In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

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

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

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

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?

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

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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?

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

Can you make the birds from the egg tangram?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

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?

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

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?

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?

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

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

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

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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?

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

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

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?

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

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.

Here is a version of the game 'Happy Families' for you to make and play.

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

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

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?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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?

You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?

These pictures show squares split into halves. Can you find other ways?

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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.

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

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

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Here are some ideas to try in the classroom for using counters to investigate number patterns.