At a glance. One lesson of about 60 minutes. It builds on column vectors and coordinates. Students find the vector between two points, a midpoint, a point that divides a line in a ratio and a missing vertex.
Calculator: no calculator. No calculator: coordinates are small integers and the skill is the structure of AB = b − a.
What students will learn
Know
- The position vector of a point is the vector from the origin.
- AB = b − a.
- The midpoint has position vector ½(a + b).
Understand
- Why end minus start is the right order.
- Why the midpoint is the mean.
- Why the ratio m : n means a fraction m over (m + n) of the way.
Do
- Find AB from two position vectors.
- Find the midpoint and a point dividing in a ratio.
- Find a missing vertex of a parallelogram.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: position vectors and the end-minus-start rule (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how P moves as the number of quarters increases.
- Criterion C. Communicating: writing each step with vector notation, such as AB = b − a.
- Criterion D. Applying mathematics in real-life contexts: meeting points, maps, a cable car and a garden plan (hook and questions 3, 4 and 6).
The one thing to do with the figure
Ask for the prediction first. Then set two quarters and check that P is the mean of A and B, and move B and ask students to predict P before reading it.
Ask: “What does P do as the number of quarters goes from 0 to 4?” Listen for: it moves in equal steps from A to B along AB.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the halfway lunch stop; collect predictions. |
| 12 min | Figure, key idea and warning card; move B and predict P. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, ask what the sum of two position vectors is geometrically. |
| 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 | (4, 3); 10; (4, 7). |
|---|---|
| Hook | The halfway point is the mean of the coordinates, (2, 3). It is the same distance from each hut. |
| Worked examples | (4, 3); (5, 3); (5, 4). |
| Faded example | Q = (5, −2). |
| 1. Midpoint | A. B forgets the half, C is the vector AB, D is half of AB only. |
| 2. Vector between two points | 4. 7 − 3. |
| 3. Midpoint coordinate | 2. 4 ÷ 2. |
| 4. Dividing a line in a ratio | 2.5. 1 + 6 ÷ 4. |
| 5. Finding an endpoint from the midpoint | 8. 2 × 5 − 2. |
| 6. Fourth vertex of a parallelogram | 4. D = (4, 7). |
| 7. Midpoint on a line | 2. k + 10 = 12. |
| 8. Dividing a line with position vectors | C. A scales the sum; B measures the fraction from B; D adds a fraction of b instead of AB. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Forgot the half, gave the vector AB, or gave half of AB. | Ask: does the answer lie between A and B? Is it a point or a journey? |
| 10 or −4 in question 2 | Added the position vectors, or subtracted in the wrong order. | Ask: from A to B, which is the end and which is the start? |
| 6 or 4 in question 3 | Gave the difference, or did not halve the sum. | Ask for the mean of the two coordinates in words. |
| 4 or 3 in question 4 | Used the midpoint, or divided by the wrong total. | Draw AB divided into four equal parts and mark P. |
| 3 or 7 in question 5 | Gave AM, or added A and M. | Ask: if M is halfway, how far past M is B? |
| 14 or 10 in question 6 | Added AB to C, or added A and C without taking away B. | Ask which side of the parallelogram is equal to AD. |
| 6 or −4 in question 7 | Gave the common coordinate, or lost the doubling. | Write the midpoint with k in it before setting up the equation. |
| A, B or D in question 8 | Scaled the sum, measured the fraction from B, or added a fraction of b instead of AB. | Ask students to draw O, A, B and N, mark the five equal parts of AB, and write the route O to A to N. |
Questions worth asking
- Why is AB equal to b minus a? Listen for: from A go back to the origin, then out to B.
- How do you find a point a third of the way along a line? Listen for: add a third of the vector AB to A.
- How do you find the fourth vertex of a parallelogram? Listen for: use a pair of equal sides, such as AD = BC.
- Why is the midpoint the mean? Listen for: it is the same distance from each end, so it is the average of the coordinates.
Differentiation
A way in for anyone stuck
- Give a labelled grid with points A and B plotted so students can see the journey before calculating.
- Provide a template with three boxes: b, a and b − a, for each vector between points.
- Give the frame: “The vector AB is … minus …, so its entries are ….”
Stretch
- Show that the point dividing AB in the ratio m : n has position vector (n a + m b) ÷ (m + n).
- Find the point that divides AB externally in the ratio 3 : 1, and say where it lies.
Students learning in English as an additional language
“Position” means where a point is, measured from the origin. “Midpoint” means the point halfway between two points. “Ratio” compares two parts. “Vertex” is a corner.
| Word | Plain meaning | Example |
|---|---|---|
| position vector | the vector from the origin to a point | OA for the point A |
| midpoint | the point halfway between two points | the middle of a road |
| ratio | a comparison of two parts of a whole | 1 : 3 |
| parallelogram | a four-sided shape with opposite sides parallel | a leaning rectangle |
| origin | the point (0, 0) | where the axes meet |
Sentence starters for explaining aloud or in writing:
- The vector AB is … minus …, which is …
- The midpoint is the mean of … and ….
- P is … of the way from A to B, so P = A + …
Two things not to say
“Subtract either way, it will be fine.” Say instead: end minus start gives AB, and the other order gives BA, which points the other way.
“The midpoint is the sum.” Say instead: the midpoint is the mean, so the sum is halved.
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.
- FindThe midpoint of (2, 6) and (10, 2).
(6, 4). - FindThe vector from (1, 5) to (4, 1).
(3, −4).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo friends live at (1, 1) and (9, 7). They meet halfway. Where?
(5, 4). - Earlier work (lesson 5)Find 2a − b for a = (1, 3) and b = (4, 2).
(−2, 4).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningP divides AB with AP : PB = 3 : 1, A = (0, 0), B = (8, 4). Find P.
(6, 3). - Working backwardsThe midpoint of AB is (4, −1) and A is (7, 2). Find B.
(1, −4).
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.