kitchen table math, the sequel: how many unknowns, part 2

Thursday, September 2, 2010

how many unknowns, part 2

gasstationwithoutpumps said:
Although Glen would never create 2 unknowns, preferring r and 12-r to r and s, I often find it easier to create multiple unknowns when initially setting up the problem, then remove the unnecessary ones. In this case, it was easier to remove (r+s) as a single unit, and never worry about manipulating 12-r.

I can't tell you all how important these threads have been to me: how much I'm learning (I hope I'm learning - !) and how rich the experience has been.

It's led me to think about the question of self-teaching a bit. Until last night, I had simply never thought about 'how many unknowns' in the way you all are talking about unknowns now. I had never thought about it because, where unknowns are concerned, the books seem to suggest that less is more.

Mind you, I don't think any math book I've used has directly stated that 12 - r is superior to r + s=12. I'm pretty sure I inferred that it was based in the fact that I don't recall any instances of r + 12 where 12 - r was a possibility.

This strikes me as the kind of thing a good math teacher would bring up in class, perhaps as an aside?

Or something that would come up in discussion?

What do you think?


gasstationwithoutpumps said...

Actually, this sort of more general discussion rarely comes up in most math classes. It does come up sometimes in problem-solving sessions for contest prep, where the goal is to learn to solve many sorts of problem quickly, rather than to master any particular subject content.

You might find the Art of Problem Solving books (Vol 1: the Basics, Vol 2: and Beyond) more useful for this sort of discussion than the usual plug-and-chug SAT prep books.
I'd still recommend stopping and reading Polya's "How to Solve It" first. You'll save time in the long run.

lgm said...

The splitting into r and 12-r might be used if a bar model a la Singapore Math was used. It wasn't necessary to go to that extent in the problem that you posted b/c the visual is already there. It was enough to just keep the relationship in mind in order to solve the problem mentally.

I think this would come up in college, probably in a physics class. High school I don't think ever talks about degrees of freedom. But maybe some readers that IRL are high school math teachers will chime in.

Do read Polya, or if you have Singapore Math NEM, read the problem solving page in the front. No - laminate it and keep it handy.

Allison said...

12 - r is not superior in this case, though, according to some (most?) of us. Writing/thinking anything other than Pi(r+s) is entirely too much computation for this problem. That's because the substitution your'e given is for r + s.

The way to solve this problem in 5 seconds is to write:
Pi r + pi s
and then substitute 12. Thinking about 12 - r just leads to mistakenly thinking about r and subtracting r, or worse, making an error and thinking you don't know how to solve for r.

What comes up in math classes more often is the more mechanical parts--labeling the radii by letters, defining the arc length with those letters, turning the RS segment length into a sum of the radii. Those are the elements that a math teacher would talk about more, I think.

Catherine Johnson said...

Off-list I've been advised that Polya isn't right for me at this point --- just mentioning as an fyi. I gather the book was written for graduate students, then made its way down to high school.

I'm still at the high school level.

I've studied the Singapore Math material on problem solving that lgm mentions. I'll re-read; maybe I'll even laminate.

But the fact that my years with Singapore Math did not lead me to solve this problem may be evidence that Polya won't necessarily be useful at this point.