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

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.

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

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?

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

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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

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

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

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?

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 make five differently sized squares from the tangram pieces?

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?

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 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.

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

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 fit the tangram pieces into the outlines of the chairs?

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

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?

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

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

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you make the birds from the egg tangram?

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

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?

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

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?

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 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?

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

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 outlines of these people?

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

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?

Can you fit the tangram pieces into the outlines of these clocks?

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

Can you fit the tangram pieces into the outline of the child walking home from school?

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?

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?