I taught how to multiply negative numbers to my 7th graders today, so I asked Twitter for ideas over the weekend. With the help of @Xer0Dynamite, @polynumera, @StefD_84, and @Quornix, via a Martin Bauer tweet (thank you!) I came up with six ways to teach it. We didn’t have time to cover them all today, so I expect I’ll return to it on Wednesday
Our question is:
“Why does (-2)(-3) = 6?”
First, let’s deal with the easy stuff.
(2)(3) = 6 because if you count the items in two groups of 3, you’ll find 6 items. That’s what we learn in first grade.
(2)(-3) = -6 because if you double anything you ought to end up with the same sort of thing, just twice as much.
(-2)(3) = -6 because we require (-2)(3) = (3)(-2), which equals -6 as just explained. (We want multiplication to be commutative.)
(-2)(-3) is harder to think about. It isn’t two groups of something, it’s -2 groups of something, and how can you have a negative group?
So why say that (-2)(-3) = 6?
Consider the alternative. Either (-2)(-3) = 6 or (-2)(-3) = -6; we aren’t going to define multiplication by negative numbers so (-2)(-3) = 7 or -15 or anything crazy like that. But if we define it as (-2)(-3) = -6, then since (-2)(3) = -6 also, we end up with -3 = 3, a contradiction that will kill all of arithmetic. So we should use (-2)(-3) = 6.
Think of rotating and stretching on a number line. When we multiply a number by 2, we stretch the number away from zero. If it starts at -3, it ends up at -6; if it starts at 3, it ends up at 6. But the sign of the number we’re multiplying by adds an extra bit of instructions: if it’s positive, just start at the same place before you stretch it to be further away from zero, so (2)(3) starts at 3 and stretches the point by 2 to get to 6, but 2(-3) starts at -3 and stretches it away from zero to get to -6.
If we start at 3 and multiply by -2, that means to rotate the point 3 by 180 degrees around 0 to get to -3, and then stretch it out by 2 to -6. If we start at -3 and multiply by -2, that means to rotate the point -3 by 180 degrees around 0 to get to 3, and then stretch it out by 2 to 6.1
Rearrange to get the negative of a negative number. The problem with multiplication is that it has both sign change and magnitude change. So let’s separate those out.
(-2) (-3) = (-1)(2) (-3) = (-1) (-6) = - (1) (-6) = - (-6) = 6.
That is first realize that -2 is the same as (-1) (2). Then, since (2)(-3) = -6 and we can do our multiplications in any order we want (commutative property again), we get (-1)(-6). But then we can separate out the negative, and make it - (1) (-6). We know from earlier than (1)(-6) = -6, so we have - (-6), and taking the negative of a negative number give us a positive one, 6.2
Use division. We’ll start from the end here. (-1)/(-1) = 1 because x/x = 1 whatever number x stands for.3 Next, let’s use LaTeX typesetting, since it represents fractions better:
Look at the pattern.
4 x -3 = -12
3 x -3 = -9
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = ?
-2 x -3 = ?
How do we keep the pattern going?
By adding 3 more at each line. So we get
0 x -3 = 0
-1 x -3 = 3
-2 x -3 = 6
Quod erat demostrandum.4
Chant “Negative, Negative, Comes Out Positive” in unison and by yourself lots of times, and you’ll know how to get the right answer, even if you don’t understand why.
The “Wipe Your Shoes” Joke.
A very dirty hillbilly walks into the store:
Lady at the counter: "Hey, wipe the mud off your shoes when you come in here."
Hillbilly: "I ain't got no shoes on."
This joke is another mnemonic for remembering the mnemoic that everyone but a hillbilly knows: "Negative, Negative, Comes Out Positive." The hillbilly says he ain’t got no shoes, but if he ain’t got ‘em, that should mean he does got ‘em.5
If you give the punch line as “What shoes?" the joke is better, but it doesn't make the math point.6
Why have all these ways of thinking about why (-2)(-3) = 6? There are several reasons. First, different people understand things differently, so if we give seven ways, maybe one of them will stick. For this reason, you do have to reassure them that it’s enough to know one reason why something is true; they don’t have to understand all of them.
There’s a different reason, though, almost the opposite: to really understand something, you need to look at it from all the angles. I like to understand what I teach. That makes even grammar school mathematics challenging. To really understand something, you need to understand it in more than one way. To teach it well, you need to know not only all the ways different people find useful to understand it, you need to know all the wrong ways they’re thinking about it too, and the wrong ways they think they understand it.
