I love Numberphile videos, but then I would, I am a Maths teacher. What really matters is if my students like them. I have shown them to various classes and mostly they have gone down pretty well. At A Level especially, they are great as an “interesting interval” in a long lesson. This one, however, is the first time I have tried to create an actual lesson out of it. I used it for the first time today with a fairly high attaining Year 9 class and it went pretty well. They quite liked it when I told them it was degree level! It took about 35-40 mins in total, but if you also get them drawing the circles it could be longer. Below is my description of the lesson with links to the relevant sections of the video and the resources I used.
Farey Addition of Fractions and Ford Circles
Start of video: It starts pretty quickly so make sure they are ready. Lots of opportunities to pause and ask questions in this first section. E.g.
Pause at 0:23. What is wrong?
Pause 1:37. What do you think he is going to say?
Pause 4:33. At this stage, get students to create their own number line. Today, I just let my students go freestyle in their books. You could give them a simple number line printed out, something like this:
Or give them something a bit more structured with lines on it like this.
Give them about 10 mins on this task. They don’t need to measure anything, they can just do it by eye. It provides good opportunity for asking questions like “is 3/5 bigger than 2/3?” “How do you know?” Put a timer on and see who can do the most in 10 minutes.
Then start the video again. He looks at all fractions where q<=234. This might be worth explaining. There are 20,000 of them, apparently. Again, might be worth touching on briefly why is it 20,000. We just look at a section of these around 1/3:
Pause 7:10. Give them out the sheet of numbers, in order to hunt the Farey additions. This isn’t the best resolution as I have screen-grabbed from the video, but it worked well enough as a print out 4 per page. Give them 5-10 mins (with calculators). If they feel like they have done enough, ask them to investigate the ones that don’t work. If they are really thinking they might get the next bit on their own.
The video then shows an explanation of examples where it doesn’t appear to work, but if you cancel them down…
Then move onto drawing circles on the number line. The explanation in the video at this point isn’t brilliant and it would be nice for students to discover that when they draw the circles they get this, with the big circles on the end just touching the other circles. However, I shied away from this as the circles soon get very small and maybe too hard to draw. But it might be good to provide a print out with the two big circles at 0 and 1 and then get them to draw the circles at 1/2, 1/3 2/3, ¼, ¾, etc. You’d have to think carefully how you scaled it. Alternatively you might want to work out how do it on Geogebra if you have access to tablets / computers. Let me know how you get on!