Longer Division

Date: 09/27/2002 at 14:12:17
From: LaDonne
Subject: Helping my elementary school daughter understand the concept 
of division

My daughter is not grasping how to do division. She is in the 5th 
grade. They are working on long division with remainders. I get 
frustrated because I show her over and over again and she IS NOT 
understanding how to divide. Please suggest something that I can do 
to help her comprehend.

Thanks.

Date: 09/27/2002 at 16:29:47
From: Doctor Ian
Subject: Re: Helping my elementary school daughter understand the 
concept of division

Hi LaDonne,

The thing about long division is that it's designed to be efficient.  
But you're not _required_ to be efficient about division! And 
sometimes it's much easier to see what's going on when you're not. 

For example, suppose I want to divide 3466 by 41.  In long 
division, I start this way:

     ______
  41 ) 3466         Hmmmm.  41 doesn't go into 3, or 34,
                    but it goes into 346.  How many times?

At this point, you start guessing - is it 8?  Is it 9? Frankly, in 
most cases, it's a pain.

In long division we have to get each choice 'right' because the 
algorithm is set up so that we can keep going from left to right, 
without having to go back. But what if we don't mind coming back?  
Then we can do something like this:

  41 * 10 is 410, so 41 goes into 3466 at least 10 times.  

   3466
  - 410      10 times
  -----
   3056

  Wow.  It goes in a _lot_ more than 10 times!  Will it go 
  in 50 times?  100 times would be 4100, so 50 times would 
  be half that, or 2050.

    3466
  -  410      10 times
  ------
    3056
  - 2050      50 times
  ------ 
    1006

  Okay, it will go at least 20 more times.

    3466
  -  410      10 times
  ------
    3056
  - 2050      50 times
  ------ 
    1006
  -  820      20 times
  ------
     186

  I can just keep subtracting now:

    3466
  -  410      10 times
  ------
    3056
  - 2050      50 times
  ------ 
    1006
  -  820      20 times
  ------
     186
  -   41       1 time
  ------
     145
  -   41       1 time
  ------
     104
  -   82       2 times
  ------
      22       remainder

Now I just count up the number of times I subtracted groups of 41:  

  10 + 50 + 20  + 1 + 1 + 2 = 84

so the answer is 84 remainder 22. 

So what's the difference between long division and what I did here 
(which we might call 'longer division')?  What's different is that I 
didn't force myself to get an '8' in the tens column on the first try!  
Instead, I broke it into 3 separate tries: 10, 50, and 20.  

Other than that, it's the same thing. Suppose I had guessed 80 the 
first time, and 4 the second time. This is what would have happened:

    3466
  - 3280     80 times
  ------
     186
  -  164      4 times
  ------
      22      remainder

Does this look familiar?  It's just long division with the numbers in 
different places.  
                                          84
                                      ______
    3466                              ) 3466
  - 3280     80 times                    328
  ------                                 ---
     186                   <==>          186
  -  164      4 times                    164
  ------                                 ---
      22      remainder                   22

There are two nice things about starting with 'longer division'. The 
first is that it's easy to understand - it's just grouping and 
subtraction, and everything is right there out in the open.

The second nice thing about it is that as you get better at guessing, 
it more or less turns into long division naturally... but you take 
that step on your own schedule, not when someone else thinks you 
should.  

Let me know if this helps. If not, we can try to think of 
something else that will. 

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