For me, there’s nothing better than discovering something for the first time. There is something exhilarating about having a tricky and/or meaningful problem, mucking around trying different things to solve it, and producing a solution. Last week a group of my yr 8 students (13/14 yr olds) showed me that they have the same drive, and in doing so discovered something that I’d never seen before.
Take 1: It all started with a challenge problem that I set them for the topic of Area. Their aim was to calculate the shaded Area of the shape to the right (1/3 of the area outside an equilateral triangle inside a circle, given a side length of a triangle). It took them a while and a few mistakes but eventually they made it to the solution.
Take 2: The next lesson I decided to push them further, and asked them to calculate the following shaded area (1/3 of the Area inside an equilateral triangle, but outside the incircle). I hadn’t really thought to much about the solution, but when I went ahead and solved it, I realised that I needed to apply the Sine rule (which they won’t learn for another two years). Much to my surprise, a student came up and showed me the same answer. When I asked him how he did it, he said that he ‘intuitively guessed’ that the height of the incircle would be 2/3 the perpendicular height of the triangle (which it turns out to be!).
Take 3: At the start of the next lesson, these students came up to me and said they wanted to combine both questions, and try and get other students to solve it (see left). I was more than happy for them to do it but only if they could produce a full solution for others to see. That was when it all started to get very interesting. One student came up to me and said “it’s really weird; the Area in those sections is almost the same as the Area of the incircle”. I asked him “is it almost the same, or is it exactly the same? Maybe you’ve got a rounding error”. He went back and discovered that in fact it was exactly the same. All of a sudden they were getting very excited as they couldn’t find anything about it anywhere on the internet (though there are sites that had info about the ratio of incircles:circumcircles). We chatted about what was going on, and decided it was time to prove it….
Take 4: The side length was removed and replaced with pronumeral l. Then we went through and produced an expression for both the Area of the incircle, and the shaded Area ((pi*l^2)/12). Boom! They were exactly the same and the kids were literally jumping in the air at this point.
Take 5: I then told them they could prove it another way considering that the radius of the incircle is 1/2 the radius of the circumcircle (something that they came across in their problem solving). Then considering the ratio of Area’s would be 1:4, the annulus would therefore be three times the size of the incircle. Considering symmetry, the shaded Area is 1/3 of the annulus, and therefore equal to the incircle!
Summary: These students investigated a problem that they designed (even though it was kind of a conglomeration of my questions), and discovered something that they weren’t expecting or even looking for that was much more interesting. I was able to use algebra to enhance their solution, and help them produce their own formula that would work for any given length (thus they learned the beauty of algebra and why it’s so useful!). At the end of the lesson they made me take a photo of them in front of their work and email it to them so they could show their parents and friends. Did anyone say new facebook dp???