Question: How can you use geometry to demonstrate that the sum of two negative numbers is positive?
Click HERE for a GSP file to explore. The GSP file begins with the configuration at the right.
Consider two lines that cross and indicate 0 as the common point. From the definition of a negative number as the opposite or additive inverse of a positive number, let each line have locations in the positive and negative directions. From the multiplicative identity, 1, indicate a unit in each direction on each line..
Any point on a line has an orientation, positive or negative, determined by a location.
Showing the product of two numbers
To demonstrate the product of two numbers p and q, locate length p on one line and a length q on the other.
Construct a segment from 1 on the line with p to the end of length q on the second line.
Construct a parallel line from p to its intersection with the line containing q.
We have similar triangles.
That is, the product pq is a length on the line containing q. In this case the demonstration is the product of two positive values.
Use this technique where one or both of the values is negative to show that
-- the product of a positive number followed by a negative number is a negative number.
-- the product of a negative number followed by a postitive number is a negative number.
-- the product of two negative numbers is a positive number.
After you have done your own explorations, go HERE to see all of these demonstrations.
There are alternative constructions. Yours may be better. The issues may be on the pedagogical considerations.