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

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

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

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?

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?

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?

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

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?

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

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

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

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

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

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

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

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

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?

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?

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

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?

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.

A game to make and play based on the number line.

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

Can you make the birds from the egg tangram?

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

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

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.

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

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

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?

Delight your friends with this cunning trick! Can you explain how it works?

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

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

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

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

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

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

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?

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?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end 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.

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?

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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.

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

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