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# Translations

Angles and shapes · Transformations

<!-- Speaker note: Students slide a triangle on the coordinate plane and describe the move by the change in each coordinate (AC9M7SP03, node 1). 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 translation slides every point of a shape the same distance in the same direction. Right adds to x, left takes away; up adds to y, down takes away. The shape does not change size or turn.

- I can translate a shape and write the new coordinates.
- I can describe a translation as so many units left or right and up or down.
- I can say what changes and what stays the same.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need to plot and read coordinates in all four quadrants (Year 6). The one idea: every point moves the same way, so the same amount is added to every x and every y. The common mistake is adding for left or down (Ravi's mistake). This is the hinge for the transformations node: a coordinate slip here carries into reflections and rotations, so check the Practise answers. Combining two translations (the last Modify question) is a wider idea. -->

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## Why it matters

> A chess piece slides across the board, a lift goes up, a character steps left on a screen. Each is a translation: every point moves the same way.

<!-- Speaker note: A **translation** slides a shape without turning or flipping it. Every point moves the same distance in the same direction, so the shape keeps its size and shape. The shape is called the **object** and its new position the **image**; a point A becomes A′ (said "A prime"). On the coordinate plane, a move right or up **adds** to a coordinate and a move left or down **takes away**. A translation of 3 right and 2 down changes $(x, y)$ to $(x + 3, y - 2)$. -->

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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 (1, 1). The triangle is translated 3 right and 2 down. Where does A land?**

- (3, 2)
- (4, −1)
- (−2, −1)
- (4, 3)

<!-- Speaker note: What happens: (4, −1). Right adds 3 to x: 1 + 3 = 4. Down takes 2 from y: 1 − 2 = −1. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask for the direction of the change in x and in y separately. -->

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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%2Ftranslations%2Fstep-2&code=widget%3A+transform%0Amode%3A+translate%0Ashape%3A+1%2C1%3B+3%2C1%3B+3%2C2%0Arange%3A+8%0Aslider+dx%3A+-8%2C+8%2C+0%0Aslider+dy%3A+-8%2C+8%2C+0%0A&fresh=1}}

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

<!-- Speaker note: The orange triangle is the image. The panel lists every corner and where it goes. -->

<!-- Speaker note: Teacher: ask what stays the same for all three corners. -->

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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?

* (4, −1). Right adds 3 to x: 1 + 3 = 4. Down takes 2 from y: 1 − 2 = −1.

<!-- 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>

## Where does B land?

**The triangle has corners A (1, 1), B (3, 1) and C (3, 2). It moves 3 right and 2 down. Where does B land?**

- (6, −1)
- (6, 3)
- (4, −1)
- (0, −1)

* Answer: (6, −1)

<!-- Speaker note: B′ = (3 + 3, 1 − 2) = (6, −1). -->

<!-- Speaker note: Teacher: ask Ravi's question again with 3 left and 2 down. -->

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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

**Ravi translates A (2, 3) by 4 left and 1 up. Click the line where his reasoning goes wrong.**

```
 1  The point A is (2, 3). The translation is 4 left and 1 up.
 2  Left is a move along the x-axis, so I add 4 to x.
 3  Up means add 1 to y.
 4  The image of A is (6, 4).
```

* Answer: line 2

<!-- Speaker note: Line 2: moving left makes x smaller, so take 4 away: 2 − 4 = −2. The image is (−2, 4). -->

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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 stays the same

**What stays the same when a shape is translated?**

- Its coordinates
- Only its size
- Its position
- Its size, its shape and the way it faces

* Answer: Its size, its shape and the way it faces

<!-- Speaker note: A translation only changes the position. The image is congruent to the object. -->

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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 move



- **Aim for A′:** Work out the change in x, then in y.
- **The fourth quadrant:** 
- **Undo the move:** 
- **Two moves:** 

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

<!-- Speaker note: What this step is for: Working backwards from where a corner should end up, and from the image back to the object, shows a translation is a reversible pair of changes. Hints, in order: Compare the x-coordinates of A and A′. / Then compare the y-coordinates. / The top corner C needs a y below 0. / A′ must be to the right of the y-axis. For students who need more support: Count squares on the grid for each move before using the sliders. For early finishers: Two translations are done one after the other. Write the single translation that does the same, and show that the order does not matter. The hard part, on purpose: Left and down mean taking away. Ask students to say, before moving, whether x and y get bigger or smaller. When students get stuck: A student adds when moving left or down. Ask: On the x-axis, does a number get bigger or smaller as you go left? Fix: Left and down take away; right and up add. / A student swaps the x and y changes. Ask: Which way does x measure, across or up? Fix: x is across, y is up; write the horizontal move first. / A student moves only corner A. Ask: Where does corner B go? Fix: Every corner moves the same way. -->

<!-- Speaker note: Teacher: for the undo question, ask what the opposite of each part is. -->

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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>

## Translations

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Ftranslations%2Fstep-5&code=widget%3A+transform%0Amode%3A+translate%0Ashape%3A+1%2C1%3B+3%2C1%3B+3%2C2%0Arange%3A+8%0Aslider+dx%3A+-8%2C+8%2C+0%0Aslider+dy%3A+-8%2C+8%2C+0%0A&fresh=1}}

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

<!-- Speaker note: What this step is for: Reading images, the move between two points and the quadrant, mixed; left and down not taking away is the planted slip; the quadrant review keeps coordinates fresh. -->

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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 x-coordinate

**C is at (3, 2). After a translation of 3 right and 2 down, what is the x-coordinate of C′?**

* Answer: 6

<!-- Speaker note: 3 + 3 = 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 y-coordinate

**With the same translation, what is the y-coordinate of C′?**

* Answer: 0

<!-- Speaker note: 2 − 2 = 0, so C′ = (6, 0). -->

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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>

## Between two points

**Which translation takes (2, 5) to (−1, 3)?**

- 1 left and 3 down
- 3 right and 2 down
- 3 left and 2 up
- 3 left and 2 down

* Answer: 3 left and 2 down

<!-- Speaker note: x goes from 2 to −1 (3 left); y goes from 5 to 3 (2 down). -->

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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: quadrants

**Review: in which quadrant is the point (−3, 2)?**

- Top right
- Bottom left
- Top left
- Bottom right

* Answer: Top left

<!-- Speaker note: A negative x is left of the y-axis and a positive y is above the x-axis. -->

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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: a move to the origin

Then write the translation in words, and as a rule for $(x, y)$.

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

<!-- Speaker note: What this step is for: Finding the translation that puts a chosen corner at the origin makes them reason from the target back to the move. Hints, in order: Corner C is at (3, 2). / Both coordinates must become 0. -->

<!-- Speaker note: Teacher: ask for the translation as a rule for (x, y). -->

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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

**A (2, 5) is translated 4 left and 3 down. Where is A′?**

* (−2, 2): x is 2 − 4 = −2 and y is 5 − 3 = 2.

<!-- 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 starts AC9M7SP03. Where it leads: reflections (next lesson), rotations, then combining transformations and the vector notation for translations in Year 8. -->