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?

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

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 activity investigates how you might make squares and pentominoes from Polydron.

Can you each work out what shape you have part of on your card? What will the rest of it look like?

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

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?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

These practical challenges are all about making a 'tray' and covering it with paper.

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

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

Can you make the birds from the egg tangram?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Can you deduce the pattern that has been used to lay out these bottle tops?

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

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

How many models can you find which obey these rules?

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

Surprise your friends with this magic square trick.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

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.

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

What is the greatest number of squares you can make by overlapping three squares?