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

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?

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

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?

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

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

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?

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

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?

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

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

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.

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?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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?

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?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

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?

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.

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?

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

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 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.

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?

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

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

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

How many models can you find which obey these rules?

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

Can you make the birds from the egg tangram?

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

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

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

Can you visualise what shape this piece of paper will make when it is folded?

Can you each work out the number on your card? What do you notice? How could you sort the cards?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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

What shape is made when you fold using this crease pattern? Can you make a ring design?