In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

This package contains hands-on code breaking activities based on the Enigma Schools Project. Suitable for Stages 2, 3 and 4.

Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?

This article for students gives some instructions about how to make some different braids.

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

Make some celtic knot patterns using tiling techniques

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?

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

Make a mobius band and investigate its properties.

Make a cube with three strips of paper. Colour three faces or use the numbers 1 to 6 to make a die.

Make an equilateral triangle by folding paper and use it to make patterns of your own.

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

Make a clinometer and use it to help you estimate the heights of tall objects.

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

Using these kite and dart templates, you could try to recreate part of Penrose's famous tessellation or design one yourself.

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

Galileo, a famous inventor who lived about 400 years ago, came up with an idea similar to this for making a time measuring instrument. Can you turn your pendulum into an accurate minute timer?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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.

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

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

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?

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

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

Surprise your friends with this magic square trick.

Did you know mazes tell stories? Find out more about mazes and make one of your own.

Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?

What do these two triangles have in common? How are they related?

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.

Follow these instructions to make a three-piece and/or seven-piece tangram.

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

What shape and size of drinks mat is best for flipping and catching?

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

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

How can you make a curve from straight strips of paper?

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

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

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

Write a Logo program, putting in variables, and see the effect when you change the variables.