Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?

Using these kite and dart templates, you could try to recreate part of Penrose's famous tessellation or design one yourself.

A game to make and play based on the number line.

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

Make a cube with three strips of paper. Colour three faces or use the numbers 1 to 6 to make a die.

Use the tangram pieces to make our pictures, or to design some of your own!

Follow these instructions to make a three-piece and/or seven-piece tangram.

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

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

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Did you know mazes tell stories? Find out more about mazes and make one of your own.

Surprise your friends with this magic square trick.

How can you make a curve from straight strips of paper?

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

Make a mobius band and investigate its properties.

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

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

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

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?

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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

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 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?

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

Can you make the birds from the egg tangram?

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

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

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?

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

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

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?

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

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

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

Can you fit the tangram pieces into the outline of this junk?

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

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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?

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?

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

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

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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