3. Demonstrating rotational symmetry

So far we have mostly looked at one or two lines of symmetry, but some objects have several lines of symmetry – a square has four: one vertical, one horizontal and two diagonally. The square also has rotational symmetry, meaning if we rotate it (turn it around) we can get the same pattern again: a square can be rotated to make the same pattern four times – it has a rotational symmetry of four. This is sometimes called having rotational symmetry of order 4.

This next part explores the idea of multiple lines of symmetry further by using objects in everyday life and searching for patterns in the shapes. Some of your pupils may be able to predict the pattern if you set up the activity so that they can work at their own pace and discuss their ideas with others.

Case Study 3: Investigating multiple lines of symmetry

Mr Namisi thought his pupils had become confident at working with one line of symmetry and he wanted to stretch them further by looking at different kinds of symmetry. He had drawn and cut out four different religious symbols (see Resource 3: Symmetry – lines and rotation), making each one as large as he could on a piece of A4 paper.

Mr Namisi held these shapes up and asked if pupils knew what each one was called. First, he asked his pupils to look for lines of symmetry. On the Cross and the Mosque, they easily found the line. With a little encouragement, they were then able to see that there were many possible lines of symmetry on the Star of David and the Dharma Wheel; the older pupils were able to count these.

Mr Namisi then put a thumbtack in the centre of the Cross, and showed that if he turned it round, it only looked the same in one position – where it started. He said this meant it had no rotational symmetry. He showed the pupils the other shapes and they tried the same rotation with each. They counted a rotational symmetry of six for the Star of David and eight for the Dharma Wheel. His class were eager to look for other shapes in real life that had multiple lines of symmetry, which pleased him.

More examples of symmetry can be found in Resource 4: Examples of symmetry in art and fabrics.

Key Activity: Explaining rotations

You will need a page of polygon shapes (see Resource 5: Polygons) for each small group of pupils.

First, ask pupils to write in their books three column headings: ‘polygon sides’ ‘lines of symmetry’ ‘rotational symmetry’. Then ask them to look at the shapes and, for each polygon, count and record:

  • How many sides it has.
  • How many lines of symmetry they can find.
  • How many orders of rotational symmetry they can find.

After the first few shapes, some pupils may begin to spot a pattern and be able to complete their table without counting; others may not see the pattern. If this happens, ask the pupils who have seen a pattern to explain how it works to those who have not.

Use questions like: ‘How many lines of symmetry would a polygon of [n] sides have? And how many orders of rotational symmetry?’ ([n] could be any whole number.)

Ask each group to complete the chart you have drawn out on a sheet of newsprint and display their charts in the classroom (see Resource 6: Recording symmetry).

2. A cross-curricular approach

Resource 1: Examples of symmetry found in nature