What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

In how many ways can you fit all three pieces together to make shapes with line symmetry?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view of the purposes and skills of visualising.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.