Properties, Defined and Illustrated

Date: 10/10/2001 at 11:25:54
From: rad2b
Subject: Property help

I need a page of information on the commutative, associative, and 
F.O.I.L. properties, to be used as a study sheet for a quiz.

Date: 10/10/2001 at 12:45:22
From: Doctor Ian
Subject: Re: Property help

Hi,

If you understand the distributive property, you can ignore FOIL:

   When FOIL Fails
   http://mathforum.org/dr.math/problems/ryan.03.22.01.html   

For that matter, if you _understand_ the commutative and associative 
properties, then you don't need a study sheet to help you 'remember' 
them. (To put that another way, if you have to 'remember' these 
properties, then you probably don't _understand_ them.) 

Take a look at the following diagrams, and construct enough other 
examples to be sure that you would be able to explain to someone else 
why the properties hold for _any_ numbers:
      
  Commutative property of addition

       3      4
     ----- -------
     * * * * * * *                   3 + 4 = 4 + 3
     ------- -----
        4      3


  Commutative property of multiplication
    
     * * *       * * * *
     * * *    =  * * * *             3 * 4 = 4 * 3
     * * *       * * * *
     * * *

  Associative property of addition

          (2+3)+4
     -----------------
        2+3
     ---------
     * * * * * * * * *              (2 + 3) + 4 = 2 + (3 + 4)
         -------------
              3+4        
     -----------------
          2+(3+4)

  
  Associative property of multiplication

        +--+--+--+--+
       /  /  /  /  /|
      +--+--+--+--+ +
     /  /  /  /  /|/|
    +--+--+--+--+ + +        (2 * 3) * 4 = 2 * (3 * 4)
    |  |  |  |  |/|/|
    +--+--+--+--+ + +
    |  |  |  |  |/|/
    +--+--+--+--+ +
    |  |  |  |  |/
    +--+--+--+--+


  Distributive property of multiplication over addition:

     2    3
    --- -----
    @ @ * * * |
    @ @ * * * | 4             (2 * 4) + (3 * 4) = (2 + 3) * 4
    @ @ * * * | 
    @ @ * * * |         
    ---------
      2 + 3


See also the Glossary of Properties from the Dr. Math FAQ:

   http://mathforum.org/dr.math/faq/faq.property.glossary.html   

I hope this helps. Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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