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:
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.