Economies of scale
Our perceptions of scale can be wildly out. A recent Year 10 lesson began with the scenario where we ordered a 20-inch pizza, but two 10-inch pizzas arrived instead because they had run out. Was that OK? The class thought so, until we drew a picture ending up with the two 10-inch substitutes sitting neatly inside the much larger 20-inch pizza.
This produced reactions such as "No way!" After realising that changing the positions of the smaller pizzas did not help, it was accepted that two 10-inch pizzas are a lot less than a 20-inch pizza. One enterprising pupil commented: "You could actually do that. Most people wouldn't realise and you could make a fortune."
Trying it with a square pizza made it clear that four 10-inch square pizzas were equivalent to one 20-inch pizza. Why four? There was a lot of confusion: "4x10=20. That's stupid, Sir." The savvy pupil decided it would be a good plan to offer customers three 10-inch pizzas, "as a gesture of goodwill", to make them happy while being cheated. Did the four come from 2+2 or 2x2? We looked at square centimetres and square metres, seeing there must be 100 rows of 100 square centimetres in a square metre (10,000) and 100x100x100 cubic centimetres (one million) in a cubic metre.
This led to the idea of an "area scale factor" being the square of the "linear scale factor" and the "volume scale factor" being the cube of it.
These are hard concepts to understand. So doubling all the lengths of something gives it four times as much area and eight times as much volume.
We imagined an enlarging machine, which doubled all your lengths, and realised that you would come out weighing eight times as much, but your feet would cover only four times the area. Would your bones break? And how hot might you get, with less surface area to dissipate body heat? The scale factor approach connected information into something that, though still a bit counter-intuitive, was beginning to make sense
Colin Foster teaches mathematics at King Henry VIII School, Coventry
Scale: useful cross-curricular resources
Books Gulliver's Travels by Jonathan Swift and Alice's Adventures in Wonderland by Lewis Carroll
Films The Incredible Shrinking Man, Fantastic Voyage and Honey, I Shrunk the Kids