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

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

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?

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

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?

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?

"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 ...?

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?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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

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?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

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

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

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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.

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

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?

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.

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. . . .

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

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

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

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

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

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?

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

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

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?

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

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

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

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?

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?

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

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

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

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

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

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?

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

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