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.

Ugh.

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?!

“Learn What You Missed Week” and MI stations

Greetings! I have recently returned from this year’s senior trip. As you can see from my wrist accessories, we went to the Rock and Roll Hall of Fame in Cleveland and the Franklin Institute in Philadelphia.

wrists

We can worry about how cool my tattoo is later. Anyway, before the trip happened we were back in class for 2 days after exams. I will be honest and say that I was not extremely in favor of this decision – I did not feel like I could come up with 2 days worth of engaging lessons on “going over the test.” Ick. Instead of trying to fix every mistake they made on the test and magically make them understand what they hadn’t just one week prior, I created an initially simple-seeming activity.

For each of my classes I identified the three most-missed objectives. In case you are curious, for math 4 they were modular arithmetic, fractals, and summation notation (sigma). For math 3 they were complex number operations, data, and exponential growth and decay. Finally, for math 2 they were factoring, similar figures, and probability. Then for each of the 3 objectives I found or created (usually created) a task for each of the 9 intelligences in Gardner’s theory. Those are: interpersonal, intrapersonal, existential, naturalistic, logical/mathematical, visual/spatial, verbal/linguistic, musical, and kinesthetic.

So that’s 3 classes x 3 objectives x 9 intelligences = 81 tasks. Luckily the art objective for exponential growth also worked for fractals so actually just 80 but still! Luckily again, these went really well, and it was all worth it.

blocks

This was the kinesthetic data task, where they had to throw the paper glob a bunch of times and record the distance, then find the mean, median, etc. I like the looks of that full page of mathematical text – that is not the work of an un-engaged math learner. This also shows the bin of algebra tiles, which I’ve never seriously used but actually really like. I experimented with them for the kinesthetic factoring task, and I think they have potential.

music

I printed the tasks on colored paper and cut them out so that they could just select one and then grab it and do it. Can you slightly see that the big one has music notes on it? I think that particular music task was one of the ones that was a bit of a stretch actually – they had to look at the notes and graph the melody on a complex plane, which probably doesn’t have musical meaning. The best ever musical task was the musical data one. I drew inspiration from this & had them record data about their favorite songs’ danceability, valence, and speechiness.

ghosts

There’s a lot going on in this picture. The picture of ghosts was for a task inspired by this which is very neat. The reason why it is ghosts is probably because of the manipulatives I created to practice multiplying complex numbers. Those are in the blue cup in the front of the picture. They are pretty much algebra tiles, only instead of “x” blocks there are positive and negative ghosts to represent i. The positive ghosts are smiling and the negative ones are frowning. It was cute, initially confusing, and potentially very effective.

I’m done teaching until the fall but I think I will still have plenty of things to write about. We shall see.