## Thursday, 14 March 2013

### Variables May Vary - Part 2

Happy Pi Day 2013! That has nothing to do with this post.

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.

### DYSFUNCTION NOTATION

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
Now we're into y=af(b[x-c])+d. Or in some cases y=af(k[x-p])+q which is simultaneously slightly better, and slightly worse. (It's definitely better than bx+c without the extra bracket.) This is the stupidest thing EVER, if you ask me.

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.

1. Math needs at least three separate alphabets, one for variables, one for placeholders of constants, one for defined constants, like e (Euler's constant) and i (square root of negative one). I can understand needing to avoid confusion, thus not touching i and j (both used for imaginary numbers), but that leaves 24 other letters in use.

Would it help to designate placeholders of constants using capital letters? The only place I can see that causing an issue is determining the period (T) of a cycle.

As an example, I'll take your classic y=ax^2 + bx +c and apply the above changes to get y=Ax^2 + Bx +C. Does that make it clearer?

In programming, you get to declare the variables at the beginning. (This is where good commenting helps, incidentally.) You can declare the user-supplied parameters A, B, and C as apple, bark, and congaline for all the computer cares, as long as the documentation explains what each one is for. This changes the above equation to f(x)=apple*x*x + bark*x + congaline. (Okay, not so clear, but it splits the variables and the constants a little.)

Maybe math needs a new grammar when it comes to letter use?

1. Nice try. Capitals are taken. Ax + By = C is the equation of a line in standard form, where A, B and C are arbitrary (albeit related to slope and an item total). Different from y = mx + b. Similarly, Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is the general form of a conic. Probably don't want to cross the streams there. (Don'tcha remember high school math? ;) )

Incidentally, where is j used as an imaginary number? Programming? I figured a case of 'looked too similar to i'. (Now, I know i, j and k are all used as unit vectors in three space. But then they have the little hats.) I will buy that computer programming has the advantage, since your variables are allowed to be more than one letter in length. More than one letter in math means invisible multiplication.

Of course, all that said, there is another alphabet available -- the greek one, and I've seen use of those symbols for transformations on the site of a UK teacher. Except from what I know of North America, the convention tends to be to use those to refer to angles only.

2. So, capitals are out, and there's still too much cross-use of lower case letters, unless there's an unwritten rule about using the letters at the end of the alphabet for real variables (x, y, and z, though one can argue that those are based on the axes on graphs) and letters at the beginning for the placeholders for constants. (Not really. It got buried by university math where Greek letters became the norm.)

The use of j as sqrt(-1) comes up in electrical engineering, for no reason I could discern at the time. It could be from the mix of i and j in some typeset. (What if you replaced the hats with berets?) It's an advantage in programming, and widely encouraged, even for counters (though you could get away with i, j, and k there. In programming, multiplication has to be explicit because the compiler isn't robust enough to tell the difference between IAmAVariable and xy (x * y, for readability here).

The Greek letters start getting used for equations in university, especially with angles. It's a case where you might not want to add a layer of confusion later. And capital Greek letters tend to be used for new types of mathematical manipulations, such as the Greek capital sigma for the sum of a series.

What if you used Japanese kana? Or letters from a made-up alphabet like Klingon or Aurabesh (http://starwars.wikia.com/wiki/Aurebesh)?