By Mike O’Hern, Center Director of Mathnasium of West Knoxville
Last time in this space we talked about integers and discussed a couple of strategies to help our students understand how they work. I know you were enjoying the exercise, so let’s carry it a bit further this time. We only talked about adding and subtracting integers, so this time why not get a bit more aggressive and do some power adding! Yes, multiplication is just “power adding” because it’s adding up a number a bunch of times in one fell swoop!
Now you recall that adding a negative is just like subtracting, right? So if we start at ten and add a negative five we’ll end up at five. 10 + (-5) = 5. And we can start anywhere with the same effect: -10 + (-5) = -15. But now I’m confronted with a different kind of problem: -5 x 7 = ? What the heck? Some would say, “That’s simple, a negative times a positive is a negative!” And they would be right. But that’s just a rule to memorize and is as easily forgotten as memorized.
To understand the process will help us get it so we won’t forget it. See, another way to say -5 x 7 is -5 seven times. So if adding negative five goes down five spaces, then doing that seven times means we’re going down 35 spaces. Yes! -5 x 7 = -35! So now the rule makes sense, since a negative times a positive will always be adding a negative so many times.
So now we can multiply a negative with a positive and know it will end up negative. But there’s one more little sticking point that we need to be ready for. What happens when we multiply and negative with a negative?
I’ll have to admit that I’ve been having a tough time coming up with a way to show this without getting all mathy on you. Fortunately, however, in her book, “Kiss My Math,” Danica McKellar came up with a really cool way to look at this: Hold up the mirror. A negative is the opposite and when you multiply by a negative one you get the opposite. -5 is the opposite of 5. So 5 x -1 = -5. If you look at your friend you’ll see that her left ear is on the right side of her head. When you stand next to her and see her in the mirror it’s on the left side of her head. It’s in the opposite position. But now put a mirror behind her. When you use both mirrors, her right ear is on your right again. One mirror makes things opposite, two mirrors make things right again!
A negative is the opposite and when you multiply by a negative one you get the opposite.
Now to our next problem: -10 x -7 = ? First, let’s stipulate (based on what we discussed above) that another way to state this problem is -1 x 10 x -7 since -1 x 10 = -10. Now take the next step to say -1 x -70 since we know that 10 x -7 = -70. We know that when we multiply by -1 we get the opposite, and the opposite of -70 is 70.
Okay, I’ll give you a minute to wrap your head around that.
Your next question is surely, “But what about division?” Good question. I don’t have the space to go into a bunch of detail, but let’s just say that it works the same way because division can be considered multiplication. WHAT?? Yes, if you want to divide by two, you could just multiply by 1/2. Divide by 273? Multiply by 1/273. Divide by (-49,722)? Multiply by –(1/49,722). Get it?
Look here now! You’re an integer expert!
Mike O’Hern, Center Director of Mathnasium of West Knoxville, earned his Bachelor’s Degree in Metallurgical Engineering at the University of Tennessee, Knoxville in 1988. He pursued graduate studies in Materials Science & Engineering while on the Research Staff at Oak Ridge National Laboratory. Mike has had a life-long love of mathematics and teaching, and feels that math is not about learning to be ready for the next math class – it’s about learning to think.