## Notes on Operations – Unary Minus

“Unary minus” is the name for the symbol in $-a$, which looks like a minus, but isn’t a subtraction of two quantities.  This simple-looking thing is associated with lots of confusion in middle school math.

We’re all familiar with “-” as the symbol for subtraction of two quantities.  When we write $6 - 3$ or $x - 1$ or $x - y$ we agree that the “-” stands for the operation of subtraction.  This operator, indicated by the “-” symbol,  is sometimes called “binary minus”.  Something is being subtracted from something else.  Two quantities are involved.

We’re also familiar with “-” as the symbol used to indicate negative numbers.  “-2” is the notation for negative two, a quantity indicated on the number line by drawing it on the other side of zero from where all the positiive numbers are.  In math text books for lower grades this symbol is often drawn higher than an ordinary minus sign.  Whether it is drawn higher or not, it is clearly regarded as somehow part of the number, a prefix.  On calculators, the way to turn a number into its negative is often through a special key marked “+/-“.  Kids using this key (which is usually pressed after entering the positive number) usually think of it as an operator that puts in the negative sign.  They may have noticed that if they press the key twice it takes the negative sign back out – and they may think of this as a convenient way to deal with having pressed the key mistakenly.  Some sophisticated calculators label this same key with “CH S”, which their manual explains stands for CHange Sign.  The calculator makers clearly think of this key as something that operates on a number, rather than something that is part of a number (like a digit key or a decimal point key).  When you enter the number “negative four”, you don’t first press “+/-” and then “4”; instead you first press 4, and then you change it into “-4” by pressing “+/-“.

The “unary minus” operation shown in $-a$ can be thought of in at least three different ways, and I suspect that this is a large part of the confusion that students have with it.  Here’s three ways:

1. We can think of $-a$ the same way the calculator people think of the “+/-” key (or the CH S key): it takes the number and then flips the sign of it: if it was positive before, it is now negative; and if it was negative before, it is now positive. $-a$ could be read as “whatever number $a$ is, but with the opposite sign”.
2. We can think of $-a$ as short hand for $0 - a$.  In other words, we can think of “-” as a subtraction after all, but a subtraction where the thing subtracted from is hidden, left unstated.
3. We can think of $-a$ as short hand for $-1 \times a$.  This may feel like it just dropped out of the sky, but at least it has an analogue in that you can think of $+a$ as short hand for $+1 \times a$.  If you think of $4a + a + 7a$ and conclude that this is the same as $12a$, you have just treated $4a + a + 7a$ the same as you would have treated $4a + 1a + 7a$.  Similar thinking would apply to $4a - a + 7a$, which you might conclude is the same as $10a$.  If so, you just treated $4a - a + 7a$ the same as $4a - 1a + 7a$.  With these examples, the relationship between $-a$ and $-1 \times a$ may not seem quite so far-fetched.

People with enough experience in math know that ultimately all these three ways of thinking about $-a$ amount to the same thing.  To the kids first encountering this, the equivalence of the three ways to approach $-a$ is not obvious at all.  What adds to the difficulty for the kids, from my observations, is that the teachers often don’t appreciate that there is a difficulty in the first place.  The teacher has often settled on one of the three ways to think about or talk about $-a$, and may miss when the students think of it differently.  In particular, teachers often think of $-a$ as $-1 \times a$, even in situations where the students quite naturally think of it differently.  For example, when solving the equation $5 - x = 9$

the teacher may show the “normal” steps, subtracting 5 from both sides $\begin{matrix} 5 & - & x & = & 9 & \\ -5 & & & & & -5 \\ & - & x & = & & 4 \end{matrix}$

and then talks about dividing both sides by -1.  What is lost is that the $-x$ came about from subtracting 5 from $5 - x$, so really appears as shorthand for $0 - x$.  And yet the teacher already thinks of the $-x$ as $-1 \times x$, hence the suggestion to divide by -1.  The student, looking at $5 - x = 9$ fails to see any multiplication.  There is simply a subtraction, x subtracted from 5, or “5 take away x”.

Indeed, even within the approved symbolic framework for solving equations (“do the same thing on both sides”), the equation $5 - x = 9$ can be solved without bringing in multiplications and divisions by -1.  The student could, for example, add x to both sides.  This would result in $5 = 9 + x$, which could be solved in the next step by subtracting 9 from both sides.

What the teacher does when talking about division by -1 is not wrong, and neither is it inappropriate.  It is simply that the student is implicitly asked to make a shift from thinking of $-x$, which arises from $0 - x$ from some kind of subtraction to some kind of multiplication, and teachers often miss that this shift is asked for.  The students are tripping over something that to the teacher doesn’t even look like a hurdle, let alone a high one.

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### 2 Responses to Notes on Operations – Unary Minus

1. hiddenhistory says:

It is very interesting that you acknowledge the role of the teacher in explaining things to students from their viewpoint. I commend you for your insight.

I have been disturbed by the fact that the United States is placing only in the top third internationally, and that Hong Kong, Singapore, Chinese Taipei, Japan, Kazakhstan, the Russian Federation, England and Latvia are scoring ahead of the US in fourth grade mathematics tests. For eighth grade math scores, again Hong Kong, Singapore, Chinese Taipei, Japan, and Korea are edging out our scholars in accomplishment.

I will be watching your blog for more information, as I believe that you may have the beginnings of a teaching revolution in your thoughts. I hope others embrace it!

Ron Harris