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

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.

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

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

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

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

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

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

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

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?

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 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?

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

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.

Follow these instructions to make a five-pointed snowflake from a square of paper.

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?

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

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

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

Can you make the birds from the egg tangram?

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?

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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

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

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

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?

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?

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

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

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

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?

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

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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?

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

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?

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

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

How many models can you find which obey these rules?

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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?

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

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

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