What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . .

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.

What's the greatest number of sides a polygon on a dotty grid could have?

Use the isometric grid paper to find the different polygons.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

What shape and size of drinks mat is best for flipping and catching?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.