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?

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

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

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?

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.

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.

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

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

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

Make some celtic knot patterns using tiling techniques

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?

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?

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. . . .

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

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

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

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

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

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

What happens when a procedure calls itself?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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?

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

Turn through bigger angles and draw stars with Logo.

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

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

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

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?

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

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

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

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

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?

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

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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?

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

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

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?

Factors and Multiples game for an adult and child. How can you make sure you win this game?

Can you make the birds from the egg tangram?