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

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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

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.

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

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

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

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

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

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

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?

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

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.

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?

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?

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?

We have a box of cubes, triangular prisms, cones, cuboids, cylinders and tetrahedrons. Which of the buildings would fall down if we tried to make them?

How many models can you find which obey these rules?

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

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

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

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

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?

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.

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?

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

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?

Make a cube with three strips of paper. Colour three faces or use the numbers 1 to 6 to make a die.

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?

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

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

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

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

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

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

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?

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?

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

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

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

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.

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

The challenge for you is to make a string of six (or more!) graded cubes.

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