Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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

This activity investigates how you might make squares and pentominoes from Polydron.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

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

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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

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.

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

Can you visualise what shape this piece of paper will make when it is folded?

Make a mobius band and investigate its properties.

Surprise your friends with this magic square trick.

Follow these instructions to make a five-pointed snowflake from a square of 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?

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

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?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.

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

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?

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

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

Here are some ideas to try in the classroom for using counters to investigate number patterns.

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

Can you make the birds from the egg tangram?

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

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?

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

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

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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

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

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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.

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

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

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

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?

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