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

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?

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

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

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

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

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

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 ...

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

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

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

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

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?

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

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

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

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?

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

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?

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

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?

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?

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?

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

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

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 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?

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 make the birds from the egg 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.

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?

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

Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?

In this activity focusing on capacity, you will need a collection of different jars and bottles.

For this activity which explores capacity, you will need to collect some bottles and jars.

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

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

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

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

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.

Here are some ideas to try in the classroom for using counters to investigate number patterns.

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

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