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

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

You will need a long strip of paper for this task. Cut it into different lengths. How could you find out how long each piece is?

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

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

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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?

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

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

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

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

Can you make five differently sized squares from the tangram pieces?

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

Can you make the birds from the egg tangram?

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

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?

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

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?

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

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

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

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

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

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

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

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

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?

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

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

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 brazier for roasting chestnuts?

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

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?

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

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?

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 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?

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

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

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?

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

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

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

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