My bottom-set students were due to be taught constructions, creating various geometric objects with only a compass and a straight edge or rule. To engage the 12- and 13-year-olds, I knew I had to do better than some horrendously contrived example, such as wanting to fly a plane equidistant between two erupting volcanoes.
I was struck by a great idea: a perpendicular bisector doesn't just show you the midway line between two points, it also highlights which parts of the map are closer to one of those points. The next task was to think of how to interest students in this fact, and before long I had come up with a map of the local area that had the primary schools they had once attended marked on it. The map was at a scale where individual streets could be recognised.
I asked the students to construct the perpendicular bisector of two post offices on the map and then asked if anyone's house was located between them. We used the bisector to decide which post office was closest to two students' houses, and I then asked what else they could see on the map that we could draw bisectors between. As if by magic, the dominoes toppled as one student mentioned the primary schools and another said, "Hang on, I don't think I went to my closest one." The lesson took off.
Everyone wanted to know whether they had gone to their local primary and they asked for a bigger map to compare secondary schools. This led to some interesting conversations about why people don't always go to their closest school, which was a nice bonus. One of my favourite things about this lesson was that the students felt they had thought up the tasks themselves, even though I had planned them that way all along.
This lesson made me re-evaluate my opinion of constructions of perpendicular bisectors. Now all I need is a similar idea for angle bisectors ...
Dave Gale is a maths teacher at Churchill Academy and Sixth Form in Somerset, England.