A Conceptual intro to Imaginaries

I feel like every single year I put in my first-draft sketchy lesson plans “conceptual intro to imaginary numbers.” Then I google that phrase and find a bunch of boring-seeming articles and dumbed-down explanations, stare at them really hard, and then end up just doing what I did the year before.


This year I actually came up with something, finally! We had a nice long discussion about the history of number systems, guided by this powerpoint:

Number Systems

This just goes through the different number systems in the order they were devised. It starts with counting numbers, and I asked them why people originally needed this type of number.

The next set of numbers is the natural numbers, which are the same just with 0 added in. Here I just told them that adding zero was a huge deal because humans used to think of it as useless but modern mathematics doesn’t function without it. We do a ton with zero later on for asymptotes, and my [totally realistic] hope is they will be remembering this moment in suspense!

The students correctly predicted that the next set was the integers, with the edition of negative numbers. I asked them why we have negative numbers and several of them independently came up with debt. I told them that one of the first uses was for pyramids, because they had structure above and below ground – this is a spoiler for that Crest of the Peacock chapter on 0. Here we started talking about infinity – the natural numbers are already infinite, because theoretically one can count forever. With negatives, the numbers are now infinite in both directions. Whoa.

They also saw rational numbers coming. I like to tie these into Mitch Hedberg’s joke about the 2-in-one shampoo/conditioner bottles, which he didn’t get since 2 could never fit in 1. Before rational numbers, there was no way to represent 1 divided by 2, and humans’ ancient minds were as boggled as Mitch Hedberg’s. I pointed out that with rational numbers, there are now an infinite amount of numbers BETWEEN all the whole numbers – embedded infinity! My brilliant and autistic student piped up here that of course, there are multiple levels of infinity, he already fully understood that concept. This did nothing to draw in my two math anxiety girls.

For irrational numbers, a term they remembered, I tied in Pythagoras’ cult, and the original discovery that the square root of 2 is irrational. I forget where I read this story but I assume somewhere reliable. Pythagoras, and therefore all of his followers, believed that every number could be expressed as a fraction. This one time, they were all on a boat, and one of the math guys was like “Pythagoras, I reallllllly don’t think that the square root of 2 can be expressed rationally,” demonstrating using a right triangle and Pythagoras’ own Theorem. This was unacceptable, and the cult threw him overboard. But now we all believe in irrational numbers (for the most part), and most of the students remembered pi as being labeled as one.

Oh also, we’d been talking about how the natural numbers are closed under addition, but not subtraction, so we had to add the integers to account for stuff like 4 minus 7. Then the integers are closed under multiplication but not division, so we had to have rational numbers to account for stuff like 1 divided by 2. The first example of an irrational number was the square root of a non-square number. I was like “so is there anything we can’t take the square root of even still?” and they were like “uhhhhhhhhhhhhh” and I was like “OK I’ll just tell you! It’s negative numbers! i is what you get when you take the square root of something negative!” I guess I could have handled that better?

As we were working with imaginary operations like multiplication and addition, I kept finding myself drawing out some ghosts to represent these imaginary numbers:

imaginary numbers

I started doing this last year. This is such an abstract concept that they sometimes feel like all bets are off when it came to things like how addition works. The imaginary number i was so associated with negatives that they would assume all imaginary parts should be written negative. They also sometimes want 2 imaginary numbers multiplied to still have i. I tried to come up with a metaphor – ghosts are pretty transparent, but when 2 of them multiply/ overlap you can see them, they’re just negative. This doesn’t really need a gimmick, though, since they just need to know that i * i = -1, the central concept. Drawing out the ghosts helped them to see that 2i + 3i would just be 5i, since you have 2 imaginary things and then 3 more imaginary things.

For the journal this week I had them write about imaginary/ totally constructed things that serve a purpose. They came up with a huge variety of things, from Santa Claus to the value of money to “basically everything.” These [obviously] got really philosophical, which I appreciated. One student, actually the one who talked about levels of infinity, said “imaginary numbers are extremely real, they just can’t be physically counted,” which I like a lot. I wanted to push back against the idea that these constructed things are purposeless since they aren’t “real.” Like what even IS real?!

Grade Narratives

At HGS we pride ourselves in truly knowing our students as people and as learners. At the end of both semesters, we do provide traditional grades, like A’s and C’s and the like. For all four quarters we also write narratives. I’m right in the middle of this process right now and not entirely eager to finish, so I though I’d share what types of observations I’m making in these write-ups.

Everyone at school strives to write these narratives using the “Oreo cookie” method of delivering bad news. First you say something positive, then whatever negative thing you need to communicate, then something else positive. It’s not exactly the best metaphor, because the white cream part of the Oreo is delicious, but it’s easy to visualize. I try really hard to make these narratives as objective and descriptive as I can, mainly noting observations about a student’s affect and behavior in class. I feel a little bit icky connecting information about behavior to an academic grade, but often it’s a great way to explain why a student was successful or struggled. Noting behavior doesn’t necessarily have to be about discipline – mainly I note things like asking questions or working smoothly in a group that will naturally help them do well in class.

I also try to be as specific as possible about their struggles and successes. Once I experimented with including test scores on particular objectives, but I didn’t like it laid out like that. If a student didn’t do very well on similar triangle proofs, it will tell them and their parents a lot more if I say “Little Jimmy struggles at times with communicating his ideas, but has a solid grasp of the mathematical processes involved in proving similar triangles,” than Similar Triangles: 44. The kids, however, know how they did on each assessment, and they are always given the opportunity to fix mistakes like that.

Here are some examples of narratives, obviously with names changed, in this case to the names of my cats:

Stanley is still doing very well in the class. He continues to take ownership over his own learning and asking as many questions as he needs. Stanley still prefers to work by himself rather than with his classmates, but he has been willing to try this quarter. In the past, Stanley has seemed intimidated by language-heavy math, but this seems to be improving. He seemed to respond well to the math words task, because although it was based on words instead of numbers, the task was asking him to discern a pattern. He also did very well with our work on fractals, where students were required to explain patterns in both algebraic expressions and words. Stanley can expect continued success in this class.

Olive is still doing well in the class. She has an excellent grasp of not just the mathematical concepts but their connection to the real world. She started the probability unit with lots of solid prior knowledge and was always willing to help her classmates, and to challenge herself to extend her knowledge. Her biggest challenge is still her self-confidence. While she is usually one of the first students to understand new material, she can tend to second-guess her answers. Moving into the end of the year, Olive has every reason to feel confident about herself as a math learner, and can expect continued success if she gains this confidence.

I wonder how I would have reacted to reading such thorough assessments of my learning in high school. It probably would have freaked me out, but maybe ultimately helped me as I moved up through college math.


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.