or “Why Two and Two Makes Fish”

Almost out of time yet again (so glad this was timed to coincide with the start of school), but I suppose part of the point is to see if you can handle reflecting as you go during the school year. My choice for week 3:

1. Introduce and show the solution to a math problem that you particularly like.

When I read this prompt, I thought of one of my favorite random teaching tangents (finally worked that in!); I have a few things I try to work in to each class whenever ~~I need to kill some time~~ a student asks a question that I can address with this idea/concept/problem. Unfortunately, these favorite mini-lessons are often not exactly tied to any particular required content, but I’ll have students a year or two later remember these and not much else.

So here’s a somewhat paraphrased, somewhat fictional account of one of my favorite problems and its solution:

Student: I liked math when it was easier; 2 + 2 = 4 is always true … why can’t algebra always work the same way?

Teacher: Well actually, you really need to remember math is completely made up by humans. I mean 2 + 2 = FISH is honestly just as valid if you know what you’re doing …

Student: Stop joking around …

Teacher: Let me show you – but you have to be willing to let me bend the rules and change the meaning of a few things. Math is a game; when you know the rules well enough, you know how to bend, break, or even make up your own rules.

I’m going to almost use normal addition, but you’re limited to combining four symbols: 1, 2, 3, and FISH (Greek alpha). First, I need to tell you that FISH is sort of like zero but not exactly. Also, I’m going to use ‘circle-plus’ since this isn’t quite normal addition …

<scribbling on board>

Student: … um, isn’t that the exact same thing as usual???

Teacher: so far yes, but how can we complete the table without using any new symbols (only FISH, 1, 2, and/or 3), and the table be consistent – it has to make sense

<student suggestions, teacher prompting/questioning, more scribbling>

Student: are you just making this up?

Teacher: I already said I’m just making it up, but it has to make sense! Okay, let’s try another one … with less numbers and multiplication instead of addition … yeah, this should do it … we’re going to use the symbol *i*; there’s a rule that by the way {yes, I can sort of use LaTex}

<scribbling, questions, more scribbling>

Student: Okay, you can play games and move around squiggles on a piece of paper just so … what’s the big deal?

Teacher: You happen to skateboard, right?

Student: Yeah, so?

Teacher: Come here … face the class; this is position zero. Show me a 180° … good, reset then show me a 90° … okay, same thing but 270° … fine, 360° … wait that’s the same as the starting position?

Student: duh

Teacher: Let’s make it interesting then … let’s start build tricks or turns on top of one another … show me a 90**°** followed by another 90° without a reset.

Student: 180° of course

Teacher: Reset, then show me a 180**°** followed by a 270° … you might want to sit there and actually work through the turns.

<student attempts, teacher helps, asks for a few other examples if necessary>

Teacher: So tell me what we just figured out …

Student: Well, you can kinda sorta add angles together but if you get to 360° you start over – it’s the same as 0° in a circle.

Teacher: Great, now consider a square in the coordinate plane … it’s basically like the skateboarding stuff we just discussed?

<scribbling>

Teacher: We’re just adding angles of rotation together, so let’s make a table … you should be getting the hang of this by now … work with a partner

<work, work, work>

Student:

Teacher: Great, notice anything yet?

Student: Not sure … this table starts over like the FISH stuff?

Teacher: Yeah! You’re getting there – can’t we really just think of these angles as multiples of 90, though?

Student: a 180 is two 90s … 270 is three … 360 resets to zero …

Teacher: You’ve just about got it … look at all three tables … color code them if you have to …

<student looks it over>

Student: The pattern is the same on each table isn’t it !?!!?!?!!?

Teacher: Awesome! The relationships are the same even though the ideas seem completely unrelated. On the first one, I used FISH instead of zero, because I wanted you focusing on relationships to get started. This problem demonstrates the basic concept of something called a * group*. You were just working on college-level math by the way – the kind for math majors even. We’re going genius-level in here …

Student: You’re kidding me – that seems easier than what we’re working on now!

Teacher: It turns out once you make it past calculus – math is mostly things you already know how to do but more abstract and even more interested in the ‘why’ than the ‘how’ … too bad so many people get turned off by the tedious calculation bits along the way

Student: This was the best thing we’ve done all year – it makes more sense than a lot of the other stuff …

Teacher: Remind me to show you the one about the empty trashcan, empty bag, and empty bottle not being empty anymore sometime … we should probably get back to factoring quadratics before class is over

Student: Yuck

Teacher: I know, I know … but this is a topic guaranteed to be on the state-sponsored high-stakes exit exam.

[Partially due to internet issues this is now over an hour ‘late’]