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### Associative, Distributive Properties

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Date: 9/5/96 at 17:26:3
From: Michael F. Biondi
Subject: Associative, Distributive Properties

We would like to know how to explain the differences between
associative and distributive properties to our eighth grade algebra
student.
```

```
Date: 9/5/96 at 20:34:9
From: Doctor Robert
Subject: Re: Associative, Distributive Properties

The associative property is needed because addition and multiplication
are really binary operations.  By this I mean that you can only do
addition or multiplication with two numbers at one time.  So, how do
you handle this question:  1+2+3=?

You can either say (1+2)+3 or 1+(2+3) where the parentheses ( )
indicate which operation is to be done first. Luckily it turns out
that you get the same answer both ways. This is the associative
property. It doesn't make any difference which numbers you associate
together.

When you have multiplication and addition in the same problem, it's a
bit different. The order of operations requires that when you
simplify 3 x (4+5), you do the operation inside the parentheses
first. However, you get the same answer if you distribute the 3 over
the addition and write it as 3x4+3x5.

This is the distributive property of multiplication over addition. It
doesn't seems like such a big deal in straight arithmetic questions,
but when variables are involved it can help one simplify an otherwise
messy situation.

For example: 3(x+5) + 2(x-4) can not be simplified since I don't
know what x stands for and the parentheses must be done first.
BUT, if I invoke the  distributive property the expression becomes

3x+15+2x-8 = 5x +7

which is quite a bit simpler than the original quantity.

I hope that this helps.

-Doctor Robert,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Elementary Definitions
Elementary Multiplication
Middle School Algebra
Middle School Definitions

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