--- title: "Rotations" subtitle: "Angles and shapes · Transformations · Lesson 14" --- # Rotations **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** The hands of a clock, a Ferris wheel, a spinning fan. Each turns about a centre by a certain angle, in a certain direction. :::teacher-note **In one sentence:** Students turn a shape about a centre and find the image of each corner for quarter and half turns (AC9M7SP03, node 3). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Picture a clock hand pointing right, turning up. :::keep-together **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) My prediction, and why: {{lines-3}} ::: :::teacher-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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## How to turn a point :::keep-together ### 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) ::: :::teacher-note **Answer:** (−4, −1) A half turn changes both signs: (−4, −1). ::: :::keep-together ### 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). ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** 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). ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** The centre, the angle and the direction The same angle about a different centre, or in the other direction, gives a different image. ::: :::typical ## Change the turn and the centre **Three quarter turns.** **A different centre.** **Clockwise.** **A full turn.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Rotations :::keep-together ### A half turn **B is at (4, 1). After a 180° turn about the origin, what is the x-coordinate of B′?** Answer: {{field-15}} ::: :::teacher-note **Answer:** -4 x changes sign: −4. The image is (−4, −1). ::: :::keep-together ### Which turn **Which turn about the origin takes (2, 1) to (−1, 2)?** ☐ 180° ☐ 90° clockwise ☐ 90° anticlockwise ☐ 270° anticlockwise ::: :::teacher-note **Answer:** 90° anticlockwise (2, 1) to (−1, 2) is the rule (x, y) → (−y, x): a quarter turn anticlockwise. ::: :::keep-together ### A quarter turn **The point (3, 1) turns 90° anticlockwise about the origin. What is the y-coordinate of the image?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 (x, y) → (−y, x), so (3, 1) becomes (−1, 3). ::: :::keep-together ### Review: reflections **Review: the point (3, 5) is reflected in the x-axis. What is the y-coordinate of the image?** Answer: {{field-15}} ::: :::teacher-note **Answer:** -5 (3, 5) becomes (3, −5). ::: ## 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? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **The point (2, 3) turns 180° about the origin. Where is the image?** {{lines-3}} ::: :::teacher-note **Answer:** (−2, −3): both signs change. :::