At a glance. One lesson of about 60 minutes. Naming a pre-image and image, translating points and shapes by a vector, and finding the vector from a pre-image and its image.
Calculator: no calculator. The numbers are small on purpose: the work is in seeing that every corner gets the same move.
What students will learn
Know
- A transformation maps a pre-image onto an image; the image of A is written A′.
- A translation adds the same vector to every point: (x, y) → (x + a, y + b).
- A translation is a congruence transformation: lengths and angles are unchanged.
Understand
- Every point of the shape moves the same way, which is why the shape is not distorted.
- The vector is image minus pre-image, across first and then up.
Do
- Translate a point or polygon by a vector, in both directions.
- Find the vector from a pre-image and its image.
- Describe a translation fully with a vector.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying a translation rule and finding a vector (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find how the “still to move” numbers relate to the guess and the target.
- Criterion C. Communicating: writing the vector with the across number first, and stating the rule as (x, y) to (x + a, y + b).
- Criterion D. Applying mathematics in real-life contexts: border patterns on walls, cloth and tiles, where one motif is slid again and again.
The one thing to do with the figure
Before moving a slider, ask for the corner. Set across 3, up −2 and ask students to say where C lands before the working appears.
On the second figure, ask: “I moved the guess across by 1. What happened to the still-to-move number across?” It goes down by 1: the guess and the number still to go always add to the target.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 3 (subtracting past zero). |
|---|---|
| 10 min | Hook, prediction for A, then the first figure. |
| 10 min | Key idea; the second figure, working backwards. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | The second quadrant; −3; −4. |
|---|---|
| Worked examples | P′ (2, −2); (−2, −2), (1, −2), (−2, 1); the vector (−6, 4). |
| Faded example | R′ (−1, 4). |
| 1. Rule from words | B, (x − 4, y + 3). A reverses both moves; C swaps the numbers; D has the wrong sign on x. |
| 2. Translate a point, across | −4. 3 + (−7). |
| 3. Translate a point, up | −3. −5 + 2. |
| 4. Find the vector | 7. 3 − (−4). |
| 5. Use a known vector | 0. Vector (−5, −5); 5 + (−5). |
| 6. What a translation keeps | A. B describes an enlargement; C a rotation; D confuses the distance moved with the size. |
| 7. Two slides in a row | −3. a = 2, b = −5. |
| 8. Vector from one corner, then apply | 8. Vector (−5, 4); 4 + 4. |
| 9. Spot the translation | B. A and D are reflections; C is an enlargement. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Reversed the moves, swapped the numbers, or used the wrong sign on x. | Ask: left or right? Then: does left add to x or take away? |
| 10 or 5 in question 2 | Subtracted the vector, or used the move up on x. | Draw P and a 7-unit arrow to the left on a number line. |
| −7 or −12 in question 3 | Subtracted the move up, or used the move across on y. | Write the two moves in a column: across on top, up beneath. |
| −7 or 5 in question 4 | Start minus end, or the change in y. | Write “image minus pre-image” above the working. |
| −3 or 10 in question 5 | Copied the other image's y, or added the wrong way. | Find the vector first and write it down, then apply it. |
| B, C or D in question 6 | Mixed up translation with enlargement or rotation. | Compare with lengths: does any distance between two corners change? |
| 13 or 5 in question 7 | Sign slip on −8, or only the first up-move used. | Add the across moves, then the up moves, on separate lines. |
| 9 or 4 in question 8 | Added R's y to P′'s y, or used only the move. | Say it aloud: R was at 4, and it moves up 4. |
Questions worth asking
- Why must the vector be the same for every corner? Listen for “otherwise the shape would be stretched or twisted”.
- If I know where one corner went, do I know where all of them went? Yes, because the vector is the same: find it from one corner.
- What is the translation that does nothing, and why does it count? The vector (0, 0); it is the starting point for what the next lessons undo.
Differentiation
A way in for anyone stuck
- Use squared paper and count squares for each move, writing the number of squares beside the arrow.
- Put the across number in a blue pen and the up number in a green pen, in the vector and in the working.
- Start with vectors whose numbers are all positive, then bring in negatives.
Stretch
- A translation maps (2, 3) to (5, 7) and a second maps (5, 7) to (4, 1). What single translation does both? (Vector (2, −2).)
- Why does a translation never fix a point unless the vector is (0, 0)? (Every point would have to move by zero, so both numbers would have to be zero.)
Students learning in English as an additional language
Translation, vector and image are new in this sense. Pair each with a picture: an arrow for vector, a stamp and its print for pre-image and image.
| Word | Plain meaning | Example |
|---|---|---|
| transformation | a rule that moves every point of a shape to a new place | slide, flip, turn, enlarge |
| pre-image | the shape before it is moved | triangle ABC |
| image | the shape after it is moved; its corners have a dash | A′B′C′ |
| translation | a slide: every point moves the same distance in the same direction | 3 right and 2 down |
| vector | the numbers that say how far across and how far up | (3, −2) |
Sentence starters for explaining aloud or in writing:
- The vector is (…, …), which means … across and … up.
- The image of the point (…, …) is …
- I found the vector by taking the image minus the …
Two things not to say
“Move the shape across and up.” It hides the negative: a vector with b = −2 moves down. Say “add b to y”.
“The shape stays the same, so nothing has changed.” The position has changed; the size and shape have not.
“The vector is just the difference.” Say which way: image minus pre-image.
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.
- TranslateTranslate (4, −1) by (−6, 5).
(−2, 4). - Earlier work, integersWork out −9 + 12.
3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Find the vectorA corner moves from (−3, −2) to (5, 1). What is the vector?
(8, 3). - ContextOn a map a ferry leaves a harbour at (12, 8) and arrives at (4, 11). Write its move as a vector.
(−8, 3): 8 left and 3 up.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTriangle ABC has A (0, 0). After a translation, A′ is (3, −4) and B′ is (8, −4). What are the coordinates of B?
(5, 0): B′ − (3, −4). - Earlier work, coordinatesWhich quadrant is (−6, −1) in?
The third quadrant.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper: draw triangle ABC, then draw it again after a slide, and write the vector on the page.