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Negative Times Negative

Date: 10/17/98 at 13:54:25
From: Julia Alaniz
Subject: Negative values

Could you provide an example for comprehending how a negative times a 
negative is a positive? - maybe on the number line. I would like to 
understand it rather than just take it as fact.

Date: 10/23/98 at 20:45:57
From: Doctor Gail
Subject: Re: Negative values

Dear Julia,

Here are a couple of examples for you:

If you are convinced that a positive times a negative is negative, then 
one way to look at negative times negative is to observe a pattern.  
Consider the following:  

     3x(-3) = -9
     2x(-3) = -6
     1x(-3) = -3
     0x(-3) =  0

so what comes next in the pattern?  Notice that in each case we are 
multiplying 3 by a number 1 smaller than the previous number. Also 
observe the pattern that is being created with the products. So the 
next logical multiplication sentence is 

  (-1)x(-3) = 3

Note that negative times negative was positive.

One of the best realistic interpretations that I have found concerns 
videotaping. Consider videotaping two students, Julia who is walking 
forward in the tape and holds a sign saying "I am walking forward."  
Also consider her friend who is walking backward in the tape and holds 
a sign saying "I am walking backward."  Now playing the tapes in the 
normal fashion would be forward times forward or positive times 
positive and Julia looks like she is walking forward (the result is 
positive). Similarly, the meaning of negative times negative might be 
interpreted as viewing "rewind" times "the friend" or backward times 
backward. What would it look like the friend was actually doing?  
Walking forward! So one can visualize backward times backward is 
forward or negative times negative equals positive.

I hope this explanations begin to convince you that negative times 
negative equals positive.

Doctor Gail, The Math Forum   

Date: 10/23/98 at 21:52:48
From: Julia Alaniz
Subject: Thanks to Docter Gail

What a most unusual example for visualizing a negative times a negative 
- the videotaped example.

I am actually going to try it since I do video training tapes. Thanks 
ever so much.

Julia Alaniz
Associated Topics:
Middle School Negative Numbers

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