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

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.

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

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

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

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.

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?

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.

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?

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

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

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

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.

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

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

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.

Surprise your friends with this magic square trick.

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

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

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

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

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

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?

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

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

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

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

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?

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

This part introduces the use of Logo for number work. Learn how to use Logo to generate sequences of numbers.

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

Follow these instructions to make a five-pointed snowflake from a square of paper.

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?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

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

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.

Turn through bigger angles and draw stars with Logo.

More Logo for beginners. Now learn more about the REPEAT command.

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

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