Teenagers want to know where the limits are - what they can get away with.
Turn that to a mathematical advantage with "possibility spaces". These are tables with different factors running horizontally and vertically, and the task is to see what is allowed and what isn't.
How far can you go before you bump into mathematical limits? For example, make one for different kinds of triangles.
Triangles can be classified according to the lengths of the sides (scalene, isosceles, equilateral) or their angles (acute, obtuse, right angle), but which of the nine boxes in your table are possible and which are impossible?
Photocopy a blank table on to A3 paper and get pupils in groups to draw an example in each space in the table. If some are impossible, pupils must justify why.
One that gets maths teachers thinking, as well as pupils from Year 7 upwards, is a table for order of rotational symmetry and number of lines of symmetry.
Pupils display great ingenuity trying to create shapes that satisfy the constraints or convincing one another that something cannot be done. There are moments when someone finds they can do one that they previously thought was impossible. And the possibility patterns in the finished tables provoke some interesting questions
Colin Foster teaches maths at King Henry VIII School in Coventry