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?

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?

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

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?

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

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?

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?

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?

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

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

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

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

A follow-up activity to Tiles in the Garden.

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?

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

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?

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

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.

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

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.

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.

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

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?

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?

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

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

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

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

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.

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

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?

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

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

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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?

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

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

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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?