Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

A triangle ABC resting on a horizontal line is "rolled" along the line. Describe the paths of each of the vertices and the relationships between them and the original triangle.

How many different symmetrical shapes can you make by shading triangles or squares?

Can you work out what kind of rotation produced this pattern of pegs in our pegboard?

Can you picture where this letter "F" will be on the grid if you flip it in these different ways?

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?

Join pentagons together edge to edge. Will they form a ring?

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 describe this route to infinity? Where will the arrows take you next?

A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

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

What is the minimum number of squares a 13 by 13 square can be dissected into?

How can you make an angle of 60 degrees by folding a sheet of paper twice?

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

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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.

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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

Can you work out what is wrong with the cogs on a UK 2 pound coin?

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?

Can you fit the tangram pieces into the outlines of the convex shapes?

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.

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

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 dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

If you move the tiles around, can you make squares with different coloured edges?

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

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

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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. . . .

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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

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!

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?