{ "schema": "khaemenes-algebra1-unit-answer-key", "version": "2026.08.21-u13-v2", "unit": 13, "question_count": 100, "notes": "20 unique questions per lesson; q0411–q0435 preserved as lesson-check IDs.", "answers": [ { "id": "q0411", "lesson": 1, "answer": 0, "answer_text": "State a precise decision/question and define the quantities", "explanation": "A model should begin with a clear purpose and measurable quantities." }, { "id": "q0412", "lesson": 1, "answer": 1, "answer_text": "How does changing weekly attendance affect projected meal demand over one month?", "explanation": "A model-ready question specifies quantities, relationship, and decision context." }, { "id": "q0413", "lesson": 1, "answer": 0, "answer_text": "So the model has a defined domain and purpose", "explanation": "Bounds keep the model interpretable and prevent unsupported extrapolation." }, { "id": "q0414", "lesson": 1, "answer": 0, "answer_text": "Assume attendance remains within 400–550 students", "explanation": "An assumption is accepted for modelling rather than directly observed as a measured value." }, { "id": "q0415", "lesson": 1, "answer": 0, "answer_text": "A parameter is treated as fixed within a model run; a variable may change", "explanation": "Parameters set model conditions while variables represent changing quantities." }, { "id": "u13x-l01-06", "lesson": 1, "answer": 0, "answer_text": "t = travel time in minutes", "explanation": "Variables should have precise meanings and units." }, { "id": "u13x-l01-07", "lesson": 1, "answer": 0, "answer_text": "They help verify meaning and dimensional consistency", "explanation": "Units can expose invalid operations and clarify interpretation." }, { "id": "u13x-l01-08", "lesson": 1, "answer": 0, "answer_text": "How can we compare two fictional transportation plans using cost and travel time without collecting personal location histories?", "explanation": "Ethical modelling minimizes unnecessary sensitive data and keeps the question proportionate to its purpose." }, { "id": "u13x-l01-09", "lesson": 1, "answer": 0, "answer_text": "Distance, walking-speed assumption, route domain, and time unit", "explanation": "A defensible model defines quantities, assumptions, units, and scope." }, { "id": "u13x-l01-10", "lesson": 1, "answer": 0, "answer_text": "What makes a good city?", "explanation": "The phrase does not identify measurable quantities, domain, or decision criterion." }, { "id": "u13x-l01-11", "lesson": 1, "answer": 0, "answer_text": "A measurable rule used to compare model outcomes", "explanation": "Criteria such as cost, time, error, or capacity support explicit comparison." }, { "id": "u13x-l01-12", "lesson": 1, "answer": 0, "answer_text": "So readers can judge whether the model applies", "explanation": "Assumptions define the conditions under which conclusions are meaningful." }, { "id": "u13x-l01-13", "lesson": 1, "answer": 0, "answer_text": "0≤x≤50", "explanation": "The domain identifies allowed inputs." }, { "id": "u13x-l01-14", "lesson": 1, "answer": 0, "answer_text": "The model ignores weather-related delays", "explanation": "A limitation describes an important real-world factor not represented or not fully resolved." }, { "id": "u13x-l01-15", "lesson": 1, "answer": 0, "answer_text": "Choose the simplest form justified by structure, data, and purpose", "explanation": "Model complexity should be earned by evidence and purpose." }, { "id": "u13x-l01-16", "lesson": 1, "answer": 0, "answer_text": "Under the same stated assumptions, which plan has lower total cost over 12 months?", "explanation": "Fair comparison applies consistent conditions and criteria." }, { "id": "u13x-l01-17", "lesson": 1, "answer": 0, "answer_text": "Different groups may experience different benefits, costs, or risks", "explanation": "A decision can be mathematically efficient yet distribute consequences unevenly." }, { "id": "u13x-l01-18", "lesson": 1, "answer": 0, "answer_text": "It produces outputs that can be compared with observations or constraints", "explanation": "Validation requires something observable or checkable." }, { "id": "u13x-l01-19", "lesson": 1, "answer": 0, "answer_text": "Unsupported extrapolation beyond the evidence