The last topic we’re covering in Math 3 will be critical points, which I’m extending to include maxima, minima, zeros, and asymptotes. We’ve already talked about zeros, and the students are all familiar with graphing polynomials based on their zeros. This week I introduced them to the idea of an asymptote, slowly and painlessly. Here’s what we did.
Remember in my math words task how I asked the students to read from The Crest of the Peacock? I found another section in the book called “The Enormity of Zero,” which was perfect for this class. I handed them the questions to preview the assignment, then I read them the two pages out loud (last time I assigned reading their reaction was “THIS IS MATH CLASS!!!” just this side of a revolt, an issue for another time). If an almost-600 page book about math is a little TL;DR for you, allow me to summarize this part. It starts with the Egyptians, who used the hieroglyph “nfr,” meaning “beautiful” or “complete” to stand in for a zero balance on budget documents. They also used nfr to stand for the ground level when constructing pyramids, using “above nfr” for the pointy pyramid part and “below nfr” for the Pharaoh-tomb part. In Mesopotamia, essentially at the same time, they had a base 60 system with no zero – they had 60 distinct symbols and words for their numbers. In ancient Greece, they actively avoided zero, and basically all arithmetic because it was too democratic and concerned with equality! In India, the Buddhist concept of “shunyata” or “emptiness” predated the number zero. This state of mind was sought after for all artistic endeavors. This emptiness or zero concept was thought of as a boundary between positive and negative. It also made place value possible, an appropriate connection to the base-10 activity these students did last week. The last line of the section is a perfect introduction to asymptotes, a quotation from an ancient text stating that dividing by zero results in a number “more infinite than the God Vishnu.”
I’m loving this because logarithms do connect to the real world and they are fascinating ways to unlock questions about actual things, but solving logarithmic equations felt to the students very rote and formulaic. Our current course of study, in stark contrast, immediately relates to history, spirituality, philosophy, and the very structure of our society.
Today in class I gave an impassioned explanation about why dividing by zero could be a very large number or a very small number, essentially teaching them about limits. I made sure to connect this back to the idea of the base-10 logarithmic scale – 6 divided by .01, then by .001, then by .0001, and so on gets bigger and bigger. They were able to quickly recognize the 60, 600, 6,000 times-10 pattern because of our previous work. Our first example of an asymptote in a graph was an exponential decay graph – something they have a lot of concrete examples of. If I drink a ton of coffee in the morning (not that I would ever do that) and it wears off during the day, I won’t really feel the effects anymore, but it won’t ever be down to exactly zero, either.
I’m super excited about this week, and I’m looking forward to reading the entire book this summer and finding a passage appropriate for every situation. I originally wrote that sentence meaning for every topic I’ll be covering, but it may come in handy in non-classroom settings, too.