From: Pat Ballew (for Teacher2Teacher Service)
Date: Jun 13, 2000 at 00:00:27
Subject: Re: Understanding abbreviation

The teacher who wrote the previous response I spoke of has pointed me to the
message, so I have copied it below.  Notice that it also includes a third
idea, the prime factorization, as part of the same approach,  a powerful
triunal concept.
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 Jeanne wrote, in answer to Larry's question about factor trees:

Hello Larry,

I agree with Gail. The orientation of the tree/root system isn't as
important as the information one obtains from them.

Factor trees have a way of "growing" in such a way that my weaker students,
especially those whose handwriting is messy, lose track of their factors.  I
teach my students factor trees and another graphic organizer which I'd like
to share with you.

For example, if my students want the prime factorization of 84, I have them
write an "upside down long division symbol" and place 84 in it.

       )_84__    (Pretend the horizontal line continues under the 84.)

Then I ask, "What prime number is a factor of 84?"  The kids usually say "2."

They record the 2 in front of the symbol and divide recording their quotient
below the 84.

      2)_84__
         42

Put another upside down division symbol around the 42.

      2)_84__
       )_42__

"What prime number is a factor of 42?"  "2."  Record the 2, divide and
record the new quotient and upside down division symbol.

      2)_84__
      2)_42__
       )_21__

Repeat until 1 is your quotient.
      2)_84__
      2)_42__
      7)_21__
      3)__3__
          1

The prime factorization of 84 in appears in a column on the left.


The greatest common factor and least common multiple can be found using the
same graphic organizer.  I teach the concepts of GCF and LCM from a variety
of perspectives.  What I am about to share is simply a method to find them.


Suppose we want to find the GCF of 12 and 18.  Set up the upside down
division symbol as follows:

     )_12__|__18__

"What factor (it doesn't have to be prime) is common to both?"  "2"
Record.

    2)_12__|__18__
        6      9

"What factor is common to both 6 and 9?"  "3"

    2)_12__|__18__
    3)__6__|___9__
        2      3

"What factor is common to both 2 and 3?"  "1"

    2)_12__|__18__
    3)__6__|___9__
    1)__2__|___3__
        2      3

Conclusion:  GCF of 12 and 18 is 2*3 or 6.

Another conclusion:  LCM of 12 and 18 is 2*3*1*2*3 or 36.

My kids really like this way of keeping things organized!



 -Jeanne, for the Teacher2Teacher service
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Hope this much better explanation helps.

 -Pat Ballew, for the Teacher2Teacher service

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