I like making my students discover ideas on their own. I like to lead them down a path and then make them put the pieces together to solve the puzzle. For all of my teaching years, it had seemed like when we finally discussed the Fundamental Theorem of Calculus (the supposed most wondrous moment of their mathematical careers up until that point), my students would just look at me and say, "And?"
The books I used made students practice taking antiderivatives by evaluating indefinite integrals way before the discussion of area ever appeared and I blindly followed along. The students were already used to the idea that integrals and antiderivatives were related, so when we found that derivatives and integrals were really inverse concepts, it was obvious because we had started off by defining them that way. Sharing that the trick to evaluating definite integrals was to use antiderivatives made them think I wasted their time making them calculate area in any other way. Wasn't that already coming? This year, I wanted them to feel the awe of the FTC.
After midterms, I had them work on taking antiderivatives by giving them differential equations and asking for the original function. I also told them to hide their books and promise not to try to read them. (Those over zealous students of mine always outdo themselves!) We worked on finding area using Reimann sums and then explored how the limit of the Reimann sum evolves into a new symbol, the integral. We practiced graphing equations to find the geometric area between the curve and the x-axis to evaluate integrals. After covering some of the basic properties of how integrals work, we worked on the first worksheet from the FTC curriculum module made available by the College Board.
This was the perfect spring board. We were able to review finding area beneath the x-axis, what happens when endpoints are "backwards", and finding area of trapezoids. Although the students understood that the integral symbol meant finding the area from a to b, this worksheet really helped them solidify that connection. Then we were able to look at the general formulas for the integral functions given and the derivatives of those equations.
When we reviewed the results from the worksheet, students began seeing "+ Cs" which happened when I began changing the lower endpoint of the integral and asking what happens to the general formula. This made them think of antiderivatives. Then they saw the relationship between the integrands and the "shortcut" formulas we were getting for evaluating the integral. They told me that to evaluate definite integrals, you must take the antiderivative. They were the ones that put together that you substitute in the upper endpoint (which in the worksheet, was the variable x) and then to find the value of the "+ C" they had told me about earlier, you take the opposite of the antiderivative evaluated at the lower endpoint. From there, they ended up working out how to take the derivative of integrals with any kind of endpoints you could imagine. They figured all of this out on their own in less than seventy minutes. It was beautiful to watch to their ideas develop and I think that they were impressed with both parts of the FTC because they saw the pattern and created it themselves.
I can't wait to do this again next year.