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

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

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?

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

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?

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

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

I cut this square into two different shapes. What can you say about the relationship between them?

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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?

What is the largest cuboid you can wrap in an A3 sheet of paper?

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

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

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

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?

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?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

How many faces can you see when you arrange these three cubes in different ways?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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?

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.

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

How many models can you find which obey these rules?

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

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

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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

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?

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

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?

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?

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

This problem is intended to get children to look really hard at something they will see many times in the next few months.

Why does the tower look a different size in each of these pictures?

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

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.