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. . . .

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

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.

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

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

Which of the following cubes can be made from these nets?

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?

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

Make some celtic knot patterns using tiling techniques

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

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

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

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

Surprise your friends with this magic square trick.

Make a mobius band and investigate its properties.

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

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.

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

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

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

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.

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

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

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

An activity making various patterns with 2 x 1 rectangular tiles.

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

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?

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?

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

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

A description of how to make the five Platonic solids out of paper.

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?

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

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?

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

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.

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

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 make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.

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?

Here is a chance to create some Celtic knots and explore the mathematics behind them.