Friday, February 16, 2018

2/13 Farey Sequences


Today was a special occasion for Math Club. Instead of just me or a vicarious video, we once again had a guest lecture from the UW Applied Math department. 




This time, Professor Jayadev Athreya came out to the middle school to give a talk on Farey Sequences. That was fairly propitious, since I had meant to get to this subject during this session: http://mymathclub.blogspot.com/2017/11/1128-egyptian-fractions.html  but quickly realized I didn't have enough time to cover even Egyptian Fractions.  So there was a good thematic fit with some of the other things we've done.

My favorite moment of the day came early on when Jayadev had each of the kids talk about why they came to math club. (I usually do this on the first session too) There were a smattering of "I like competitive math" responses but then we reached a girl who roughly said "I don't know why I came originally but I like it so I keep coming." That's victory in my book!

What's also interesting here is a chance to more closely observe all the kids and another person's teaching style.  Jayadev's basic structure was fairly similar to what I might have done.



  • Start with having the kids map out all the reduced form fractions where the denominator was less than or equal to 10 and then arrange them by size from smallest to largest. He then graphed this on a number line as a group.





      • Closely investigate the numerators of the fractions (in suitable common denominator form) when comparing them to notice a trend: they always differed by one.






      • Do a formula proof that the mediant is always 1 apart from its generators if they are 1 apart.






      A lot of this was structured as group work at the tables with discussions after a few minutes where things were consolidated.  I always find these transitions a bit tricky to time so it was useful watching someone else.  I would also probably have done a few of these parts as group work on the whiteboard itself and then gallery walked for the discussions but there was good work done at everyone's seats.

      Like last time, I also noticed some unexpected hesitancy with operating on fractions. It took a bit more time to draw out the kids and have them explain how to compare fractions with different denominators. Although create common denominators was mentioned as well as variety of numeracy instincts ("for unit fractions, the fraction become smaller as the denominator increases" or "you can also create common numerators to do informal comparisons").   Again, I felt like this was a useful practice/review for some.  Alternate hypothesis: the kids were more reluctant to volunteer at points which was more social rather than indicating any gaps.  If this is the case, I'd like to work on activities to bring out more questions. One idea I have toyed with in the past is, is selecting one student to be "the skeptic" during any demo and come up with at least one question about the logic. If I do go this way, I'll probably start with having them do this with something I discuss and depending on how it goes try it also during all whiteboard discussions.

      Overall I was really pleased. We now have an invitation to visit the Applied Math Center on the UW campus. I have to investigate whether the logistics are workable.

      1 comment:

      1. By coincidence this showed up:

        http://voices.norwich.edu/daniel-mcquillan/2018/02/25/fun-with-fractions-from-elementary-arithmetic-to-the-putnam-competition-the-first-1-2/

        The putnam problem at the end is a really interesting extension and this demonstrates an alternate buildup approach.




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