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

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?

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.

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

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

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?

You can use a clinometer to measure the height of tall things that you can't possibly reach to the top of, Make a clinometer and use it to help you estimate the heights of tall objects.

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

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Make an equilateral triangle by folding paper and use it to make patterns of your own.

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

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.

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?

Galileo, a famous inventor who lived about 400 years ago, came up with an idea similar to this for making a time measuring instrument. Can you turn your pendulum into an accurate minute timer?

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?

Can you make the birds from the egg tangram?

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?

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

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

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

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

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?

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

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?

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

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

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

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?

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

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

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

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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

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

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Here is a chance to create some Celtic knots and explore the mathematics behind them.

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

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

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

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?

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?

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

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