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Q&A #1733


Long division

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From: Gail (for Teacher2Teacher Service)
Date: Jul 08, 1999 at 06:48:28
Subject: Re: Long division

I agree with Suzanne that there are different methods.  I have found that
some elementary students do just fine with the traditional method, but some
need a little help going from the concrete step to that abstract algorithm.

I ask my students to think about what they think the answer should be.  For
example, if the problem is 478 divided by 4, about how many groups of 4 do
you think there are in 400?  How many in 500?

The answer to the problem should be somewhere between those two amounts.
Using base ten blocks, and making groups of four (which is tedious), or four
equal groups, using the blocks, is one way to make this hands on.  Students
can arrange the blocks to find that the quotient is somewhere between the
number of groups in 400 and the number of groups in 500.

Then, I like to start with an alternate algorithm for division, rather that
the traditional method.  I have found that the transfer to the traditional
is very smooth, and this alternate method gives students who are still
struggling to learn the multiplication facts, a chance to be successful,
even if they only know the 1's, 2's, 5's and 10's. You might consider an
alternate algorithm:

It is easier for them because they do not have to have the prefect factor
each time, just one that will work.  They can continue finding factors and
dividing until they have a remainder that is smaller than the divisor (in
this case, 7).  When they have found all the factors, they just add them up
(not the 7, but the other one each time).  That is their quotient, and what
is left is the remainder, which can be put in fraction form (in this case, 2
out of 7, or 2/7) , or they can place a decimal, and add zeroes to continue
dividing.

Besides the help this gives a student, it treats the dividend (in this case,
478) as a whole amount, instead of looking at 4 hundreds, then 48 tens, as
the traditional algorithm does.

         ____
     7 /478     7 X 10
       - 70
       _____
        408     7 X 10
        -70
       ____
        338     7 X 10
        -70
       ____
        268     7 X 10
        -70
       ____
        198     7 X 10
        -70
        ____
        128     7 X 10
        -70
        ___
         58     7 X 2
        -14
        ___
        44      7 X 2
       -14
       ___
       30       7 X 2
      -14
       ____
       16       7 X 2
      -14
       ___
        2

     10 + 10 + 10 + 10 + 10 + 10 + 2 + 2 + 2 + 2= 68

See, this student was able to finish the problem without using any of the
facts except 7 X 10 and 7 X 2.  This could be of great assistance to someone
who understood why we needed to divide, but did not have the facts mastered
yet.

I try to move my students to try larger "chunks" once we have that first step
mastered.
        ____
     7 /478     7 X 30
       -210
       _____
        268     7 X 30
       -210
       ____
         58     7 X 5
        -35
       ____
         23     7 X 3
        -21
       ____
          2

     30 + 30 + 5 + 3 = 68

Then we have a challenge to try to find the one "best chunk" for each place
value.  In the next example, there is one "chunk" for the tens, and one for
the ones.  (There is no hundreds chunk, because 7 X 100 would be more than
478).
         _____
     7 /478     7 X 60
       -420
       _____
        58     7 X 8
       -56
       ____

          2

     60 + 8 = 68

You will probably notice that that is very similar to the traditional
algorithm.  It is just a small nudge away for most students, at that point.
:-)

 -Gail, for the Teacher2Teacher service

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