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

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?

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.

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?

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

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?

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 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?

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?

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?

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

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?

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

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?

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

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?

How many models can you find which obey these rules?

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

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?

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

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?

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

Make a flower design using the same shape made out of different sizes of paper.

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

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

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Can you visualise what shape this piece of paper will make when it is folded?

Can you make the birds from the egg tangram?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

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

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?

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

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 problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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

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

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

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

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.

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

Here are some ideas to try in the classroom for using counters to investigate number patterns.

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.

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

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?