Imagining the First Day of Calculus

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: $2x^2-56x+1=0$

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!

$2x^2-56x+1=0$

$2x^2-56x=-1$

$x^2-28x=-\frac{1}{2}$

$x^2-28x+196=-\frac{1}{2}+196$

$(x-14)^2=\frac{391}{2}$

$(x-14)=\pm \sqrt{\frac{391}{2}}$

$x-14=\sqrt{\frac{391}{2}}$ and $x-14=-\sqrt{\frac{391}{2}}$

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

$2x^2-56x+1=0$

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

$\int_{-3}^{3} \sqrt{9-x^2}dx$

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.

1. David P says:

I like the idea. I tried something similar (although less concrete) in our first few days this year. You might want to hit some of the more “visual” (and/or geometric) minded students (who may be the ones picking up the calculator) and do something similar with geometry that you’ll use this year. Maybe like, “Find the area of this shape” where the shape has rectangles glued on to semicircles, triangles, etc. “You took a complicated shape and cut it into smaller pieces. We’ll be doing something similar in finding areas for the second half of this course.”

1. I did this at the beginning of the second half of the course, last year. But maybe I’ll do it early on this year.

You’ve reminded me that one thing I should stop trying to do is making everything creep up like a surprise — and go voila! See how magical and amazing that is? They don’t get the surprise. By the time we got there, we’ve lost the forest for the trees. I am going to try to remember to give the end result, or where we’re heading in the course, first and THEN we’ll work up to it.

For most things.

I don’t ever want to give up the surprise of the antiderivative being the same as the (signed) area under the curve.

Not that I get oohs! and aahs! when we do it. But there’s just some magic there that I bet at least one kid will enjoy.

2. Mr. K says:

But we’ll turn it into something we are more familiar with.

This is a a pretty good paraphrase of one of the steps of George Polya’s problem solving strategy.

1. I’ve wanted to read that book forever. I am trying to see if I can use our “professional development time” to form a reading group to read and discuss it. I doubt my school will agree though.

3. I love this idea! I am struggling to figure out what to do with my Calculus students on our first day, Wednesday, besides going over expectations and the syllabus. I was thinking some groupwork/icebreaker/competition type thing but I like this idea much more as an introduction to a new/exciting/scary class! Thanks!

1. Thanks! I think, especially since my kids are not taking the AP course, that they have some trepidation of what they’re embarking on. I want it to make it sound like we’re going to be working on a puzzle all year. Not that it will in actuality be like that, because I’m just not that good a teacher yet, but maybe pockets of it will be that.

4. 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.

That’s important, but you will likely get cookie cutter answers. I have a sense students will go right for the exact solution. If they wanted to check whether they got the correct answer, they will either re-verify their steps or plug solution into quadratic to see if the equation still holds. Which is fine. But they are missing out on the interconnectedness of various topics in math.

This may be a good opportunity for them to put their pre-calculus skills to use.

2x^2-56x+1=0

If they don’t quite catch it, ask for solutions to
x^2-28x=0
(With any luck, it should be 0 and 28)

Then ask for the solutions to
2x^2-56x=0
(Hopefully, they still answer 0 and 28)

How does the graph of x^2-28x compare to 2x^2-56x?
(the latter is vertically stretched by a factor of 2, both have zeroes at 0 and 28)

2x^2-56x-2
2x^2-56x-1
2x^2-56x
2x^2-56x+1
2x^2-56x+2

A parabola that faces upward is shifted up by 1. Are the new zeroes closer to the axis of symmetry or farther from the axis of symmetry? What are 2 good guesstimates for solutions?
(a number slightly above 0 and slightly below 28, how do these guesses compare to their answers? how long did it take? how long would it take if they had this “intuition”?)

Getting the right answer if students remember to follow the right procedures is good. But this may be a good opportunity to connect skills learned in a prior class to promote their math sense.

1. I see what you’re saying, but my hope/expectation is for them to NOT remember all of this stuff — and I believe they will go straight for their calculator or the quadratic formula. But the point of the exercise isn’t to show them “all about parabolas” and remind them how to solve ’em. It’s to show them (a) they know a lot, but (b) they are probably missing the bigger picture, and (c) how what they’re going to do in this class is work on building the bigger picture.

You’re DEFINITELY right about your approach helping refresh knowledge of parabolas/quadratics. And it reinforces the interconnectedness of what they’ve learned. I might actually do that when we get to it. But the way I see it, it is less about refreshing skills and more about giving them an *allegory* — to make a larger point.

5. Great stuff, Sam, BOTH posts! I think I’m going to start using that mantra more in my Alg 2 and Calc classes…those are the ones where the “newness” of all the information crammed into the short time we have can easily overwhelm. When we look at it as building upon what we know already, it’s so much easier!

Also, in one of your comments you mentioned showing them the forest instead of all the trees and then the big “voila!” moment. That’s actually research-sound. Brain based research is showing that students need the big picture experience first, then they need to know the individual parts. Thanks for giving us something to chew on.