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

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

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

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

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

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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

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?

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?

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

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.

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

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Here are many ideas for you to investigate - all linked with the number 2000.

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

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

How many different sets of numbers with at least four members can you find in the numbers in this box?

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

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

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

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

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

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?

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?

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?

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

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

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

In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

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?

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

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

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?

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?

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?

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

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?

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

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

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

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

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?