Mathematicians like finding different proofs of the same theorem for this reason. To really understand the number pi, you need to understand all the different equations that describe it.7 Here, we really need LaTeX to pretty up the notation:
This idea that important ideas must be approached from a variety of angles8 is why there’s a book called The Pythagorean Theorem and Its Many Proofs containing 367 proofs, and why I wrote up three pages of 25 formulas for Pi for my students.9 For an example more profound but also suitable for a preschooler, take a look at the classic poem,“The Blind Men and the Elephant”, which is really a poem about the Trinity.10
I’ll leave you, percipient reader, with another example I may explore more fully at a later time. Why are unblinded wise men looking at a cone like blind men feeling an elephant?
The rotate-and-stretch approach is nice because it’s also the way to understand imaginary numbers. There, multiplying 3 by 2i, the square root of -1, rotates you 90 degrees, taking you into a second dimension, the imaginary one, vertically up to 3i, and then the 2 part stretches you up to 6i. If you multiply by 2i again, you rotate by 90 degrees again, taking you back to the real axis, but now you’re on the negative side of zero, and then it stretches you by 2 again, so you end up at -12, a real number. The connection to the square root of -1 is that (3) (2i)(2i) = (6i)(2i) = 12*i*i = 12 (-1) = -12.
Should I have done that using numbered equations and lots of lines with explanations in between? I might try writing it up that way sometime and see which is more understandable.
Except for zero, I realized as I was teaching this today. 0/0 is undefined. We have two maxims in conflict here: “Anything divided by itself equals 1” and “Anything divided by zero is something we can’t define”. Unfortunately, the second maxim wins, unsatisfactory though it is. First, 0*0/0 = 0/0 = 1 if we use the first maxim. But 0*0/0 = 0*(0/0) also, which equals 0*1, which equals 1. So we get acontradiction: 0 = 1, if we use the first maxim. The problem is deeper than that. You can fit one nothing into nothing evenly, to be sure, which is the first maxim. But you fit lots of nothing into nothing too, because lots of nothing amounts to nothing. So how many nothing can you fit into nothing? Zero, infinity, or anything in between. We’d best leave it undefined. (And this doesn’t even get into the problem of taking limits to 0/0, e.g. limit as N goes to infinity of N/N^2.
That’s a bit of a cheat. True “-2 x -3 = 6” is what is to be demonstrated. But did we prove it? No. We just showed a pattern. We used scientific induction (extrapolation, actually) instead of mathematical deduction (or mathematical induction, which is totally different from induction generally).
This is reminiscient of another “Got them shoes” joke.
Boy: Mister, I bet you a dollar I can tell you where you got them shoes. I can tell you the city, and the street.
Tourist: OK, you’re on.
Boy: You got them shoes right here on the sidewalk of Bourbon Street, New Orleans.
I was thrilled to be in New Orleans for the American Economic Association conference (best conference city ever, actually) and have a little boy come up and ask me that. I knew the joke, but I played along anyway. See “FOUR COMMON NEW ORLEANS SCAMS TO AVOID.”
You can tell this as a Kentucky or a West Virginia joke instead of a hillbilly joke if you prefer the insult to be against a state instead of a class.
See Peter Borwein’s The World of Pi website and “The Amazing Number Pi” (2000).
Just in case you don’t met it [sic] that’s what I’m doing in this paragraph with the idea about ideas.
See also how to prove Pi is between 2 and 4, which is the start of how Archimedes calculated it using polygons with more and more sides, and a crude but effective Python program for calculating Pi two different ways using two different formulas to compare their use of computer time, and Olivet’s Pi. Maybe I should Substack on Pi separately someday.
Aquinas was very good on the point that some things can be proved and something cannot, and how “A man’s gotta know his limitations.” The Blind Men poem goes
It was six men of Indostan, to learning much inclined,
who went to see the elephant (Though all of them were blind),
that each by observation, might satisfy his mind.
The first approached the elephant, and, happening to fall,
against his broad and sturdy side, at once began to bawl:
"God bless me! but the elephant, is nothing but a wall!". . .
So, oft in theologic wars, the disputants, I ween,
tread on in utter ignorance, of what each other mean,
and prate about the elephant, not one of them has seen!
First, thank you for teaching my seventh grader!
Second, when I try to think about what I think of multiplying negatives, I think of negatives as "anti." Like your rotation explanation and the hillbilly w/o shoes.
If going 'forward' is to the right on the number line, 2*-3 means to go backward three times twice. Adding another negative (-2*-3) means to do the opposite -- basically go anti-backwards three times twice. And of course, anti-backwards is forwards. I'm not sure that adds anything to your post. But thanks for helping me think.
So, I didn't notice that you did it by showing that (-1)(-1) = +1. Once you've got that plus associativity and commutativity, it's a walk in the park:
(-2)(-3) = (-1)(2)(-1)(3) = (-1)(-1)(2)(3) = 1*2*3 = +6.
Since -1 * -1 = taking a step backward and the reversing what you just did...that brings you back to your starting place = 1 (multiplicative identity).