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Q&A #12670

Teachers' Lounge Discussion: Dividing fractions

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From: Loyd <Loydlin@aol.com>
To: Teacher2Teacher Public Discussion
Date: 2006081207:36:47
Subject: Why we invert!

On 2006081123:20:06, Paige wrote:
>
>
>So please explain the reason behind the algorithim. Why do we invert
>and multiply? If I have 1/6 divided by 1/5, isn't that saying how
many
>times will 1/5 divide into 1/6? but 1/6 is smaller than 1/5. I know
>the answer is 5/6, but I don't know how to explain why. I've never
>thought about it before because I just learned the rule.
>

If you have ever used Pascal, Fortran, Basic etc. you will probably
never see the symbol for division used in elementry school.  It is
used on some calculators but division is really a fraction. The symbol
for division doesn't appear on most computer keyboards.  Here is what
I wrote on my last post.  

<<<<<"The culprit in this division topic is the symbol that is used
for
division, that is, the the dot over and under a bar.

 .
---
 .

Division is really best represented by a fraction.  Thus when we use
the symbol above, we should let the students know that we still have a
numerator and a denominator.  

If you have noticed, the division symbol used in the lower grades
dissapears from text books in higher level mathematics. >>>>>

It often helps to ask the question, "How many 1/5s are in 1/6?  This
way, you can visulize that there has to be less than one.  In your
problem of 1/6 divided by 1/5 the kids become confused when you invert
the 1/5.  But if you wrote it as a fraction as:

1
--
6
----
1
--
5
it would be easier to see that if you multiply the numerator and
denominator by 5/1 you wouldn't change the value of the fraction
because 5/1 divided by 5/1 is "one."  Over and over in math, this
"multiplying by one" method enables us to solve math problems. In the
current problem, multiplying by the recipical converts the denominator
to one.  

One other problem is that kids in the lower grades have not had
experience with fractions, so the division symbol was invented just
for those kids.  But after they have learned fractions, the division
symbol should slowly disappear; and it does usually by the time the
students are in highschool. 

Rule:  If you multiply the numerator and denoninator by the same none
zero quantity, the value is not changed because you have multiplied by
"one."
 
This rule explains the reason why we can change fractions such as 3/4
tto 6/8 because we multiplied by one.  (That is 2/2).



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