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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

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

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.

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

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

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.

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

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.

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

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

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

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?

A challenging activity focusing on finding all possible ways of stacking rods.

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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?

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

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?

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

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?

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?

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

This article for teachers suggests ideas for activities built around 10 and 2010.

An investigation that gives you the opportunity to make and justify predictions.

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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

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

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

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?

Investigate the number of faces you can see when you arrange three cubes in different ways.

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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.