---
marp: true
theme: default
html: true
---

# Rotations

Angles and shapes · Transformations

<!-- Speaker note: Students turn a shape about a centre and find the image of each corner for quarter and half turns (AC9M7SP03, node 3). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

---

## Today

**We are learning to** a rotation turns a shape about a centre by an angle in a direction. About the origin, a quarter turn anticlockwise changes (x, y) to (−y, x) and a half turn changes it to (−x, −y).

- I can rotate a shape 90°, 180° or 270° about the origin.
- I can describe a rotation by its centre, angle and direction.
- I can use the pattern in the coordinates to check an answer.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need translations and reflections (previous lessons). The one idea: a rotation needs a centre, an angle and a direction. About the origin, an anticlockwise quarter turn is $(x, y) \to (-y, x)$; the half turn changes both signs. The common mistake is only swapping the coordinates (Lee's mistake). The board turns anticlockwise only; clockwise turns are the questions in Modify and Make (90° clockwise is 270° anticlockwise). Rotation is the hardest transformation to reason about: let students trace and turn before using the rule. The rule is a check on the answer, not the method. Turning about another centre (the second Modify task) is done by the picture: the rule does not apply. -->

---

## Why it matters

> The hands of a clock, a Ferris wheel, a spinning fan. Each turns about a centre by a certain angle, in a certain direction.

<!-- Speaker note: A **rotation** turns a shape about a fixed point, the **centre of rotation**. To describe one, give the **centre**, the **angle** and the **direction** (clockwise or anticlockwise). The shape keeps its size and shape. About the origin, an anticlockwise turn of 90° changes $(x, y)$ to $(-y, x)$; 180° (a half turn) changes it to $(-x, -y)$; 270° changes it to $(y, -x)$. -->

---

<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). The triangle turns 90° anticlockwise about the origin. Where does A land?**

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

<!-- Speaker note: What happens: (−1, 2). A is right of the centre, so it turns up and to the left. Its distance from the origin stays the same. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask students to point a pencil and turn it a quarter. -->

---

<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%2Frotations%2Fstep-2&code=widget%3A+transform%0Amode%3A+rotate%0Ashape%3A+2%2C1%3B+4%2C1%3B+4%2C2%0Arange%3A+6%0Aslider+t%3A+0%2C+360%2C+0%2C+90%0A&fresh=1}}

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

<!-- Speaker note: The black dot is the centre of rotation. The sliders turn the shape anticlockwise. -->

<!-- Speaker note: Teacher: ask what stays the same after each turn (the distance to the centre). -->

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

---

<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?

* (−1, 2). A is right of the centre, so it turns up and to the left. Its distance from the origin stays the same.

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

---

<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>

## A half turn

**B is at (4, 1). It turns 180° about the origin. Where does B′ land?**

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

* Answer: (−4, −1)

<!-- Speaker note: A half turn changes both signs: (−4, −1). -->

<!-- Speaker note: Teacher: ask how Lee could test his answer. -->

---

<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

**Lee turns P (3, 1) by 90° anticlockwise about the origin. Click the line where his reasoning goes wrong.**

```
 1  The point P is (3, 1). Lee turns it 90° anticlockwise about the origin.
 2  A quarter turn is a big move, so P goes round to the next quadrant.
 3  A quarter turn just swaps the coordinates.
 4  So the image of P is (1, 3).
```

* Answer: line 3

<!-- Speaker note: Line 3: a quarter turn also changes a sign. P turns up and to the left, so the image is (−1, 3), not (1, 3). -->

---

<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 describes a rotation

**What three things do you need to describe a rotation?**

- The angle only
- The centre, the angle and the direction
- The centre and the size of the shape
- The axis and the angle

* Answer: The centre, the angle and the direction

<!-- Speaker note: The same angle about a different centre, or in the other direction, gives a different image. -->

---

<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>

## Change the turn and the centre



- **Three quarter turns:** 
- **A different centre:** 
- **Clockwise:** 
- **A full turn:** 

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

<!-- Speaker note: What this step is for: Choosing an angle to reach a target, then moving the centre, shows the image depends on the centre; clockwise and anticlockwise turns are linked. Hints, in order: A is at (2, 1). / Three quarter turns anticlockwise is 270°. / The centre is now (1, 0), not the origin. / A is one right and one up from the centre. A quarter turn takes it to one left and one up. For students who need more support: Use tracing paper: trace the shape, pin the centre and turn the paper a quarter at a time. For early finishers: Show that reflecting in the x-axis and then the y-axis gives the same image as a 180° rotation about the origin. Which two reflections (one in the line y = x) make a 90° rotation? The hard part, on purpose: Picturing the turn. Use tracing paper pinned at the centre, or have students point a pencil at the shape and turn it. When students get stuck: A student swaps the coordinates without a sign. Ask: Which quadrant should the image be in? Fix: Picture the turn first, then use the rule to check. / A student turns clockwise. Ask: Which way do clock hands go? Which way do we turn here? Fix: The board turns anticlockwise; clockwise 90° is 270° anticlockwise. / A student forgets the centre. Ask: What point does the shape turn about? Fix: Always name the centre, the angle and the direction. -->

<!-- Speaker note: Teacher: for the centre (1, 0), ask where A is measured from. -->

---

<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>

## Rotations

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Frotations%2Fstep-5&code=widget%3A+transform%0Amode%3A+rotate%0Ashape%3A+2%2C1%3B+4%2C1%3B+4%2C2%0Arange%3A+6%0Aslider+t%3A+0%2C+360%2C+0%2C+90%0A&fresh=1}}

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

<!-- Speaker note: What this step is for: Half turns, finding the turn between two points and a reflection review, mixed; swapping without a sign is the planted slip; the review keeps mirrors fresh. -->

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

---

<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>

## A half turn

**B is at (4, 1). After a 180° turn about the origin, what is the x-coordinate of B′?**

* Answer: -4

<!-- Speaker note: x changes sign: −4. The image is (−4, −1). -->

---

<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 turn

**Which turn about the origin takes (2, 1) to (−1, 2)?**

- 180°
- 90° clockwise
- 90° anticlockwise
- 270° anticlockwise

* Answer: 90° anticlockwise

<!-- Speaker note: (2, 1) to (−1, 2) is the rule (x, y) → (−y, x): a quarter turn anticlockwise. -->

---

<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>

## A quarter turn

**The point (3, 1) turns 90° anticlockwise about the origin. What is the y-coordinate of the image?**

* Answer: 3

<!-- Speaker note: (x, y) → (−y, x), so (3, 1) becomes (−1, 3). -->

---

<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: reflections

**Review: the point (3, 5) is reflected in the x-axis. What is the y-coordinate of the image?**

* Answer: -5

<!-- Speaker note: (3, 5) becomes (3, −5). -->

---

<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: turn to a target

Then describe the rotation fully: the centre, the angle and the direction. How else could you say it, turning clockwise?

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

<!-- Speaker note: What this step is for: Finding the turn that puts a chosen corner on a target applies the rule and then states the full description: centre, angle and direction. Hints, in order: Corner B is at (4, 1). / Try each turn and read B′. -->

<!-- Speaker note: Teacher: ask for the description with the direction. -->

---

<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 (2, 3) turns 180° about the origin. Where is the image?**

* (−2, −3): both signs change.

<!-- 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 completes AC9M7SP03. Where it leads: combining transformations and tessellations (Year 8); rotational symmetry. -->