At a glance. One lesson of about 60 minutes. It builds on position vectors and the triangle rule. Students prove the midpoint theorem, test parallel and collinear with scalar multiples, and distinguish an example from a proof.
Calculator: no calculator. No calculator: the algebra is with letters and small integers.
What students will learn
Know
- Parallel vectors are scalar multiples.
- Collinear points: parallel vectors with a shared point.
- AB = b − a from base vectors.
Understand
- Why letters prove for all cases and numbers do not.
- Why the midpoint line is half the third side.
- Why parallel alone does not give collinear.
Do
- Write a vector in terms of base vectors.
- Show vectors are parallel or points are collinear.
- Write a short vector proof with reasons.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the definitions of parallel and collinear and the triangle rule (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students conjecture the relationship before the proof and test it for several corner positions.
- Criterion C. Communicating: a proof with each line justified, and the conclusion stated in words.
- Criterion D. Applying mathematics in real-life contexts: a path across a field (the hook), a courtyard path and beacons on a straight line (questions 5 and 6).
The one thing to do with the figure
Ask for the prediction first. Then move each of the four sliders in turn and ask students to say what happens to AB and what happens to MN.
Ask: “Can you find a position of A and B where MN is not half of AB?” Listen for: no, and then “So how do we prove it for every position?”.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the field path; collect predictions. |
| 12 min | Figure, key idea and warning card; then build the proof on the board. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask whether two parallel railway lines are one line. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | (5, −6); (4, 4); every length is doubled. |
|---|---|
| Hook | Yes: the path MN is always parallel to AB and half as long, for every triangle, because MN = ½AB. |
| Worked examples | MN = ½AB; collinear (BC = 2AB); the midpoints agree at (4, 2.5). |
| Faded example | Not collinear: PQ = (3, 3), QR = (4, 5). |
| 1. Parallel is not collinear | C. A claims collinear without a shared point; B inverts the factor; D ignores the factor. |
| 2. Midpoint vector | −1. ½(2 − 4). |
| 3. Scalar multiple | 0.5. 5 ÷ 10. |
| 4. Collinear points | 4. (12, 16) = 4(3, 4). |
| 5. Diagonal of a parallelogram | −5. 1 − 6. |
| 6. Using collinearity | 7. k − 1 = 6. |
| 7. A fraction along two sides | 0.2. PQ is a fifth of AB. |
| 8. Building a vector from base vectors | B. A is the centre, C halves the wrong side, D reverses AB. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Treated parallel as collinear, inverted the multiple, or ignored it. | Ask: is there a point that both AB and CD contain? What do you multiply AB by to get CD? |
| 1 or −2 in question 2 | Reversed the vector, or gave AB instead of MN. | Ask for OM and ON first, then subtract in the order end minus start. |
| 2 or 1 in question 3 | Inverted the multiple, or assumed equal vectors. | Ask: what do I multiply AB by to get MN? |
| 9 or 0.25 in question 4 | Subtracted entries instead of dividing, or inverted the multiple. | Ask for PQ and PR first, then: 3 times what gives 12? Does it work for 4 and 16? |
| 7 or 5 in question 5 | Found the other diagonal, or reversed the direction. | Draw the parallelogram, label B and D, and trace B to D. |
| 6 or 9 in question 6 | Forgot to add back the start, or scaled a coordinate. | Ask: is k the coordinate or the entry of the vector AC? |
| 0.8 or 5 in question 7 | Used PA instead of OP, or inverted the multiple. | Ask: what fraction of OA is OP? Then write PQ in terms of a and b. |
| A, C or D in question 8 | Found the centre, halved the wrong side, or reversed AB. | Sketch OABC, mark M, and write the route O to A to M. |
Questions worth asking
- What is the difference between an example and a proof? Listen for: an example is one case; a proof uses letters and works for every case.
- Why do we need a shared point to say collinear? Listen for: parallel lines never meet, so parallel vectors on different lines do not make points collinear.
- What base vectors would you choose here? Listen for: two sides from one corner, since every other vector is a combination of them.
- Is the diagonal sum or difference? Listen for: sum for the diagonal from a common corner, difference between the far corners.
Differentiation
A way in for anyone stuck
- Give a proof frame with blank steps: OM = …, ON = …, MN = …, AB = …, so ….
- Provide numerical vectors to test the claim first, then replace them with a and b.
- Give the sentence frame: “MN is a multiple of AB, so they are ….”
Stretch
- Prove that the point two thirds of the way along each median of a triangle is the same point for all three medians.
- Prove that the diagonals of any parallelogram bisect each other, using letters for two adjacent sides.
Students learning in English as an additional language
“Prove” means to show that something is always true, with a reason for each step. “Collinear” means on the same straight line. “Parallel” means going in the same direction and never meeting. “Bisect” means to cut into two equal parts.
| Word | Plain meaning | Example |
|---|---|---|
| proof | an argument showing a result is true in every case | using letters for the sides |
| collinear | lying on one straight line | three towns on one road |
| parallel | in the same direction, never meeting | railway lines |
| bisect | to cut into two equal parts | a line through the middle |
| base vectors | two vectors from which others are built | OA and OB |
Sentence starters for explaining aloud or in writing:
- Let OA = a and OB = b. Then …
- So MN is a scalar multiple of AB, which means …
- Because they share the point …, the points are collinear.
Two things not to say
“It works for these numbers, so it is proved.” Say instead: numbers test a claim, letters prove it.
“Parallel means collinear.” Say instead: parallel vectors are collinear only if they share a point.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FindAB = (4, 6) and PQ = (2, 3). Are they parallel?
Yes: AB = 2PQ. - FindM is the midpoint of OA, OA = (6, 8). Find OM.
(3, 4).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA fence runs from O to A(6, 4) and another from O to B(2, 10). A path joins the midpoints. What is its vector?
(−2, 3). - Earlier work (lesson 6)Find the midpoint of (4, 7) and (10, 1).
(7, 4).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA, B, C have AB = a and BC = 3a. Are they collinear? What is AC?
Yes, they share B. AC = 4a. - Working backwardsAB = (2, k) and BC = (6, 15) are parallel. Find k.
k = 5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.