Rotations

Objectives Be able to rotate a shape by the given angle about a centre of rotation. Keywords: Rotation, centre of rotation, clockwise, anti-clockwise

Think about the following activities. Opening a door? Walking up stairs? Bending your arm? Opening your mouth? Opening a drawer? Anything that is fixed at a point and turns about that point is an example of a rotation. This is true even if a complete rotation cannot be completed, such as your jaw when opening your mouth. Which of the above , you think are examples of rotation? Can you suggest any other examples?

Describing a rotation A rotation occurs when an object is turned around a fixed point. To describe a rotation we need to know three things: The angle of the rotation. For example, ½ turn = 180° ¼ turn = 90° ¾ turn = 270° The direction of the rotation. For example, clockwise or anticlockwise. The centre of rotation. This is the fixed point about which an object moves.

Direction of a rotation Sometimes the direction of the rotation is not given. If this is the case then we use the following rules: A positive rotation is an anticlockwise rotation. A negative rotation is an clockwise rotation. For example, A rotation of 60° = an anticlockwise rotation of 60° This is probably the opposite to what most people would expect. Explain that it is a convention that has been agreed between mathematicians all over the world. It’s just that, mathematically speaking, the hands of a clock turn in a negative direction! Establish equivalent rotation can always be found by changing the sign and subtracting the angle from 360º. Ask pupils to give examples of other equivalent rotations. A rotation of –90° = an clockwise rotation of 90° Explain why a rotation of 120° is equivalent to a rotation of –240°.

Rotating shapes If we rotate triangle ABC 90° clockwise about point O the following image is produced: B object 90° A A’ image B’ C C’ Explain that if the centre of rotation is not in contact with the shape, we can extend a line from the shape to the point. A line extended from the corresponding point on the image will meet the centre of rotation at an angle equivalent to the angle of rotation. O In rotation, each point/vertex is rotated about the centre at the given angle. A is mapped onto A’, B is mapped onto B’ and C is mapped onto C’. The image triangle A’B’C’ is congruent to triangle ABC.

An example of a rotation center of rotation A ROTATION MEANS TO TURN A FIGURE around a point.

This is another way rotation looks The triangle was rotated around the point. center of rotation A ROTATION MEANS TO TURN A FIGURE.

If a shape spins 360, how far does it spin? ROTATION If a shape spins 360, how far does it spin? 360 All the way around This is called one full turn.

Half of the way around ROTATION If a shape spins 180, how far does it spin? Half of the way around 180 This is called a ½ turn. Rotating a shape 180 turns a shape upside down.

If a shape spins 90, how far does it spin? ROTATION If a shape spins 90, how far does it spin? One-quarter of the way around 90 This is called a ¼ turn.

ROTATION Describe how the triangle A was transformed to make triangle B A B Triangle A was rotated right 90 Describe the translation.

y x Rotate shape A 90 degrees clockwise with the centre (0, 0). 5 Trace out the shape A to be rotated. 4 3 2 A 1 x - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 900 - 1 - 2 Rotate the shape clockwise. - 3 - 4 - 5 - 6 - 7 Trace out the new position of the shape.

Rotation y x Rotate shape A clockwise about the centre A 1 2 3 4 5 6 7 8 x - y A Trace out the point/object/shape to be rotated. Hold your pencil on the coordinate of the centre of rotation. Rotate the shape clockwise. Trace out the new position of the shape.

Rotation A Rotate shape A anticlockwise centre 1 2 3 4 y - x 5 6 7 A Rotate shape A anticlockwise centre Trace out the shape to be rotated. Hold your pencil on the coordinate of the centre of rotation. Rotate the shape anticlockwise. Trace out the new position of the shape.

Rotation A Rotate shape A anticlockwise centre 6 2 3 4 y - x 5 1 7 A Rotate shape A anticlockwise centre Trace out the shape to be rotated. Hold your pencil on the coordinate of the centre of rotation. Rotate the shape anticlockwise. Trace out the new position of the shape.

ROTATION Describe how the arrow A was transformed to make arrow B B A Arrow A was rotated right 180 Describe the translation.

Rotation X

X y Rotate triangle ABC 90° clockwise about the origin. Rotate triangle ABC 180° anti-clockwise about the origin. Rotate triangle ABC 90° clockwise about the origin. 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 A B C A B C A B C 2 2 X x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 2 2 2 2

0 1 2 3 4 5 6 7 8 9 x 1 9 8 7 6 5 4 3 2 y

Rotations on a coordinate grid The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). 7 A(2, 6) 6 5 B(7, 3) 4 3 C’(–4, 1) 2 Rotate the triangle 180° clockwise about the origin and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 C(4, –1) –3 Pupils should notice that when a shape is rotated through 180º about the origin, the x-coordinate of each image point is the same as the x-coordinate of the original point multiplied by –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point multiplied by –1. –4 What do you notice about each point and its image? B’(–7, –3) –5 –6 A’(–2, –6) –7

Rotations on a coordinate grid The vertices of a triangle lie on the points A(–6, 7), B(2, 4) and C(–4, 4). 7 B(2, 4) 6 5 C(–4, 4) 4 3 B’(–4, 2) 2 Rotate the triangle 90° anticlockwise about the origin and label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 –3 Pupils should notice that when a shape is rotated through 90º anticlockwise about the origin, the x-coordinate of each image point is the same as the y-coordinate of the original point multiplied by –1. The y-coordinate of the image point is the same as the x-coordinate of the original point. –4 What do you notice about each point and its image? C’(–4, –4) –5 –6 A’(–7, –6) –7

Rotation - Summary When the centre of the rotation is the origin Rotation at 90°: P(x, y) P’(-y, x) Rotation at 180°: P(x, y) P’(-x, -y) Rotation at 270°: P(x, y) P’(y, -x)