A couple of weeks ago the mathematician, author and all-round nice chap, Rob Eastaway tweeted a question which fitted perfectly with the next topic for my Year Seven top set:
“What happens if you give an irregular triangle to a class & ask: can anyone find the middle of this by tomorrow? Can somebody test it out?”
Now it being half-term, ‘by tomorrow’ wasn’t going to happen but I offered to do it anyway and get back to him. This blog is by way of a rather long answer to his question.
The aim of the topic is to extend the pupils’ ability with ruler and compasses by tackling more interesting work than the usual fare. The end point is a construction known as Napoleon’s Theorem but we started with a recall/introduction to bisectors.
I then asked them to draw an irregular triangle, think about how they could find the middle of it then share their ideas with their neighbours. We finished the lesson by briefly reviewing the different thoughts and they left with homework being to choose one of those methods, or a different one, and write it up.
There was a pleasing mix of answers, some more intuitive than mathematical, and they found four different centres (I think!).
Centroid or Centre of Gravity
Aimee has described perfectly how to find the centre of gravity; if we were to cut out her triangle, it should balance on this centre. She will be annoyed when she sees that her lines do not go perfectly into the corners! To be fair, she does also appear to have done the next method and the two seem to have become overlaid. Several pupils used this method for their Napoleon construction.
The pupils managed to find this in two different ways. Firstly, they reasoned that if you find a point equidistant from each corner, that must be the middle.
Here is Abi’s explanation with Abbie’s drawing which includes the Circumcircle that just touches all three vertices.
[I use the word ‘eek’ for the curved mark made when using a compass which harks back to the ‘good old days’ and using the cumbersome teacher’s compasses on a blackboard when the chalk would indeed go ‘eek’!]
Nella managed to find this in both ways! She started with a sketch where she shows the middle being a point equidistant from the three sides by drawing the Inradii. She then, with a little prompting, went on to find the same point by bisecting the angles. Her drawing also shows that this middle is the centre of the Incircle which just touches all three sides.
Not quite the Fermat Centre!
Josh was one of several pupils to use the method he describes. If he had constructed equilateral triangles on each side then he would have found the Fermat Centre. Instead he has created three isosceles triangles where the two outer sides of each triangle are all equal. If extended this would create an infinite series of triangles and a concomitant series of centres. I have not managed to find this in the literature!
Here are several examples of Napoleon’s Theorem, using the different centre-finding methods described above.
Finding the centre of a triangle was an interesting exercise. It added an element of pupil input into the activity that I hadn’t really had before and encouraged them to describe their method. It was also salutary to see how many pupils did not connect, and therefore use, the bisecting skills to the triangle challenge. The Napoleon Theorem then becomes a more logical extension exercise. Hopefully Rob, and other visitors, will find our work as interesting and enjoyable as we did!