## It Makes a Difference

Posted by mark schwartz on January 7, 2012

**IT MAKES A DIFFERENCE**

In Milne’s 1894 *Elements of Algebra, *he commented about operations with signed numbers. He noted that difference between two numbers on a number line is really the distance between them. I thought about this a while and reflecting on his discussion, developed it a little further.

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

The point is that the math operation is subtraction. If you showed 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. But if the difference (or the distance) is always, always taken starting from the second term (the term that’s being subtracted), something interesting happens. And, if in doing this, the student moves down the thermometer the difference is a negative value, while moving up gives a positive value. Take a moment to reflect on these 2 sentences.

Using this idea, look at the problem 7 – (–3). If these 2 values are located on the thermometer, the difference between them 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 value being subtracted, find the difference to the other value. Try for example, − 4 – (+3). In this case, since it’s 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)? Two paragraphs above it was found that 7 – (–3), is equivalent to 7 + (+3). Think about this. Subtraction is commonly defined in current texts as “adding the opposite”. Given this and what was demonstrated above is that its converse is also true — to add, subtract the opposite! So, in this problem − 4 + (+3) becomes − 4 − (−3). Since it’s subtraction, put both values on the thermometer and starting at (−3), the distance to minus four is minus one. Remember, moving down gives a negative value. So, − 4 + (+3) = −1. Also, somewhat obvious, is that − 4 − (−3) really says “if you have 4 negatives and subtract 3 negatives, you have one negative left.”

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.

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