Distributive Property over Addition vs. Multiplication

Date: 12/11/2002 at 02:25:05
From: Kathy O'Neill
Subject: Distributive property over addition vs. multiplication

Please help me explain the following to my child:

His math problem was (3x3) = (4+5)

He understands that both sides of the equation equal nine.

Then he was asked to multiply both sides by 2.

Although 2(3x3) = 18 and 2(4+5) = 18 he wanted to use the distributive 
property. But then he got 36 equals 18. Why can't you use the 
distributive property when the integers in the parenthesis are being 
multiplied if it works when you ADD inside the parenthesis?

Date: 12/11/2002 at 08:55:19
From: Doctor Peterson
Subject: Re: Distributive property over addition vs. multiplication

Hi, Kathy.

The simple answer is that there is only a distributive property of 
multiplication over addition, not a distributive property of 
multiplication over multiplication. The closest you can get to the 
latter is the associative property:

    2*(3*3) = (2*3)*3

because we can move the parentheses around.

You might picture it this way. The distributive property says that 
multiplying a sum multiplies each part of the sum:

    * * * * * * * * *     * * * *   * * * * *
    * * * * * * * * *  =  * * * * + * * * * *
    \_______________/     \_____/   \_______/
         2*(4+5)       =    2*4   +    2*5

But if we double two numbers, and multiply them together, we don't 
just double the product; we double it twice:

    * * * * * *    * * *   * * *
    * * * * * *    * * *   * * *
    * * * * * *    * * *   * * *
    * * * * * *    
    * * * * * *    * * *   * * *
    * * * * * *    * * *   * * *
                   * * *   * * *
    \_________/    \___________/
     (2*3)(2*3)  =  (2*2)(3*3)

That is, using the associative property,

    (2*3)*(2*3) = 2*3*2*3 = 2*2*3*3 = (2*2)*(3*3) = 4*(3*3)

which is not 2*(3*3).

Does that help?

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