Volume XXXIV, No. 14 | April 27, 2012

It Makes a Difference

Let’s start with the question “What’s the difference between 7 and 3?” The answer is 4. If you ask students how they got that answer, they will say something like, “I took 4 from 7,” or “I subtracted them.”

The point is that the math operation is subtraction. Demonstrating this on a thermometer (as a matter of personal preference, I make the number line vertical), the difference between 7 and 3 is also the distance between those two points on the thermometer. So when students are asked to find the difference between two numbers, they are also being asked to find the distance between those two values on a thermometer. The distance could be measured from the 7 or the 3. Always, always take the difference (or the distance) from the second term, or the term that is being subtracted. In some cases, the student will have to move down the thermometer (down gives a negative value), while moving up gives a positive value.

Using this idea, look at the problem 7 – (-3). If these two values are located on the thermometer, the distance between them (the “difference”) starting at (-3) is 10. This makes it seem as though – (-3) results in +3, which it does. Being more specific, it appears that a +3 has been added. So, the original problem 7 – (-3), can be written as 7 + (+3). This is consistent with a current textbook definition of subtraction, which is “to subtract, add the opposite.”

It seems then that the key to addition is subtraction. When subtracting, put both values on the thermometer; then starting at the term being subtracted, find the distance to the other value. Try, for example, -4 – (+3). In this case, since it is subtraction, start at the +3 and move down to the negative four. This gives a “difference” of -7. This is consistent with using the rule of adding the opposite.

What about -4 + (+3)? As noted above, 7 – (-3) can be written as 7 + (+3). The nice thing is that subtraction is commonly defined as “adding the opposite.” Given this and what was demonstrated above is that its converse is also true—to add, subtract the opposite! So, -4 + (+3) becomes -4 − (-3). Since it is subtraction, put both values on the thermometer, and starting at (-3), the distance to minus -4 is -1. So, -4 + (+3) = -1.

I haven’t yet tested this out with students. This visualization, however, captures all the rules for operations with signed numbers. It just takes a “different” approach.

Mark Schwartz, Adjunct Instructor, Math

For further information, contact the author at Southern Maine Community College, 2 Fort Road, Portland, ME 04106. Email Author.