregate summaries cannot resolve.
For the recurrence x(n+1)=r x(n)(1−x(n)), a numerical report conjectures that every starting x(0) in [0,1] remains in [0,1] whenever r is positive. Prove a sharp parameter range for this interval-invariance claim and give a counterexample outside it. Compute the first three iterates for r=2, x(0)=1/4 using exact fractions. Design checks that distinguish an implementation error from a false conjecture and from floating-point sensitivity. Specify what must be archived for another person to reproduce an experimental sweep, and explain why checking a million starting values is not a proof of the universal statement.
A hypothetical source proves that x(n+1)=Tx(n) converges to zero for every initial vector when the spectral radius ρ(T)<1. A draft cites it as proving that every matrix with determinant below 1 has decaying iterates, and illustrates the claim with T=diag(2,1/5). Audit the claim, calculate its determinant, spectral radius and the first three iterates from (1,1), and repair the citation and conclusion. Write a short replacement paragraph that distinguishes the cited theorem from your calculation. Supply a review-response record identifying the error, the replacement evidence and a new regression check. Explain what bibliographic and proof details must be checked in the real source before citing this hypothetical theorem.
Prepare a bounded investigation proposal and defense for the synthetic retention process R(t)=R0 exp(−kt), with t measured in days. Exact pilot observations are R(0)=18 and R(2)=8. (a) Fit k and predict R(4). Compare a constant-rate linear alternative matching the same pilot data; calculate its R(4) and identify a physical-domain problem. (b) Design a new measurement that distinguishes these candidates, including precision, timing, replication and a predeclared comparison rule. (c) Supply a deliverable plan linking a mathematical derivation, reproducible computation, uncertainty/validation evidence, a limitation statement and reviewer questions. (d) Defend what can and cannot be concluded from a future measurement near one prediction. Explain why completing this public task does not itself establish an independently reviewed capstone or a novel research result.
