Groups in Multiplication

Date: 12/04/2002 at 23:27:21
From: Leslee A. Richardson
Subject: Multiplication and placement of factors

This may seem like a ridiculous question, but the resources I've used 
to determine the answer contradict each other.  I would like to know 
when multiplying factors vertically, which factor represents the 
number of groups? Is it the top factor or the bottom factor next to 
the multiplication symbol?  

Some resources show it as being read from the top down, and others 
from the bottom up. For example, when multiplying factors where the 
groups are nine and the amount in each is eight, would the nine 
(groups) be placed on the top or bottom next to the sign? It is simple 
when the direction is horizontal - 9x8.  The factor for groups is the 
first one (on the left).  

I would sincerely appreciate your help as it makes a big difference if 
you have to draw out the multiplication facts.  

Thank you.
L.R.

Date: 12/05/2002 at 08:46:07
From: Doctor Peterson
Subject: Re: Multiplication and placement of factors

Hi, Leslee.

You're not the first to have asked this! You can read some previous 
answers to similar questions here:

   Defining Multiplication
   http://mathforum.org/library/drmath/view/61066.html 

   Multiplicand, Multiplier
   http://mathforum.org/library/drmath/view/58567.html 

Here is how I would answer the specific question as you express it:

When we write something like

     835
    * 24
    ----

in order to multiply two numbers, we are not representing a physical 
problem on paper. Rather, we have already determined that in order to 
solve a problem, we must multiply these two numbers together; and we 
know certain techniques for doing so. One of the things we know is 
that the order doesn't affect the result; we can either multiply 835 
by 24, or multiply 24 by 835. So when we come to doing the actual 
multiplication, we can choose whichever way is easier. In this case, 
I would always put the 24 on the bottom, because it is smaller. So 
the fact that 24 is on the bottom has absolutely nothing to do with 
whether I have 24 groups, or groups of 24.

Even when written as 835 * 24 (or 24 * 835) the order does not really 
reflect the problem. As you'll see in one of the answers cited, I 
myself vacillate on how to interpret it. That's because, again, it 
makes no difference at all. It may be of use to choose one meaning 
when you first introduce multiplication to children, and stick with 
that yourself for the sake of consistency; but you must not insist 
that they learn that as the only correct order, but rather should 
emphasize from the start that the order is unimportant, and that this 
gives them the freedom to see the same multiplication in two ways, 
and to change views at will if it makes anything easier.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/