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2 This is an excerpt from the “Rotation” presentation in Boardworks Maths for Australia, which contains 129 presentations in total.

3 Rotations Teacher notesBefore playing the video, ask “ If the far left cog turns clockwise, what direction will the other cogs turn in?” From left to right: Clockwise Anti-clockwise Discuss what the centre of rotation is and how each point remains the same distance from this centre when turned. Ask pupils how they could determine how far each cog had turned in a given time period, for example, by marking one point and comparing start and stop positions. Video credit: © YAKOBCHUK VASYL 2010, Shutterstock.com 3

4 Describing a rotation A rotation occurs when an object is turned around a fixed point. Descriptions of rotations involve three different pieces of information: The angle of the rotation. For example, ¼ turn = 90° ½ turn = 180° 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.

5 Rotating images

6 Rotating shapes Teacher notesExplain 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.

7 Rotations on a coordinate gridTeacher notes 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 if a rotation maps (x, y) to (x’, y’) then: for a 90° rotation about the origin (x’, y’) = (–y, x) for a 180° rotation about the origin (x’, y’) = (–x, –y) for a 270° rotation about the origin (x’, y’) = (y, –x)

8 Finding the centre and angle of rotationTeacher notes Two lines are sufficient to define the centre of rotation. If required, a third line can be used to check the position. The angle of rotation here is 124°.

9 Combining transformationsWhen one transformation is followed by another, the resulting change can often be described by a single transformation. Shape A is reflected in the line y = x to give its image A’. y = x y x A’ is rotated through 90° about the origin to give the image A’’. A A’’ What single transformation will map shape A onto A’’? A’ Map shape A onto shape A’’ by a reflection in the y-axis.

10 Combining transformationsTeacher notes Select two transformations to be performed on shape L. Ask a volunteer to use the pen tool to draw the position of L after the given pair of transformations before revealing the solution. Investigate which pairs of transformations are commutative. Investigate which pairs of transformations are equivalent to a single transformation.