One of the ways I identify the strength of a student’s foundation in mathematics is to give them a simple equation:
–5 – 3 = ?
What I usually get back from the student is the question “Is that a minus 3 or a negative 3?”
To which I reply “Huh?”
You see, I couldn’t understand why students today think that a “minus” and a “negative” are different things. To me they are the same thing and are only called differently depending on where they appeared. For instance if the symbol “–“ appeared in front of a number that stood alone, we called it a “negative”. If it appeared between numbers in an equation, we called it a “minus”. But truthfully, “minus” and “negative” act the same way. So, although the above equation is properly read: Negative 5 minus 3 it could just as easily be read: Minus 5 Minus 3 or even Negative 5 Negative 3.
My confusion got to the point that in one of my high school Algebra classes, several students wanted to argue that they were different. You know what they used as evidence that they are different things? They showed me their calculators.
The high-tech calculators of today have a – sign and a (–) sign. The first is called a “minus” and is used as the “operation” command. The second is called a “negative” and is used to denote a number with a “negative” value.
Wow, what do I say to that? We have to realize that calculators simply aren’t that smart. You’ve heard the term “trash in trash out”. That means that the accuracy of a calculator or any computer is only as good as the information you give it. The newer calculators have 2 symbols because in higher math the order of operations can get confusing when using a – sign such as negative exponents.
So, my counter argument is that addition and subtraction is simply the movement along a number line. Therefore, the – symbol simply denotes the location or the direction on that number line.
For instance our sample equation can be illustrated as follows:
Although we started at -5 (negative 5), we actually started at 0 and first subtracted the 5 to get to the starting point of negative 5. Then we subtracted 3 more to get to -8.
Before you get to comfortable with that, let me point out that the – symbol means to move “in the opposite direction”. It does not mean “to move left”. We read left to right, so the default direction is right. The – direction would be left. So if we have the following equation:
–5 – –3 = ?
We again start at negative 5, but the first – means left and the one after that means move in the opposite direction or to the right. So we get:
To me, this is simple to understand, but for students it can be confusing. So, here are “rules” for addition and subtraction problems that the students seem to grasp.
- If the signs of both numbers are the same, add the numbers together and keep that sign.
- If the signs of the numbers are different, find the “difference” and keep the sign of the larger number.
- Keep, change, change. My students use this a lot to deal with subtraction problems. It means to keep the sign on the first number, change the subtraction to addition & change the second number to a negative. So the equation 5 – 3 becomes 5 + -3. For some reason 5 plus negative 3 is easier for them to understand.
Here are a couple examples for these rules:
Rule 1: 5 + 3 = 8 (same sign add & keep sign)
-5 – 3 = -8 (same sign add & keep sign)
Rule 2: -5 + 3 = -2 (Diff. signs, find diff, keep sign of larger number)
5 – 3 = 2 (Diff. signs, find diff, keep sign of larger number)
Rule 3: 5 – 3 becomes 5 + -3 = 2 (keep, change, change)
-5 – 3 becomes -5 + -3 = -8 (keep, change, change)
5 – – 3 becomes 5 + + 3 = 8 (keep, change, change) Corrects the double negative
Dealing with negative numbers should be second nature to a student at least by the time they are in 7th grade. I’ve seen students that can be taught Algebra, but then get poor grades because they aren’t handling negative numbers correctly.
If you aren’t sure about your student, then ask them to solve this equation:
–5 – 3 = ?
If they can’t give the answer instantly & correctly, then it might be time to get a tutor.