I had wanted to do a unit about fractals for a long time, and this year I finally got the opportunity! I had originally envisioned my Math Four class as a traditional fourth course through more of a visual lens. Fractals involve some complicated mathematics, mainly exponentials and infinite sums, but also some complicated intricate designs.

We started with an exploration task. I would have assigned this in groups but I definitely burned them out of group work earlier in the year – live and learn. It started with Sierpinski’s Carpet, an increasingly-dense set of squares with an easily-recognized exponential growth pattern.

They then explored the Koch curve, finding the proportion of the area of triangles added in successive iterations.

As a group, students created a big giant Sierpinski’s Pyramid! Credit where credit’s due, I got the idea from this school in Winston Salem. Excellent idea!

I kept tripping on it because my room is not exactly large enough to accommodate a large pyramid, so I moved it into the administrative building. There, it encountered this guy and met an untimely demise.

To wrap up content for the entire year (?!!), students used grids to draw their own fractals. First we used triangle grids to draw the Sierpinski arrowhead curve, then regular square grids for the Dragon Curve. Vihart has an excellent video about that – a few students finished early and I had them watch it.

Next week is review week and then it’s exams – I’m starting to reflect on what I’ll be changing about this open-ended course next year, but mainly to avoid thinking about writing and grading exams! Although I will admit I secretly do enjoy exams in some ways.

# Tetrahemihexahedrons and other things that violate Euler’s Criterion

In math 2 we’re working with three-dimensional figures. I like to start with this problem, inspired by this sign spotted outside of Labyrinth in Capitol Hill:

I LOVE Labyrinth, and think this is an excellent way to get the students to start thinking about the connection from two-dimensional shapes to three-dimensional objects.

Later I gave them a task involving toothpicks and marshmallows. I handed them each a measured-out quantity of each, as well as a table with two out of the three qualities in Euler’s Criterion* filled in – the number of edges (toothpicks), faces, and vertices (marshmallows). The class struggled with but eventually succeeded in creating cubes, icosahedra, and dodecahedra with this limited information about what they were supposed to look like. If we hadn’t run out of time I was going to challenge them to create a tetrahemihexahedron.

I was only going to ask them to try it because it would almost definitely be impossible with toothpicks – not all of the edges can be the same length for it to work. I’d love to have them build a net for one on the group part of the final though – definitely more exciting that a rectangular prism.

I’ll probably be back tomorrow; fractals have been very exciting lately, plus I made a sweet quadrilateral flow chart.

*For all convex solids, V – E + F = 2. For the Tetrahemihexahedron it yields 1 instead.

# Zero

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.

# What ARE logarithms, though?

After their exponential growth and decay studies, my Math 3 class moved onto logarithms, a natural progression. Watching ViHart’s How I feel About Logarithms video inspired me, possibly more so than some of her others, so I had them watch it as an introduction. I love that her explanation compares logarithms (impossible!) to basic counting (easy!), along the way making points about the flawed way math is traditionally taught in classrooms. Watching this video, checking for understanding, and clarifying the meaning of logarithms takes a full class period, but I see this as worth it.

I created a companion handout for this film where students are asked to use the logarithmic scale to answer questions, and to interpret the different parts of a logarithmic expression. In their feedback to me this week, most said they had no trouble with the concept of logarithmic scale, but were confused about converting from log to exponent. The most surprising feedback that I received from multiple students is that this math seemed different from the things we had studied previously. To me, logarithms are such a natural sequel to exponentials because each function family is just asking a different question about the same topic. Clearly that connection is the part that is getting lost.

I should pause to say that it’s not just on this week with the complicated-seeming logarithms that I seek feedback. I have the students write a weekly journal, asking them to let me know how their week went in the class, then asking a follow-up about the math. I stole this idea from a teacher I had in high school, and encourage you to steal it from me! This week, they are really shaping my plans moving forward. On Monday we’ll focus on connecting logarithms to exponentials. They have already learned about inverses, and they have explored the concept of function families, so I’ll rely heavily on their prior knowledge.

Last week, just in case the logarithmic scale concept hadn’t been driven home quite yet, we did an activity designed to show logarithms’ connection to real life. I handed each student three slips of paper, each listing some length. These ranged from very large, like the distance from the earth to the moon, to very small, like the radius of a red blood cell. I asked them to put their slips in order, biggest to smallest. Then, I asked them to try to combine their list with a partner’s. This is not always immediately obvious – I wouldn’t have known that the Amazon River was longer than the diameter of the moon. Secretly, I wrote the size of these items in meters (all found here, I should point out) on the back. They flipped the slips over, converted these numbers into scientific notation, then placed them on this giant outdoor log-10 scale!

I’m totally using this again, especially if it’s as beautiful outside that day. I’m resolving to go outside more.