This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

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. . . .

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?

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

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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?

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

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.

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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

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?

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

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.

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

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

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!

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.

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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

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?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

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

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

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

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

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

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

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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 outline of Little Ming and Little Fung dancing?

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

A package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on “visualising” and is designed to meet the needs. . . .

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?

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.

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

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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

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

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

Square It game for an adult and child. Can you come up with a way of always winning this game?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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.

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?