1. G17 Combining transformations
Boardworks KS3 Maths 2009
G17 Combining transformations
G17 Combining transformations
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2. G17.1 Combining transformations
Boardworks KS3 Maths 2009
G17 Combining transformations
G17.1 Combining transformations

3. Combining reflections
Boardworks KS3 Maths 2009
G17 Combining transformations
Combining reflections
An object may be reflected many times.
In a kaleidoscope mirrors are placed at 60° angles.
Shapes in one section are reflected in the mirrors to make a pattern.
How many lines of symmetry does the resulting pattern have?
Ask pupils to tell you how many lines of symmetry and the order of the rotational symmetry the pattern will have.
Encourage pupils to realise that if mirrors are placed like this would always produce a pattern with three lines of symmetry and rotational symmetry of order 3.
What would happen if mirrors were placed at 45º angles?
Ask pupils to design a kaleidoscope pattern with either 60º angles between the mirror lines (using isometric paper) or 45º angles between the mirror lines (using squared paper).
Does the pattern have rotational symmetry?

4. Boardworks KS3 Maths 2009 G17 Combining transformations
Parallel mirror lines
What happens when an object is reflected in parallel mirror lines placed at equal distances?
Ask pupils to describe what has happened in terms of other transformations that they know. Establish that the object is translated as a result of being reflected and then reflected again.

5. Boardworks KS3 Maths 2009 G17 Combining transformations
Parallel mirror lines
Suppose we have two parallel mirror lines M1 and M2.
We can reflect shape A in mirror line M1 to produce the image A’. A A’
A’’
We can then reflect shape A’ in mirror line M2 to produce the image A’’.
How can we map A onto A’’ in a single transformation? M1 M2
Reflecting an object in two parallel mirror lines is equivalent to a single translation.

6. Perpendicular mirror lines
Boardworks KS3 Maths 2009
G17 Combining transformations
Perpendicular mirror lines
Suppose we have two perpendicular mirror lines M1 and M2.
We can reflect shape A in mirror line M1 to produce the image A’. A’ A M2
We can then reflect shape A’ in mirror line M2 to produce the image A’’.
A’’
How can we map A onto A’’ in a single transformation? M1
Reflection in two perpendicular lines is equivalent to a single rotation of 180°.

7. Boardworks KS3 Maths 2009 G17 Combining transformations
Combining rotations
Suppose shape A is rotated through 100° clockwise about point O to produce the image A’. A
Suppose we then rotate shape A’ through 170° clockwise about the point O to produce the image A’’.
100°
A’’ O A’
170°
How can we map A onto A’’ in a single transformation?
To map A onto A’’ we can either rotate it 270° clockwise or 90° anti-clockwise.
Two rotations about the same centre are equivalent to a single rotation about the same centre.

8. Combining translations
Boardworks KS3 Maths 2009
G17 Combining transformations
Combining translations
Suppose shape A is translated 4 units left and 3 units up.
Suppose we then translate A’ 1 unit to the left and 5 units down to give A’’. A’
How can we map A to A’’ in a single transformation? A
A’’
We can map A onto A’’ by translating it 5 units left and 2 units down.
Point out to pupils that we can quickly translate a shape by considering just one of its points (in this case, the vertex shown by an orange cross).
Discuss how a translation 4 units left and 3 units up followed by a translation 1 unit left and 5 units down can be combined to form a single translation 5 units left and 2 units down.
4 units left + 1 unit left = 5 units left
3 units up + 5 units down = 2 units down
Two or more translations are equivalent to a single translation.

9. Transformation shape sorter
Boardworks KS3 Maths 2009
G17 Combining transformations
Transformation shape sorter
The object of this activity is to use a combination of transformations to fit each shape into its matching hole in as few moves as possible.
Reflections can be through any horizontal or vertical line, rotations can be 90°, 180° or 270° around a chosen point and translations can be a given number of units right (negative for left) or up (negative for down).