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

What can you see? What do you notice? What questions can you ask?

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

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?

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

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

Can you fit the tangram pieces into the outlines of the candle and sundial?

Can you fit the tangram pieces into the outlines of the workmen?

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

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

Which of these dice are right-handed and which are left-handed?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outlines of these people?

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 Little Ming playing the board game?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

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

Can you fit the tangram pieces into the outlines of these clocks?

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

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of Granma T?

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

Can you fit the tangram pieces into the outline of Little Ming?

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

This article for teachers describes a project which explores thepower of storytelling to convey concepts and ideas to children.

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?

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

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

An activity centred around observations of dots and how we visualise number arrangement patterns.

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

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

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?