Explaining Fractions

Date: 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/