Next year, for the first time in 15 years, I am going to be teaching high school full-time. While I have enjoyed writing and working primarily with teachers for the last 15 years, I’m looking forward to “going back to the classroom” as they say. It looks like I’ll be teaching BC calculus and Algebra 1 – two of my favorite classes. I’m very positive about that.
With that in mind I have been thinking of some of the things I want to be sure I get right in the Algebra 1 classes to get the kids off to a good start. So a few of my blogs in the coming year may be on Algebra 1 topics with the view of having students do things right from the start and not having to relearn things when they get to calculus.
So here is the first thing I want to be sure to work on: the m-dash also known as the minus sign.
According to Wikipedia:
The minus sign (−) has three main uses in mathematics:
- The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.
- Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5.
- A unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2.
Using the same symbol understandably can confuse beginning math students. I am not going to invent new symbols so I will just have to be careful with what I say and let the kids say. And I have to say it right , if I expect them to.
When used between two numbers or two expressions with variables the symbol means subtraction. That’s pretty easy to spot and understand in context. But when used alone in front of something the minus sign means different things.
The m-dash may always be read “opposite.” So “–a” is read “the opposite of a” and not “negative a.” Likewise, –5 is read “the opposite of five.”
There is only one instance where the m-dash may be read “negative.” When it is used in front of a number it indicates a negative number so “–5” is also correctly read “negative five.” This is the only time the m-dash should be read “negative.” Things like “–a” should always be read “the opposite of a” and never read “negative a.”
There was a time when the folks who write math books tried to make the distinction by using a slightly raised dash to indicate negative number so negative 3 was written “–3.” This has carried over into calculators where the key marked “(–)” is used for “negative” and “opposite.” and is printed on the screen as a shorter and slightly raised dash. The subtraction key is only used for subtraction.
Oh, if it were only that simple. What do you do with –(–5)? Not really a problem the “opposite of the opposite of 5” and the “opposite of negative 5” are both 5.
I’ll know I’ve succeeded when everyone can get 100% on this little True-False quiz:
- The opposite of a number is a negative.
- –x < 0
- x > 0
- | x | = x
- |– x |= x