domain", "explanation": "Prediction far beyond observed conditions requires additional justification." }, { "id": "u13x-l01-20", "lesson": 1, "answer": 0, "answer_text": "Question, quantities, units, domain, assumptions, criteria, and relevant ethical constraints", "explanation": "A complete setup makes later reasoning auditable." }, { "id": "q0416", "lesson": 2, "answer": 0, "answer_text": "A time-series line graph", "explanation": "Ordered time data are naturally shown with time on the horizontal axis." }, { "id": "q0417", "lesson": 2, "answer": 0, "answer_text": "Equations and a shared coordinate graph", "explanation": "Equations support exact comparison and a shared graph shows intersections and relative cost." }, { "id": "q0418", "lesson": 2, "answer": 0, "answer_text": "Meaning and unit", "explanation": "A symbol is interpretable only when its quantity and unit are defined." }, { "id": "q0419", "lesson": 2, "answer": 0, "answer_text": "Document the mismatch and decide whether adjustment or exclusion is justified", "explanation": "Data provenance and comparability affect model validity." }, { "id": "q0420", "lesson": 2, "answer": 0, "answer_text": "Cost per rider calculated from total cost/riders", "explanation": "A derived quantity is calculated from other variables." }, { "id": "u13x-l02-06", "lesson": 2, "answer": 0, "answer_text": "Different forms reveal different features and allow cross-checking", "explanation": "Tables, equations, and graphs can expose structure and errors differently." }, { "id": "u13x-l02-07", "lesson": 2, "answer": 0, "answer_text": "A linear model with slope 5", "explanation": "Constant first differences support a linear relationship." }, { "id": "u13x-l02-08", "lesson": 2, "answer": 0, "answer_text": "Exponential", "explanation": "A constant multiplicative factor suggests exponential structure." }, { "id": "u13x-l02-09", "lesson": 2, "answer": 0, "answer_text": "Quadratic", "explanation": "A parabola can model a single turning point with roughly symmetric change." }, { "id": "u13x-l02-10", "lesson": 2, "answer": 0, "answer_text": "A travel time of −12 minutes", "explanation": "Negative travel time is physically implausible in the stated context." }, { "id": "u13x-l02-11", "lesson": 2, "answer": 0, "answer_text": "Expressing quantities relative to a common scale such as per person", "explanation": "Normalization can support fair comparisons across different-sized groups." }, { "id": "u13x-l02-12", "lesson": 2, "answer": 0, "answer_text": "Missingness can affect estimates and bias conclusions", "explanation": "The pattern and extent of missingness influence validity." }, { "id": "u13x-l02-13", "lesson": 2, "answer": 0, "answer_text": "Using inconsistent axis scales between comparison panels", "explanation": "Scale changes can create misleading visual comparisons." }, { "id": "u13x-l02-14", "lesson": 2, "answer": 0, "answer_text": "Fixed starting cost when x=0", "explanation": "The intercept is the predicted output at zero input." }, { "id": "u13x-l02-15", "lesson": 2, "answer": 0, "answer_text": "Output change per one unit of x", "explanation": "Slope is rate of change." }, { "id": "u13x-l02-16", "lesson": 2, "answer": 0, "answer_text": "Median and IQR", "explanation": "Median and IQR resist influence from extreme values." }, { "id": "u13x-l02-17", "lesson": 2, "answer": 0, "answer_text": "A value known to come from a broken sensor with documented failure", "explanation": "Exclusion should be based on defensible data-quality criteria, not desired conclusions." }, { "id": "u13x-l02-18", "lesson": 2, "answer": 0, "answer_text": "Inflation adjustment or explicit nominal-dollar limitation", "explanation": "Price-level changes can make nominal amounts non-comparable." }, { "id": "u13x-l02-19", "lesson": 2, "answer": 0, "answer_text": "Clarify geometry, relationships, and assumptions before equations are built", "explanation": "A schematic can make structure and constraints visible." }, { "id": "u13x-l02-20", "lesson": 2, "answer": 0, "answer_text": "At least two complementary forms with consistent units and labels", "explanation": "Complementary representations improve interpretation and verification." }, { "id": "q0421", "lesson": 