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 students gives some instructions about how to make some different braids.

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?

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

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

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

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. . . .

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.

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 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?

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

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

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

What happens when a procedure calls itself?

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?

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

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

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?

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

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

Turn through bigger angles and draw stars with Logo.

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

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

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

Can you make the birds from the egg tangram?

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?

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?

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

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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.

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.

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

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?

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