When teachers teach how to multiply and divide negatives, they often take a shortcut - right off the edge of a cliff - by teaching students the “rule” that “two negatives make a positive.” What’s wrong with this “rule”? The fact that it only works for multiplication and division. When you add two negatives, the answer is more negative, and when you subtract a negative from a negative, the answer can be negative, positive, or zero (for the simple reason that subtracting a negative is the same thing as adding a positive). Kids are terrible with distinctions like these because they involve additional layers of complexity, and thus will tend to take the (off the cliff) shortcut of applying the double-negative “rule” to all situations involving two negative numbers.

Key takeaway so far: never start with the rules. Math is not a series of rules to be applied; it’s more like a bunch of patterns to discover and exploit.

The best way to reveal the pattern involved in multiplying and dividing negatives, bar none, is through the kid-friendly concept of Opposite Day. This concept is so educational that there are claims it helps students learn essential skills in math, science and reading, and compares it to a children’s “philosophy course” for the way it encourages kids to think. The best way to introduce the concept of Opposite Day is through the ever-brilliant Calvin and Hobbes:

Once the kids get the gist of Opposite Day (if they haven’t gotten it already), the concept can be built upon to reveal a fundamental pattern regarding opposites:
Opposite Day = Opposite Day
The opposite of Opposite Day = normal day
The opposite of the opposite of Opposite Day = Opposite Day
The opposite of the opposite of the opposite of Opposite Day = normal day
The opposite of the opposite of the opposite of the opposite of Opposite Day = Opposite Day
and so on…

Teachers should only introduce this pattern, of course. Students should continue it themselves until they can both identify and describe it - and until they can use it to answer questions like, “You’re very clever on the opposite of the opposite of Opposite Day. Did I just insult you?”

Now that they’re down with Opposite Day, the students are ready to multiply and divide negatives. All they need to know before they begin is that any number can be turned into its opposite by multiplying it or dividing it by -1. (They may also need to be reminded, via the number line, that opposites always sum to zero, and that zero is its own opposite.)

The connections to Opposite Day should suggest themselves:

The rule for multiplying and dividing negatives (each pair of negatives “cancels itself out,” or even number of negatives: positive; odd number of negatives: negative) should also suggest itself:

The extrapolations should be child’s play:

From here on out, it’s just practice, practice, practice. As with all math practice, there should be fully worked-out example problems to refer to, and each problem should come with instant feedback in the form of the worked-out solution. There should also be scrambled practice involving addition, subtraction, multiplication, and division of negatives, of course, to ensure that students can switch back and forth between the operations accurately. (Follow me using the Twitter link below for my thoughts on “see it, do it, check it” activities and other math topics).

Too often over the course of my career, I’ve heard math teachers jump in and correct a student’s incorrect answer by saying “You mean negative” or “You mean positive” in a tone suggesting it’s not that big of a deal. It’s an epic deal; a failure to master negatives is a failure to ever master math.

Today our students are failing with negatives.

May tomorrow be Opposite Day.