Sunday, 11 October 2015

Vertex Form and Sucking

To be clear, this post will not say the vertex form of the parabola sucks. Namely because it doesn’t (it’s the y=mx+b of the quadratics world). Instead, it will discuss how teaching an aspect of it sucks, why squares suck, and why sometimes I feel like I suck.



I was at a professional development session on a Thursday night (October 1st, put on by our local math organization) which was quite good. One of the main ideas was that algorithms don’t equal engagement or understanding. It’s better to make things more visual, and to leverage student ideas - to move from a task towards the abstract, rather than hoping an abstract procedure will allow time for problem solving later on.

And what did I do the very next day? I went into my class on Friday and showed them a couple algorithms for turning standard form of a parabola into vertex form. The disconnect could not be more obvious. Particularly when it kind of blew up in my face.


MISSING THE POINT?


Let's first cast an eye towards reality, starting with “real world” explanations for why I went algorithmic right out of the gate. Because if you’re a teacher, I’m sure you’ve been there too, thinking “I really should do ‘x’, yet I’m not”. I’ll leave it to you to decide (in the comments?) if my “excuses” are valid.


Expanding is a thing...
One of the main reasons is that I wasn’t going to be there on Monday. (Due to extra curriculars on my part, there would be a substitute.) So while I could have done expanding first, and moved on to show how vertex form creates that literal square next... I wouldn’t be there for that day of going backwards. And it didn’t seem fair to leave that with a substitute, since it could have blown up in their face instead of mine.

That’s an excuse because maybe I could have found something else in the interim, and followed up on Tuesday.

The second reason is because, honestly, “completing the square” ONLY has a use in high school at THIS particular instant in time. (We don’t do conics in Ontario aside from polynomial parabolas.) A majority of these students aren’t going to see (or need) to do this sort of thing again, so why explore it? Plus there’s a work-around for it, namely that the axis of symmetry is -b/2a.

That’s an excuse because maybe there’s some reasoning coming out of this (particularly as related to algebra tiles below) that could be useful.

Finally, there’s the obvious point that I didn’t really have time to adjust my whole timetable for the unit in a single evening. Add to that the fact that I am really, REALLY not a task guy, and, well, we got what we got. Which is an excuse because I think I had the pieces for something there, if only I’d put a bit more thought into it.

It’s a matter of convincing the students to make a square, rather than use all the pieces. Right? Well, here's what DID happen:


ALGEBRA TILES


I may not be a fan of tasks, but I DO like concrete models. Hence I am a fan of algebra tiles. For the uninitiated, here’s a summary of how the various models of a quadratic work:
A tile is named by it's area. Positive is red, negative blue.

So here’s where I try to redeem myself, in that I was not merely showing the algorithm, I was walking through it in the broader context. (And if you want to go 3D with this sort of idea, check out Al Overwijk.) Granted, the next logical step would be playing around BEFORE the algorithm, not DURING it.

Here’s the interesting piece about the situation which drove me to blog, which is something to bear in mind if you want to try such an activity yourself: The symmetry of squares is a pain in the ass.

The first example I used included x^2 + 2x, and to complete the square you need a single unit chip. (x^2 + 2x + 1) Minds were blown (possibly because I had an ‘a’ value one step before, derp). Then I offered up x^2 + 4x, and again we need to complete the square, and I circulated around a little.

The interesting bit is students did not make the square (x+2)(x+2). They made the square (1+x+1)(1+x+1), in other words, the tiles you see on the LOWER part of the image below, not the UPPER part. And this was two students, completely independently of each other (while most of the other students weren’t using the tiles).


Huh. Squares suck.

I completely get why they did that. And it didn’t disrupt from the fact that you need +4 and -4 for completeness. The question is, where does one go with that?

Should x^2 + 2x have been completed by putting HALF a tile on each side? Eventually, they will have to deal with fractional tiles... but it’s a little trickier to complete the square that way (four one quarter components). Or is it?

On the flip side, it’s important to recognize that the upper model is also a square, because you’re extending in the two dimensions you have... you don’t need to extend FOUR directions, a square is not a four dimensional shape. Or is that important?

