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Which vector equals (2,−1), regardless of its drawn start?

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

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高一必修 第二册(A版).pdf · 6.1 · PDF 9 / printed page 2

TOPIC 01

Vectors as displacements

Translate, add and scale arrows; distinguish a journey from its net displacement.

What you will be able to explain

  • Represent a displacement independently of its starting point.
  • Use head-to-tail addition and absolute scaling of lengths.
  • Choose a velocity that meets a directional constraint.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Which vector equals (2,−1), regardless of its drawn start?

Skills and prerequisite lessons

Scope: JM02

  1. Option A(2,−1)(2,-1)
  2. Option B(−2,1)(-2,1)
  3. Option C(1,−2)(1,-2)
  4. Option D(2,1)(2,1)
Working and explanation
BUILD THE REASONING
Hint 1
Equality preserves both components.
Hint 2
A translation changes neither component.
Worked solution
  1. Compare horizontal and vertical displacements.
    (2,−1)=(2,−1)(2,-1)=(2,-1)
  2. Only A preserves both signs and lengths.
A.
Checks and common pitfalls: The position of a free arrow is irrelevant; orientation is not.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5Prediction and diagnosisCompare two translated arrows and vote on equality.
8Concept derivationDerive head-to-tail addition and negative scaling using a closed route.
10Guided applicationWork the translated-arrow example; pairs explain the length counterexample.
12Transfer and explanationSolve the constrained river-crossing task and discuss when it fails.
5Exit evidenceWrite and check the exit statement about displacement.
Question sequence
  • If I slide this arrow to another desk, what changes?
  • Can travelling farther leave you closer to your start?
Board development
  • Draw east 3, north 4 and the single diagonal; label distance 7 and displacement 5.
  • Display boat + current = ground velocity, then impose the zero horizontal component.
Expected correct and incorrect responses
  • A student gives −6 as a scaled length. Ask them to locate a negative distance on a ruler.
  • Correct: “The length is 6; the negative multiplier reverses the arrow.”
  • Incorrect: “The length is −6 because the multiplier is negative.”
Follow-up questions
  • If the current is faster than the boat, can a due-north path still exist? Explain with a squared component.
Remediation
  • Use two opposite paper arrows, then separately measure the two lengths and the zero resultant.
Exit check
  • State one quantity preserved by translating an arrow and one route property it does not encode.
Assessment checklist and sample responses
  • Credit the chosen orientation, signed components and a nonnegative magnitude separately. Sample: (−3,4)+(3,0)=(0,4), so crossing time is 20/4.
    Correct: “The length is 6; the negative multiplier reverses the arrow.”
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