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

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

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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 ...

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

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.

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

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

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?

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

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?

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

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

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

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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

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?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

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?

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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

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

Can you make the birds from the egg tangram?

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?

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

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

How many models can you find which obey these rules?

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

We have a box of cubes, triangular prisms, cones, cuboids, cylinders and tetrahedrons. Which of the buildings would fall down if we tried to make them?

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?

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

Here are some ideas to try in the classroom for using counters to investigate number patterns.

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?

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?