Rotations

Classroom resources

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Students will be able to

• 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.

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 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.

Running the lesson

1. Predict: ask students to point a pencil and turn it a quarter.
2. Run: ask what stays the same after each turn (the distance to the centre).
3. Investigate: ask how Lee could test his answer.
4. Modify: for the centre (1, 0), ask where A is measured from.
5. Make: ask for the description with the direction.

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

Answers to the checks

1. a quarter turn: Point A is at (2, 1). The triangle turns 90° anticlockwise about the origin. Where does A land?
   (−1, 2). A is right of the centre, so it turns up and to the left. Its distance from the origin stays the same.
2. turn the triangle: Turn the triangle 90° anticlockwise. Was your prediction right?
   
   A (2, 1) becomes A′ (−1, 2): the new x is the old y with its sign changed, and the new y is the old x.
3. A half turn: B is at (4, 1). It turns 180° about the origin. Where does B′ land?
   (−4, −1)
   A half turn changes both signs: (−4, −1).
4. Find the mistake: Lee turns P (3, 1) by 90° anticlockwise about the origin. Click the line where his reasoning goes wrong.
   line 3
   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).
5. What describes a rotation: What three things do you need to describe a rotation?
   The centre, the angle and the direction
   The same angle about a different centre, or in the other direction, gives a different image.
6. Three quarter turns: Turn the triangle so that A′ lands at (1, −2).
   
   A turn of 270° anticlockwise changes (2, 1) to (1, −2).
7. A different centre: The centre of rotation is (1, 0). Turn the triangle 90° anticlockwise. Where does A land?
   
   A is (1, 1) from the centre; after a quarter turn it is (−1, 1) from the centre, which is (0, 1).
8. Clockwise: A turn of 90° clockwise ends in the same place as a turn of how many degrees anticlockwise?
   270°
   A quarter turn clockwise is three quarter turns anticlockwise: 270°.
9. A full turn: What does a rotation of 360° do?
   Nothing: the shape ends up where it started
   A full turn brings every point back to where it started.
10. A half turn: B is at (4, 1). After a 180° turn about the origin, what is the x-coordinate of B′?
    -4
    x changes sign: −4. The image is (−4, −1).
11. Which turn: Which turn about the origin takes (2, 1) to (−1, 2)?
    90° anticlockwise
    (2, 1) to (−1, 2) is the rule (x, y) → (−y, x): a quarter turn anticlockwise.
12. A quarter turn: The point (3, 1) turns 90° anticlockwise about the origin. What is the y-coordinate of the image?
    3
    (x, y) → (−y, x), so (3, 1) becomes (−1, 3).
13. Review: reflections: Review: the point (3, 5) is reflected in the x-axis. What is the y-coordinate of the image?
    -5
    (3, 5) becomes (3, −5).
14. turn to a target: Turn the triangle so that B′ is at (1, −4).
    
    A turn of 270° anticlockwise about the origin changes (4, 1) to (1, −4).

Solutions

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

change the turn and the centre › Three quarter turns

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

change the turn and the centre › A different centre

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

change the turn and the centre › Clockwise

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

change the turn and the centre › A full turn

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

turn to a target

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

Stretch and support

Stretch: 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?

Support: Use tracing paper: trace the shape, pin the centre and turn the paper a quarter at a time.

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 "Rotations".

The idea: Rotations about a given point: I can Students rotate a set of points about a given point (origin or other) by 90°, 180° and 270° and describe the rotation using coordinates.

What understanding looks like:
- Correctly rotates points by 90° and 180° about the origin.
- Records the new coordinates and identifies the pattern.
- Specifies direction of rotation (clockwise/counterclockwise) when describing.

Mistakes students often make here:
- A student swaps the coordinates without a sign.
- A student turns clockwise.
- A student forgets the centre.

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.
- If I seem to be missing the idea that comes before this one (Reflections on axes), check it quickly, then come back.
- 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 "Rotations".

Exit ticket

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

(−2, −3): both signs change.

What each step is for

• a quarter turn: Committing to the image of a point after a quarter turn shows whether they swap coordinates or picture the turn.
• turn the triangle: Turning the triangle in quarter steps and reading the corners shows how each coordinate changes for 90°, 180° and 270°.
• how to turn a point: A quarter turn swaps and changes one sign, depending on the direction (Lee only swaps); a rotation needs a centre, an angle and a direction.
• change the turn and the centre: Choosing an angle to reach a target, then moving the centre, shows the image depends on the centre; clockwise and anticlockwise turns are linked.
• rotations: 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.
• turn to a target: Finding the turn that puts a chosen corner on a target applies the rule and then states the full description: centre, angle and direction.

Where it leads

This completes AC9M7SP03.

Where it leads: combining transformations and tessellations (Year 8); rotational symmetry.