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?

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

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?

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?

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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.

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 possible answers?

Which way of flipping over and/or turning this grid will give you the highest total? You'll need to imagine where the numbers will go in this tricky task!

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

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?

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

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 number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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?

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

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

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.

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?

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

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?

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

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. . . .

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

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 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?

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

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

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

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?

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

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?

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

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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?

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

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

A follow-up activity to Tiles in the Garden.

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