At one point during my first class, I want to drive home a point. You guys know a lot, and I want this course to help it all hang together.
I’m going to ask them to spend a few minutes minutes solving the problem:
Then I’m going to go around and have students explain how they got their answer, why they think they’re answer is the answer, what they know about the question, whatever.
Who wants to bet that 100% of them graph it on their calculators or use the quadratic formula?
I’m counting on it. I will then show them an example of their graphing calculator lying to them (there are a million of ’em), and then say “why does the quadratic formula work? why are we allowed to use it?”
We’ll then take a moment and say “what do we need to know to solve this problem? I’ll start by throwing something out: we have to know what a variable is.”
So we’ll throw a bunch of things on the board: variable, number, exponent, addition, square roots, etc.
Then I’m going to reveal the big secret to mathematics, the secret that all teachers have kept from them until now: all we’re doing to solve problems is to turn something we don’t know into something we do know.
This is the rant I have playing in my head…
So when we first learned quadratics we didn’t know how to solve ’em. We had only seen baby linear equations. But guess what? When we learned to solve quadratics two years ago, we turned these horrible grossities (quadratics) into beautiful nice-ities (linear equations). Watch!
I didn’t know how to solve the quadratic, but I do know how to solve (two) linear equations!
This procedure is completing the square. I know y’all remember it — vaguely. I know y’all hated doing it. But why did we evil math teachers foist it upon you? Because what this lengthy, arduous, annoying process did was took something gross, and turned it into something nice. The process wasn’t super nice, I know, but taking a step back and looking at the forest for the trees, it did that magic little math secret: turned what you didn’t know how to solve into something you did.
And look at all you needed to know about in order to make this happen. Numbers, addition, exponents, square roots, positive and negative, linear equations, variables. In that one equation
is a whole universe of knowledge! And you KNOW that knowledge. And in this class, we’re going to time and time again see equations that we might not think we know how to solve. They’ll look scary and unfamiliar like
But we’ll turn it into something we are more familiar with. Just don’t lose the forest for the trees. Don’t get stuck in the muck and mire of the procedure and not forget about why we’re embarking on that particular path, or what ground the path is built upon.
Our mantra: take what we don’t know and turn it into what we do. Math is an art. The creative aspect of it is finding the right path to turn what we don’t know into what we do know. Therein lies the puzzle, the beauty, and yes, the frustration.
PS. This post is basically a recap of this previous post. I just think it will be fun to talk about on the first day.