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

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

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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

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

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.

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

Make some celtic knot patterns using tiling techniques

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

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

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.

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?

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

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.

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?

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

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Turn through bigger angles and draw stars with Logo.

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

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

Learn to write procedures and build them into Logo programs. Learn to use variables.

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

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

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?

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

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

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?

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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?

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

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.

A game in which players take it in turns to choose a number. Can you block your opponent?

A jigsaw where pieces only go together if the fractions are equivalent.

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?

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

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

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

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

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

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