How do you provide an opportunity for your students to figure out the relationship between differentiation and antidifferentiation?
We have used the Calculus Nspired activities The First Fundamental Theorem of Calculus and The Second Fundamental Theorem of Calculus for several years now to improve our understanding of the relationship between a function and its accumulation function. I actually do print the student handouts for these activities and give students time during class to make sense of the relationship between differentiation and antidifferentiation.
It was time for our whole class discussion.
We defined the accumulation function using a definite integral. What do we know?
Students had figured out earlier that the definite integral of f(x) from a to a would be 0 and concluded that F(a)=0.
Students recognized that the value of the definite integral of f(x) from a to d would be F(d) and that the value of the definite integral of f(x) from a to c would be F(c).
Suppose we want to calculate the definite integral of f(x) from c to d. They could tell area-wise that was equivalent to finding the definite integral of f(x) from a to d and subtracting the definite integral of f(x) from a to c, which of course gives us F(d)-F(c).
And what does F(x) have to do with f(x)?
They could tell from the exploration that F(x) is the antiderivative of f(x).
Really? You mean we don’t have to do the limit-sum-infinite-number-of-rectangles every time? Really. You’ve earned the Fundamental Theorem of Calculus.
We talked for a little while about average value using the Calculus Nspired activity MVT for Integrals, and then checked in on their understanding.
As we moved into the second part of the Fundamental Theorem of Calculus, I posed a question to see how they would answer. (Remember that at this point, they’ve been using the FTOC for about 20 minutes.)
I was excited about a few students getting it right. Without discussing the correct responses with the whole group (I showed their answers but had Show Correct Answer deselected, I sent another question, which unearthed their misconception and revealed my initially bad question.
The students who got the answer correct in the first question had gotten it correct the wrong way, but their mistake wasn’t revealed because sin(-π)=0.
By now we were past the bell, and so we started over the next lesson with the second part of the Fundamental Theorem of Calculus.
It’s always exciting to find both the right questions to ask (the ones that reveal student misconceptions) and the wrong questions to ask (the ones that hide student misconceptions) so that I can continue asking the right ones and discontinue asking the wrong ones. In this lesson, I found at least one of each. And so the journey finding and asking the right questions continues …