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 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. 🙂

I haven’t graded the tests yet, but over half my class says they expect to need a retake. So the pull-back cars may become a big out-of-class relearning exercise. (I’m still hoping my son will come through on providing those.)

I’m seeing students’ understanding of slope get deeper as they work on these things.