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

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

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

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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.

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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?

Here are many ideas for you to investigate - all linked with the number 2000.

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

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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

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

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?

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

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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

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?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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?

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

A follow-up activity to Tiles in the Garden.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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

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

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

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

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

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