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?

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?

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?

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

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?

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

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?

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

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

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

What is the greatest number of squares you can make by overlapping three squares?

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?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

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?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

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

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?

How many pieces of string have been used in these patterns? Can you describe how you know?

How many loops of string have been used to make these patterns?

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

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

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. . . .

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

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

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

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

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

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

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

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?

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

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

Can you fit the tangram pieces into the outline of the telescope and microscope?

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

Here's a simple way to make a Tangram without any measuring or ruling lines.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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.

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

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

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. 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.