Explore the triangles that can be made with seven sticks of the same length.

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?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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 fit the tangram pieces into the outline of this junk?

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?

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

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?

Can you make the birds from the egg tangram?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

Can you see which tile is the odd one out in this design? Using the basic tile, can you make a repeating pattern to decorate our wall?

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

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?

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

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

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outlines of the candle and sundial?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you fit the tangram pieces into the outline of the child walking home from school?

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

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

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

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?

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?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

These pictures show squares split into halves. Can you find other ways?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

How many models can you find which obey these rules?

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

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

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

You will need a long strip of paper for this task. Cut it into different lengths. How could you find out how long each piece is?

In this activity focusing on capacity, you will need a collection of different jars and bottles.