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

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

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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.

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

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.

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

This article for teachers suggests ideas for activities built around 10 and 2010.

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?

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

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

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?

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

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?

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?

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

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

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.

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?

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

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

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

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?

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

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

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

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

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?

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

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

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

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

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!

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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?

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

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?

A follow-up activity to Tiles in the Garden.