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

Build a scaffold out of drinking-straws to support a cup of water

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Make some celtic knot patterns using tiling techniques

What shape would fit your pens and pencils best? How can you make it?

As part of Liverpool08 European Capital of Culture there were a huge number of events and displays. One of the art installations was called "Turning the Place Over". Can you find our how it works?

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

Can Jo make a gym bag for her trainers from the piece of fabric she has?

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

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

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.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.

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

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. . . .

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

Turn through bigger angles and draw stars with Logo.

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

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

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.

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

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

Learn about Pen Up and Pen Down in Logo

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

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

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?

What happens when a procedure calls itself?

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

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

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

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

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

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

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

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

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

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

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

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

Can you make the birds from the egg tangram?

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?

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

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?

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

Here's a simple way to make a Tangram without any measuring or ruling lines.