At a glance. One lesson of about 60 minutes. Reflecting points and shapes in the axes, y = x and lines parallel to the axes, and finding a mirror line from a point and its image.
Calculator: no calculator. The numbers are small on purpose: the work is in the distance on the other side of the mirror.
What students will learn
Know
- A reflection flips a shape over a mirror line; each point and its image are the same distance from it.
- In the x-axis (x, y) → (x, −y); in the y-axis (x, y) → (−x, y); in y = x (x, y) → (y, x).
- A reflection is a congruence transformation; points on the mirror line do not move.
Understand
- The coordinate rules come from the equal-distance idea, so they extend to lines such as x = 3.
- A reflection keeps lengths and angles but reverses the order of the corners.
Do
- Reflect points and polygons in the axes, y = x and lines x = c or y = c.
- Find the mirror line from a point and its image.
- Describe a reflection fully by naming the mirror line.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the reflection rules and the distance method (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to state a rule linking the gap to the distance from the mirror.
- Criterion C. Communicating: naming the mirror line as an equation, and justifying an image by a distance.
- Criterion D. Applying mathematics in real-life contexts: symmetric patterns in tiles, lanterns and printed cloth.
The one thing to do with the figure
Before pressing a button, ask for the image. Ask for A′ in the line y = x before the working appears, and let them argue before you reveal it.
On the second figure, ask: “I moved the mirror by 1. How far did the image move?” It moves 2: the distance changes by 1 on each side. This is the key to Question 8.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 3 (distance across zero). |
|---|---|
| 10 min | Hook, prediction for C, then the first figure. |
| 10 min | Key idea; the second figure, keeping the distance. |
| 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 | (−1, 1); 6; 7. |
|---|---|
| Worked examples | (3, 5); (−1, 1), (−4, 2), (−2, 5); (1, 5); (2, 3). |
| Faded example | S′ (−4, 3). |
| 1. Reflect in the x-axis | B, (−6, −4). A is the y-axis reflection; C swaps; D changes both signs. |
| 2. Reflect in the y-axis | −7. x changes sign. |
| 3. Reflect in y = x | 5. The image is (8, 5). |
| 4. Reflect in x = c | 7. 2 + 5. |
| 5. Reflect in y = c | 2. 4 − 2. |
| 6. Find the mirror line | B, x = 5. The midpoint of 2 and 8. |
| 7. Two reflections | −3. Final image (−4, 1). |
| 8. Distance and gap | 5. 10 ÷ 2. |
| 9. Reflect in y = x | A. B is the y-axis; C the x-axis; D changes both signs. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Changed the wrong sign, swapped, or changed both. | Ask which direction the x-axis flips: up and down, so which coordinate? |
| 7 or −3 in question 2 | Used the x-axis rule, or moved y into x. | Plot (7, −3) and its mirror image on the opposite side of the y-axis. |
| 8 or −8 in question 3 | Left y alone, or used the sign change. | Swap the coordinates and check that the pair is equally far from y = x. |
| 3 or 5 in question 4 | Reflected in the y-axis, or used only the 3. | Draw the line x = 2 and mark the distance to Q. |
| −6 or 10 in question 5 | Reflected in the x-axis, or added. | Count the distance to y = 4 on a vertical number line. |
| A, C or D in question 6 | Chose a mirror that is not halfway. | Find the midpoint of the two x-values. |
| 5 or −5 in question 7 | Applied only one reflection. | Write each point after each reflection. |
| 10 or 20 in question 8 | Used the whole gap, or doubled it. | Draw the mirror halfway between the two points. |
Questions worth asking
- Why does a point on the mirror not move? Listen for “it is zero distance from the line, so its image is itself”.
- Is reflecting in the x-axis the same as changing the sign of x? No: the x-axis flips up and down, so the sign of y changes. Many students expect the opposite.
- How can you find a mirror line from a point and its image? It is the perpendicular bisector: halfway between them, at a right angle to the line joining them.
Differentiation
A way in for anyone stuck
- Fold squared paper along the mirror line and prick through to find the image.
- Draw a short arrow from each corner across the mirror and count squares each side.
- Start with mirrors that are the axes, and bring in x = c when those are secure.
Stretch
- A point and its image are (1, 5) and (7, 5). What is the mirror line? (x = 4.)
- Reflect (a, b) in the line y = x, then in the y-axis. What is the final image? ((−b, a).)
Students learning in English as an additional language
Reflection, mirror line, image and distance are new in this sense. Pair each with a picture: a mirror, a fold line, and a double-headed arrow for distance.
| Word | Plain meaning | Example |
|---|---|---|
| reflection | a flip over a line so the shape faces the other way | a face in a mirror |
| mirror line | the line a shape is flipped over; also called the line of reflection | the y-axis, or x = 3 |
| image | the shape after it is flipped; its corners have a dash | A′B′C′ |
| perpendicular | at a right angle (90 degrees) | the dashed lines meet the mirror perpendicularly |
| congruent | exactly the same size and shape | a shape and its reflection |
Sentence starters for explaining aloud or in writing:
- The mirror line is …, and the point is … units from it.
- The image is on the other side, so its x-coordinate is …
- I know it is a reflection because the distances are …
Two things not to say
“A reflection changes the sign.” It is only true for the axes. Say which coordinate and which axis.
“Flip it over.” Over what? Always name the mirror line.
“The picture is the same.” The shape is the same, but the order of the corners is reversed.
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.
- ReflectReflect (−5, 2) in the x-axis.
(−5, −2). - Earlier work, translationsTranslate (1, 4) by (−3, −6).
(−2, −2).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Reflect in y = cReflect (3, −2) in the line y = 1.
(3, 4): the point is 3 below the line, so the image is 3 above. - Find the mirror(−4, 1) reflects onto (2, 1). Find the mirror line.
x = −1.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA shape is reflected in y = x. A corner at (4, 4) does not move. Why?
It is on the line y = x, so its distance to the mirror is zero. - Earlier work, absolute valueHow far is −5 from 3 on a number line?
8.
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: draw triangle ABC and a mirror line, then fold along the line and trace.