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

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?

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

You have a set of the digits from 0 – 9. Can you arrange these in the 5 boxes to make two-digit numbers as close to the targets as possible?

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?

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

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

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

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

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

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

In this activity focusing on capacity, you will need a collection of different jars and bottles.

For this activity which explores capacity, you will need to collect some bottles and jars.

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. . . .

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?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you split each of the shapes below in half so that the two parts are exactly the same?

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 models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

Make some celtic knot patterns using tiling techniques

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

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?

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

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?

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.

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

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 make the birds from the egg tangram?

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

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

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

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?

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

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?

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?

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?

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?

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?