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

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

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

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.

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

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

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

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.

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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

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

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?

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

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?

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

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?

Make a flower design using the same shape made out of different sizes of paper.

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

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

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.

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

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. . . .

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you visualise what shape this piece of paper will make when it is folded?

How many different triangles can you make on a circular pegboard that has nine pegs?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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?

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?

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!

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. . . .

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?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Can you fit the tangram pieces into the outlines of these people?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Can you fit the tangram pieces into the outlines of these clocks?

Can you logically construct these silhouettes using the tangram pieces?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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?