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

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?

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?

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

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?

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

A follow-up activity to Tiles in the Garden.

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

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?

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?

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

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

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

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

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 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?

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

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

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

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

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?

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

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.

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

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?

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.

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?

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

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

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

A description of some experiments in which you can make discoveries about triangles.

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

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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?

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

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?

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?

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!

A challenging activity focusing on finding all possible ways of stacking rods.

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?

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?

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

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?

In how many ways can you stack these rods, following the rules?

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.