Throughout school, I took what I have come to think of as a guess-and-test approach to math word problems. This was never the 'official' approach, so I eventually had to learn to dress my answers up to seem more official. But they were derived from guess-and-test.
Simple example. There are two school buses, A and B. A has 18 more seats in it than B does. Together, they have 96 seats. Find the number of seats in each.
Guess-and-test. Hmmm. Sixty is a nice round number lower than 96. Let us see if the big bus could have 60 seats. That would mean the small bus has 18 fewer seats than that. Hmmmm.
60 + 42 = 102 NOT 96.
Okay, each bus has to have fewer seats than that. Let us guess that the big one has 55. Does that work?
55 + 37 = 92 NOT 96.
Oops. But we do now know that the right answer is higher than 55 yet lower than 60.
57 + 39 = 96. Aha! 96. This means 57 and 39 is the right answer. You'll be surprised how often this works better than the official proceeding.
So you won't think me ignorant. The official way is something like this:
B + 18 = A.
A + B = 96. (These two steps simply formalize the problem).
(B + 18) + B = 96
2B + 18 = 96
[96 - 18 = 78, so]
2B = 78
B = 39.
39 + 18 = 57.
A = 57.
But in no sense is the official approach a better one.
Many years later than my use of that approach to word problems: I studied a little calculus (as part of an adult education course that was supposed to help me at work). I was delighted to discover that something much like my old guess-and-test approach has an official name in calculus. It is called the Newton-Raphson method for finding the root of a real valued function.
I won't try to explain what the root of a real valued function is. I will only say that one quite official way of looking for it (named for THE Newton, as well as another fellow with the posthumous bragging rights of having his name forever linked with that of THE Newton) -- one quite official way of looking for the root of a real valued function is to start by guessing, run through the results of that guess, and -- if the guess is wrong -- allowing that fact to determine your next guess.
The above illustration is of a bond once signed by Joseph Raphson. I use it instead of a portrait of him because there don't seem to be any portraits of him. God bless him. His existence, and the above bond, prove that Newton-Raphson can work for everyone, not just world-historical geniuses.
Christopher, I would have started as you did, by guessing 60, but then, after seeing that 60 + 42 = 102 NOT 96, I'd have said, "96 is 6 less than 102," so subtract 3 from each, and I'd have 57 + 39. My second step was not a guess, but neither was it algebra, which you call "official" approach.
ReplyDeleteOne of the great things about guess-and-test is that it doesn't require purism.
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