3, "answer": 0, "answer_text": "Comparing model output with data, constraints, or an independent method", "explanation": "Validation tests whether a model performs acceptably for its purpose." }, { "id": "q0422", "lesson": 3, "answer": 0, "answer_text": "−2", "explanation": "Residual=50−52=−2." }, { "id": "q0423", "lesson": 3, "answer": 0, "answer_text": "Systematic patterns can reveal model misspecification", "explanation": "A curve or trend in residuals suggests missing structure." }, { "id": "q0424", "lesson": 3, "answer": 0, "answer_text": "Consider a nonlinear model", "explanation": "A curved residual pattern suggests a linear form may be inadequate." }, { "id": "q0425", "lesson": 3, "answer": 0, "answer_text": "Substitute the result into the original relationship", "explanation": "Independent verification should test the original condition." }, { "id": "u13x-l03-06", "lesson": 3, "answer": 0, "answer_text": "Revisit domain/constraints or model structure", "explanation": "Outputs violating physical constraints indicate a model or domain problem." }, { "id": "u13x-l03-07", "lesson": 3, "answer": 0, "answer_text": "Premature rounding can accumulate error", "explanation": "Exact intermediate results preserve precision for later calculations." }, { "id": "u13x-l03-08", "lesson": 3, "answer": 0, "answer_text": "Overfitting", "explanation": "An overly tailored model can memorize noise rather than general structure." }, { "id": "u13x-l03-09", "lesson": 3, "answer": 0, "answer_text": "Adjusting parameters so model output aligns with relevant observations", "explanation": "Calibration estimates model parameters from evidence." }, { "id": "u13x-l03-10", "lesson": 3, "answer": 0, "answer_text": "Mean absolute error", "explanation": "Absolute errors retain the response variable's units before averaging." }, { "id": "u13x-l03-11", "lesson": 3, "answer": 0, "answer_text": "Is 2 units acceptable for the decision purpose?", "explanation": "Model adequacy depends on decision tolerance, not just numerical fit." }, { "id": "u13x-l03-12", "lesson": 3, "answer": 0, "answer_text": "Flag the prediction as unsupported or expand validation before using it", "explanation": "Use outside the validated domain is extrapolation and needs justification." }, { "id": "u13x-l03-13", "lesson": 3, "answer": 0, "answer_text": "Using a second valid method or representation to verify a result", "explanation": "Independent methods reduce risk of shared calculation error." }, { "id": "u13x-l03-14", "lesson": 3, "answer": 0, "answer_text": "The predicted demand exceeds the capacity constraint by 50", "explanation": "A model should preserve and interpret binding constraints rather than silently clip results." }, { "id": "u13x-l03-15", "lesson": 3, "answer": 0, "answer_text": "Change model form after residuals show systematic curvature", "explanation": "Revision should respond to diagnostic evidence." }, { "id": "u13x-l03-16", "lesson": 3, "answer": 0, "answer_text": "Potential sensitivity or instability", "explanation": "Large output change from small input change signals sensitivity." }, { "id": "u13x-l03-17", "lesson": 3, "answer": 0, "answer_text": "Data reserved for testing model performance after fitting", "explanation": "Holdout data can assess generalization beyond the fitting sample." }, { "id": "u13x-l03-18", "lesson": 3, "answer": 0, "answer_text": "To create an auditable trail of what changed and why", "explanation": "Version history supports reproducibility and accountability." }, { "id": "u13x-l03-19", "lesson": 3, "answer": 0, "answer_text": "Validation requires comparison with data, constraints, or independent reasoning", "explanation": "Software can execute an incorrect model perfectly." }, { "id": "u13x-l03-20", "lesson": 3, "answer": 0, "answer_text": "When errors, constraints, assumptions, or instability make it unfit for purpose", "explanation": "Models are judged relative to purpose and acceptable evidence standards." }, { "id": "q0426", "lesson": 4, "answer": 0, "answer_text": "How results change when inputs or assumptions change", "explanation": "Sensitivity analysis evaluates dependence of outputs on model inputs." }, { "id": "q0427", "lesson": 4, "answer": 