Vertex Form considered Harmful

A parabola in its natural setting. Notice the air of smugness it gives off.

This is a general audience question: Do you still remember anything about algebra? Do you remember parabolas?

Do you remember specifically how to find the vertex form of a parabolic equation?

I don’t. Or at least, I don’t have it memorized. I’m an engineering and compsci major – I study crap like this all day, every day. But the thing is, it never seemed helpful to memorize long equations. For one, there’s just too many things to memorize. And even then, even if you knew all of the equations, it doesn’t really help you solve problems. You gotta know what to use when where why. So what we study is never the final equations, it’s how to set everything up to find the final equations. That’s the whole point of calculus, algebra, trigonometry, everything really. That’s why they’re called mathematical tools – they’re designed to be used. And hey, sure, there are still a ton of things that we do need to memorize – but the reason I’m writing now is because so many of the equations that you, me, my friends memorized back in high school were completely useless. Let me tell you a story.

I’ve been tutoring for a while. My first regular job was back in 11th grade, as a Kumon assistant. I ‘specialized’ in higher-level math – what that means is that I love higher level math, so any time someone had an advanced math question I would jump to help them. When I go back to Brasil I work as an English teacher with my aunt Katia. And here at college, I’ve been tutoring in the local Oberlin schools as part of a class called Practicum in Tutoring.

This is what me and my friends like to do.

Practicum in Tutoring is a program set up by Booker Peek, a kindly Oberlin professor Emeritus. It is (as the name implies) entirely based on tutoring in the local school system. The premise is a very fair deal: the school system has the chance to bring undergraduate students into classes at no cost, and we receive some credit for tutoring. It seems to be one of the best-intentioned classes offered, regardless of it not being a directly academic pursuit.*[1]

So, today I went to the high school as usual – never sure what to expect, but we work with what life gives us. The boy I was working with was a good student with some questions about parabolas. He was unsure about completing the square, he didn’t remember exactly how to find the ‘vertex form’ of a parabola.

Any time someone mentions completing the square I flash back to 9th grade, and all the feelings of frustration and anger at seemingly arbitrary algebraic rules floods back. I hated those damn exercises, and I never understood why we did them. They seemed pointless but, our teacher said, these will be important in the future, you need to know how to do them. Interestingly enough, the only time that I ever really encounter a ‘complete-the-square’ problem is when a 9th-grader asks me to help them understand how it works.

An Aside: There is an extremely fine line that I’m walking here. Most of the things we did in high school math were and still are extremely important – I still use nearly everything involving parabolas (factoring, solving, finding from points, etc.) today – literally. There are at least four problems on my Classical Mechanics problem set (due Friday!) that use parabolic functions to approximate more complex ones. Parabolas are insanely useful and they gotta be taught. The must be taught. Hell, parabolas might be the most useful function in all of physics (Taylor approximations anyone?). What I’m arguing here is that there are still some select items in the modern math curriculum that I encounter over and over, which seem to be there purely to make kids jump through a hoop. They always frustrate and confuse, and most importantly, they never appear anywhere outside of a textbook. This drives me up the wall, and I want it to stop. Hence, this blog post.

So, Mr. Kid asks about completing the square, and I’m treated to a brief trip through the past few years of my mathematics career – hating the exercise in high school, beginning to understand what was going on in precalc, suddenly having the sky open up in calculus – learning differentiation and integration, which completely changed the way I understand functions and greatly simplified finding the maxima and minima of any function, not just the parabola. Coming to college and taking more advance mathematics, beginning to really admire the beauty and simplicity of these very powerful mathematical tools. This point right now, where we’ve come full-circle from differential equations with no analytic solutions, to turn around and say “Hey okay, so we can’t actually solve this equation. But we still need an answer. So what to we do? We approximate. What do we approximate with? A parabola.”*[2]

Taylor Approximation of sin(x) at Pi/2. This should make you blush.*[6]

I snap back. Okay, I think, completing the square is reasonable. I don’t really remember using it, but I know there are some rare cases when it could be done and it’s not such a bad trick. Though, I don’t know why you wouldn’t just use the quadratic formula. Anyway, we do some exercises, trying very very hard to be honest and clear about all these things. I write out the steps as we go, in english and in math-lish.*[4] He’s nodding, he’s answering my questions, but I can see that he still doesn’t really get it – he hasn’t grokked it.

