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

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

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!

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

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

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.

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?

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.

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

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.

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

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

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?

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

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

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?

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

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

Here are shadows of some 3D shapes. What shapes could have made them?

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

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

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?

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

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.

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?

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?

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

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

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

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.

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

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

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 ...

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

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

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

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

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

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?

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

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

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

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

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?