Wednesday, 13 March 2013

Variables May Vary - Part 1


Using variables in math can be a tricky business. In my opinion, two aspects of this have the potential to be especially problematic. Those would be: (1) The fact that they have different uses, and (2) The fact that certain 'letters' get special traits attached to them (which can then be hard to break).

I've decided to consider each issue in a separate entry, because, I don't know, is a few short posts better than one long?


USING THE VARIABLE: CONTEXT IS IMPORTANT


VARIABLE MEANING


That's a double entendre. Let me grab the most standard example, the equation of a line: y = mx + b. It's full of variables... but their meaning is different.
I'M A VARIABLE.
DEAL WITH IT.
y : dependent variable
m : slope
x : independent variable
b : y-intercept

Notably, some variables are used as variables, taking on ANY value, while others just stand in for specific values we don't know yet. So if you want the equation of a line, you find the slope and y-intercept and write y = 2x + 1.  You know, as opposed to choosing a point on the line and writing 7 = m3 + b.

So why don't we do it the second way?  Oh, wait, we DO, when we know one of slope or intercept and want to find the other... yeah... but then the key variables revert back to x and y in the end.  Easy, right?

The issue is that x and y are VARIABLE variables, the values we're actually studying. Meanwhile m and b are CONSTANT variables, placeholders until we figure out what constant value matches the situation we're actually studying.

Of course, they're placeholders with a meaning on the graph. That makes them different from a variable in an equation, like 2x + 1 = 7, where x is now the CONSTANT of 3; that's the only value which makes it work. (Yes, it relates to intersection of two lines, my "point" is that x is not seen to be varying in this context.)

Starting to see the distinction?  Let's go one step further.


AFTER THE LINE



Contrast those situations with the parabola equation a year later, y = ax2 + bx + c, where now c is the y-intercept, a is the leading coefficient or stretch factor (not the slope, where are you getting those crazy ideas?), and b... well, b is a CONSTANT variable, like a and c, yet it doesn't really have any specific graphical meaning. (Yes, it relates to the axis of symmetry, but it's not itself the axis of symmetry.)

You can't solve for b either, not unless you're given more information. But more than that, to solve for the x in a parabola means setting the equation equal to zero - instead of setting the equation equal to x, as was the case before! In one year, the rules for solving equations have changed!

Confused yet?

If not, it probably means you're able to free associate meanings to 'variables' depending on context. But if so, you're probably in good company with students who can't figure out when they're supposed to be solving or interchanging values for so-called 'variable' letters, and when they're supposed to actually stay, you know, VARIABLE.

You'll also notice that the meaning of 'b' changed from line to parabola. That's the lead in to Part 2. I fear I'll be ranting a bit more in that entry.

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