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?

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?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Can you work out what kind of rotation produced this pattern of pegs in our pegboard?

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

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

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

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

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

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 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 picture where this letter "F" will be on the grid if you flip it in these different ways?

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

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

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

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

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

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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

Make a cube out of straws and have a go at this practical challenge.

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Reasoning about the number of matches needed to build squares that share their sides.

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

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 visualise what shape this piece of paper will make when it is folded?

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.

What shape is made when you fold using this crease pattern? Can you make a ring design?

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

Can you fit the tangram pieces into the outlines of the convex shapes?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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

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?

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

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

Can you fit the tangram pieces into the silhouette of the junk?

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

Can you fit the tangram pieces into the outline of the playing piece?

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

Can you fit the tangram pieces into the outline of Granma T?

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

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

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

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