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

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?

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

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

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?

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?

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

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

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

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?

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

Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

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

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

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

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

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

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

What do these two triangles have in common? How are they related?

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?

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

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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?

What shape is made when you fold using this crease pattern? Can you make a ring design?

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

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you put these shapes in order of size? Start with the smallest.

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

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

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other 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?

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

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?

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?

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.

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?

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.

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

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

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

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

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

Can you make the birds from the egg tangram?

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?