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.

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

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.

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

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

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

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.

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?

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 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 some celtic knot patterns using tiling techniques

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

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

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. . . .

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?

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

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

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

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

Learn about Pen Up and Pen Down in Logo

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.

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 happens when a procedure calls itself?

Turn through bigger angles and draw stars with Logo.

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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.

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?

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

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

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

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

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?

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

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?