Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

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?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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.

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

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

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?

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?

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

You have a set of the digits from 0 – 9. Can you arrange these in the 5 boxes to make two-digit numbers as close to the targets as possible?

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

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 investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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?

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

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?

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

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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

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?

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?

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

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

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

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

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?

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

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

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

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

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

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

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

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?

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

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 Wai Ping, Wah Ming and Chi Wing?

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

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?

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

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

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

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?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?