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Setting Up a Division Calculation Correctly

Date: 12/16/2005 at 22:57:34
From: Pat
Subject: division help please

I need math help, please.  What exactly does "divided by" mean?  I 
never know what number to divide by.  I get the two numbers mixed up.  
Is there an easy way to remember what number goes where?  So when they 
say divide this number by that number, what number goes in the 
calculator first?  And what number goes on the outside or the inside 
if I am doing it on paper?

What does the word "divide" mean?  Maybe that will help with the 
problem?  Thank you for your time in advance.

Pat



Date: 12/16/2005 at 23:29:43
From: Doctor Peterson
Subject: Re: division help please

Hi, Pat.

This is a very common problem!

Let's pick a simple example to work with, that involves whole numbers.
We know that 2*3=6 (I'm using "*" for the multiplication symbol.)  So
if we divide the product, 6, BY either of the factors (2 or 3) we get
the other one:

  6 / 2 = 3

  6 / 3 = 2

(I'm using "/" as the division symbol; that's what it actually means
in a fraction.)

You can see that division is basically "undoing" a multiplication:
starting with the result of the multiplication and one of the two
numbers you started with, you can find the other.

When we enter the problem into a calculator, we put the numbers in the
same order we write on paper: 6 / 2, "this" "divided by" "that".
"This" is called the dividend (the number being divided), and "that"
is called the divisor (the number it is divided BY).

When we do a division by hand, we put the divisor on the left, and the
dividend under the bar:

    __3_
  2 ) 6

This is what confuses a lot of students, who read division problems
the wrong way, as if this were 2 divided by 6.  You might find it
helpful to think of the bar as an operating table, and the patient
lying on it is being "divided" by the surgeon, who stands next to it.
(I suppose the organs being taken out are put on top.)  The 6 is
divided BY the 2.

A division can be interpreted in several different ways.  The most
general way to talk about it is what I just did: division means 
finding what number you would have to multiply the divisor by (the 2) 
in order to get the dividend (the 6).  In practical problems, we might
divide 6 by 2 to find out how many piles I can make from 6 objects if
I put 2 objects in each pile; or how many objects will be in each pile
if I "divide" 6 objects into 2 piles.  The basic idea here (and the
origin of the word) is that we are "dividing" a group into parts.
Incidentally, the word "quotient", which is used for the answer to a
division, is essentially the Latin word for "how many?".

Of course, once you stop talking only about whole numbers, some of
these ideas disappear; we can't really talk about "dividing 2 objects
into 3 piles", and it gets even worse if we want to divide a fraction
by a fraction.  But these images are the original ideas that motivate
division; for deeper understanding of division, we move on from there
to the idea of undoing a multiplication, which I started with.

I hope that helps.  If you have any further questions, feel free to
write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Division
Middle School Division

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