Details of the algebra notwithstanding, what in the world did you think I meant? "Better would be to have them express - and then graph - - - volume of resulting cylinders as a function of side length of rectangles of fixed area and find the maximum. Nice algebra and it greases the skids for eventual calculus. What a concept."
At 01:43 PM 5/28/2013, Louis Talman wrote: >On Mon, 27 May 2013 19:54:08 -0600, Wayne Bishop <firstname.lastname@example.org> >wrote: > >>If they really know the formulas, they would start by using the given >>information to compute the volumes based on the evidence presented. > >How very interesting. > >Wayne prates incessantly about the necessity of avoiding math avoidance by >teaching kids the power of algebra. And then he suggests the weakest >possible mathematical strategy for arriving at a correct solution to this >problem. And it's a solution that uses algebra in name only, substituting >numbers into a memorized formula in order to compare numerical results. >But let's give him some credit: This is consistent with his rote approach >to word problems ("word problems by type"). You get a few points for >consistency, Wayne, but none for mathematics---which looks at the >*context* of a problem, and not just its answer. > >If Wayne's approaches aren't algebra avoidance, I don't know what is. And >Robert acquiesces, suggesting that "a student of algebra" would never >avoid algebra---leaving us to guess that he agrees that Wayne's strategy >is Real Algebra. > >(In fact, students of algebra generally avoid algebra as much as they can: >Learning new ways of thinking involves work that they'd rather avoid.) > >Here's a *real* algebraic approach: The volume of a cylinder is Pi r^2 h, >where r is the radius and h is the height. Let's begin with a sheet of >paper of length L and width W = k L, where k is some positive real number >that's at most one. If we roll the paper up along an axis parallel to the >L side of the paper, the radius of the resulting cylinder is k L/(4 Pi), >so the volume we've formed is > >V_L = Pi [k L/(4 Pi)]^2 L = k^2 L^3/(16 Pi). > >If, on the other hand, we roll the paper up along an axis parallel to the >W side of the paper, the radius is L/(4 Pi), and the volume of the second >cylinder is > >V_W = Pi [L/(4 Pi)]^2 k L = k L^3/(16 Pi). > >So V_L = k V_W. > >Now we've answered the question---but we've done much more than that. >We've explained the answer in a way that applies in more general >circumstances. Moreover, we're building an understanding of the >relationship between measurements of length, area, and volume. >Substituting the given numbers into the cylinder's volume formula >accomplishes none of these things. > >What Wayne's sermons and Robert's acceptance of them give us is something >less than a half-measure. > >But the two of them are right in one respect. Without what I've just done >here, the activity isn't complete. That doesn't mean that it's useless, >though. Understanding of a phenomenon is based on repeated examination of >the phenomenon in a variety of circumstances, of which this activity >provides one. It should be part of a progression that leads students to >ask "Why?" And that leads, ultimately, to the analysis I've given above. > >--Lou Talman > Department of Mathematical & Computer Sciences > Metropolitan State University of Denver > > <http://rowdy.msudenver.edu/~talmanl>