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 ...

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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

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

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.

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

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?

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?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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

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

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

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

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

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?

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?

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?

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

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.

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

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?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Can you fit the tangram pieces into the outline of the butterfly?

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of the candle?

Can you fit the tangram pieces into the outline of the telephone?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

Can you fit the tangram pieces into the outline of the sports car?

Can you fit the tangram pieces into the outlines of the telescope and microscope?

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 fit the tangram pieces into the outline of this teacup?

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?

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

Can you fit the tangram pieces into the outline of the house?

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

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

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?

Can you find a way of counting the spheres in these arrangements?

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

Can you fit the tangram pieces into the outlines of the convex shapes?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?