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?

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

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

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?

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?

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

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

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.

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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

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

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?

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?

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 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.

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

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?

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. . . .

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.

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

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

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

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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?

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

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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 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. . . .

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

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

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 fit the tangram pieces into the outline of these convex shapes?

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

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

Can you fit the tangram pieces into the outline of the rocket?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Can you fit the tangram pieces into the outline of this plaque design?

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

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

Can you fit the tangram pieces into the outline of these rabbits?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?