Subtracting Integers
Objective Learn how to subtract both positive
and negative integers.
In this lesson, you will study how to do arithmetic with
integers (both negative and positive). We will be studying
subtraction in this lesson. The most difficult part of this topic
is the confusion that is sometimes experienced regarding
“where to put the minus sign”.
Opposites
Let's begin by recalling that the opposite of any positive
integer is the negative integer that is the same distance from 0
on a number line. So, the opposite of 6 is 6, and the opposite
of 25 is 25. Conversely, the opposite of any negative integer is
the positive integer that is the same distance from 0 on a number
line. Thus, the opposite of 8 is 8, and the opposite of 31 is
31.
Subtracting Integers
How do we subtract a negative integer from another integer? Have
you any ideas about how to subtract a negative integer from a
positive integer? And what about subtracting a negative integer
from another negative integer?
Remember that subtraction and addition are opposite
operations.
Key Idea
Subtracting a negative integer gives the same result as adding
the opposite positive integer.
Let's demonstrate why this is true using shaded squares in the
following way. Think about collections of gray (positive) and
black (negative) squares. Then, subtracting a negative integer
from another negative integer can be modeled by taking a group of
black squares that model the first negative integer and removing
the number of black squares (if there are enough in the model)
that represent the second negative integer. When there are not
enough black squares in the model, a sufficient number of pairs
of gray and black squares need to be added to the model so that
the required number of black squares can be removed. When a
negative integer is subtracted from a positive integer there will
not be any black squares in the model for the first integer. In
this case, pairs of gray and black squares must be added to the
model to begin the subtraction process.
Let's take a look at the following examples, some using models
to find the solution and others done by direct calculation. One
example of each type is given here.
Example 1
What is the value of 6  (2)?
Solution
To model 6  (2), first represent 6 as a collection of 6 gray
squares. Since there are not 2 black (negative) squares that can
be removed to model subtracting 2, add two pairs of squares, one
black and one gray in each pair. (Remind students that we removed
pairs like these because they “canceled” each other;
therefore, we can also add such pairs because adding them will
not change the final result.)
There are now 8 gray squares and 2 black squares in the model.
So we are able to show the subtraction of 2 by removing the 2
black squares, leaving 8 gray squares.
Therefore, the model shows that 6  (2) = 8.
Example 2
Find 5 ( 2).
Solution
Use the fact that subtracting a negative integer gives the
same result as adding the opposite positive integer. This means
that 5  (2) is equivalent to 5 + (+2) = 3.
To see this in terms of a model, we can represent 5 as a
collection of 5 black squares and then simply remove 2 black
squares. The result is a model showing 3 black squares, which
represents 3.
Try to solve subtraction problems involving negative integers
by using shaded squares, colored counters, or some other type of
manipulatives. Then do the subtractions directly by using the
idea that subtracting a negative integer is the same as adding
its opposite.
