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# Reflections

Angles and shapes · Transformations

<!-- Speaker note: Students reflect shapes in the x-axis and y-axis and state which coordinate changes sign (AC9M7SP03, node 2). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** a reflection flips a shape over a mirror line. In the x-axis, (x, y) becomes (x, −y); in the y-axis, (x, y) becomes (−x, y). Each point lands as far on the other side as it started.

- I can reflect a shape in the x-axis or the y-axis.
- I can say which coordinate changes sign for each axis.
- I can check a reflection by its distance from the mirror.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need translations (previous lesson). The one idea: a point and its image are the same distance from the mirror on opposite sides. In the x-axis the point crosses up or down, so y changes sign; in the y-axis it crosses left or right, so x changes sign. The common mistake is the swap (Mia's mistake). Reflecting in other lines (such as y = x) is wider and is left for the stretch. Seeing both reflections close together is deliberate: the Run board lets students flip between them. -->

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## Why it matters

> A face in a mirror, a butterfly's wings, a lake showing a mountain. The mirror line sits exactly halfway between each point and its reflection.

<!-- Speaker note: A **reflection** flips a shape over a mirror line. Each point and its image are the same distance from the mirror, on opposite sides, and the mirror cuts the line joining them at right angles. - Reflect in the **x-axis**: the point moves up or down, so $y$ changes sign: $(x, y) \to (x, -y)$. - Reflect in the **y-axis**: the point moves left or right, so $x$ changes sign: $(x, y) \to (-x, y)$. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Point A is at (2, 1). It is reflected in the x-axis. Where does A′ land?**

- (−2, 1)
- (1, 2)
- (2, −1)
- (−2, −1)

<!-- Speaker note: What happens: (2, −1). The point flips over the x-axis, to the same distance below it, so only y changes sign. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask which way the point travels across the mirror. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Freflections%2Fstep-2&code=widget%3A+transform%0Amode%3A+reflect%0Ashape%3A+2%2C1%3B+4%2C1%3B+4%2C3%0Arange%3A+6%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/reflections/?full=step-2#step-2)

<!-- Speaker note: Press the buttons to choose a mirror, and read the list of corners. -->

<!-- Speaker note: Teacher: ask what stays the same for each mirror. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* (2, −1). The point flips over the x-axis, to the same distance below it, so only y changes sign.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which axis

**B is at (4, 1). It is reflected in the y-axis. Where does B′ land?**

- (4, −1)
- (−4, 1)
- (1, 4)
- (−4, −1)

* Answer: (−4, 1)

<!-- Speaker note: In the y-axis, x changes sign: (−4, 1). -->

<!-- Speaker note: Teacher: ask how Mia could check her answer with the picture. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Mia reflects P (4, 3) in the x-axis. Click the line where her reasoning goes wrong.**

```
 1  The point P is (4, 3). Mia reflects it in the x-axis.
 2  The x-axis is horizontal, so she changes the x-coordinate.
 3  She changes 4 to −4.
 4  The image of P is (−4, 3).
```

* Answer: line 2

<!-- Speaker note: Line 2: reflecting in the x-axis moves the point up or down across it, so y changes sign. The image is (4, −3). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## On the mirror

**A point is on the mirror line, for example (0, 3) reflected in the y-axis. Where is its image?**

- At (3, 0)
- In the same place: (0, 3)
- At (0, −3)
- At (−3, 0)

* Answer: In the same place: (0, 3)

<!-- Speaker note: It is zero distance from the mirror, so it does not move: −0 is still 0. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Choose the mirror



- **Reach (4, 1):** 
- **Reach (−2, −4):** 
- **Distance:** 
- **Both mirrors:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/reflections/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Choosing a mirror to reach a target and comparing the two together shows the rule for each axis; two reflections together make a half turn. Hints, in order: The triangle is on the left of the y-axis. / A has x = −4. Flip it across the y-axis. For students who need more support: Use tracing paper folded along the axis and press through the corners. For early finishers: Reflect in the line y = x. What happens to the coordinates? What about two reflections in a row, in the x-axis and then the y-axis: which transformation is that? The hard part, on purpose: Which coordinate changes. Ask students to say which way the point moves across the mirror, up and down or left and right. When students get stuck: A student changes the wrong coordinate. Ask: Does the point cross the mirror by going up and down, or left and right? Fix: The coordinate that measures the distance to the mirror changes sign. / A student changes both signs. Ask: How many mirrors did you use? Fix: One mirror changes one coordinate; two mirrors change both. / A student puts the image on the same side. Ask: Which side of the mirror is the image on? Fix: The image is on the opposite side of the mirror. -->

<!-- Speaker note: Teacher: for the two mirrors, ask what single move does the same. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Reflections

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Freflections%2Fstep-5&code=widget%3A+transform%0Amode%3A+reflect%0Ashape%3A+2%2C1%3B+4%2C1%3B+4%2C3%0Arange%3A+6%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/reflections/?full=step-5#step-5)

<!-- Speaker note: What this step is for: Rules for both axes and a translation review, mixed; changing both signs is the planted slip; the review keeps left and down fresh. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## y-axis

**The point (−4, 6) is reflected in the y-axis. What is the x-coordinate of the image?**

* Answer: 4

<!-- Speaker note: x changes sign: −4 becomes 4, so the image is (4, 6). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## x-axis

**The point (−4, 6) is reflected in the x-axis. What is the y-coordinate of the image?**

* Answer: -6

<!-- Speaker note: y changes sign: 6 becomes −6, so the image is (−4, −6). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The rule

**Which rule describes a reflection in the y-axis?**

- (x, y) → (y, x)
- (x, y) → (−x, y)
- (x, y) → (−x, −y)
- (x, y) → (x, −y)

* Answer: (x, y) → (−x, y)

<!-- Speaker note: The y-axis is the vertical mirror: x changes sign and y stays. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: translations

**Review: the point (2, 3) is translated 4 left and 1 up. What is the x-coordinate of the image?**

* Answer: -2

<!-- Speaker note: 2 − 4 = −2, so the image is (−2, 4). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: an arrow's mirror image

Then write the rule for each mirror in your own words, and say which coordinate stays the same.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/reflections/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing the mirror that puts the point of an arrow in a named place combines the rule and a check on each corner. Hints, in order: The point of the arrow is at (4, 2). / To get x = −4, change the sign of x. -->

<!-- Speaker note: Teacher: ask for the rule written for both axes. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**The point (5, −2) is reflected in the x-axis. Where is the image?**

* (5, 2): y changes sign.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This continues AC9M7SP03. Where it leads: rotations (next lesson); reflection and rotational symmetry in Year 8; combining transformations. -->