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### 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/
```
Associated Topics:
High School Basic Algebra

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