Symmetry Challenge

'Symmetry Challenge' printed from https://nrich.maths.org/

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Symmetry Challenge

In this activity, we are going to shade the squares of this grid with one colour to make different designs.

Square grid formed from 9 smaller squares

There are a few rules that our designs need to follow:

Whole squares have to be shaded, rather than parts of squares.
Each design must have at least one line of symmetry.
Each new design you come up with must be different from all of the others. If two designs would look the same if you rotated or reflected them, they do not count as different designs in this task.

Have a go at making some designs. How many can you find? You might like to shade your designs on squared paper or on this sheet of blank grids.

If you wanted to find every possible design, how would you do this?

Why do this problem?

The 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 solution.

Possible approach

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 adopted.

Key questions

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?

Possible extension

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.

Possible support

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.