I don't like math because sometimes it likes to do things that are difficult just for the sake of being difficult. Let's take quadratics for example. Kids are familiar with standard form ax^2+bx+c = 0 and can usually manage to do things like factor them, or throw them through the quadratic formula if necessary. But a more useful form for describing the behavior of this function is to examine the vertex form, or a(x-h)^2+k. It quickly points out the minimum/maximum, can tell you how it's modified from the parent function, and when you go deeper, gives you everything you need to graph it by hand (once you add the concept of the focus). It's also the easiest form to use when you're approximating a model for a quadratic scenario (like I do in class). However, there are some real asinine ways to get there.

completesquare.png

Easy enough when a=1. I have taught it this way and it is a struggle. The simultaneous add/subtract thing isn't a problem, it's the apparent magic trick you do by taking something like x^2+5x+6.25 and reforming it as (x+2.5)^2. When a!=1, it gets a little gross. You have to divide through by a, there's a guarantee you get funky decimals, and then how do you decide to replace a at the end? These are things that you and I, the experienced mathematician can determine. But a 16 year old? OMG the torture. There's no lesson I have more trouble with than conversion to vertex form, even in Pre Cal.

Here's a simpler way:

minmax.png

We're supposed to like vertex form because as the name says, it quickly tells you the vertex. Calculating the vertex of ANY quadratic is much simpler than completing the square. And the value of a doesn't matter. I was skeptical when I first stumbled upon this the other day, but the method is solid. The expansion will always yield the given, regardless of the a value. Showing this to the kids went 1000% better than anything I've done before. It gives me hope for second semester Algebra II.

So what are we trying to prove by doing it the crazy first way? There's no bonus for degree of difficulty people.

Posted
AuthorJonathan Claydon