Explaining, convincing and proving

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What sizes of rectangle contain exactly 100 squares? Can you find them all?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Here's a chance to work with large numbers...

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Can you sketch graphs to show how the height of water changes in different containers as they are filled?