At a glance. One lesson of about 60 minutes. It assumes reflections, rotations and translations. Students find the single transformation made by two reflections, choose mirrors for a given slide or turn, use three mirrors, and test the order of a combination.
Calculator: no calculator. Every distance and angle is exact and small, so no calculator is used.
What students will learn
Know
- Parallel mirrors a gap d apart give a translation of 2d.
- Mirrors meeting at an angle θ give a rotation of 2θ.
- The order of a combination can change the result.
Understand
- The slide depends only on the gap, because the distances from the mirror cancel.
- Any isometry is at most three reflections.
- A turn and a slide commute only if one of them is trivial.
Do
- Combine reflections and name the single transformation.
- Choose two mirrors for a given slide or turn.
- Decide with evidence 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 and combined transformations (questions 2 to 8).
- Criterion B. Investigating patterns: the what-if, a slide along or across the mirror.
- Criterion C. Communicating: the proof in the key card and the clearly ordered steps in the worked examples.
- Criterion D. Applying mathematics in real-life contexts: the hall of mirrors and patterns in tiling.
The one thing to do with the figure
Ask for the slide before moving the sliders. Set the mirrors x = −2 and x = 1 and ask for the slide and where A ends, then press the swap button and ask which way the slide goes.
In the second figure set mirrors 60° apart and ask for the turn before reading it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the vase. Collect predictions for how far the second image is. |
| 12 min | The figures with predictions; the key idea and the proof; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the picture of the two crossings. |
| 18 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 | (−3, 5); (8, 1); (3, 4). |
|---|---|
| Hook | The second image is twice the gap away from the vase, on the far side. |
| Worked examples | (9, 4), a slide of 6 to the right; any mirrors 45° apart; the images are different. |
| Faded example | (11, 2). |
| 1. Slide from two parallel mirrors | B. A uses the gap, C the other order, D thinks two flips make one. |
| 2. Image in two parallel mirrors | 15. 2(9) − 3. |
| 3. Rotation from two mirrors | 70. 2 × (55 − 20). |
| 4. Choosing a second mirror | 80. 15 + 65. |
| 5. Turn then slide | −1. Image (−1, 3). |
| 6. Working back through two mirrors | 14. 30 − 16. |
| 7. Three mirrors | 55. t goes to 110 − t. |
| 8. Reflection then rotation | −3. Image (−3, 0). |
| 9. Does the order matter | C. A, B and D give different images when the order changes. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the gap, reversed the order, or thought two reflections make a reflection. | Follow one point through both mirrors: x goes to −2 − x, then to x + 12. |
| 8 or −13 in question 2 | Added the gap, or used the mirrors in the other order. | Reflect one mirror at a time. |
| 35, 150 or −70 in question 3 | Used the angle between, the sum, or the other order. | Subtract, then double. |
| 145, 65 or −50 in question 4 | Added the whole turn, gave the half turn, or went the wrong way. | Halve the turn, then add. |
| −5, −4 or 7 in question 5 | Wrong order, left out the slide, or turned clockwise. | Write the order from left to right. |
| 46 or 22 in question 6 | Added the slide, or used the gap. | Find the slide then undo it. |
| 85, 115 or 110 in question 7 | Gave the last mirror, moved the wrong way, or doubled. | Follow a point at angle t through the three mirrors. |
| 5 or 0 in question 8 | Wrong order, or left out the turn. | Do the mirror first and write the point. |
| A, B or D in question 9 | A: thinks a turn and a slide commute. B: thinks a reflection and a quarter turn commute. D: forgot that swapping two mirrors reverses the turn. | Test with a numerical example. |
Questions worth asking
- Why is the slide twice the gap? A point reaches the first mirror, goes to the far side, then reaches the second and goes beyond again: the distances add to twice the gap.
- Can two reflections give a reflection? No: two reflections give a slide, a turn, or no move if the mirrors are the same.
- Why can three mirrors give a glide reflection? A glide is a reflection plus a slide along the mirror, and the slide is two more reflections in parallel mirrors.
- Does the order of two translations matter? No: the vectors add.
Differentiation
A way in for anyone stuck
- Use tracing paper: trace the shape, flip it over the first mirror, then flip that over the second.
- Provide a fill-in table: mirror 1, image 1, mirror 2, image 2, single move.
- Start with mirrors that are the axes before using other positions.
Stretch
- Prove that two reflections in intersecting mirrors at an angle θ give a rotation of 2θ, using a triangle with one side on the first mirror.
- Show that a rotation of 180° about O is the same as a reflection in the x-axis followed by a reflection in the y-axis, in either order.
Students learning in English as an additional language
“Reflect” means flip over a mirror line. A “translation” is a slide and a “rotation” is a turn. “Commute” means give the same result in either order. A “glide reflection” is a reflection with a slide along the mirror. Act out each one with a sheet of paper.
| Word | Plain meaning | Example |
|---|---|---|
| isometry | a transformation that keeps all lengths and angles the same | a slide, a turn or a flip |
| translation | a slide of every point by the same vector | a tile moved along a wall |
| rotation | a turn about a point through an angle | a clock hand moving |
| glide reflection | a reflection followed by a slide along the mirror | footprints in sand |
| commute | give the same result when the order is swapped | 3 × 5 and 5 × 3 give the same; putting on socks then shoes does not |
Sentence starters for explaining aloud or in writing:
- Reflecting in the first mirror sends A to …, and reflecting in the second sends it to …
- The single transformation is … because the mirrors are …
- The order matters / does not matter because …
Two things not to say
“Two reflections always give back the shape.” Only the same mirror twice does.
“Order never matters in maths.” Test it: a slide then a turn is not a turn then a slide.
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.
- MirrorsReflect (0, 0) in x = 2 and then x = 5. Where is the image?
(6, 0). - TurnTwo mirrors through O are 25° apart. What is the single transformation?
A rotation of 50° about O.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tile is slid across a floor by reflecting it in two parallel grout lines 12 cm apart. How far does it move?
24 cm. - Earlier work (Year 9)Reflect (−2, 7) in the x-axis.
(−2, −7).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA glide reflection reflects in the x-axis and slides 4 to the right. Find the image of (1, 3).
(5, −3). - ProofShow that two reflections in parallel mirrors give a translation by finding the image of (x, y).
(2q − 2p + x, y), which is a translation by (2(q − p), 0).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed. Every answer is reached by hand.