At a glance. One lesson of about 60 minutes. Performing two transformations in order, describing a transformation fully, and deciding between congruence and similarity.
Calculator: no calculator. The numbers are small on purpose: the work is in the order, and in testing every point.
What students will learn
Know
- To combine, apply the second transformation to the image of the first.
- A full description of a translation needs its vector, a reflection its mirror line, a rotation its centre, angle and direction, an enlargement its centre and scale factor.
- Translations, reflections and rotations are congruence transformations; an enlargement is a similarity transformation.
Understand
- The order of two transformations can matter.
- One point is not enough to identify a transformation: all points must agree.
Do
- Perform two transformations in a stated order.
- Identify and fully describe a single transformation from a pre-image and its image.
- Decide whether a transformation is congruent or similar.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: combining transformations and identifying them (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to find when the two orders agree and to say why.
- Criterion C. Communicating: writing a full description, and showing the intermediate point.
- Criterion D. Applying mathematics in real-life contexts: tile and cloth patterns built by repeating one motif with several moves.
The one thing to do with the figure
Before pressing the second button, ask for the answer. With the first order showing (5, −1), ask: “Where will A end up if I swap the order?” Let them calculate before it appears.
On the second figure, ask: “Which one changed the lengths? How do you know from the numbers?”
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 2 (a quarter turn clockwise). |
|---|---|
| 10 min | Hook, prediction for the two orders, then the first figure. |
| 10 min | Key idea; the second figure, naming what you see. |
| 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 | (12, −8); (4, −1); (3, −4). |
|---|---|
| Worked examples | (−1, −1); (−1, −3); reflection in the y-axis; rotation 90° anticlockwise about the origin. |
| Faded example | (4, 1). |
| 1. Two moves in order | B, (3, −5). A reverses the order; C omits the reflection; D subtracts the vector. |
| 2. Translate then reflect | −5. Final image (−5, 2). |
| 3. Translate then rotate | −3. Final image (2, −3). |
| 4. Identify from two points | C. Test every point; D works for the first point only. |
| 5. Enlarge then translate | 2. Final image (2, 8). |
| 6. Describe fully | B. A is true but incomplete; C changes both signs; D works for P only. |
| 7. Area through three moves | 20. 5 × 4. |
| 8. Work back | 1. (1, 3) → (3, 1) → (5, 0). |
| 9. Order matters | A. (−3, 2) to (2, 3) to (3, 2). B leaves out the reflection; C reverses the order; D leaves out the turn. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Reversed the order, skipped a move, or subtracted the vector. | Write the point after each move on its own line. |
| 5 or 3 in question 2 | Skipped the reflection, or reversed the order. | Underline “first” and “then” in the question. |
| 3 or −1 in question 3 | Skipped the turn, or turned the original point. | Write the translated point first, and turn that. |
| A, B or D in question 4 | Tested one point only, or chose the wrong rule. | Test all given points against each rule. |
| −6 or 10 in question 5 | Reversed the order, or subtracted the vector. | Write the point after the enlargement, then add. |
| A, C or D in question 6 | Incomplete description, or the wrong move. | Ask: which line? Which centre? Which angle? |
| 10 or 5 in question 7 | Area multiplied by k only, or not at all. | Draw a 2 by 3 rectangle, enlarge by 2, and count squares. |
| −1 or 0 in question 8 | Sign slip, or forgot the translation. | Write the point with a in it after each move. |
Questions worth asking
- Does reflecting then translating give the same result as translating then reflecting? No, in general. Ask for a pair of orders that do agree.
- How many points do you need to check to name a transformation? Enough to be sure: at least two, ideally all of the corners.
- Why is “a reflection” not a complete answer? Listen for “you have not said which mirror line”.
Differentiation
A way in for anyone stuck
- Use two colours: pre-image, first image, second image.
- Make a one-page table with the four types and what each needs to be fully described.
- Use one point and one move at a time before whole shapes.
Stretch
- Reflect in the x-axis and then in the y-axis. What single transformation is that? (A half turn about the origin: (x, y) to (x, −y) to (−x, −y).)
- Do a translation by (a, b) and then its reverse (−a, −b). What single transformation is that? (The identity: nothing moves.)
Students learning in English as an additional language
Combined, order, congruent and similar are used precisely here. Pair each with a picture: two arrows in a row for combined, and a numbered list for order.
| Word | Plain meaning | Example |
|---|---|---|
| combined transformation | two or more transformations one after another | reflect, then translate |
| order | which move comes first | the order matters |
| congruent | exactly the same size and shape | a shape and its translation |
| similar | the same shape, possibly a different size | a shape and its enlargement |
| describe fully | give every detail needed to repeat the move | reflection in the line y = x |
Sentence starters for explaining aloud or in writing:
- First I …, and the point moved to …; then I …
- It is a … because … and the mirror line / centre / vector is …
- It is congruent / similar because the lengths …
Two things not to say
“It does not matter which you do first.” It can.
“It is just a reflection.” Which line? A full description names it.
“Both points match, so it is right.” Check all of them.
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.
- CombineReflect (2, 6) in the y-axis, then translate by (1, −4).
(−1, 2). - Earlier work, enlargementEnlarge (−2, 3) by scale factor 5 from the origin.
(−10, 15).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Identify(1, 2) goes to (3, 4) and (0, 5) goes to (2, 7). Describe the transformation.
A translation by (2, 2). - OrderTranslate (0, 0) by (4, 0), then enlarge by 2 from the origin. Then do it in the other order. Compare.
(8, 0) and (4, 0): different.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA square is reflected, rotated and translated. Does its perimeter change? Why?
No: all three keep lengths. - Earlier work, rotationsTurn (−3, 2) through 90° anticlockwise about the origin.
(−2, −3).
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 button or 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 and three colours: pre-image, first image and final image.