Date: Dec 15, 2012 2:27 PM
Author: Joe Niederberger
Subject: Re: Some important demonstrations on negative numbers > a MACS<br> syllabus
Clyde Greeno says:
>By failing to so develop the concept of "slope" from the multiplication tables for the whole numbers,
Is that true? I'm not exactly who you are referring to, but I agree that the 4 quadrant graph and viewing mult. as a family of lines through the origin is very desirable, a great unifying picture. They can tie together multiplication, division concepts, (whole number, fractional, real number, and signed,) similar triangles, rates of change, and later, variable rates of change. Slope seems to get stuck in fairly at some point as an almost separate topic that comes with a formula for calculation, and is just a precursor to "formulas for lines in a plane."
For example, it is easy to illustrate the distributive law with such a picture. Most often, the distributive law is pictured as an N x M grid divided into two smaller rectangular grids. Now, I've not seen the corresponding picture (based on sloped lines) used at all. A Google search for images related "distributive law" does not show this picture: the 1st quadrant (or all 4), a line of slope 3 representing the values of "3x" for any x, and two of what I call "knights moves" (over and up). So, 3(3+2) can be illustrated as two such moves (over 3 and up, over 2 more and up) and of course you come to the same place as if you had moved over 5 and up originally.
It can be shown pictorially then, if we want the distributive law to work in all 4 quadrants, that "multiplication" needs to be extended to keep the nice straight lines going as they are.
So, I'm with you that those kinds of illustrations should perhaps be used more and earlier, augmenting standard "number line" pictures as soon as multiplication is developed.
On the other hand, I wouldn't call such aids "a common sense understanding" of the sign rules. I'm not convinced there is any common sense understanding of the signed number system (integers), if common sense means "readily understandable how to map to everyday concerns". But that's OK, its not as if its a truism that everyone should have a common sense understanding of how a watch works. They don't, and they need not. But, a watch can be taken apart piece by piece and an appreciation generated for how the various simple elements work together. A problem with the integer system is that, unlike a watch, its not even clear to most people what the composite object is for! As Jonathan Crabtree has pointed out, its not really necessary for everyday bookkeeping duties.
>What now are commonly called "negative" numbers are more
accurately called "negator numbers."
How about "anti-numbers" similar to anti-matter?
When we add like quantities of opposite sign, we get a kind of a annihilation. That's nice, but with the sign rules and multiplication, we also get a new capability: reversal, with our new "toggling operator" (-1). Now
that *is* a somewhat common sense notion, of sequentially alternating between opposites; night and day, love and hate, etc. Its wrapped up in some strange formalism, but it is recognizable.