[Note: I’m in a little rush to get up to Boston for Brute Squad practice today, so I’m just getting the videos up now and will expand this blog entry either later today after practice or later in the week. I haven’t even proof read this, but it was so fun I just wanted to get it out there!]
Today I told the boys that we could cover whatever they wanted for today’s Family Math project and they chose infinite series as the topic. In particular they wanted to talk about
(1) Fibonacci Numbers,
(2) Pascal’s triangle,
(3) the sum 1/2 + 1/4 + 1/8 + 1/16 + . . . ., and
(4) “the -1/12 series”
I’m not sure that I could have been more excited about this list of topics!
We started with the Fibonacci numbers. The idea here was to review the idea of how you create the list of Fibonacci numbers, see what the boys remembered about this sequence, and show them how the Fibonacci numbers arise in a simple continued fraction. The boys remembered that you could use the numbers to make a spiral, so we spent a little bit of time talking about the spiral, too.
I wanted to show the continued fraction example because the Fibonacci numbers occur in both the numerator and the denominator of the continued fraction convergents, but the numbers are shifted over 1 in the numerators. That shift of an infinite sequence will come into play in our last videos when we discuss “the -1/12 series”
The next topic was Pascal’s Triangle, which turns out to be an absolutely perfect next step by luck. We started by reviewing how you create the triangle and then moved on to looking at some other sequences that are hiding in the triangle. We found several fun patterns hiding in the triangle including some patterns that describe some fun geometry. At the end I showed them that even the Fibonacci numbers are hiding in the triangle in sort of a sneaky way. I wanted to talk more about this but a bee flew into the room, oh well . . . :
The third topic was the sum 1/2 + 1/4 + 1/8 + . . . and why this series sums up to be 1. This was also really fun and I got a nice surprise as each kid had a slightly different geometric way of showing why this series summed to 1. I showed them a 3rd slightly different idea and then showed them a second neat series that also sums to 1 -> 1/4 + 2/8 + 3/16 + 4/32 + . . . Patrick Honner gave a really cool visual proof of this fact here and show them how his visual proof works.
The last topic is the “-1/12 series” made famous by this Numberphile video:
After an introductory talk about this series and the seemingly (or perhaps, “actually”) crazy sum, I backed up a little by talking about a question that seems to be a tiny bit easier -> does 0.999…. = 1? Following the line of reasoning in Jordan Ellenberg’s “How Not to be Wrong” I showed that the standard way of proving this also can produce some strange results. I really like Ellenberg’s description of this standard proof as “algebraic intimidation” and you can see how that algebraic intimidation plays out in the next two videos as both kids really don’t believe that the original sum is -1/12, but also seem to be convinced by the math that it does.
Finally, I followed the ideas in the Numberphile video above and showed how you get the result that the sum of 1 + 2 + 3 + 4 + . . . . = -1/12. I love that this result seems to actually physically bother my younger son.
This was a super fun project. Shows the fun you can have when you let the kids pick the topics 🙂