What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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?

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

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

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.

What can you see? What do you notice? What questions can you ask?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

What is the greatest number of squares you can make by overlapping three squares?

Exploring and predicting folding, cutting and punching holes and making spirals.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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.

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

How many loops of string have been used to make these patterns?

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

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

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!

Can you fit the tangram pieces into the outline of this sports car?

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of these convex shapes?

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

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

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Which of these dice are right-handed and which are left-handed?

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Here are shadows of some 3D shapes. What shapes could have made them?

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

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.

Can you fit the tangram pieces into the outlines of the candle and sundial?

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?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

Here's a simple way to make a Tangram without any measuring or ruling lines.