Copyright © University of Cambridge. All rights reserved.
'Symmetry Challenge' printed from http://nrich.maths.org/
Why do this problem?
requires learners to recognise and visualise the transformation of a 2D shape, and invites them to work systematically in a spatial environment. It is a problem that is accessible to most pupils even if they need support in organising and presenting their ideas and ensuring the completeness of their
You could start by displaying these two shaded grids
on the board to simulate a discussion about reflection symmetry. It might also help to have some blank $3$ by $3$ grids on the board for learners to shade as they talk.
Invite each pupil to produce one symmetrical shading on a copy of this sheet of blank grids
. Ask the pupils to put themselves into groups according to the number of squares they have shaded on their grid and answer the following questions:
- Are the patterns you have all symmetrical?
- Which are the same and which are different?
- Are there any more with the same number of squares shaded?
Bring the group together to talk about some of the issues, which may include talking about what counts as the same and how they went about finding all the possibilities. Introduce the task and encourage each group to work on it as a whole.
The plenary should focus on persuasion that the approach adopted by particular groups will yield all solutions. It is important not to get too 'bogged down' with listing all the possibilities. Pupils could share out all the possibilities between the group and produce one or more image each which could then be displayed in an organised fashion to emphasise a system that has been
Try shading $1$ square only. What symmetries are possible?
If you try shading $2, 3, 4 \ldots$squares, what symmetries are possible now?
How will you record your findings?
The $3$ by $3$ grid has four lines of symmetry and rotational symmetry of order $4$. How might this help?
The problem can be extended to discuss larger square lattices, e.g. $4$ by $4$ and whether there are any differences between even and odd lengths of side. The activity Shady Symmetry
is also an extension possibility.
Some children might find it helpful to start with a $2$ by $2$ grid so that they feel confident that they understand the requirements of the problem.