Sharing Numberphile’s “Amazing Graphs” video with kids

Last week Numberphile put out a fantastic video featuring Neil Sloane:

For today’s project we explored the sequence described in the first half of the video. Namely, the sequence that begins with a_1 = 1 and then continues with a_{n+1} depending on the greatest common divisor of n and a_{n}. See either the Numberphile video or the first video below for the full formula.

To introduce the boys to the sequence, I had them calculate the first 10 or so terms by hand:

Next we wrote (off camera) a Mathematica program to calculate many terms of the sequence, and studied what the graph of those terms looked like:

Finally, I asked the boys to watch the Numberphile video and the describe what they learned. They were both able to give a nice explanation of why the sequence eventually repeated:

I love math projects that allow kids to play with really interesting math and also sneak in some k-12 math practice. The first sequence in the Numberphile video is a perfect example of this kind of project!

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