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?

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

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

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

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

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

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

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 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.

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

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

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?

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

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?

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

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

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?

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

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

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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

Can you make the birds from the egg tangram?

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

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.

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.

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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?

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.

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

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

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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

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 this junk?

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?

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

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?

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