[There’s a lengthy update to this post, published 1/5/17.]
Huston, we have an inconsistency.
In a slope ratio, the dependent variable is always compared to the independent, y:x or more commonly, y/x. If you’re making a right triangle on a graph to determine the slope, this y to x comparison needs to remain intact: changes in y are vertical: changes in x are horizontal. You can train kiddos to always move vertically first to draw one leg of the triangle, then horizontally, always to the right. Yes, I know this is a trick to figure out the right slope, but bear with me, I’m trying to get to understanding.
You can also train your students to create a table that lists the independent variable in first column, dependent in the second. Very easy to then grab the ordered pairs straight from the table and plot those points. The independent variable axis is always horizontal, you tell them. The first value in the ordered pair is the independent variable, which means to first move (from zero) horizontally. Followed by the 2nd value, and the 2nd move, which is vertical. This gets confusing to students because for plotting points, its horizontal, vertical, but graphing slope is vertical, horizontal.
If you are lucky, they wonder why.
I’ve been wondering why for a long time, too, and I’m stuck. I figure this is a safe place to admit that. What do you say to a curious student who is trying to keep track of (x,y) and y/x and asks, “Why is it different?” I want to be able to give them a better answer than, “Its convention.” I really do want to know why the ratio is written dependent/independent, and not the other way around. It’s inconsistent.
Here’s my thinking so far: For the sake of
sanity argument, let’s agree that the placement of independent and dependent variables on the X and Y axes, respectively, are convention. Also, that slope is a ratio that describes the steepness of a line; comparing point A to point B, point B is both higher and to the right of point A.
If a table lists the coordinates in this order (ind. var., dep. var.), then plotting points on the graph requires that same order, using the appropriate axis. First distance from zero horizontally, then distance from zero vertically. As long as a person understands which variable is which, and which variable goes with which axis, then the table could be reversed (dep. var., ind. var.) and the plotting would be accurate as long as the same order is followed: first vertical distance from zero, then horizontal distance. The key is consistency with the order. Rather than memorizing convention, you need to apply an understanding of the co-variance of independent and dependent variables and make sense of horizontal and vertical changes. Perhaps it is just convention (you tell me!) that guides us to always list the independent variable first in tables and coordinates.
Yet when I experiment with the order within the ratio, inverting it to x/y, I get the same line. Example: Suppose y/x = 3/2. That means x/y = 2/3. These look like completely different slopes. But they are not, because both the values and the variables are inverted. So, to correctly graph a y/x = 3/2, even if I have inverted it and written x/y = 2/3, everything will be super groovy AS LONG AS I understand which variable is which and which variable gets a vertical or horizontal move. Moving first horizontally a distance of 2, then vertically a distance of 3, creates the same slope as before. Huh. The x/y ratio fits nicely with (x,y).
So why do we use y/x instead? Does it all really boil down to convention? I’m sure there are people who can explain the reasoning for all this to me– please do, it’s bugging me. What do you say to your students when they notice the inconsistencies? What lessons and activities do you use to get at the heart of this?
UPDATE: Do we use y/x instead of x/y because we want to associate a higher ratio with a steeper line? That only works for y/x.