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Q&A #12670

Teachers' Lounge Discussion: Dividing fractions

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From: David Chandler <david@mathhwithoutborders.com>
To: Teacher2Teacher Public Discussion
Date: 2007101316:58:35
Subject: Why Invert and Multiply

"Invert and multiply" is by far the easiest way to do division of
fractions.  It is also important for students to be clear on this when
they get to algebra. However most people (myself included) were simply
taught "invert and multiply" as a mantra and were never shown why it
works.  Rather than go all around the barn to find some other tricky
(and more difficult) way to do it, why not just learn why it works? 
Here goes.

I will use simple algebra to do the explanation, but that is because
I'm presumably explaining it to to adults.  The simplest way to
explain this to kids (of an appropriate age) might be to teach them
the tiny fragment of algebra needed to understand the proof.

When we divide, we are looking for a missing factor:

For instance,  12 divided by 4 means ...   4 times (what) = 12, or in
algebraic notation,  4 x = 12.

In the same way,  5/8 divided by 2/3 means ...  (2/3) x  = 5/8

To solve for x, we have to cancel out the 2/3 that is "cluttering up"
the left side of the equation.  Therefore we multiply both sides by
3/2.

Thus     (3/2)(2/3) x = (5/8)(3/2)

or                  x = (5/8)(3/2)

Notice that we end up multiplying 5/8 by the reciprocal of 2/3.

More generally,   a/b  divided by c/d  means ...  (c/d) x = a/b

Multiply both sides by the reciprocal of c/d and you get

x = (a/b)(d/c).  Therefore to get the answer to a/b divided by c/d,
multiply a/b by d/c.

In algebra division is DEFINED as multiplying by the reciprocal. 
Therefore avoiding the topic or finding some other way to do the
problems is ultimately counterproductive.

--David Chandler

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