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. . . .

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

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

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

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?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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 recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

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

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

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. . . .

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

Can you mark 4 points on a flat surface so that there are only two different distances between them?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

See if you can anticipate successive 'generations' of the two animals shown here.

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees?

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

Can you find a way of representing these arrangements of balls?

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

What is the shape of wrapping paper that you would need to completely wrap this model?

When dice land edge-up, we usually roll again. But what if we didn't...?