### Why do this problem?

This problem provides a simple starting point creating
attractive patterns using rotations, with the potential to go much
further - exploring rotational symmetry, understanding the
relationship between coordinates when rotating through multiples of
$90^{\circ}$, and practising accurate construction with ruler,
compasses and protractor are just a few ideas.

Work on this problem could provide an excellent opportunity
for forging cross-curricular links with Art and Design
departments.

### Possible approach

Begin by showing learners

this image, and give them the
chance to discuss in pairs anything they notice about it. Share
ideas, and ask learners to suggest how they think the image might
have been created.

Once they have had a chance to consider the symmetry of the
image, show them the

second
image, which shows how the first image was made. Ask learners
to create similar images of their own based on rotations through
multiples of $90^{\circ}$ or through other angles using a
protractor.

While they are constructing their own images, ask them to
think of mathematical questions prompted by their images - these
might include some of the lines of enquiry suggested below. Bring
the class together during the lesson to share what they are
choosing to work on, and at the end of the lesson to share their
findings.

### Key questions

How can you use a coordinate grid to help you to rotate each
point through multiples of $90^{\circ}$ around $(0,0)$?

What is the relationship between the coordinates of the points
as they rotate through multiples of $90^{\circ}$?

Can you use an isometric grid to rotate a shape through
multiples of $60^{\circ}$?

What do you notice about the rotational symmetry of images
based on rotations such as $30^{\circ}$ or $72^{\circ}$?

### Possible extension

What is the rotational symmetry of your final image if you
rotate through multiples of $80^{\circ}$ or $135^{\circ}$?

Can you explain why?

Can you suggest other angles which would give similar
effects?

### Possible support

Learners could begin by rotating single points through
multiples of $90^{\circ}$ using the technique suggested
below: