A Week of Adding and Subtracting and Subtracting by Adding

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Integers are wonderful things. To add and subtract Integers all you need are three simple rules. The first two rules, which I discussed in my previous post, Same Sign Sum and Different Sign Difference, are used for both Integer Addition and Integer Subtraction. However, for Integer Subtraction, on additional rule needs to be applied first.

Theoretically, when it comes to Integers, there is no such thing as Subtraction. In order to subtract one Integer from another, one instead adds the opposite of the Integer being subtracted to the Integer being subtracted from. To help students remember to do this, I use the acronym KCC, which means Keep, Change, Change.

Keep, Change, Change reminds the student to Keep the first Integer in the expression the same, change the minus operation sign to a plus sign, and change the second integer to its opposite. For example:

        -5 – (-7) when KCC is applied becomes -5 + 7.

The student then applies the appropriate Integer Addition rule, in this case, DSD because two Integers with different signs are being added. The difference between the Absolute Values of (-5) and 7 is 2. [|7| - |-5| = 7 – 5 = 2] {Note: The bars on either side of the Integer are known as Absolute Value Bars.} Since the Absolute Value of the Positive Integer is greater than the Absolute Value of the Negative Integer the sum, 2, will be Positive.

Let’s look at another example.

        4 – 9

In this example we are attempting to subtract a larger number from a smaller number. In the set of Whole Numbers this would not be possible. Since Integers include all the Whole Numbers and their opposites (the Negative Numbers) this subtraction can be done using Integers.

4 – 9 = 4 + (-9) = (-5) because |-9| - |4| = 9 – 4 = 5 and since the Absolute Value of (-9) is greater than the Absolute Value of 4, the sum, 5, will be Negative, or (-5).

Both of these examples result in using the rule Different Sign Difference to determine the answer. Now we’ll look at an example that uses Same Sign Sum.

-18 – 14. {Note: The 14 being subtracted is Positive.} Use Keep, Change, Change to rewrite this expression as -18 + (-14). Since both Integers are Negative, we take the sum of their Absolute Values |-18| + |-14| = 18 + 14 = 32, and we give the sum the same sign as the original pair of Integers, (-32).

Students have more difficulty with Integer Subtraction than Integer Addition mostly due to them wanting to change both terms in the expression as well as the operation. To counter this, we practice Integer Subtraction more than Integer Addition. Of course, by practicing Integer Subtraction we are getting additional practice in Integer Addition. But don’t tell my students.

As always, I remain,

The Exhausted Educator