We have seen "adding" and "subtracting" of integers using a number line.
Let's take a look at another way to visualize working with integers as signed numbers.
On this page, we are going to be using "counters" to visualize adding and subtracting of integers.
Teachers will recognize this process as a modification of using Algebra Tiles.
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The red counter "chip" represents a "negative" one.
The green counter "chip" represents a "positive" one.
 One red chip with one green chip equal zero. + 1 + -1 = 0
The red and green chips are "opposites". They cancel each other out.
Addition: Show 3 + (-5) = - 2
The process of addition is the combining of the two quantities.
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The process of SUBTRACTION is the removal of a certain quantity from another quantity.
When working with counters, it will be the removal of a certain number of chips from the first number in the problem. If there are NO negative (or positive) chips in the first number of the problem to be removed,
the needed chips can be introduced using "pairs" of opposites.
This will add "zero" to the problem, thus not changing the values.
Subtraction: Show 5 - (-2) = 7
The process of subtraction is the removal of -2 (or two red chips).
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Since positive 5 contained NO negative chips to be removed, it was necessary to rewrite 5 by introducing the two needed negative (red) chips (but not changing the value of 5). The quantity of zero (one red and one green) was introduced twice into the problem to accomplish this task.
Understanding the Subtraction RULE:
This manner of viewing the process of subtracting integers, shown above,
is justification of "WHY" we use the "Rule" for subtracting signed numbers.
RULE: "To subtract signed numbers,
change the sign of the number being subtracted
and then follow the rules for addition."
You can now "see" that changing the sign of the number being subtracted is actually just using its "opposite" which is left behind after the subtracted number was "taken away".
And, the "follow the rules for addition" is the introduction of a plus sign when the zero quantity was combined with the first value.
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