The Joy of X: A guided tour of math, from one to infinity by Steven Strogatz. Copyright ©2012 Steven Strogatz. Reprinted with permission from Houghton Mifflin Harcourt." data-pin-do="buttonBookmark">
The Joy of X: A guided tour of math, from one to infinity by Steven Strogatz. Copyright ©2012 Steven Strogatz. Reprinted with permission from Houghton Mifflin Harcourt. Image: Houghton Mifflin Harcourt
Reprinted from The Joy of X: A guided tour of math, from one to infinity by Steven Strogatz. Copyright ©2012 Steven Strogatz. Reprinted with permission from Houghton Mifflin Harcourt.
Every decade or so a new approach to teaching math comes along and creates fresh opportunities for parents to feel inadequate. Back in the 1960s, my parents were flabbergasted by their inability to help me with my second-grade homework. They’d never heard of base 3 or Venn diagrams.
Now the tables have turned. “Dad, can you show me how to do these multiplication problems?” Sure, I thought, until the headshaking began. “No, Dad, that’s not how we’re supposed to do it. That’s the old-school method. Don’t you know the lattice method? No? Well, what about partial products?”
These humbling sessions have prompted me to revisit multiplication from scratch. And it’s actually quite subtle, once you start to think about it.
Take the terminology. Does “seven times three” mean “seven added to itself three times”? Or “three added to itself seven times”? In some cultures the language is less ambiguous. A friend of mine from Belize used to recite his times tables like this: “Seven ones are seven, seven twos are fourteen, seven threes are twenty-one,” and so on. This phrasing makes it clear that the first number is the multiplier; the second number is the thing being multiplied. It’s the same convention as in Lionel Richie’s immortal lyrics “She’s once, twice, three times a lady.” (“She’s a lady times three” would never have been a hit.)
Maybe all this semantic fuss strikes you as silly, since the order in which numbers are multiplied doesn’t matter anyway: 7 × 3 = 3 × 7. Fair enough, but that begs the question I’d like to explore in some depth here: Is this commutative law of multiplication, a × b = b × a, really so obvious? I remember being surprised by it as a child; maybe you were too.
To recapture the magic, imagine not knowing what 7 × 3 equals. So you try counting by sevens: 7, 14, 21. Now turn it around and count by threes instead: 3, 6, 9, . . . Do you feel the suspense building? So far, none of the numbers match those in the sevens list, but keep going . . . 12, 15, 18, and then, bingo, 21!
My point is that if you regard multiplication as being synonymous with repeated counting by a certain number (or, in other words, with repeated addition), the commutative law isn’t transparent.
But it becomes more intuitive if you conceive of multiplication visually. Think of 7 × 3 as the number of dots in a rectangular array with seven rows and three columns.
If you turn the array on its side, it transforms into three rows and seven columns—and since rotating the picture doesn’t change the number of dots, it must be true that 7 × 3 = 3 × 7.
Yet strangely enough, in many real-world situations, especially where money is concerned, people seem to forget the commutative law, or don’t realize it applies. Let me give you two examples.
Suppose you’re shopping for a new pair of jeans. They’re on sale for 20 percent off the sticker price of $50, which sounds like a bargain, but keep in mind that you also have to pay the 8 percent sales tax. After the clerk finishes complimenting you on the flattering fit, she starts ringing up the purchase but then pauses and whispers, in a conspiratorial tone, “Hey, let me save you some money. I’ll apply the tax first, and then take twenty percent off the total, so you’ll get more money back. Okay?”