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

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

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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?

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 work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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

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

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

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

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?

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

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?

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

What does the overlap of these two shapes look like? Try picturing it in your head and then use some cut-out shapes to test your prediction.

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

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

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

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

Move just three of the circles so that the triangle faces in the opposite direction.

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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

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

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

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.

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

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?

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

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

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.

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 ...

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

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

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?

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

Why do you think that the red player chose that particular dot in this game of Seeing Squares?

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

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

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

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

How many loops of string have been used to make these patterns?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

How many pieces of string have been used in these patterns? Can you describe how you know?

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?