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?

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?

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

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?

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

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

A follow-up activity to Tiles in the Garden.

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

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?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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?

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!

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?

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?

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?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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

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

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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

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

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.

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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?

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

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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?

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?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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

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

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?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?