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

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

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

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

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

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

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

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.

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

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?

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

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?

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

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

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

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different 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?

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?

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

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

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

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

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

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?

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

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 shape is made when you fold using this crease pattern? Can you make a ring design?

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?

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

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?

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?

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

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?

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

How many models can you find which obey these rules?

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

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

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

Can you make five differently sized squares from the tangram pieces?

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

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

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

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?

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