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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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

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?

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

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?

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

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?

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

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 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?

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?

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

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

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

A follow-up activity to Tiles in the Garden.

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

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 happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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.

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!

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?

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

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

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.

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?

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

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

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?

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.

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?

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

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

Investigate these hexagons drawn from different sized equilateral triangles.

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?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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