1. Rotations

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

3. 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.

4. Determining the 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°.

5. Rotation Which of the following are examples of rotation in real life?
Opening a door?
Walking up stairs?
Riding on a Ferris wheel?
Bending your arm?
Opening your mouth?
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.
Opening a drawer?
Can you suggest any other examples?

6. 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
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.

7. Rotating shapes The centre of rotation can also be inside the shape.
For example,
90° O
Rotating this shape 90° anticlockwise about point O produces the following image.

8. Rotation X

9. 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 2 2 2 2

10. x 1 9 8 7 6 5 4 3 2 y

11. Inverse rotations
The inverse of a rotation maps the image that has been rotated back onto the original object.
For example, the following shape is rotated 90° clockwise about point O.
90° O
To rotate a shape that has been rotated through 90º clockwise back onto itself we can either rotate it through 90º anticlockwise or we can rotate it through another 270º (360º – 90º).
What is the inverse of this rotation?
Either, a 90° rotation anticlockwise,
or a 270° rotation clockwise.

12. Inverse rotations The inverse of any rotation is either
A rotation of the same size, about the same point, but in the opposite direction, or
A rotation in the same direction, about the same point, but such that the two rotations have a sum of 360°.
What is the inverse of a –70° rotation?
Remind pupils that a –70º rotation is equivalent to a 70º rotation clockwise. Establish that to rotate a shape that has been rotated through –70º back onto itself we can rotate it through +70º (that’s 70º in the opposite direction) or – 290º (360º – 70º).
Either, a 70° rotation,
or a –290° rotation.

13. 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

14. 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

15. Rotations on a coordinate grid
Remind pupils that, unless stated otherwise, positive rotations are always taken as anticlockwise. Demonstrate each rotation, dragging the vertices to change the shape and dragging on the shape to change its position.
Investigate the relationship between the coordinates of the object and its image for 90°, 180° and 270° rotations. This can be done by revealing the coordinate of A and A’ say, moving the shape (or the point) around the grid and observing the change in the coordinates.
Pupils should notice that when a shape is rotated through 90º about the origin, the x-coordinate of each image point is the same as the y-coordinate of the original point × –1 and the y-coordinate of the image point is the same as the x-coordinate of the original point.
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 × –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point × –1. In other words the coordinates are the same, but the signs are different.
When a shape is rotated through 270º about the origin, the x-coordinate of each image point is the same as the y-coordinate of the original point and the y-coordinate of the image point is the same as the x-coordinate of the original point × –1.

16. Finding the centre of rotation
Find the point about which A is rotated onto its image A’. A A A’
Draw lines from any two vertices to their images.
Two lines are sufficient to define the centre of rotation. If required a third line can be used to check the position.
The perpendicular can be drawn using a set square.
Mark on the mid-point of each line.
Draw perpendicular lines from each of the mid-points.
The point where these lines meet is the centre of rotation.

17. Finding the angle of rotation
Find the angle of rotation from A to its image A’. A A A’
126°
This is the angle of rotation
Join one vertex and its image to the centre of rotation.
Use a protractor to measure the angle of rotation.

18. 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)