Kenneth ChengMathematical modeling
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A learner fits a straight line through two observations and claims the model is valid forever. Evaluate the claim.

Read the idea, work independently, then explain what changed.

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高一必修 第一册(A版).pdf · 3.4 · PDF 100 / printed page 93

Revisit first: Power functions

TOPIC 01

Applications of functions I

Build understanding of applications of functions i through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Interpret and use applications of functions i with domain checks.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

A learner fits a straight line through two observations and claims the model is valid forever. Evaluate the claim.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Distinguish fitting from validation.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Check further observations and the mechanism generating the data; state a justified range of use.

  3. A model is conditional on assumptions and evidence.

Two observations determine a line but do not validate unlimited extrapolation.

Checks and common pitfalls: A model is conditional on assumptions and evidence.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • What must be true before using the main rule for applications of functions i?
  • Which representation makes this task easier, and why?
  • Change one assumption. Does the conclusion survive?

Board plan

  • Interpret and use applications of functions i with domain checks.
  • Modelling: State variables, units, assumptions and feasible inputs.
  • Validation: A fitted formula needs contextual checks and further evidence.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: State variables, units, assumptions and feasible inputs.
  • Expected reasoning: A fitted formula needs contextual checks and further evidence.
  • Expected correction: Average speed is not generally the average of two speeds.

Assessment checklist

  • 1: identify the givens and required quantity.
  • 1: choose a valid definition, representation or method.
  • 1: present connected, correct reasoning.
  • 1: check conditions and explain the result.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