Down At The Bottom Of The Garden

What with the outstanding early returns in Calc I, I’m even more convinced that “students working up front in the classroom” is something I should be doing in every class (at least once per quarter; I’m far from saying “every class meeting” [unless in a class of about three]). So the plan is to make part of the “quiz” grade depend on doing a presentation.

Today I drew a circle centered on a co-ordinate frame and filled it in. “Use calculus to find the area bounded by the unit circle”.

That is, though I didn’t say so, find the area of the unit disk. Since I’m feeling expansive and don’t always dare to introduce such sidetrackers in lecture—or even in “problem solving” sessions—the unit circle is known in the pros as S^1 (the “One-Sphere”… the n-sphere generally is the set of \langle x_1, x_2, ... , x_n\rangle such that x_1^2 + x_2^2 + ... + x_n^2 = 1 [i.e., points at distance 1 from 0 (the zero vector]). If the unit ball has a cool symbol of its own I’ve evidently forgotten it but of course it’s the set of points at distance 1or less from zero. Back to today’s class.

Our first volunteer then wrote out y = \sqrt{1-x^2} and observed that since the bit in the first quadrant was obviously a quarter of the whole, we were looking at 4\int_0^1  \sqrt{1-x^2} dx. Okay, good work… how do you do the integral? A brave stab at a “u-substitution”; I jump in (much too soon I imagine): “We want a trig substitution here…” and, when he doesn’t know right away how next to act, I allow as how it’d be okay to pass the chalk (really whiteboard marker; ugh) to somebody else. Who works through it pretty well, though with a few gentle reminders from me (again probably premature… of course I want corrections to come from the other students… and there were some of these) about, in particular, changing the limits on the integral when integrating with respect to the new variable. So things are going swimmingly.

“Now somebody do it in polar co-ordinates!” (Along with a little mini-rant by me to the effect that I should’ve already—in the first lecture about polar co-ordinates—have mentioned that [of course] “radius equals constant” gives the equation of a circle [centered at the origin] and “theta equals constant” [theta—again of course—here denotes the angle in a point given by polar co-ordinates (r, \theta)] is the equation of a line through the origin [one is supposed to say “pole” here and probably soon I’ll have that habit]. I’d like to be able to take it for granted that these are glaringly obvious [when pointed out] but actually they’d make darn good quiz questions in pre-calculus and might slip few minds even around here. To continue.)

And somebody does. For better or worse, it’s the student doing much the most participation during the others’ presentations (so this is probably way too easy for him); I’d hinted that I intended to make the problems get harder to encourage early volunteers (this appears to’ve been a tactical error). Anyhow, he knocks it off flawlessly… one could wish for a teacher-like running commentary I suppose but anyhow the written work required no corrections (and he wasn’t entirely silent as students often will be with boardwork). Part of the point here is how much easier the calculation is—\int d\theta could make a pretty good claim to be the world’s simplest integral (\int 0 d\theta is of course its only rival).

The work has essentially been done by choosing the right notations (and having already developed the “formula” A = {1\over 2}\int_{\theta_1}^{\theta_2} r^2 d\theta; I took the opportunity to remind everyone—anything you only say once in a lecture series, you might as well never have said at all [or so it sometimes seems]—that the “one-half” comes about because of triangles in a certain drawing… and that remembering the reason it’s there is a good way to remember that it is there).

Then a textbook exercise: slope of a parametric curve at a point. More good student work; some more hints from me about connections with stuff we’ve looked at before. Next exercise, the area bounded by the same curve gets no takers and I’m content to look the others in the eye and say they all owe me boardwork.

There’s plenty of other stuff I want to do in our two-hours-and-change today; more than I’ll ever be able to do… and this is progress. Still, it’s beginner’s progress indeed for me. I’m so doggone uncomfortable asking anybody to do anything they’re uncomfortable with… and it’s so easy to just wimp out and just lecture…I’d be deeply ashamed of this if so many of my colleagues weren’t even worse (at drawing students out by voice… at insisting on homework and times and dates and whatnot I’m probably one of the biggest wimps there is).

So the rest is mostly me talking about the other mid-course correction I’ve just instituted: “Project” style homeworks. I’d mentioned a couple problems as potential “project” problems; today I opened it up and said “pick any two from the ‘P.S.’ sections of chapters 9 and 10″… and went on to talk about what I‘d done with the two I’d singled out on Monday (without giving too much away; these can still be finished for credit).

Then went on and (quickly) worked on still another topic (Taylor Series… the centerpiece for half the course). But the point is that with the student boardwork and the substantial problems, I feel I’m one step closer to the course as it ought to be.

At which point there ought to be a rant about the textbook. I’ve got quite a bit of material prepared. But that’s it for tonight. Thanks for your kind attention.

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Posted in 153. 3 Comments »

3 Responses to “Down At The Bottom Of The Garden”

  1. Sue Says:

    This is Calc II, right? When did your semester start? (I thought I read a post recently in which you discussed the first exam?)

    You do polar before infinite series, I see. I guess I switch it around from the book’s order because I want plenty of time for infinite series, and I don’t mind if the chapter with polar gets crunched.

  2. vlorbik Says:

    right. what i’m here calling “153”
    is a calc iii class (that was on my schedule
    *first*… i’m also doing a “151”–
    a calc i– but this blog is named
    “calciii.wordpress.com” because
    i didn’t *know* i’d be doing it
    when i started the darn thing up).

    i’m doing the geometry and the series
    as two “strands” (as it were simultaneously;
    each two-hour-and-change lecture
    typically includes about an hour of each…

    the series [chapter 9 of larson (et al)])
    will run all quarter; we’re wrapping up the
    parametric curves; next week we’ll begin
    the vector stuff. (3D is calc iv around here
    and i’ve never done it. i *ought* to but
    god help the students if i do it during a *busy*
    quarter since i’ll have to work much harder
    at learning “new” stuff [that i’ve seen before
    but never understood in much depth] than
    what i’m used to [even though i’ve been
    more productive in the past couple of quarters
    than i’ve been in many years]).

    the polar as handled by the version of the course
    laid out by the course co-ordinators here is already
    pretty badly crunched (skipping the conics at the end
    of the chapter… which appears to be much of the
    real payoff since “cycloids” and “n-petal roses” and
    suchlike easy-to-work-with stuff occur pretty naturally
    in *mathematics” but ellipses and parabloas
    and that lot occur naturally in *nature*).
    there are good reasons… there’s already too much
    for a ten week course as it is… but it’s just one more
    reason we’re *obviously* using the wrong book…
    there’s a section 10.1 where they review conics
    for rectilinear co-ordinates; this *is* on our syllabus
    (and shouldn’t be). here are further remarks
    from last year’s version.

  3. Sue Says:

    (dang, i found a way to ‘subscribe to your blog, but it doesn’t point out the comments…i’m just now seeing this.)

    I’ve never taught calc III (our 3d) for about the same reason. The first time I taught differential equations, I didn’t do as good a job as I’d like, because it took too much learning. The second time ,it went great. I want to be much better prepared before I teach Calc III the first time.


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