You may recall last time I remarked on how 'x' is a VARYING variable that can be anything, while 'm' is more of a CONSTANT variable, in that slope is a ratio which we simply may not have figured out yet. On to my second issue, variables with 'special traits'. This contains a personal pet peeve, which for all I know is unique to the Ontario educational system. But we'll get to that in a moment.
USING CONSTANT VARIABLES
The first thing I have to address is whether you, dear reader, even use y = mx + b, or whether you use: y = ax + b (like in France, or on the TI calculators), or if it's y = mx + c (as in the UK), or y = mx + d (Japan) or perhaps y = kx + m (Sweden). I'm getting that information from this site, as sent in by readers.
|ALSO DO YOU USE A PC OR... WAIT...|
Seems the world is actually pretty standard on the VARIABLE variables (Cartesian axes), but on the CONSTANT variables, we're all over the place. Which again is not a problem if you're free associating the meaning of variables. However, first picture that as something you struggle with, and now look at the following equations (all seen in North America):
x = a; y = ax + b; y = ax2 + bx + c... and now ask yourself again why students keep saying parabolas have slope.
Though my personal pet peeve is how variables are used in function transformations.
In Grade 10, we teach vertex form of a parabola is y = a(x-h)2+k. Where (h, k) is the vertex. WHY? I have no flipping idea, but it's all over the internet, so not just Ontario. Best guess 'h' is for horizontal, then 'k' because... 'i' and 'j' were taken? It's RIDICULOUS. Why? Because in Grade 11, we have function transformations. But do we teach y = af(g[x-h])+k? NO!
|HORIZONTAL HAIR COMPRESSION|
Because given a parabola, while 'a' has the same context, suddenly 'b' isn't an intercept or connected to the axis of symmetry, it's a horizontal compression - and the vertex is now (c, d)!!
Now, am I saying the variable must always have the same context? Of course not. It's important to understand how meaning can change based on situations. But why oh WHY do we teach (h, k) when we NEVER use that context AGAIN? It's like putting someone on the edge of a rug just to see if they fall when we yank it out from under them. (You didn't fall? Great, you understand function notation.)
Now, y=af(k[x-p])+q is slightly better, and is the one I use. It means I can use 'p' as a phase shift come trig, and 'q' as an asymptote come exponentials, while 'a' of course has the same context it always does, and we don't run into letter confusions with cubics, y=ax3+bx2+cx+d. Even the 'k' would be pretty good (and is what my textbook uses, incidentally) - if it weren't for (h, k) having been vertex form!! (Which may be my real pet peeve: 'k' is used to represent unknowns a lot more than 'q', so why the devil don't we use something sensible for parabolas like y=a(x-p)2+q...? AGH.)
This was really brought home to me this term, since our lower level level students don't learn function notation. So they keep using y = a(x-h)2+k in Grade 11. Meaning I have to change the posted formulas, depending on the period, to be consistent!
CALM BLUE OCEAN
Right. Sorry about that. (Is it just me? Tell me it's not just me.)
My point though is that switching up variables can be REALLY problematic for someone who's weak at maths. I know of someone in my family who studied for a math test, and figured they had it. Then they went into the exam, and instead of (x, y, z) were confronted with (a, b, c) and suddenly had no idea what they were doing anymore. (I'm generalizing slightly, but you get the idea.)
Variables may vary. Or they may be constants we don't know. But depending on CONTEXT, they may not be the constants we're looking for, or even the ones we remember from last year.
This is an area where people can struggle, and I think we at least need to be aware of it. Though, seriously, someone needs to get on changing - or explaining - that (h, k) thing.