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

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?

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

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

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

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

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?

The challenge for you is to make a string of six (or more!) graded cubes.

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

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.

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

Can you each work out what shape you have part of on your card? What will the rest of it look like?

What shape is made when you fold using this crease pattern? Can you make a ring design?

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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?

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

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

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?

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?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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.

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

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

How can you make a curve from straight strips 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?

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

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

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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

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

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

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Can you visualise what shape this piece of paper will make when it is folded?

Make a flower design using the same shape made out of different sizes of paper.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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.

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

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

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

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

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?

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

Turn through bigger angles and draw stars with Logo.

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