The whole time I’m thinking about a TED talk that I watched, of a man who describes the same sort of situation in his Biology class. We’re doing this the wrong way around – this kid shouldn’t be learning to complete the square, I should be learning to complete the square. As something that’s far more likely to appear in an undergraduate physics course, I would gladly take an hour of my day to learn this and then immediately apply it to a problem. But to this kid, it’s just one more unnecessary procedure that he doesn’t really get and isn’t going to use any time soon. I’ll use the example from the TED talk:

“If a young learner thinks that all viruses have DNA, that’s not going to ruin their chances of success in science. But if a young learner can’t understand anything in science and learns to hate it, [then] that will.” – Tyler DeWitt

If a 9th grader isn’t taught how to complete the square of a parabola or put it in vertex form, it’s not the end of her math career. Sure, it’s good stuff to see and know it exists. But if she’s continually frustrated by inane formulas, it will end her math career. This needs to stop.

So we work through some examples and I can see he’s gotten a better idea of what’s up. Enter vertex form of a parabola.

Vertex form of the parabola is the following:

$f(x) = a(x-h)^2+k$, where h is the x-coordinate of the center and k is the height of the parabola at its center.

And to top it off, these kids need to complete the square to transfer between standard form and vertex form. I did some cursory googling/youtubing (as I was writing this post, not as I was tutoring) to try and find a motive behind this and the best I got was “so you can see where the center of the parabola is”.

Really. So the expectation is that kids are going to memorize this useless piece of shit transform on top of the standard-form equations? And while we’re at that, who the fuck chose these variables? Why the fuck would you choose the letter ‘h‘ (as in height) to represent the x-coordinate of the center? Is this a joke? Is the goal to siphon off the kids who have trouble memorizing arbitrary variables?*[5]

In other words, What the fuck?!

So I look at this shitty ass textbook and see a greasy-haired motherfucker of an example (Example 7). Take a minute to follow that link and read the example.

Now is the time to act. This kid doesn’t need to learn this shit. Why? Because it is literally shit. I have very little experience compared to the world, but I’ve been in this kid’s shoes and I know exactly where this worksheet is going after the teacher hands it back. No, this kid doesn’t need to learn the vertex form of a parabola. He has a budding interest in math, and that needs to be fostered. If I try to reinforce this textbook, I risk squashing the little flame that many of my friends had back in 7th grade. We hear so many complaints about algebra in schools and the ‘necessity of math’, whether or not to cut it, reduce the passing grades, anything to appear more competent as a nation.*[3] But then at the very same time, we’ve got the most boring curriculum there could be. How can anyone have fun learning this crap? Here is this boy who could go on to become a great mathematician. He could love math as my friends, my professors, myself love math. He could feel the joy! I’m sitting here, next to the future of mathematics. How could I dare to look at him and say that he’s going to need to know this? No. What am I going to do? I’m going to show him the easy way. You know, the way people actually solve this problem in the real world:

We’ve got this shitty parabola: $y = -16t^2+96t+3$

Notice that the first number is negative, which means the parabola opens toward the negative y-axis. Negative in font $\rightarrow$ think negative.

Now, to solve parabolas there is only one equation you need to know. The quadratic formula. An ancient gift from the Babylonians, being passed on to you. Like all best friends, it’s kinda awkward looking when you first meet it, but you grow on each other and eventually realize that you’re inseparable. Why? Because this thing will solve any standard-form parabola anywhere no matter what. Complex roots? No problem. Can’t factor? NBD. Don’t give a fuck about shitty textbook exercises and want to do something productive with your life? Write the thing on your arm. Fuck that, get a notepad and keep it on the inside cover. Write a program on your calculator to do it for you. Don’t stress about memorizing, just let the familiarity flow into you. Next time you see a parabola you should blush or somethin’, cuz that shit is already solved.

It loves you

$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

That’s cute. Notice that the whole thing is explicit – it’s the solution of $ax^2+bx+c=0$ for x. You put in the numbers, it will give you the two x solutions to that parabola (called the roots of the parabola). I’m gonna rewrite this a little different for convenience:

$x=\frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}$

Coming back to our greasy piece-of shit example. We want to find the center of the parabola. What? Parabola? We solved that already.

