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?

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.

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.

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

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?

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

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

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 ...

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

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

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?

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

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?

Here is a version of the game 'Happy Families' for you to make and play.

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?

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?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

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

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 many models can you find which obey these rules?

These practical challenges are all about making a 'tray' and covering it with paper.

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.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

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

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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.

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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 you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

A game in which players take it in turns to choose a number. Can you block your opponent?

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

How can you make an angle of 60 degrees by folding a sheet of paper twice?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Use the tangram pieces to make our pictures, or to design some of your own!