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.
Bisectors
Here is Harry’s bisection of an angle, also showing that if you make the compass marks further out, it still results in the same bisector.
This is Evie’s bisection of a side. Notice that both pupils didn’t trust me that the angles would be equal, so they measured them just to make sure, and also that both pupils bisected perfectly!
Triangle Challenge
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.
Circumcentre
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.
Alternatively they constructed perpendicular bisectors like Alex did here.
[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’!]
Incentre
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!
Napoleon’s Theorem
Here are several examples of Napoleon’s Theorem, using the different centre-finding methods described above.
In the first one, Sol has bisected two angles in each triangle and added the third bisector by inspection!
Jude has found the midpoints and drawn to the opposite corners as Aimee did above.
Evie’s very colourful explanation features the equidistant method of finding the circumcentre that Abbie and Abi explained earlier.
Alex bisected two sides of each triangle, using the same method as he described above.
Summary
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!
I think this challenge was very interesting and I am very pleased that Mr H put two pieces of my work on the blog. I think we all loved doing this challenge and we all had lots of fun along the way. So thank you to Rob Eastaway for the fun lessons your challenge gave us.
Thanks!!!
I think it is very interesting. I did this with Mr H. It was very fun but difficult at the same time. I think the whole class enjoyed it just like I did. Thank you Rob Eastaway for a great challenge.
The challenge was hard but when you got the hang of doing the ‘eeks’ it was fairly easy.
I very much enjoyed reading this blog, especially as it’s a very long time (33 years really) since I was expounding such stuff in the classroom. A feeling of nostalgia swept over me! It struck me that enterprising pupils might be interested to press on further to investigate Nine Point Circles and Euler’s Line.
Thanks for testing it out, it’s very interesting to see the different ways that children approach this problem.
These were ‘top set’ children, and it sounds like they were primed with some instruction on bisectors. How would middle (or even lower) set children tackle this without any lead from the teacher, done as a team challenge maybe? Perhaps pose it as a treasure map: “The pirates say they have buried the treasure in the exact centre of this triangular island. Who can find the treasure?” (Note that the pirates need to have made an arbitrary decision about which centre they mean, but don’t tell the children this!) It would be interesting to eavesdrop on the children’s conversations. I would expect a starting point for finding the middle would be either a guess like pinning the tail on the donkey (“that looks about right”) or a basic measurement (“half way along the bottom and half way up”). Then see how they respond when told that the pirates didn’t guess, they measured it, and that halfway along/up no longer looks like the middle if you rotate the triangle.
I think there’s a nice article for the MA (or ATM) to be had in this.
I love a good three-sided argument!