Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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?

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?

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

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

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?

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

Here is a version of the game 'Happy Families' for you to make and play.

Can you make the birds from the egg tangram?

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 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?

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 ...

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

Can you deduce the pattern that has been used to lay out these bottle tops?

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

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

Surprise your friends with this magic square trick.

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.

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?

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

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

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

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?

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

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

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

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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

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

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?

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Delight your friends with this cunning trick! Can you explain how it works?

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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