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?

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.

Learn about Pen Up and Pen Down in Logo

This article for students gives some instructions about how to make some different braids.

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

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

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.

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

Turn through bigger angles and draw stars with Logo.

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

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

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?

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

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

Make some celtic knot patterns using tiling techniques

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.

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?

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

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

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

What happens when a procedure calls itself?

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. . . .

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

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

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?

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

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?

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

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

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

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.

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?

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

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

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

These practical challenges are all about making a 'tray' and covering it with paper.

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

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?

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?