Distributive Property with Negatives

Date: 10/13/2004 at 21:26:37
From: Albert
Subject: Distributive property with negatives

I am a grade eight student enrolled into a grade nine math program.

When using the distributive property with negative numbers, I don't
understand when, why, and how the operator changes (from addition to
subtraction, or the other way).  Sometimes it changes, sometimes it
doesn't.

It gets me extremly frustrated.  I never get it.  Can you please
explain it when to change it and why?

Date: 10/14/2004 at 12:50:47
From: Doctor Ian
Subject: Re: Distributive property with negatives

Hi Albert,

Here's the thing.  If you try to learn a lot of rules for what to do,
you're going to get them mixed up, and forget them when you're under
pressure (e.g., during a test).  I suggest that you start by always
converting subtractions to equivalent additions:

For example, suppose I have

    (-3 + 4) - (7 - 2)

  = (-3 + 4) - (7 + -2)             Change to addition

  = (-3 + 4) + -1(7 + -2)           Change to addition

  = (-3 + 4) + (-1*7 + -1*-2)       Distribute the -1

  = (-3 + 4) + (-7 + 2)

and now all I have are additions.  So I can use the associative and
commutative properties to do them in any order I want:

  = -3 + 4 + -7 + 2

  = -3 + -7 + 4 + 2

  = -10 + 6

  = -4

The same thing works with variables:

    (-3 + x) - (2x - 2)

  = (-3 + x) - (2x + -2)             Change to addition

  = (-3 + x) + -1(2x + -2)           Change to addition

  = (-3 + x) + (-1*2x + -1*-2)       Distribute the -1

  = (-3 + x) + (-2x + 2)

  = -3 + x + -2x + 2

  = -3 + 2 + x + -2x

  = -1 + -x

Does this make sense?    

This might seem like a lot of extra trouble to go through, but if you
do the extra steps, you'll get the right answers slowly, which is 
better than getting the wrong answers quickly.  And as you get 
practice, you'll start forming your own set of rules for shortcuts,
which will make sense to YOU--and that's better than trying to learn a 
set of rules that someone else made up, because they made sense to 
HIM. 

Does this help? 

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

Date: 10/14/2004 at 16:46:21
From: Albert
Subject: Thank you (Distributive property with negatives)

Thank you very much.  It absolutely works!  I dunno if I passed the
test, but if I do, it's thanks to you!