[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”

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 🙂

# Fibonacci Numbers and basic modular arithmetic for kids

Last fun little Family Math of the summer and I thought it would be fun to play around with the Fibonacci numbers since they were already on my mind from this weekend:

Fibonacci Factorials

We started with a quick review of the Fibonacci numbers and then I explained that today’s project was going to be looking at the remainders when you divided the Fibonacci numbers by different integers.  I picked 2, 3, 5, and 11 because the first three have patterns that aren’t too hard to understand and 11 has a bit of a surprising pattern.

Having decided to make a chart for all of our remainders, we started looking at the remainders when you divide by 2. They found a pattern relatively quickly and thought that pattern would continue forever. We also talked about what fraction of Fibonacci numbers are even? This question caused a little bit of difficulty, but we got it straightened out eventually.

Next we moved on to looking at the remainders when you divide by 3. As with the remainders when you divide by 2, the pattern in the remainders here isn’t too hard to see. We also talked through the proportion of Fibonacci numbers that are divisible by 3. This proportion idea still gave them a little bit of trouble, but I thought they were starting to understand it a little better by the end of this part. They had a little trouble explaining why the pattern they saw would continue, though.

Next we moved on to looking at the remainders when you divide by 5. This one is slightly more difficult because the pattern takes a little longer to repeat and, in particular, longer than the number of rows that we have in our table. I was really pleased by the curiosity that they showed in this section while trying to figure out what the pattern was going to be.

Finally we look at the remainders when you divide by 11. I didn’t know until this weekend that you don’t get all possibly remainders modulo 11. I thought it would be neat to look at 11 specifically to show them this interesting difference from what we saw when we divide the Fibonacci numbers by 2,3, and 5.

This was a really fun project, and something that I think many kids would enjoy. It was especially fun to see them realize that they’ve seen modulo arithmetic with clocks already in the last video. Number theory has so many easy to understand projects, and I’m hoping to do a few more number theory projects with them in the upcoming school year.

# Ben Vitale’s Triangle puzzle

I’m on vacation right now or I’d write something much more thorough longer, but I wanted to point out a really interesting puzzle about triangles posed by Ben Vitale on twitter yesterday:

Since I never know if the link’s from twitter will be preserved in the embedding, here’s a direct link to his blog:

He gives a couple of examples and challenges the reader to find some more.  One of his examples solutions is the  triangle with sides $\sqrt{61}$, $\sqrt{153}$, and $\sqrt{388}$ which has an area of 21.

Following the hint on Ben’s blog, I played around a little bit and found that the triangle with side lengths of $\sqrt{801}$, $\sqrt{932}$, and $\sqrt{3461}$ has an area of 3.   Here’s my work, where you’ll see the Fibonacci numbers hiding in the middle of the page! I really like this problem, and as I mentioned to Ben via Twitter last night if we weren’t on vacation I’d be working through this problem straight away with my kids.  This problem shows a beautiful connection between geometry and arithmetic, and the Fibonacci numbers come up in my example from a neat connection to continued fractions.  What a wonderful problem.  Thanks Ben!

# *The quadratic formula and Fibonacci numbers

I knew ahead of time that I was going to have a busy week of work this week and was looking for something fun to cover with my older son in the limited amount of time that we would have.  We were supposed to be covering properties of functions so I was looking for a topic that would at least be tangentally related to that, but I also wanted to get him a little review with solving quadratic equations.  Diving into the formula for the Fibonacci numbers seemed to fit the bill quite nicely . . .

We started with a short talk about the Fibonacci numbers that focused on thinking about the usual recurrence relation definition:

I didn’t do a great job with the graph at the end, and we spent another 10 minutes after the video talking through the graph and comparing it to $y = x^2$ and $y = 2^x.$  I was pretty happy with how that talk about the graphs went after the first video and wanted to reinforce some of those ideas the next day.  With that in mind, the next talk begins by discussing how the Fibonacci numbers grow and then considers what happens if the Fibonacci numbers could be written in the form $F(n) = C * \lambda^n$.

I’ve always found this technique for finding the closed form for the Fibonacci numbers to be really beautiful.  Turns out it is a great little tool for algebra review, too!

After finding that the Fibonacci numbers somehow related to the numbers $\frac{1 \pm \sqrt{5}}{2}$ at the end of the last video, in this one we finish up the calculation and write down the closed form for each Fibonacci number.  Lots of great algebra review for kids in these calculations, too!

With the formula for the Fibonacci numbers now in hand, I wanted to play around with the formula so we jumped over to Wolfram Alpha.  The first neat thing I wanted to go through was how we could now easily calculate each Fibonacci number without reference to the prior two numbers.  Fun!  The second thing I wanted to show was that the second term in the formula doesn’t play much of a role for the larger Fibonacci numbers.  It is pretty amazing to see how well $\frac{1}{\sqrt{5}} * (\frac{ 1 + \sqrt{5}}{2})^n$ approximates the larger Fibonacci numbers.  The meat of the formula is all in the first term!  I thought this would be an especially fun fact to show him since the two terms look so similar when you write them down.

We’ve talked a little bit about the Fibonacci numbers previously, but not with this level of math.  I’d chosen this topic because I thought the math is really interesting – which it was – but all of the algebra review turned this in to a nice learning opportunity, too.  For what was supposed to be just a  little diversion way from the book during a busy week of work for me, this topic turned out to be really fun.