Kenneth ChengMathematical modeling
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Can side lengths 1,1,3 form a triangle to which the cosine rule applies? Explain.

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

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高一必修 第二册(A版).pdf · 6.4 · PDF 45 / printed page 38

Revisit first: Basis theorem and coordinate representation

TOPIC 01

Applications of plane vectors

Build understanding of applications of plane vectors through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Connect geometric and algebraic forms of applications of plane vectors.
  • 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 / Standard#Your turn

Can side lengths 1,1,3 form a triangle to which the cosine rule applies? Explain.

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

Working and explanation

BUILD THE REASONING

Hint 1
Check geometric feasibility before applying formulas.
Hint 2
The two shorter sides must have sum exceeding the longest side.
Worked solution

  1. Check geometric feasibility before applying formulas.

  2. Calculate or simplify this relation.

    cos⁡C=1+1−92=−7/2∉[−1,1]\cos C=\frac{1+1-9}{2}=-7/2\notin[-1,1]

  3. An impossible cosine exposes inconsistent side data rather than a valid angle.

No: 1+1<3 violates the triangle inequality.

Checks and common pitfalls: An impossible cosine exposes inconsistent side data rather than a valid angle.

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 plane vectors?
  • Which representation makes this task easier, and why?
  • Change one assumption. Does the conclusion survive?

Board plan

  • Connect geometric and algebraic forms of applications of plane vectors.
  • Geometry: Vectors yield metric relations, sine and cosine rules, and triangle centres.
  • Physical quantities: Add forces as vectors; calculate work with a dot product.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: Vectors yield metric relations, sine and cosine rules, and triangle centres.
  • Expected reasoning: Add forces as vectors; calculate work with a dot product.
  • Expected correction: An SSA triangle may have two valid configurations.

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 ↗