0, "answer_text": "High sensitivity", "explanation": "Large output response to a small input change indicates instability/sensitivity." }, { "id": "q0428", "lesson": 4, "answer": 0, "answer_text": "A quantified or described limit on what is known or predicted", "explanation": "Uncertainty reflects incomplete knowledge, variability, or estimation limits." }, { "id": "q0429", "lesson": 4, "answer": 0, "answer_text": "It can communicate plausible variation or uncertainty", "explanation": "A range can better reflect uncertainty than false precision." }, { "id": "q0430", "lesson": 4, "answer": 0, "answer_text": "Comparing outputs under several coherent sets of assumptions", "explanation": "Scenarios show how conclusions depend on plausible conditions." }, { "id": "u13x-l04-06", "lesson": 4, "answer": 0, "answer_text": "The decision is not robust to that assumption", "explanation": "A robust conclusion should not hinge on tiny reasonable changes without disclosure." }, { "id": "u13x-l04-07", "lesson": 4, "answer": 0, "answer_text": "Who receives benefits, burdens, errors, or exclusions under the model", "explanation": "Fairness includes distribution of model consequences across relevant groups." }, { "id": "u13x-l04-08", "lesson": 4, "answer": 0, "answer_text": "Report and investigate the disparity", "explanation": "Aggregate performance can conceal subgroup harm." }, { "id": "u13x-l04-09", "lesson": 4, "answer": 0, "answer_text": "One that remains reasonable across plausible changes in assumptions", "explanation": "Robustness concerns stability under plausible uncertainty." }, { "id": "u13x-l04-10", "lesson": 4, "answer": 0, "answer_text": "Natural day-to-day demand variability", "explanation": "Some uncertainty arises from inherent variability rather than lack of knowledge alone." }, { "id": "u13x-l04-11", "lesson": 4, "answer": 0, "answer_text": "Measurement uncertainty", "explanation": "Improved instruments or data collection can reduce some epistemic uncertainty." }, { "id": "u13x-l04-12", "lesson": 4, "answer": 0, "answer_text": "Extreme unrealistic inputs can distract from decision-relevant behavior", "explanation": "Analysis should focus on credible variations relevant to the decision." }, { "id": "u13x-l04-13", "lesson": 4, "answer": 0, "answer_text": "Plan A is unlikely to be cheaper under the stated plausible range", "explanation": "The threshold lies outside the stated plausible scenario range." }, { "id": "u13x-l04-14", "lesson": 4, "answer": 0, "answer_text": "An input value where compared outcomes become equal", "explanation": "Threshold analysis identifies where a preferred decision may switch." }, { "id": "u13x-l04-15", "lesson": 4, "answer": 0, "answer_text": "Projected cost is $12,000–$14,000 under the stated demand scenarios", "explanation": "The statement connects uncertainty to explicit scenarios." }, { "id": "u13x-l04-16", "lesson": 4, "answer": 0, "answer_text": "Transparent trade-off analysis rather than claiming universal improvement", "explanation": "Fairness decisions can involve competing objectives that should be disclosed." }, { "id": "u13x-l04-17", "lesson": 4, "answer": 0, "answer_text": "Which inputs have the largest effect on an output over chosen ranges", "explanation": "Ranked bars can visualize relative output sensitivity to inputs." }, { "id": "u13x-l04-18", "lesson": 4, "answer": 0, "answer_text": "Test multiple plausible scenarios including unfavorable but credible ones", "explanation": "Balanced scenario analysis reduces cherry-picking." }, { "id": "u13x-l04-19", "lesson": 4, "answer": 0, "answer_text": "State the unresolved issue and keep it in the decision process", "explanation": "Not every ethical trade-off can be reduced to one scalar metric." }, { "id": "u13x-l04-20", "lesson": 4, "answer": 0, "answer_text": "Plan B is preferred under tested ranges, but the choice reverses if demand falls below 120 units", "explanation": "The statement communicates preference, scope, and a decision threshold." }, { "id": "q0431", "lesson": 5, "answer": 0, "answer_text": "The decision question and main conclusion", "explanation": "Audience orientation starts with purpose and the answer being defended." }, { "id": "q0432", "lesson": 