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
This is a typography observation, not really about math. I’ve always typed the symbol in this post as an en dash (–) rather than an em dash (—). Looking at how they appear in this comment box, the en dash looks right to me, and like what you typed in your post (which you refer to as an “m-dash.”)
Do you know which one is right, or if there’s an authoritative standard?
(The origins of those names are from typography; the en dash is traditionally the width of the lower case N in the font, and the em dash the width of the lower case M.)
LikeLike
Peggy
I believe in the original post I typed all m-dishes (which you can do from the keyboard with CTRL+ (the minus sign from the number pad)).
I write in Word and then paste it into the online WordPress word processor which then changes things. This is what looks right to me. As I mentioned in the post I’ve seen the raised n-dash used to represent negatives, but not opposites.
In my reply to Liz Warner, I agree that they all look like n-dashes. (hyphens). I’m not sure what I typed, but I meant m-dashes. It may be that the WordPress word processor for replies does not make the distinction. So an experiment:
This is an n-dash –
This is an m-dash Nope didn’t work. All it did was make the type smaller. So that must be the explanation. I can get an m-dash from the Character map: — 3 Too much trouble, and it now looks too long. I’m typing this in a font called Calibri but I’m pretty sure it will change to Georgia when I post it, which may change the length.
Anyway, I don’t think it really matters which you use. The point is do you call it “opposite” or “negative.” I think “negative” should only be used with a number following it. A variable, or even a number in parentheses, should by read “opposite.” Of course, that’s just my opinion.
Thanks for writing.
LikeLike
My Honors Pre-Calculus class went through this discusison as we were working on limits involving absolute value. They have never really gotten a good feeling for “-x” being a positive number or the concept of absolute value, if the expression contains a variable. I, too, am working on the habit of calling “-x”, “the opposite of x” rather than “negative x”.
All of your statements are false and I hope my students would get that right after our 15-20 minutes of discussion a couple of weeks ago.
LikeLike
I have to admit that I don’t fully understand the distinction. Why can’t “-a” be caleld negative a, even if a is negative? I supposed I never learned the difference. So, I am guessing: first three false, last two true?
LikeLike
Liz
Thanks for the question.
Of course “-a” may be called “negative a” even if a is negative. Everybody does it; I’ve done it. I for one am going (try) to stop doing it.
I think “negative” should be reserved strictly for numbers that are negative (< 0) and with a variable "a” or “-a” you really cannot tell if the expression is really less than zero. And often it is not. I hope that if we, as teachers, are more careful about when we say “negative” it will be less confusing to the students.
Sorry to say, but your answers to the quiz are not all correct. All 5 statements are false.
|x| = x only if x is greater then or equal to zero. I like to read this as “the absolute value of a number is the same number” which is only true when the number is not negative.
|-x| = x read “the absolute value of the opposite of a number is the number” is true only if x < 0.
Said another way: if x < 0 then |x| = –x. This is true. The absolute value of a negative number is its opposite, which of course is positive even though –x “looks” negative.
Absolute values are the first place where all this seems to make a difference. The difference persists for a long time. Kids never seem to remember that if x represents some number, there may be a “-” inside the x, where we cannot see it. So that x may stand for a positive number or a negative number and –x may also stand for a positive number or a negative number. By reading –x as “the opposite of x” I hope students will pause a second and realize –x could be positive or negative and then look around for some additional information before they say “negative.”
LikeLike