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

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?

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. . . .

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?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

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

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

A description of some experiments in which you can make discoveries about triangles.

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.

Numbers arranged in a square but some exceptional spatial awareness probably needed.

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 three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

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

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?

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

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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?

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!

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

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

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?

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?

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?

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

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

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

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

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?

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?

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.

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

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?

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?

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?

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

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

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

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.