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

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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

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

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

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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.

What happens when you try and fit the triomino pieces into these two grids?

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

Exchange the positions of the two sets of counters in the least possible number of moves

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.

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!

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

Can you make a 3x3 cube with these shapes made from small cubes?

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

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

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?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

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

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?

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

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

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

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

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

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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

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 outlines of the chairs?

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

Can you find ways of joining cubes together so that 28 faces are visible?

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

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?