KKhaemenes PrecalculusOpen-Age · Printable Analytical Lesson
Unit 13Course HomeTeacher Key

Unit 13 · Lesson 02 · Week 35 · Advanced Modelling Capstone & Comprehensive Synthesis

Selecting Variables, Assumptions & Function Families

Formulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. In this lesson, the central focus is selecting variables, assumptions & function families. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

  • Interpret the defining quantities and notation used in selecting variables, assumptions & function families.
  • Construct or analyze at least two representations related to selecting variables, assumptions & function families.
  • Show symbolic work with correct domains, units, intervals, or angle measures.
  • Explain why the selected method is valid and check the result against another representation.

Essential question

How can a learner select, build, validate, communicate, and defend a multi-representation mathematical model?

Multiple-representation model

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy

Formulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations.

residual=observed-predictedR^2=1-\frac{SS_{res}}{SS_{tot}}relative\ error=\frac{|observed-predicted|}{|observed|}

Worked reasoning

Residual checkA positive residual means the observed value is above the model prediction.
Model limitationA model fitted on 0≤t≤12 should not be extrapolated far beyond 12 without a contextual argument and new validation data.

Misconception check: A high numerical fit does not prove causation or guarantee valid extrapolation beyond the observed domain.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Selecting Variables, Assumptions & Function Families

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U13 · L02
Name
Date
1

Name the response variable in a model predicting height from age.

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy
2

What does a positive residual mean?

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy
3

Why document assumptions?

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy
4

What is interpolation?

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy
5

What is extrapolation?

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy
6

What should a model defense include?

Modeling visualFormulate, fit, compare, validate, communicate, and defend mathematical models with explicit assumptions and limitations. xy
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