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

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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?

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

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.

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?

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 work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Can you split each of the shapes below in half so that the two parts are exactly the same?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

Can you fit the tangram pieces into the outline of Mah Ling?

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

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

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?

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

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

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

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

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

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

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

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.

Make a flower design using the same shape made out of different sizes of paper.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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 outlines of the convex shapes?

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?

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

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?

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

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?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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

Can you logically construct these silhouettes using the tangram pieces?

What is the best way to shunt these carriages so that each train can continue its journey?

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?

Read about the adventures of Granma T and her grandchildren in this series of stories, accompanied by interactive tangrams.

Can you fit the tangram pieces into the outlines of Wai Ping, Wu Ming and Chi Wing?

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

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

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