A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?
Students have a fundamental understanding of integration in all its forms: approximate area through Riemann sums, abstract area through geometric breakdowns, and algebraically through antiderivatives.
My approach is unique, as far as I can tell. In all the Calc AB material I've seen, integration enters the picture really late, after the idea of curve relations (f, f prime, and f double prime). In my opinion this does students a disservice.
Discuss the Riemann sum methods: left, right, midpoint, and trapezoid.
Place emphasis on an integral as the approximation of area under the curve. At this point in time I will use functions defined only with a set of data. I show them definite integral notation and how our Riemann sum values are one option.
The biggest lesson of all needs to happen right here. WHY would a data table defined with x-units of seconds and y-units of meters/second have an integral value measured in meters? Don't walk away from the concept until they see that the dimensions used in your sum approximations have units attached. Multiplying those dimensions to create an area have consequences for the resulting units. This is how your April fortunes are won and lost.
Before moving on, take the opportunity to discuss antiderivatives. Keep the examples simple. Reverse power rules and definition based trig functions without limits of integration. The word "antiderivative" is the next key.
Return to the concept of a definite integral. Review the idea of a Riemann sum. With arbitrary data, we can make approximations of an integral's value. What if we had a function defined solely with a picture?
Knowing your final learning target matters here. Can students recognize when to interpret a function value as an area? In the graph shown here, what, exactly, is that integral symbol implying about the relationship between g(x) and f(t)?
This type of integration scenario immediately follows f, f prime, f double prime relations and the curve sketching exercises associated with the topic. If you're students know about antiderivatives in the midst of curve sketching, you'll be amazed at how quickly they can draw a curve family AND discuss the transition between curves using the concepts of derivation and integration.
Right before you break for winter, briefly return to the antiderivative idea mentioned in October. Using the same simple scenarios (reverse power rule, base trig functions), add limits of integration. Where have we seen those before? What do those imply? Are we still talking about area even though I have no picture? Could I make a picture? What does my algebra demonstrate on that picture?
If you find students struggle with implementing the fundamental theorem, see what happens if they know what an integral is months before you introduce it.
Pat yourself on the back because the heavy conceptual lifting is over. Now you can move to integrals that require chain rule concepts (or u-substitution if that's your thing, it's not mine). A huge problem for Calculus is battling student understanding of skills while at the same time realizing they have no conceptual idea of what's going on. April is the wrong time to realize that students might be able to parrot all the appropriate methods associated with integrals, but can't think their way around the concept at all.
Students need time with this stuff. Cramming the many many facets of integration (we didn't even talk about volume!) into 4 weeks is incredibly strenuous, and I think it breeds a student who can do everything you say but has no idea what they're doing.
Don't make your students become math teachers before they really get this stuff.