Can you each work out what shape you have part of on your card? What will the rest of it look like?

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

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

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

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?

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

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

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?

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?

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

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

What is the greatest number of squares you can make by overlapping three squares?

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?

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

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

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

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

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

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

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?

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

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

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

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

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

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

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?

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

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

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

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

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 outline of this junk?

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

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?

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.

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

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

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.

Can you make the birds from the egg tangram?

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

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?

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?

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