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.

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

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

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.

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

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?

Learn about Pen Up and Pen Down in Logo

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

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

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.

What happens when a procedure calls itself?

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

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

Turn through bigger angles and draw stars with Logo.

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

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

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

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

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

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

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.

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

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?

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

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

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

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

Can you deduce the pattern that has been used to lay out these bottle tops?

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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

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

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

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?

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

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

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

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