### Why do this problem?

This problem is a good activity for the visualisation of symmetry, and for encouraging students to work systematically. There are many different avenues for exploration and extension, and learners' work can be used to brighten up the classroom walls.

### Possible approach

These printable resources may be useful: Shady Symmetry,

As the class come into the room, display the two patterns from the problem at the front for everyone to see. The patterns are available on this PowerPoint Slide.

Ask students to discuss the two images in pairs, focusing on what they notice about the two pictures, what is the same and what is different. Then bring the class together to share their ideas.

Explain that the challenge will be to explore symmetrical patterns drawn on grids of triangles or squares, and give students a little longer with their partners to come up with some lines of enquiry to explore. Collect their ideas together on the board at the front (some suggestions are made in the problem if more ideas are needed).

Now allow pairs or small groups to choose one of the ideas to work on, and hand out some of these square and triangular grids. Make the class aware that at the end of the time spent on this (it could be over several lessons) they will be expected to display their work in a way that will convince others that they have considered every possible symmetrical pattern for their chosen question. While students are working on the task, there may be opportunities to share what people are thinking about through mini-plenaries, particularly to draw attention to those who are working in a systematic way.

### Key questions

• What different types of symmetry do the initial grids exhibit?
• If you colour a triangle or square here, what else must be coloured in to keep it symmetrical?
• What are the possible symmetries of a finished pattern?
• How can you be sure you have found all the symmetric patterns?

### Possible extension

The problem can be extended to be done on these 4 by 4 square and triangular grids, and of course there are opportunities to extend into three dimensions...

### Possible support

Encourage students to begin by looking at all the patterns that can be made by first colouring in just one cell, then two, then three and so on.

Have tracing paper available if required.

Encourage students to number the cells of their grid to help them to list shadings in a systematic way.

Draw a big table on the board with
column headings: 0, 1, 2, 3, more lines of symmetry,
row headings: order 1, 2, 3, more for rotational symmetry.
Fill in "not symmetrical" in the top left cell, then ask students to stick their solutions up in the correct cell on the table as they complete them.