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?

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?

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?

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

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?

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

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

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

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?

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

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?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?

Can you make the birds from the egg tangram?

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?

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?

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

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

Here is a version of the game 'Happy Families' for you to make and play.

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?

How many models can you find which obey these rules?

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

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

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.

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?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

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

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

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.

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?

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

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 Wai Ping, Wah Ming and Chi Wing?

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

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

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 Little Fung at the table?