Explaining FractionsDate: 11/06/2001 at 10:10:11 From: Ellen Roddy Subject: Explaining fractions Dr. Math, I am a practicum student in a 8th grade math class. The teacher and I don't know how to explain to the kids why you can do this manuever. c = d/g c/d = g d/c = g Please help. Date: 11/06/2001 at 11:11:20 From: Doctor Greenie Subject: Re: Explaining fractions Hello, Ellen - You can't do that maneuver, as you have shown it. Your first and last equations are equivalent; the middle one is different. You can get from the first form to the last form with the following middle step: c = d/g c/d = 1/g [divide both sides of previous equation by d] d/c = g [take reciprocals of both sides of previous equation] Compare this middle equation with the middle equation as you showed it.... I hope this helps. Write back if you have any further questions on this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 11/08/2001 at 15:33:41 From: Doctor Ian Subject: Re: Explaining fractions Hi Ellen, It may be easier to understand what's going on here by going back to basics. The following are three different ways of expressing the very same fact: 1) cg = d 2) g = d/c 3) c = d/g For example, 1) 3*4 = 12 2) 4 = 12/3 3) 3 = 12/4 This is, in fact, how we _define_ 'division': If ab = c, then c/a = b and c/b = a. So this is _why_ you can transform equations (2) and (3) into each other. As for _how_ you go about transforming them, Dr.Greenie showed you one way. I like to get rid of denominators whenever I can, so I would have done it somewhat differently: c = d/g cg = d [Multiply both sides by g] g = d/c [Divide both sides by c] In each case, we followed the basic rule of algebra, which is that you can do anything (except divide by zero) to both sides of an equation without changing the truth of the equation. So that's another way of justifying _why_ the transformation works. To sum up, you can do this transformation because: 1) It follows directly from the definition of division. 2) There exists a sequence of multiplications and divisions that leads from either equation to the other. Students may differ on which explanation they find more convincing. I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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