Making Guesses, Correcting Mistakes

Date: 06/07/2001 at 18:59:09
From: Julian Marrero
Subject: Division

When I was doing my divison homework, I was on a division problem I 
couldn`t figure out:
      ______
    6 )$9.12

This is when I got stuck:

    _0______
   6)$9.12
     -0
    --------
      9

I couldn`t "bring down" because 9 is higher than 6.

Date: 06/07/2001 at 22:34:21
From: Doctor Peterson
Subject: Re: Division

Hi, Julian.

This is the kind of question I love to see: you've told me just what 
you did so I can see just where your problem is. And it's a common 
problem!

One of the keys to doing long division is to be able to correct 
mistakes, because when you get to larger divisors you will have to 
make guesses, and can expect to do it wrong sometimes. There are two 
mistakes you can make in choosing a digit for the quotient: it can be 
too large, or too small. If it's too small (as your 0 is), then the 
product (0) is too small, making the remainder (9) too large, as you 
discovered. So when you see that 9 is bigger than 6, you know that you 
have to increase the quotient from 0 to 1:

      __1____
    6 ) 9.12
       -6
        -
        3

That looks better, doesn't it? This time I'm subtracting a bigger 
number, so the remainder is small enough to work.

Now I'm going to deliberately make the wrong guess for the next digit, 
just so I can show you what happens when your guess is too large. 
Suppose I try 6:

      __1.6__
    6 ) 9.12
       -6
        -
        3 1
        3 6
        ---

Do you see the problem? The 36 is larger than the 31, so I can't 
subtract. That tells me that the 6 is too large, and I should try 5 
instead:

      __1.5__
    6 ) 9.12
       -6
        -
        3 1
        3 0
        ---
          1

With these two tools under your belt, you can divide with confidence, 
because you'll never get stuck. When you do get stuck, you can just 
back up and try again, knowing what to do differently.

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