Unit 06 · Lesson 07 · Open-Age
Comparing Linear, Quadratic & Exponential Models
Linear models have constant differences, exponential models have constant ratios, and quadratic models have constant second differences. Model choice should be evidence-based.
Learning objectives
- Explain the central structure of comparing linear, quadratic & exponential models using precise mathematical language.
- Apply the governing relationship and verify results with an alternate representation or substitution.
- State domain, unit, precision, and context restrictions whenever they affect the result.
- Analyze a misconception and communicate a correction with evidence.
Essential question: How do repeated multiplication, percent change, and continuous growth create exponential patterns and decisions?
Governing relationship
Compare structure, residuals, domain, and interpretability.
Linear models have constant differences, exponential models have constant ratios, and quadratic models have constant second differences. Model choice should be evidence-based.
Worked examples
Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.
Guided and independent practice
- Classify a table by differences or ratios.
- Compare residuals for two models.
- Explain why extrapolation differs across model families.
- Create one original example involving comparing linear, quadratic & exponential models and solve it completely.
- Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
- Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.
Differentiated pathways
Foundation
Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.
Core
Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.
Extended
Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.
Application laboratory
Compare competing models for the same dataset and document a defensible choice.
Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.
Misconception and correction
Common misconception: Choosing the model with the largest output rather than the best structural fit.
Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.
Mastery evidence
- Accurate symbolic work with preserved equivalence.
- Correct domain, interval, unit, and precision statements.
- At least one independent verification.
- A correction explaining why an incorrect approach fails.
- A clear application conclusion that distinguishes mathematical result from recommendation.
Reflection
What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?