Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

Move just three of the circles so that the triangle faces in the opposite direction.

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

What happens when you try and fit the triomino pieces into these two grids?

Exchange the positions of the two sets of counters in the least possible number of moves

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Can you make a 3x3 cube with these shapes made from small cubes?

An activity centred around observations of dots and how we visualise number arrangement patterns.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

This article for teachers describes a project which explores thepower of storytelling to convey concepts and ideas to children.

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you find ways of joining cubes together so that 28 faces are visible?

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outline of Little Ming?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Can you fit the tangram pieces into the outline of the rocket?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Can you fit the tangram pieces into the outline of this telephone?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Can you fit the tangram pieces into the outline of this plaque design?

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?

How many pieces of string have been used in these patterns? Can you describe how you know?

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.