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Prove A,B,C are collinear and locate C relative to AB.

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

TOPIC 01

Coordinates, bases and division

Convert arrows to coordinates, recover unknown endpoints and test when a basis actually determines coordinates.

What you will be able to explain

Subtract endpoints in the correct order.
Use affine weights for a midpoint, division point and centroid.
Distinguish an independent basis from parallel generating vectors.

Try it. Leave your reasoning visible.

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

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Worked example

Prove A,B,C are collinear and locate C relative to AB.

A=(0,0),B=(2,3),C=(4,6)

Skills and prerequisite lessons

Scope: JM02

Working and explanation

BUILD THE REASONING

Hint 1

Compare AC with AB.

Hint 2

A factor greater than one places C beyond B.

Worked solution

The displacements are proportional.
$AC=(4,6)=2AB$
Since 2>1, C lies on the ray AB beyond B.

Collinear; B is the midpoint of AC.

Checks and common pitfalls: Parallel displacement vectors through A imply collinearity; equal lengths alone do not.

Reasoning checklist · self / teacher assessment

Collinear; B is the midpoint of AC.
Parallel displacement vectors through A imply collinearity; equal lengths alone do not.

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

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).”