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 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.

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

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

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?

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

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.

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?

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

Learn about Pen Up and Pen Down in Logo

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.

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 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?

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

Turn through bigger angles and draw stars with Logo.

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

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

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

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

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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

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

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

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.

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?

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.

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

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

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

Can you make the birds from the egg tangram?

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

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

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.

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?

What happens when a procedure calls itself?