These are obviously questions best asked of the individual students, but this was 2 people out of a class of 25, and I still had to show partial factoring for the ones in the room who were curled up in their chairs, weeping quietly. (Hyperbole!) So I moved on, in part because I couldn’t see how extending this squares idea to 23 other people would be more than added frustration. If the others weren’t even making squares, how would different squares be helpful?

It did gave me something to think about though. Maybe it's given you something to think about. If not, here's one last item to ponder:


METAPHORS


In my mind, a lot of the individual exploration being used in classes these days works best with, well, individuals. Is groups really a way to share opinions? Or are they more a way to force deciding on a "best" method? Even with “open questions” (like ‘what do you notice?’) you ultimately have to narrow the focus of the class, and honestly? I hate that. The good of the many can outweigh the good of the few. Or the one.

Which educators can also do to themselves. Making me feel like my thinking sucks.


When vertex meets factored.
To wit, I have a song I parodied about vertex form. It’s one of my favourites, it’s call-and-response, I do it as a wrap-up... and I have at least one student who recorded it and showed it to his parents (yeah, um, okay then). And YET what seems to be a more and more common refrain at the Professional Development I go to? Educators who say: “Yeah, you can toss in a silly song, but that’s not learning, here’s a better way.”

Okay, yeah, but songs are my way, so... so I guess I'll be over here... apparently not helping, only entertaining...

Oh sure, I can talk to myself about “You’re a Good Teacher”, and don’t take things personally, and the speaker means silly raps not what I’m doing (right?)... but even though I KNOW songs aren’t valued by the mainstream educating community, does the mainstream REALLY have to keep poking at it to get a chuckle out of the majority of the crowd?

I guess what I’m saying is, one size doesn’t fit all, even in the adult world. Which I saw in a class setting while doing something as simple for me as completing the square.

WrapUp: Despite my hesitation about activities, if I WERE to create one around “making vertex form”, maybe I’d need to use values divisible by four to start. Then from there, see where it goes. Though maybe expanding first would drum that “four sides to square” thing out of students? (The x squared is always in the upper left corner for expanding, and they really, REALLY want to use those unit tiles in their tile charts, even though they’re unneeded for completing the square.)

I'm curious for more thoughts on any of this.

For further vertex reading: Variables May Vary
For further math viewing: My Math Webcomic
For more about my mental states: The Fringe of Depression

2 comments:

  1. As I mentioned on Twitter, I have a beautiful, metaphor rich, meaningful way of teaching complete the square with algebra tiles, and the students forget it the next day. I do teach the concept, but I no longer do it with algebra tiles. I wish I knew why the kids promptly forget it.

    But vertex form is equivalent to point slope, not y=mx+b.

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    1. Figured I'd wait for the Twitter storm to die down before recapping here. My offhand remark about "vertex form" being the "slope-intercept" form of the quadratics world was mostly geared towards it's importance -- after all, who remembers "standard form" of a line? No, it's usually "y=mx+b" that students know, and in my mind, one should also remember "vertex form" above the "standard" for quadratics.

      My logic is as follows: Vertex form is needed to give the range, and the max/min. You can see the transformations, and the axis of symmetry. You can easily solve for the zeroes, and in a more intuitive way (solving for x, no messy formula to remember). Meanwhile, what does standard form give? Uh, well, the "y-intercept" is "c"... but set x=0 and you find that pretty quick for both. There was an argument that you need it to get from factored to vertex, but no, you really don't. Standard form is, in my mind, a real pain, not unlike standard form of a line. (Which I know has uses! I wrote a song about it, after all.)

      Now, it has since been pointed out to me that the quadratic "standard form" has applications in projectile motion, namely "c" as initial position, "b" as velocity, and half "a" as acceleration. (Which would be why most word problems use -5, or -4.9 there.) Okay. Valid point. I will grant it's more useful than I gave it credit for. (But vertex still gets you a graph faster.)

      As to the algebra tiles, yeah, I wish I knew why that didn't stick either.

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