At a glance. One lesson of about 60 minutes. It assumes the descriptions of the four transformations from lesson 5. Students replace two reflections by a single translation or rotation, combine transformations by coordinates, and decide whether the order matters.
Calculator: no calculator. The lesson is done by hand on a coordinate grid, so it carries no calculator label. The figure shows decimals only for turns that are not multiples of 90°.
What students will learn
Know
- Reflections in parallel lines x = a, then x = b make a translation of 2(b − a) perpendicular to them.
- Reflections in lines that meet at an angle θ make a rotation of 2θ about the meeting point.
- Every isometry is a reflection, translation, rotation or glide reflection, and the order of transformations can matter.
Understand
- Two flips give back the orientation, so the result cannot be a reflection.
- The order matters because each transformation acts on the output of the one before.
Do
- Replace two reflections by a single transformation.
- Find the image of a point under two transformations in a given order.
- Decide whether the order matters.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: two reflections, combined transformations and the order of transformations (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure (slide is twice the gap, turn is twice the angle) and the second (when the order does not matter).
- Criterion C. Communicating: naming which transformation comes first in every description (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the border pattern in the hook.
The one thing to do with the figure
Settle the hook with two mirrors. Set a = 0 and b = 1, then a = 0 and b = 2. Ask what happened to the distance moved when the gap doubled.
In the second figure, ask for any p and q other than 0 for which the two images coincide. There are none, and students can say why from the formulas.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the border. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook. |
| 8 min | Second figure; worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the gap. |
| 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 9. |
The answers
| Warm-up | (−4, −3); (3, 7); 110°. |
|---|---|
| Hook | A slid copy, moved by twice the gap. |
| Worked examples | 10 units to the right; 70°; (3, −2) and (3, −4). |
| Faded example | (7, 4). |
| 1. Two parallel reflections | D. A is the gap itself; B adds the positions; C needs mirrors that meet. |
| 2. The slide from two parallel mirrors | 8. 2 × 4. |
| 3. Reflect in two axes | −3. (−3, −4). |
| 4. The turn from two meeting mirrors | 50. 2 × 25. |
| 5. Turn then slide | 5. (5, −5). |
| 6. Slide then turn | −7. (1, −7). |
| 7. Rotation of 180° about another centre | −1. (−1, −2). |
| 8. Angle between mirrors from a rotation | 65. 130 ÷ 2. |
| 9. Does the order matter | C. A, B and D are the three combinations that give different images in different orders. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Gave the gap, added the positions, or called it a rotation. | Mark the two mirrors and measure the gap between them. |
| 4 or 6 in question 2 | Gave the gap, or added the positions. | Count the squares between the mirrors, then double. |
| 3 or −4 in question 3 | Did not change the sign of x, or gave y. | Write the point after each reflection on its own line. |
| 25 or 65 in question 4 | Gave the mirror angle, or used the complement. | Ask: is the turn half or double the angle between the mirrors? |
| 1 or 2 in question 5 | Did the slide first, or only the turn. | Write “first” and “second” above the two transformations. |
| −5 or 1 in question 6 | Did the turn first, or gave the slide alone. | Compare with question 5: the same two moves, the other order. |
| −5 or 7 in question 7 | Turned about the origin, or added the centre to the point. | Mark the centre and the point and count to the centre. |
| 130 or 260 in question 8 | Gave the rotation, or doubled it again. | Work forwards: mirror angle, then double. |
| A, B or D in question 9 | Thought any two transformations commute. | Test with the point (1, 2) both ways, as in the feedback. |
Questions worth asking
- Why are two reflections never a reflection? Each reflection reverses orientation; two reverse it back, so the result keeps orientation.
- What happens if the two mirrors are the same line? The shape returns to where it began: the identity.
- Does the order matter for two reflections in parallel lines? Yes: swapping the mirrors reverses the direction of the translation.
- When does the order not matter? For example for two translations, or two rotations about the same centre.
Differentiation
A way in for anyone stuck
- Give tracing paper and a ruler: students make each reflection physically.
- Provide the frame: “The mirrors are … so the result is a … of …”.
- Use only mirrors through the origin or only vertical mirrors until the pattern is clear.
Stretch
- Write a glide reflection as three reflections. What are the three mirrors?
- Two rotations of 90° clockwise, about different centres, are applied one after the other. What single transformation is that?
Students learning in English as an additional language
“Then” means “after that”, so “reflect in A then in B” has A first. “Parallel” lines never meet. “Glide” means to slide smoothly. Say “twice the gap” and “twice the angle” aloud as a pair.
| Word | Plain meaning | Example |
|---|---|---|
| combined transformation | one transformation followed by another | a turn then a slide |
| parallel | never meeting, always the same distance apart | railway lines |
| glide reflection | a reflection followed by a slide along the mirror line | footprints in sand |
| identity | the transformation that moves nothing | turning 360° |
| orientation | which way round a shape is: clockwise or anticlockwise | the order of the corners A, B, C |
| commute | give the same result in either order | putting on a hat and a scarf: either order gives the same result |
Sentence starters for explaining aloud or in writing:
- The mirrors are … so the result is a …
- The gap between the mirrors is … so …
- The order matters because …
Two things not to say
“It does not matter which one you do first.” Sometimes it does: show the counter-example.
“Two reflections cancel.” They cancel only if the mirrors are the same line.
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.
- SlideA shape is reflected in x = 0 and then in x = 3. Describe the single transformation.
A translation of 6 units to the right. - Earlier work (lesson 5)Reflect the point (2, 7) in the line y = 4.
(2, 1).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tile in Marrakesh is reflected in x = 2 and then in x = 7. By how far has the motif moved?
10 units. - Working backA translation of 12 units right is made from two reflections. One mirror is x = 1. Find the other.
x = 7 if x = 1 is the first mirror (2 × (7 − 1) = 12); if x = 1 is the second, the first is x = −5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why a rotation of 90° anticlockwise about the origin followed by a rotation of 180° about the origin is the same as a rotation of 270° anticlockwise.
Angles add for rotations about the same centre. - OpenFind two transformations for which the order does not matter and two for which it does, and justify each.
For example two translations commute; a reflection in the x-axis and a rotation of 90° do not.
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. “Show me the steps” gives the same states in words.
No device? Tracing paper, squared paper and a ruler are enough.