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

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

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

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?

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

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

Can you each work out the number on your card? What do you notice? How could you sort the cards?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Can you deduce the pattern that has been used to lay out these bottle tops?

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

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

Here is a chance to create some Celtic knots and explore the mathematics behind them.

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?

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

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?

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?

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

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

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

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

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

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

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

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

Can you make the birds from the egg tangram?

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

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

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

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?

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

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

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

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

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

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.

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

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

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

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?

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?

How many models can you find which obey these rules?

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

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

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?