How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

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.

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

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

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?

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?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

Make a clinometer and use it to help you estimate the heights of tall objects.

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?

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

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

Make an equilateral triangle by folding paper and use it to make patterns of your own.

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.

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

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.

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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?

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

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

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

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?

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

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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?

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

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?

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?

Make a cube out of straws and have a go at this practical challenge.

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

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?

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

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 make the birds from the egg tangram?

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 squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

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

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.

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?

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?