5, "answer": 0, "answer_text": "A calculation linked to data, assumptions, and validation", "explanation": "Defence requires a traceable evidence chain." }, { "id": "q0433", "lesson": 5, "answer": 0, "answer_text": "Acknowledge it and explain its effect on interpretation", "explanation": "Transparent limitations strengthen rather than weaken defensibility." }, { "id": "q0434", "lesson": 5, "answer": 0, "answer_text": "The conclusion applies to the tested 0–50 unit domain under the stated cost assumptions", "explanation": "Scope connects conclusions to domain and assumptions." }, { "id": "q0435", "lesson": 5, "answer": 0, "answer_text": "One proportional to the evidence and explicit about uncertainty", "explanation": "Evidence strength should bound claim strength." }, { "id": "u13x-l05-06", "lesson": 5, "answer": 0, "answer_text": "They make quantitative claims interpretable and checkable", "explanation": "Values without units can be ambiguous or misleading." }, { "id": "u13x-l05-07", "lesson": 5, "answer": 0, "answer_text": "Another reviewer could follow the data, assumptions, and calculations to recreate the result", "explanation": "Reproducibility depends on documented methods and inputs." }, { "id": "u13x-l05-08", "lesson": 5, "answer": 0, "answer_text": "A labeled graph directly supporting the decision with readable scales", "explanation": "Visuals should clarify evidence without distortion." }, { "id": "u13x-l05-09", "lesson": 5, "answer": 0, "answer_text": "The observed structure and residual diagnostics supported linear behavior over the decision domain", "explanation": "Model choice should be justified by structure, diagnostics, and purpose." }, { "id": "u13x-l05-10", "lesson": 5, "answer": 0, "answer_text": "Correct it, revise the conclusion if needed, and document the change", "explanation": "Good modelling treats correction as part of the evidence process." }, { "id": "u13x-l05-11", "lesson": 5, "answer": 0, "answer_text": "Let reviewers trace data and claims to their origins", "explanation": "Source transparency supports auditability." }, { "id": "u13x-l05-12", "lesson": 5, "answer": 0, "answer_text": "I can select methods, justify them, verify results, and explain limits", "explanation": "Readiness includes reasoning, verification, and communication—not procedure alone." }, { "id": "u13x-l05-13", "lesson": 5, "answer": 0, "answer_text": "A correct calculation can still support an invalid contextual claim", "explanation": "Mathematics and contextual inference require separate justification." }, { "id": "u13x-l05-14", "lesson": 5, "answer": 0, "answer_text": "What worked, what changed, remaining limitations, and the next skill to strengthen", "explanation": "Reflection should identify learning evidence and future improvement." }, { "id": "u13x-l05-15", "lesson": 5, "answer": 0, "answer_text": "Name a specific assumption, threshold, or new evidence that would alter the result", "explanation": "A defensible recommendation identifies conditions under which it would change." }, { "id": "u13x-l05-16", "lesson": 5, "answer": 0, "answer_text": "Round to justified precision and explain uncertainty", "explanation": "Reported precision should reflect input quality and model uncertainty." }, { "id": "u13x-l05-17", "lesson": 5, "answer": 0, "answer_text": "No; data quality can undermine an otherwise correct calculation", "explanation": "Correct computation cannot repair a systematically flawed evidence base." }, { "id": "u13x-l05-18", "lesson": 5, "answer": 0, "answer_text": "Test whether the modeller understands choices, evidence, and limits", "explanation": "Defence questions probe reasoning rather than rote reproduction." }, { "id": "u13x-l05-19", "lesson": 5, "answer": 0, "answer_text": "Include meaningful revisions and explain why discarded approaches were rejected", "explanation": "A revision trail demonstrates modelling judgment." }, { "id": "u13x-l05-20", "lesson": 5, "answer": 0, "answer_text": "Within the stated domain and tested assumptions, the model supports Plan A; uncertainty and threshold analysis identify when that choice could reverse", "explanation": "The claim matches evidence while preserving scope and uncertainty." } ] }