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

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

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

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?

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

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?

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?

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

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

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.

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

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 ...

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

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

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?

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

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

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?

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

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

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

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

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?

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?

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

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 shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Can you cut up a square in the way shown and make the pieces into a triangle?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

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?

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

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

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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?

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

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

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?

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?

Why do you think that the red player chose that particular dot in this game of Seeing Squares?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.