In Number Theory, we're hunting for treasure, honing our proof-writing skills, and discovering mathematical delights.
Our Introduction to Number Theory class is off to a fast start. We’ve covered an alarming percentage of the advertised topics (given that we’re only two weeks into a six-week block), including divisibility tests for small integers, counting divisors, perfect numbers, number bases, divisibility tests in other bases, prime factorization, GCD and LCM, and the Euclidean Algorithm. We’re planning to use the extra time on hand to delve more deeply into square numbers, irrational numbers, and especially congruences during the last two weeks.
Each of my courses this year has quickly developed a distinct character, in terms of class format, and number theory has been no exception. It’s become a tradition to hold quiz review “treasure hunts,” in which the class tackles a set of questions, then figures out how to interpret the answers as an encoded message, which then leads to the final goal. Number theory has a long history in the service of encryption, so this practice seems especially suitable for our class.
I’m also implementing a new means of incorporating writing into the mix. Approximately once a week we discuss the strategy for proving a number theoretic statement as a class, then each student writes out a short paragraph proof on a sheet of looseleaf. This paper travels back and forth between myself and the student, accumulating feedback and new drafts as often as necessary until the proof is beautifully polished, at which point it receives a “squircle-star” seal of approval and is filed in their folder.
One of the perks of teaching Number Theory is that it is such a delightful subject; kids who love math tend to gravitate to the clever and unexpected results that await us each week to an even greater degree (if possible) than in our other math classes. For instance, one afternoon I presented them with the first four perfect numbers; namely, 6, 28, 496, 8128, and challenged them to detect the pattern and find the fifth such number. A merry hunt ensued, which eventually led to the correct value of 33,550,336. We then made and proved a conjecture about how to build perfect numbers, which dates back to Euclid.
We still have a ways to go this block, of course, but I’m very pleased with the foundation that has been laid thus far. I believe that I'm speaking for everyone when I say that we're all looking forward to the last four weeks of math in the classroom this year.