At a glance. One lesson of about 60 minutes. It builds on column vectors and translation. Students add and subtract columns, find scalar multiples, test for parallel vectors and use the triangle rule.
Calculator: no calculator. No calculator: the entries are small integers and the skill is structural.
What students will learn
Know
- Add and subtract the entries; multiply every entry by a scalar.
- AB + BC = AC.
- Two vectors are parallel when one is a scalar multiple of the other.
Understand
- Why addition is the same as following one movement then the next.
- Why a negative scalar reverses the direction.
- Why the order of addition does not matter.
Do
- Add, subtract and scale column vectors.
- Use the triangle rule to find a vector between two points.
- Find a missing entry for parallel vectors.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: operations on column vectors (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how the sum changes as the multiples change and why the dashed route meets the full one.
- Criterion C. Communicating: writing a vector equation such as AC = AB + BC with the legs in order.
- Criterion D. Applying mathematics in real-life contexts: drones, a ferry, a survey walk and a warehouse robot (hook and questions 2, 4 and 7).
The one thing to do with the figure
Ask for the prediction first. Then set m = 1 and n = 1, and move n to a negative value and ask what happens to the second leg of the route.
Ask: “Why do the solid route and the dashed route meet at the same point?” Listen for: both end with the same sum, m times 2 plus n times 1 across, and the order of adding does not matter.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: two drones, opposite orders; collect predictions. |
| 12 min | Figure, key idea and warning card; ask what a negative multiple does. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, ask them to write the working for each entry. |
| 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, 3); −2; (4, −6). |
|---|---|
| Hook | Yes. The sum is (3, 4) in either order, so both drones arrive at the same place. |
| Worked examples | (2, 6); (7, 0); parallel, scalar 3. |
| Faded example | (−5, 19). |
| 1. Subtracting vectors | B. A is the sum, C is b − a, D subtracts one entry only. |
| 2. Adding vectors | 4. −1 + 5. |
| 3. Scalar multiples and subtraction | 8. 6 + 2. |
| 4. Triangle rule | −9. −3 − 6. |
| 5. Finding a scalar | 5. 10 ÷ 2. |
| 6. Parallel vectors | −4. 2 ÷ 3 of −6. |
| 7. Using a parameter | 4. Both entries are 10. |
| 8. Reversed route | C. A is AC itself; B reverses a − b instead of a + b; D is a difference of vectors from a common point. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Added instead of subtracting, reversed the order, or changed only one entry. | Ask them to write a − b entry by entry, with brackets round the negative. |
| −6 or 5 in question 2 | Subtracted when the sum was asked, or gave only one vector's entry. | Underline the operation in the question, then write each entry's working. |
| 4 or 9 in question 3 | Dropped a negative sign, or multiplied the whole difference by one scalar. | Compute 3a and 2b as separate columns first, then subtract. |
| 3 or −3 in question 4 | Subtracted the legs, or gave only the first leg. | Ask for the vector from A to B and then from B to C in order. |
| 8 or −5 in question 5 | Subtracted the entries, or lost a sign. | Ask: what number times 2 gives 10? Then check it on the bottom entry. |
| 4 or −9 in question 6 | Lost the sign, or used the scalar the wrong way round. | Ask: what number times 9 gives 6? Use the same number on −6. |
| 8 or −2 in question 7 | Forgot to divide by 2, or reversed a sign when forming the equation. | Write the top and bottom entries in terms of t before setting them equal. |
| A, B or D in question 8 | Gave the forward route, reversed a difference instead of the sum, or used a difference of vectors from a common point. | Draw A, B, C with arrows and ask which way each arrow points. |
Questions worth asking
- What does a negative scalar do to a vector? Listen for: it reverses the direction and scales the length by the size of the scalar.
- Why is a + b equal to b + a? Listen for: the entries are added in either order, and the two routes make a parallelogram.
- How do you test whether two vectors are parallel? Listen for: divide matching entries and get the same number.
- What does AB + BC = AC say about a triangle? Listen for: going along two sides is the same movement as going along the third.
Differentiation
A way in for anyone stuck
- Give squared paper and ask students to draw each vector tip to tail before adding entries.
- Provide a two-row table with a column for each vector, a column for the sum and a column for the difference.
- Give the frame: “The top entry is … + … = … and the bottom entry is … + … = ….”
Stretch
- Show that if p and q are not parallel, then m p + n q = 0 only when m = n = 0.
- Find the vector t that satisfies 2t + (1, 3) = (7, 9) and then find a vector parallel to t with magnitude 5.
Students learning in English as an additional language
“Scalar” means an ordinary number (not a vector). “Parallel” means pointing the same way or the opposite way. “Tip” is the end with the arrow and “tail” is the start. “Resultant” means the single vector that does the work of several.
| Word | Plain meaning | Example |
|---|---|---|
| scalar | a number that multiplies a vector | the 3 in 3a |
| parallel | pointing in the same or opposite direction | one is a multiple of the other |
| sum | the vector found by adding two vectors | a movement then another |
| negative vector | a vector with the same size and the opposite direction | the reverse journey |
| tip to tail | joining vectors end to start so they follow one another | a route in two legs |
Sentence starters for explaining aloud or in writing:
- The top entry is … because …
- The vectors are parallel because one is … times the other.
- AC = AB + BC because the journey goes …
Two things not to say
“Just change the sign.” Say instead: subtracting a vector changes the sign of both of its entries.
“A bigger number means a longer vector.” Say instead: the length depends on both entries together, by Pythagoras.
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.
- Find(3, 1) + (2, 6).
(5, 7). - Find2 × (4, −3).
(8, −6).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA ferry goes (12, 5) km and then (−4, 9) km. What single movement is that?
(8, 14). - Earlier work (lesson 4)Find the magnitude of (8, 6).
10.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that (2, 6) and (−3, −9) are parallel.
(−3, −9) = −1.5 × (2, 6). - Working backwardsa + b = (7, 2) and a − b = (1, 6). Find a and b.
a = (4, 4) and b = (3, −2).
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.