One activity I use with my Year 7 maths class early in the year involves a square of card. I give pupils a 4-centimetre square and ask them to draw a line from a corner to the midpoint of a side. Then they cut along the line to make two pieces. This is their two-piece jigsaw.
They then join the two pieces along sides of equal lengths to make a new shape. I ask them to draw the shape and name it. For example they might join the two shorter sides to make a right-angled triangle. This is a popular activity because it is accessible to all pupils, and it gives me the opportunity to go round and talk to them, gaining some insight into their mathematical vocabulary. After a few minutes of exploratory work I ask one or two to show their shapes on the OHP. There is no shortage of volunteers.
Once they have got the idea of what to do, ask them to work in pairs, preparing an OHT so they can talk about their results at a plenary at the end of the lesson. Taking the right-angled triangle example, pupils have mentioned that it has been made by rotating the triangle around the midpoint so the two angles still form a straight line (now the hypotenuse) and that the two right angles come together to make a straight line on the other side. For homework, I ask them to find another shape and write a letter to a friend about its properties, and how they can be deduced from the original square.
I use this activity with older pupils to develop their geometrical reasoning. Asking for reasons why the shape they made is what they say it is encourages logical reasoning and it is interesting to note that often they are far better at verbal reasoning than written reasoning.
Peter Ransom, leading mathematics teacher, the Mountbatten School and Language College, Romsey