Late April means we dabble in a little Calculus during Pre-Cal. Traditionally we look at evaluating limits with a table, limits by direct substitution, limits at infinity, and the limit definition of a derivative. It's always been ok. It suffers from the malaise associated with end of the year content.
I went a different direction this year by setting up the calculus with a bump of sequences and series to later in the year. Normally these are shoved in between vectors and conics. It never felt right. Instead I went vectors, polar equations, then conics but the polar form of conics. When we concluded I felt better calling it the end of Pre-Cal, classifying sequences and series as the beginning of our calculus discussion.
We spend a week or two on sequence generation, finite sums, and sums of infinite sequences (would it be easier to call those infinite su--nah).
Then, the real calculus connection.
Do we make a turkey?
Take some grid paper. Instruct the student to slap down their hand and trace it.
After discouraging making actual turkeys, make a request: how many squares are taken up by your hand?
Some groaning. Some suspicion that this is a trick. You don't really expect me to count all these squares?
In fact, I don't really care what the totals are. If some manage to figure it out I'll have them share. Mostly, I'm watching to see who takes a second to think about what I've asked them to do. Lots will immediately embark on counting the hundreds of little squares. But somewhere, a clever kid has realized that yes, it was a trick:
You hold a few of these up and the groans will get louder. I should have known!
Clever Math Transition
Discuss the problem a little bit. There is no formula for the area of a hand (though one kid honest to goodness tried to google it). So what was easy to do? The area of a square/rectangle is easy to find quickly. Approximating a hand as a series of rectangles will get you close.
Now to generalize what this has to do with calculus. Any arbitrary area can be broken into a series of rectangles. What's impressive here is the cleverness that will surface. Students will want to chop things up into rectangles and triangles. And why should the rectangles be the same width anyway? If only they knew how close they were to the definition of an integral.
The extent to which I cover this topic concludes with iterating the right hand sum of an area manually. We do a first progression together:
Now I messed up a little here. The right most rectangle has no area because of the way I cropped the functions. Fundamentally this is not a problem, but a weird quirk to include when it's the first exposure. I fixed this later. This activity helps introduce the idea of how the slice up a function. Annoyingly some students were confused about how to compute the sum (add up the area of a bunch of rectangles? Are you kidding me!? April, everybody). We conclude with a discussion about the merits of this method. Is it flawed? Would it ever be accurate to slice something into only two slices?
A day or two later I had them demonstrate the entire process. I created a set of 12 irregularly bounded areas. Students chose two. One iteration set in their notebook. One for display. Chop the area into 2, 4, and 8 slices. Make a conclusion about what the method is attempting to do.
If you look super closely at the red one you'll spot some misconceptions (the number of slices is assumed to be the width of the rectangle, a couple slices aren't bounded correctly). This student wasn't quite finished yet and will be able to fix the problems, but it reveals how strangely complex the idea can be.
This progression leads into limits. When you return to the concept of infinite rectangles, you reach the limit of the area calculation: the true value of the area.
As a sacrifice we probably won't get to derivatives, but that's ok. I felt the progression raised the presence of summations and served as an example of the simplicity behind some calculus concepts.