I was stumped, where was I going wrong? There I was standing in front of my class and I couldn't get the maths to work. There was nothing wrong with my thinking, there was nothing wrong with my calculation, but yet it still wasn't working. It was then that it struck me, I looked at the question and realised that the maths wasn't working because the question I had put up wasn't a triangle. If the longest side was 9, then there was no way that sides of 3 and 4 would form a point above that side, they wouldn't even come close to joining. I felt like an idiot, I wanted to crawl under the desk and hide.
On that day, I had ran through my carefully prepared examples and they wanted more, so I made the questions up on the spot, I drew a triangle, put some numbers against them and tried to work it out. This is something I had done on other occasions without incident, but this time I picked wrong. It was a valuable learning experience for me as a young teacher and this moment still sticks with me, even 12 years later.
Late last year I was doing some work on mathematical reasoning at the school, I gave the maths teachers the question above and gave them the time to figure it out. Not surprisingly they were all able to figure out that B and C were definitely not right angled, however they all used Pythagoras' Theorem to determine the answer, which was fine because I put no restrictions on it.
So my next question to them was "If you knew nothing about Pythagoras' Theorem, how would you approach this question, how could you eliminate at least some of the possibilities?" This was an interesting process, to look at at how they now approached the problem. It wasn't until I asked them to draw triangles B and C that they realised that they were impossible, they didn't need to draw them, it was just that stimulus that got them thinking about it in that way.
I began to realise that knowing that the sum of the two smaller sides must be bigger than the largest side is not a frequently discussed property of triangles, at least from our high school perspective of shape, I really don't think I knew much about it until I was standing in front of that Year 10 class and trying to figure out why the maths wasn't working. We are often told that triangles have got three angles and three straight sides, we also are told sometimes that the sides must be connected, however I never remember exploring the conditions under which those sides would be connected and how this property extends to polygons with more than three sides. We are told about angle sum in polygons, but don't explore ideas around the sum of sides.
This experience has taught me a few things
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
Opinions in this blog are my own and do not necessarily represent the views of my employer.