Changing Decimals to Whole Numbers when Dividing

Date: 12/10/2001 at 09:40:09
From: Joann Calabrese
Subject: Dividing by decimals

I am trying to think of ways to explain to my students why a decimal 
needs to be a whole number before one can divide.

Date: 12/10/2001 at 10:52:30
From: Doctor Ian
Subject: Re: Dividing by decimals

Hi Joann,

You might start by considering why you think that's the case.  Are you 
saying that I can't divide 3.6 by 2.4 without converting the 2.4 into 
a whole number? I'm pretty sure I can.  

Maybe you're having a hard time thinking of ways to explain it, 
because it isn't true.  Could that be the case? 

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

Date: 12/10/2001 at 18:53:22
From: Joann Calabrese
Subject: Re: Dividing by decimals

Maybe.  What about 2.003 divided by .02?

Thanks.

Date: 12/10/2001 at 22:46:39
From: Doctor Ian
Subject: Re: Dividing by decimals

Hi Joann,

Well, I'd look at that and think: 0.02 goes into that at least 100 
times, right?  And

  2.003 - (100)(0.02) = 0.003

So the answer is 100 + something. Now, 1/10 of 0.02 is 0.002, and 2/10 
of 0.02 is 0.004, and 0.003 is halfway between, so the something must 
be halfway between 1/10 and 2/10, which is 0.15. 

So 2.003 divided by 0.02 is 100 + 0.15.  

Of course, I wouldn't normally do it that way.  I'd normally do it 
this way:
             2003
             ----
  2.003      1000      2003   100    2003
  ----- = ---------- = ---- * --- = ------ = 200.3 / 2 = 100.15
   0.02        2       1000    2    10 * 2
              ---
              100

Now, in a sense, I _have_ converted the decimals to integers, although 
I really converted them to fractions. I suppose the distinction is 
somewhat academic. 

However, the fraction trick shows why the trick of 'converting to 
integers by moving the decimal points' doesn't change the result of 
the division.  

Now that I think about it, I would probably just do this: Dividing by 
2/100 is the same as multiplying by 100/2, or 50; so

  2.003 divided by 0.02 = 50 * 2.003

                        = 50*2 + 50*0.003

                        = 100 + 0.150

But the important point is that there are lots of different ways to 
divide one number by another. We tend to teach one particular 
algorithm (long division), without worrying too much about whether the 
students understand what's going on; and that particular algorithm 
seems to work best when we manipulate the decimal point out of the 
divisor, but it's not _necessary_ to do that. It's just that lots of 
people find it _convenient_, because they find it hard to multiply 
anything except integers in their heads.  

A few examples should help your students see that it _is_ more 
convenient. But you'll never be able to convince them that it's 
_necessary_, because it isn't.  

Does this help? 

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

Date: 12/11/2001 at 12:28:46
From: Joann Calabrese
Subject: Re: dividing by decimals

Thanks! I couldn't agree more about the more than one way idea! I like
to be able to justify to them why someone decided one algorithm makes 
more sense or is easier to work with than another.

Joann