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

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

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

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

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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

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?

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

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

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 greatest number of squares you can make by overlapping three squares?

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

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

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?

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

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

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

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.

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 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?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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

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?

How many different triangles can you make on a circular pegboard that has nine pegs?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

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

Why do you think that the red player chose that particular dot in this game of Seeing Squares?

How can you make an angle of 60 degrees by folding a sheet of paper twice?

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?