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

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

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

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

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

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

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

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?

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

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

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

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?

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

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?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Can you make the birds from the egg tangram?

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.

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?

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

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

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?

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?

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?

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.

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

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

Can you lay out the pictures of the drinks in the way described by the clue cards?

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

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?

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

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?

How many models can you find which obey these rules?

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

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?

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?

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?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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?

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

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

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?