Use the half arrow shape as a starting block to produce your own wallpaper friezes. Put this shape into the grid to make a repeating pattern.

How many different repeating patterns can you make?

There is a great deal of mathematics in everyday objects. Walls of rooms are no exception. Wallpaper friezes exploit reflections, translations and rotations. Producing their own friezes will give students the opportunity to explore all of the basic transformations of the plane.

There are three transformations:

Translations (or shifts)

Rotations (or turns)

Reflections

### Problem

Use the half arrow shape as a starting block to produce your own wallpaper friezes. Put this shape into the grid to make a repeating pattern.

How many different repeating patterns can you make?

**Teaching sequence**

- Introduce the problem by looking at some wallpaper friezes – discuss the symmetry that they have.
- Introduce the problem shape and the grid to develop the frieze in.
- As a class develop one of the patterns.
- Pose the problem.
- Let the students investigate the problem in small groups using a grid and at least 8 of the half-arrow shapes. Remind them they must be put in the grid so that there is a pattern that will repeat indefinitely.
- As the students solve the problem encourage them to record their findings so that they can share them later. This requires them to either draw or describe the pattern made. Ask questions that focus on the symmetry of the pattern made:

*How did you create the pattern?*

Have you used reflection in your pattern? Show me?

Have you rotated the shape? Show me?

Have you shifted the shape? Show me how? - Share patterns made.

*Have we found them all?*

How do you know?

#### Extension to the problem

You might try to find actual wallpaper friezes that match up with the patterns found in the problem.

#### Solution

There are 7 different wallpaper friezes that can be made using the basic block in the picture. We list them all here.

Note: Possible patterns such as:

have the same symmetry as one of the 7 listed. For instance 8 = 1 and 9 = 5.