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

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?

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

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

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

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

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?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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?

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

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?

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

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

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

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?

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?

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

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.

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

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.

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.

Numbers arranged in a square but some exceptional spatial awareness probably needed.

A follow-up activity to Tiles in the Garden.

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

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?

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

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

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.

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

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

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

How many models can you find which obey these rules?

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

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

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

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?

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

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!

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