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?

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

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 work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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

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

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Can you split each of the shapes below in half so that the two parts are exactly the same?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Can you cut up a square in the way shown and make the pieces into a triangle?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Can you make the birds from the egg tangram?

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

Make a cube out of straws and have a go at this practical challenge.

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?

Exploring and predicting folding, cutting and punching holes and making spirals.

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 ...

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Make a flower design using the same shape made out of different sizes of paper.

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Use the tangram pieces to make our pictures, or to design some of your own!

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

Follow these instructions to make a three-piece and/or seven-piece tangram.

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.

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

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

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

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

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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?

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

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

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?

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?

Reasoning about the number of matches needed to build squares that share their sides.

Can you make five differently sized squares from the tangram pieces?

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

A game in which players take it in turns to choose a number. Can you block your opponent?

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.

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

Can you lay out the pictures of the drinks in the way described by the clue cards?

Ideas for practical ways of representing data such as Venn and Carroll diagrams.