Distributive Property, Illustrated

Date: 09/28/2001 at 22:35:48
From: Tom
Subject: Math is hard

I need to show 12 times 2 + 12 times 3 using the distributive property 
and I don't know where to start.

Date: 09/29/2001 at 09:56:08
From: Doctor Ian
Subject: Re: Math is hard

Hi Tom,

Suppose you have some cookies, and you arrange them in a 12 by 2 
rectangle:

  @ @ @ @ @ @ @ @ @ @ @ @
  @ @ @ @ @ @ @ @ @ @ @ @

And suppose you have some other cookies, and you arrange them in a 12 
by 3 rectangle:

  * * * * * * * * * * * *
  * * * * * * * * * * * *
  * * * * * * * * * * * * 

If you put these two rectangles together, you get this:

    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * *    = (12 * 2) + (12 * 3)
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 

The number of items in each rectangle is the height times the width.  
This includes the larger reactangle. So we have two different ways 
that we can compute the number of cookies without counting them all: 

    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * *    = (12 * 2) + (12 * 3)
    @ @ * * * 
    @ @ * * *    = 12(2 + 3)
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 
    @ @ * * * 

And this is the distributive property. It just says that instead of 
doing the two separate multiplications, you can do an addition and a 
multiplication. It's usually written with letters instead of with 
particular numbers, to let you know that it works for _any_ numbers at 
all:

  a(b + c) = ab + ac

Sometimes using the property to rewrite an expression makes things a 
lot easier to compute. For example, which of these would you rather 
compute?

  1. (210 * 196) + (210 * 54) = ?

  2. 210(196 + 54) = 210 * 250 = ?

Sometimes you can use it to turn one hard operation into two easy 
ones:

  210 * 196 = 210(200 - 4)

            = 210*200 - 210*4

            = 42000 - 840

But where the distributive property really becomes important is in 
algebra, where you'll be using it about every 30 seconds to simplify 
expressions like

                    ________________________
                   |                        |
                   |                        v
              ----------               -----------
  3(x + 2y) + 5(2x + 4y) = (3x + 6y) + (10x + 20y)
  ---------                ---------
      |                        ^
      |________________________|


                         = 3x + 10x + 6y + 20y
                           ---------  ----------
                               |          |  
                               v          v
                           ---------   ---------
                         = x(3 + 10) + y(6 + 20)

                         = 13x + 26y
                           ---------  
                               |       
                               v        
                           ---------    
                         = 13(x + 2y)

Each one of the arrows represents an application of the distributive 
property. 

Trying to learn algebra without having a real understanding of the 
distributive property is like trying to learn to cook without pots and 
pans. It's really worth spending some time to make sure you get it. If 
you have questions about it, write back, or bug your teacher, or get a 
friend to explain it to you... but do _not_ think that you can keep 
going in math without being able to apply the distributive property in 
your sleep. 

I hope this helps. 

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