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

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.

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.

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.

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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.

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

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

It starts quite simple but great opportunities for number discoveries and patterns!

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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

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. . . .

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

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

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

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

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?

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?

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

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?

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?

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

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?

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?

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Investigate what happens when you add house numbers along a street in different ways.

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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!

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?

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

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

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?

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"?

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?

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. . . .

What statements can you make about the car that passes the school gates at 11am on Monday? How will you come up with statements and test your ideas?

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.

What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?