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

Sort the houses in my street into different groups. Can you do it in any other ways?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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

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.

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

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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

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

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?

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

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

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?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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?

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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

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

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Can you find ways of joining cubes together so that 28 faces are visible?

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?

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

Investigate the different ways you could split up these rooms so that you have double the number.

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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 we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

How many models can you find which obey these rules?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

In how many ways can you stack these rods, following the rules?

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

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

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

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.