Posts Tagged 'calculus'

Introducing differentials

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The lesson inside the lesson

I love my calculus class. They are mostly seniors, and seniors are busy people. I know that my predecessor’s greatest peeve was kids missing class and not taking responsibility for catching up, and I know that in the past many seniors have had a less than rigorous final academic year. This week I confronted these issues head-on.

We are on the block, and meet Tuesday, Thursday. Friday (today) is a professional day, so no class. Tuesday, five students were gone to help with an elementary school campout. This is a neat opportunity to mentor young kids, and I encourage it. They came to me the week before with a planned absence form. I let them know that makeup work would be available on the website, and when they mentioned they’d be out camping Tuesday I promised to post it a day early. The lesson they missed was about sums, products, and composites of functions, and finding the domains. It went pretty well; kids were getting it and I assigned some additional practice. Thursday, folks came to class and promptly took a quiz on the subject. And? those who had attended the previous class did well; those who did not attend… Their faces told the story. I let everyone know this was a formative assessment, and not high stakes. Still, these are the school’s high-achieving kids and they are not used to bombing quizzes.

So, we moved on to the lesson. Their task was to extend from our current understanding – derivative of power functions, and function composition – to differentiating a general polynomial. They did the work in groups; there was a worksheet to provide scaffolding; I circulated. They struggled mightily, because of the abstract level we were working at, but ultimately everyone got it. It was a great hour.

I thanked everyone for their hard work. I mentioned that three students were gone today (for peer mediation training; again, a worthy cause!) And I think that it hit them. This lesson had been good, and hard. It would be really difficult to make it up outside class. I let them know that my goal was to make every class worthwhile; I asked them to please think carefully about their absences, to take responsibility, and to work very hard to make up for each one. This was the lesson. It felt like a step in the right direction.

Working our way towards the derivative…

First, we played with pullback cars and calculated average velocity, and realized that a smaller interval could yield a more accurate estimate of instantaneous velocity.

Then we went to boot camp and got really familiar with the polynomial functions and their qualities. (Also reviewed slope, for good measure.)

Next we talked briefly about reflection (with a detour to pool tables) and scientists’ quest to understand curved mirrors and lenses.

Then it was Geogebra in the lab. Given a polynomial curve, place a point on it where you would like to know the “slope.” It is cool that if you zoom in far enough, the curve looks linear. Place another point on the curve, make a line that connects them, and apply the slope tool. Now, zoom in, and move the second point ever closer to the first. Watch the slope measure change. You will be able to estimate the slope right at that point. This exercise felt like both a good introduction to some new GGB tools, and a really visceral experience of calculating a one-point difference quotient.

Next, we did the algebra to find the slope of the tangent to y=x^2 at (1,1). (We are working without the formal definition of limits. We have a variable h for the interval, which “gets really small” and we are left with a numerical value for slope. It is great how they get this, but think it is fishy. It is fishy. Once they get more comfortable with it, I will pull the rug out with a rant about how messed up everything gets if you allow division by zero. And so, hopefully, motivate limits. I don’t really know how else to motivate limits.) Homework: Choose three more points on the same curve; use this technique to calculate the slope of the tangent at each of those points.

So, having done all this, I believe hope they will be ready for the real difference quotient definition on Thursday. The big hurdle remaining is understanding that we are deriving one function from another – that is, that the derivative gives us that numerical slope for any x value we input.

As a side note: I have three exchange students from Beijing in my class, who all took calculus previously, in China. They seem fascinated; I think their previous experience of learning math was very different. The hometown kids are fascinated, too, by the algebra prowess of the exchange students. They can factor quadratics just like that! One of the host parents approached me this week, and said his guest student is really enjoying math. “He told me he learned something new on the very first day!” That was the day we played with cars.  🙂


We are all born into this world, and at some point we will die and that will be that. In the meantime, let’s enjoy our minds and the wonderful and ridiculous things we can do with them. I don’t know about you, but I’m here to have FUN.
-Paul Lockhart

In theory, theory and practice are the same. In practice, they are not.

-Yogi Berra