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

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!

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

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.

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

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?

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.

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?

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?

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

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

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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

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

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

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"?

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

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

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?

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?

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

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

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.

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

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 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?

A follow-up activity to Tiles in the Garden.

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?

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?

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

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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

Investigate these hexagons drawn from different sized equilateral triangles.

How many different sets of numbers with at least four members can you find in the numbers in this box?

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?

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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

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

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