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

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?

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

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?

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

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?

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?

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

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?

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?

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

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

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

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?

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

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

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 involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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

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?

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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?

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

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!

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 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?

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?

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

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

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

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?

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

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

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

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