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

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?

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?

Can you make the birds from the egg tangram?

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?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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?

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

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

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?

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

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?

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

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

Here is a version of the game 'Happy Families' for you to make and play.

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

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

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

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

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

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

Surprise your friends with this magic square trick.

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.

Make a mobius band and investigate its properties.

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

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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?

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

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

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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?

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

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?

Make a flower design using the same shape made out of different sizes of paper.

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.

How many models can you find which obey these rules?

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