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?

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

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

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

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

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

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?

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

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

A follow-up activity to Tiles in the Garden.

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

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

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

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?

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

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 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?

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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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?

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

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?

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

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

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Investigate the different ways you could split up these rooms so that you have double the number.

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

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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.

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?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Sort the houses in my street into different groups. Can you do it in any other ways?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

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

How many models can you find which obey these rules?

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

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.