$x=\frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}$

But wait. Look at that equation. Do you see how there’s

$x=\frac{-b}{2a}$

and then

$\pm\frac{\sqrt{b^2-4ac}}{2a}$

That’s because the roots of the parabola are both the same distance from the center of the parabola. As in, the center of the parabola is the first part of the quadratic equation.

$x=\frac{-b}{2a}$

Which means that we can solve this example in two steps:

Find the center of the parabola:

$\left. t_{center}=\frac{-b}{2a} \right|_{a=-16;b=96}\rightarrow t_{center}=3$

So we have (3, ?). What’s the height?

Plug t into the original equation to find the height:

$\left. y_{max}=-16t^2+96t+3 \right|_{t_{center}=3} \rightarrow y_{max}=147$

Write the answer and go do something else:

The vetex of the parabola is at (3, 147).

Come on guys, we gotta cut the shit. You want kids to love what they’re learning? Show them the easy ways to do math. Let them mess around with Mathematica and write some program that crashes their computer. Let them have fun! Recognize that there’s a place for these methods, but with a different audience. Save vertex form and completing the square for the nerdy kids – and wait until you’re really gonna need it. We love this kind of thing. Pulling out something like this in the middle of a twisty proof could be as sexy as using Euler angles to compute vector rotations. But for the kids who’re still getting grips on the reigns, give them time. Give them space. Show them the nifty tricks early, and build off of those. Hell, you could teach any 7th grader the power rule and they’d be finding the critical points of a polynomial in no time – no formulas needed. Let’s recognize the good and the bad, and adapt. Move out of the dark ages. Let’s start teaching fun.

Also, If I ever find the person that wrote this example I’m gonna kick their teeth in.

__

*[1] This is one of those commonly held ideas – but I don’t think Practicum is really a filler class. In teaching anything you walk out with a better understanding of the student, yourself, and especially the material. I find it to be a nobler fulfillment of my social science credits than pretending to care about an Intro to Anthropology class – but that’s just me. If I need to have social science credit to graduate, I don’t want them to be wasted on material that doesn’t interest me.

*[2] Or, if we’re in compsci mode, we toss all of that out the window and numerically solve the thing in Mathematica. Who cares about analytic functions when you have a numeric ODE solver?

*[3] It is necessary. Do not think otherwise. The problem is not math, it’s that the teachers don’t have room to do anything with the administrators, school board, PTA, politicians, etc. constantly breathing down their backs and changing the rules every other month. Stop fucking with the teachers. You want kids to learn? Put them in schools with teachers and then go do something else. Don’t tell the teachers how to do their job.

*[4] If you want to know, here’s how completing the square works:

$ax^2 + bx + c = 0$

For some reason, we’ve got to solve this thing by completing the square. I don’t know why. What’s the first thing we do? We divide by a:

$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

Now we move anything without an x to the other side:

$x^2 + \frac{b}{a}x = -\frac{c}{a}$

This is the sneaky bit. We need to figure out something to add to both sides so that the left-hand side can be factored into something like $(x + h)^2$ for some dummy variable h. Well, that’s not bad. All that we need to do is look at how that multiplies out:

$(x + h)^2 = x^2+2h+h^2$

To get from $2h$ to $h^2$, we divide h by 2 and square it. This is the key. But h is just a placeholder for whatever’s in front of x in the original quadratic equation, which in this case is $\frac{b}{a}$.

So we’re going to add $(\frac{b/a}{2})^2$ to both sides:

$x^2 + \frac{b}{a}x + (\frac{b/a}{2})^2 = -\frac{c}{a} + (\frac{b/a}{2})^2$

It appears that this equation can be factored. We’re done.

$x^2 + \frac{b}{a}x + (\frac{b/a}{2})^2 \rightarrow (x+\frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b/a}{2})^2 = \frac{b^2-4ac}{4a^2}$ which is just the discriminant-side of the quadratic formula, squared.

Useless technique.

*[5] While we’re on the topic of bad variable choices:

Mathematicians, Physicists,

What the is going on with the spherical coordinate systems? Why are Phi and Theta reversed between two entire disciplines?

*[6]

I done a math.

3 thoughts on “Vertex Form considered Harmful”

1. Kotchetkoff says:

Great man! I also I’m a tutor and this is exactly what I think!

2. Pingback: Experiments in Coding | Kip Price