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Q&A #4130

Teachers' Lounge Discussion: European long division

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From: annie <niedbala_a@union133.org>
To: Teacher2Teacher Public Discussion
Date: 2008072520:36:29
Subject: Re: a German algorithm

Actually, there is more structure to the German style beyond the
multiplication approach you specified.

First off, the German method follows the conventions of all other
calculations where we write down the operation in the order we
complete it in. In the US the long division is a very disturbing
exception to this rule and so far I did not find a logical
(historical?) explanation for this. The "corner" very much resembles a
square root for the unsuspecting German eye seeing the US long
division algorithm for the first time.

12785 divided by 5 equals? (In lieu of the international division
symbol I'll use the backward slash):

12785 / 5 = _______

It will look exactly like the US long division and follows the same
lay out of subtracting and bringing down digits! Only the operation
itself is written out differently to reflect the flow of thinking in a
much more logical way.

You'll see the simple beauty of executing the operation German style
by following a circular sweeping motion first from left to right doing
the simple division and then from right to left completing the reverse
operation (multiplication) to figure out any remainders.

1 divided by 5? 5 does not go into 1, skip this and bring down next
digit

12 divided by 5? Yes, that would be two and 2 left over:

Here is the circular motion in action:

 from left to right operating above the equation you divide 12 by 5
(cover the other digits to make it obvious) and write down the first
result next to the = sign like this:
           to       
(left)   12XXX / 5 = 2   (right)

Now continue your circular motion right to left below the equation
with multiplying (reverse operation) 2 X 5 = 10 like this:
               
(left)    10  =   5   X   2    (right) 
               to
and writing the product down underneath the dividend we started with,
lining it up nicely, preferably using graph paper


12785   /   5    = 2557
10                  do right to left multiplication 10   = 5  x  2
-____
 27   pull down 7 to get next dividend and divide by 5 (left to
right)! 
 25            do right to left multiplication 25   = 5  x  5 
-_____
  28      pull down 8 to get next dividend and divide by 5 !
  25          do right to left multiplication 25   = 5  x  5
-______
   35    pull down 5 to get next dividend and divide by 5
   35           do right to left multiplication 35   = 5  x  7
-_______
    0


 
       

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