Now that the polar coordinate system is behind us, it's time for the end game material in Pre-Cal. That means sequences, summations, and limits. Over the years I have been spending less and less time with the sequences and summations portion. Primarily because I want to use it as a foundation for Calculus, discussing the problem of arbitrary area. A secondary goal is using the idea of a summation for some financial literacy.
In the short time I spent with sequences and summations, here is the random order of events:
- Discussed sequences as an idea, everyone remembers them from some time in the past, no big deal. I wrote a series of random sequences on the board (linear, quadratic, etc) and had them generate terms. Then they added them together. No sigma notation yet.
- Then I did a magic trick. Well, a Nowak trick. Students create a random linear pattern, generate the first 20 terms and add them together. Have someone verify your sum. They did this on little half index cards and I collected them and quickly called out sums like some sort of genius. This drove a few kids crazy trying to come up with theories.
- Next we go Dan Meyer with some penny pyramids:
- This group of kids weren't too intimidated by this. Many of them figured out the pattern to the number of stacks and were able to get the correct answers manually. Several thought the answer might be possible through the magic trick I demonstrated before.
- At this point I show them sigma notation and how to use it on the calculator. They compute a few for practice.
- Then it's Fawn's turn and I mix in some Visual Patterns. I picked pattern 2, 18, and 19. Two of those are simple, 19 is a fun kind of frustrating.
- Next step is converging and diverging series. I pick two sequences: 4*(0.74)^n and 4*(1.02)^n and have them compute the sum of each for 10, 50, 100, 500, and 1000 terms. This is some good calculator practice and gets them to observe a few things. Why does the answer start repeating for the first one? Are these giant numbers I'm getting for the second one right? I give them some hints about the things possible in calculus when I mention that there is a simplification for computing the sum of an infinite diverging series. "We're going to learn that!?"
- Then a summary assignment. Generate some terms, generate some terms and sum them manually, compute sums on the calculator, and identify convergence and divergence.
All of this builds towards approximating area under curves using left, right, midpoint and trapezoidal sums. Students who went through last year's version remembered it well now that I have some of them for Calculus.
Most Pre-Cal curriculum I see sticks this stuff in between vectors and polar coordinates and I find that a little awkward. Putting it right before whatever Calculus material I intend to cover has worked better for me.