Modeling Multiplying Two Negatives with Number Lines

Date: 11/13/2006 at 07:36:28
From: Joan
Subject: -2*-3= 6???

I'm trying to use a number line to figure out why -2 times -3 makes 6,
and I can't do it.  If I start at -2 and move to the right three times
I wind up at 4, and if I move to the left I wind up at -8.

What is the logic behind the rule of a negative times a negative makes
a positive, and how do I demonstrate it on a number line?

Date: 11/13/2006 at 08:28:45
From: Doctor Rick
Subject: Re: -2*-3= 6???

Hi, Joan.

You can find a variety of ways to think about multiplication of 
negatives, and also ways to *prove*, more or less formally, that a 
negative times a negative is positive, in the Dr. Math FAQ:

  Negative X Negative = Positive
    http://mathforum.org/dr.math/faq/faq.negxneg.html 

It's hard to demonstrate signed multiplication on a number line, but 
there is a way to do it using *two* number lines.  See what you think 
of this:

Draw two number lines that cross at the origin on both lines--the 
angle between the lines doesn't matter.  To multiply a number x by 
another number y, first draw a line through 1 on the first line and 
x on the second line:

               /
              /
             /
            /
           /
        x /
         *
        /
       /
      /
  ---0--*-----------
    /   1
   /
  /

I can't draw that line, you'll have to do it on paper.  Then draw 
another line, parallel to this one and passing through the number y 
on the first number line.

               /
            z /
             *
            /
           /
        x /
         *
        /
       /
      /
  ---0--*--*--------
    /   1  y
   /
  /

You can easily prove by similar triangles that 

  x/1 = z/y

so that

  z = xy

That is, the position of z on the second number line is the product 
of x and y.

Now, use this same graphical method to multiply -2 by -3:

                               /
                              *-2*-3=6
                             /
                            /
                           /
                          /
                         /
                        /
                       /
                      /
                     /
                    /
                   /
                  /
  --+------------0----+-------------
   -3           /     1
               /
              /
             *-2
            /
           /

Draw a line connecting 1 on the first (horizontal) number line with -2
on the second number line.  Draw a line parallel to this line, and
passing through -3 on the first number line.  You will see that it
intersects the second number line at 6, which is therefore the product
of -2 and -3.

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