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.

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

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

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?

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.

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

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

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 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?

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?

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

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

A follow-up activity to Tiles in the Garden.

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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

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

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?

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

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?

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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

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

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

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

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

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?

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.

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

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!

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?

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 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.

These pictures show squares split into halves. Can you find other ways?

"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 ...?

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?

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

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. . . .

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

How many faces can you see when you arrange these three cubes in different ways?

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

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

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

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

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

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?