The penny finally dropped.
I blogged on March 1 of last year about the apparent inconsistencies between coordinates and tables organized in one order (x, y) and rate/slope in another, as y/x. I played around a bit with writing slope as x/y (which feels more intuitive to students) and found that when graphing, you get the same graph as long as you keep the change in x horizontal. This post is a lengthy update to that one.
In a predominately DI classroom where students are shown How and expected to memorize processes, they tend to not raise questions about Why. I certainly never did. Like others, I might’ve notice the inconsistency and accepted it because, well, math is just confusing and you have to just memorize and follow rules in order to pass a test to get an A and look/feel smart. Many practices in education not only encourage this approach but also award those who can use it well.
I started wondering about the Why of (x, y) vs y/x last year when I started regularly volunteering in an 8th grade classroom. Students were being show a whole lot of How, with plenty of examples to copy, and then given HW to practice. When I heard one student mumble mostly to himself, a little to me, about why you move horizontally first when graphing coordinates and vertically first when determining slope from a graph, I was delighted by him yet frustrated with me because I had no response for him other than I was wondering about that, too.
Which bugged me, so I kept picking at it. I even got brave and posted my musings. I really hoped I would figure it out and get back to that curious student. I let him down.
Fast foward ten months to yesterday. I woke up early, and for some reason, it popped into my head WHY rate is written y to x and not the other way around. Isn’t that weird? I suspect seeing this “Silent Solution” video the previous day on writing rate from a table lit up that part of my brain again.
It was in front of me all along, I already knew this, so it was my Duh moment.
If there is relationship between two things that co-vary, then you can see a pattern and develop generalized equations. The three equations you can write for a proportional relationship are:
Dependent Variable = rate • Independent Variable
IV = DV / Rate
Rate = DV / IV <<<——Whomp, there it is! Why (algebriacly) m = y/x and not the other way around. (I still am leaning toward convention as to the order of coordinates. I does feel more intuitive to list the IV first.)
Students should be empowered to explore ideas like rate and slope in such a way that they do the sense-making. I am confident that they are fully capable of understanding Why, that the How can emerge from them and be owned by them. And it will be glorious.
What do opportunities for student-generated sense-making look like in your classrooms? What do you do to nurture curiousity and facilitate learning? What do you struggle with?
PS. I am also pondering the duality of the way rate is represented. As a ratio, it is a comparison of two values, a quantitative description of a c0-varying relationship. A change of two in This for every change of three in That. We even need two number lines to graphically represent This and That’s relationship. When rate gets put into a proportional or linear equation, it suddenly behaves like a single value, locateable on a single number line: two-thirds. What’s with that?