Translations

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min

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.

Running the lesson

1. Predict: ask for the direction of the change in x and in y separately.
2. Run: ask what stays the same for all three corners.
3. Investigate: ask Ravi's question again with 3 left and 2 down.
4. Modify: for the undo question, ask what the opposite of each part is.
5. Make: ask for the translation as a rule for (x, y).

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

Answers to the checks

1. where does A land?: Point A is at (1, 1). The triangle is translated 3 right and 2 down. Where does A land?
   (4, −1). Right adds 3 to x: 1 + 3 = 4. Down takes 2 from y: 1 − 2 = −1.
2. slide the triangle: Slide the triangle until A′ is at (4, −1). Was your prediction right?
   
   A moves 3 right and 2 down, so A′ is (1 + 3, 1 − 2) = (4, −1).
3. 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)
   B′ = (3 + 3, 1 − 2) = (6, −1).
4. Find the mistake: Ravi translates A (2, 3) by 4 left and 1 up. Click the line where his reasoning goes wrong.
   line 2
   Line 2: moving left makes x smaller, so take 4 away: 2 − 4 = −2. The image is (−2, 4).
5. What stays the same: What stays the same when a shape is translated?
   Its size, its shape and the way it faces
   A translation only changes the position. The image is congruent to the object.
6. Aim for A′: Slide the triangle so that A′ is at (−3, 2).
   
   From (1, 1) to (−3, 2) is 4 left and 1 up.
7. The fourth quadrant: Slide the triangle into the fourth quadrant (right of the y-axis, below the x-axis).
   
   For example 2 right and 4 down: every corner has a positive x and a negative y.
8. Undo the move: A shape is translated 5 right and 3 down. Which translation moves the image back to where it started?
   5 left and 3 up
   Undo each part: 5 right becomes 5 left, and 3 down becomes 3 up.
9. Two moves: A shape is translated 2 right, then 3 up. Another shape is translated 3 up, then 2 right. Do they end up in the same place?
   Yes: the order of the two moves does not matter
   Both change $(x, y)$ to $(x + 2, y + 3)$, so the order does not matter.
10. The x-coordinate: C is at (3, 2). After a translation of 3 right and 2 down, what is the x-coordinate of C′?
    6
    3 + 3 = 6.
11. The y-coordinate: With the same translation, what is the y-coordinate of C′?
    0
    2 − 2 = 0, so C′ = (6, 0).
12. Between two points: Which translation takes (2, 5) to (−1, 3)?
    3 left and 2 down
    x goes from 2 to −1 (3 left); y goes from 5 to 3 (2 down).
13. Review: quadrants: Review: in which quadrant is the point (−3, 2)?
    Top left
    A negative x is left of the y-axis and a positive y is above the x-axis.
14. a move to the origin: Slide the triangle so that corner C′ lands exactly on the origin (0, 0).
    
    C goes from (3, 2) to (0, 0): 3 left and 2 down.

Solutions

One possible solution for each task. Other correct programs are fine.

choose the move › Aim for A′

Open task: any program that does what the step asks.

choose the move › The fourth quadrant

Open task: any program that does what the step asks.

choose the move › Undo the move

Open task: any program that does what the step asks.

choose the move › Two moves

Open task: any program that does what the step asks.

a move to the origin

Open task: any program that does what the step asks.

Stretch and support

Stretch: Two translations are done one after the other. Write the single translation that does the same, and show that the order does not matter.

Support: Count squares on the grid for each move before using the sliders.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Translations".

The idea: Translations using coordinates: I can Students translate a set of points on the Cartesian plane and describe the translation using coordinate changes.

What understanding looks like:
- Correctly translates a set of points and records the new coordinates.
- Describes the translation in terms of horizontal and vertical movement.
- Verifies that the shape is congruent to the original after translation.

Mistakes students often make here:
- A student adds when moving left or down.
- A student swaps the x and y changes.
- A student moves only corner A.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Translations".

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.

What each step is for

• where does A land?: Committing to the image of one point under 3 right and 2 down shows whether they add to the right coordinate in the right direction.
• slide the triangle: Sliding the triangle and reading each corner's change shows that the same amount is added to every x and to every y.
• reading the move: Right and up add, left and down take away (Ravi adds for left); every corner moves the same; the shape stays the same size.
• choose the move: 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.
• translations: 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.
• a move to the origin: Finding the translation that puts a chosen corner at the origin makes them reason from the target back to the move.

Where it leads

This starts AC9M7SP03.

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