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A=(0,0), B=(9,0), AP:PB=2:1 internally. Choose Pₓ.

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

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高一必修第二册(A版).pdf · 6.3 · PDF 32 / printed page 25

Revisit first: Vectors as displacements

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Prediction and diagnosis	Diagnose reversed endpoint subtraction.
8	Concept derivation	Derive division and centroid expressions using displacements.
10	Guided application	Solve one independent and one dependent basis system.
12	Transfer and explanation	Recover a triangle from its centroid and verify it is nondegenerate.
5	Exit evidence	Check a proposed basis counterexample.

Question sequence

Which subtraction gives an arrow pointing from A to B?
Why do the coefficients of a point division sum to one?

Board development

Mark A, P, B on a number line before substituting into the division formula.
Put the independent and dependent basis systems side by side and compare determinants.

Expected correct and incorrect responses

Students reverse the ratio weights. Check whether the longer AP places P nearer B.
Correct: “P is two thirds of the way from A to B, so P=(6,4).”
Incorrect: “AP:PB=2:1 means P=(A+2B)/2.”

Follow-up questions

Translate every vertex by the same vector. How does the centroid move?

Remediation

Use A=0 and B=9 to locate a 2:1 internal division before doing two coordinates.

Exit check

Give a dependent pair and a vector that it cannot represent.

Assessment checklist and sample responses

Require the endpoint order, correct affine weights and a uniqueness check where a basis is used. Sample: G=(A+B+C)/3 implies C=3G−A−B.
Correct: “P is two thirds of the way from A to B, so P=(6,4).”