By Mike O’Hern, Center Director of Mathnasium of West Knoxville
Do you ever wonder why in the world we use number lines that are horizontal? How arbitrary is that? Right is positive and left is negative? Why? Sure, we’ll need to use horizontal when we have multiple axes, but for a simple number line to start learning how numbers work, I think the number line should be vertical. If we add up and subtract down that seems a bit more intuitive to me, and it will really come in handy when we start to learn about negative numbers.
Negative numbers are not too difficult to grasp if we find various ways to talk about them. We can start with the swimming pool, of course. If your head is seven feet above the water when you’re on the diving board, then you jump up three feet (with a good bounce, of course!), how high is your head now? Simple: 7 + 3 = 10. But that jump ends up putting your head under water eventually. From your peak height you fell down 13 feet, so where’s your head now? Okay, it’s ten feet to the water, so you have three more feet to go. Your head is now three feet under the water. Since the surface of the water is our reference point, we’ll call that zero. So 7 + 3 = 10, then 10 – 13 = -3.
The set of numbers that includes all of the whole numbers (0, 1, 2, 3, …) and their negative counterparts (-1, -2, -3, …) is called integers. Working with integers can be a challenge for a lot of students until it finally clicks. That’s why we need multiple ways to think about them.
Some seem to get it when we talk about money. If I have ten dollars and I want to buy something for 20, I can borrow ten from you, right? (Please…) So when I’ve bought my thing I now have not zero dollars, but I have negative ten dollars (because I owe it to you). If I earn five dollars how much do I have? Negative five, because I paid you back half of what I owe you. Now if I earn 12 dollars, how much will I have? Well, I’ll take five from the 12 to pay you the balance leaving seven for me. 10 – 20 = -10, -10 + 5 = -5, -5 + 12 = 7. Here’s the fun part. On that second step we actually paid back some of the debt, so instead of adding five, we subtracted negative five, right? That would be -10 – (-5) = -5. There’s more we could do here, but let’s move on to think of another strategy.
There is a staircase in a castle. There’s a landing at the main level (we’ll call that zero), and the stairs go up to the tower in one direction and down to the dungeon in the other. As the maiden goes onto the landing, she can be in a positive mood or a negative one. When her mood is positive, she looks up toward the tower. When it’s negative, she looks down toward the dungeon. In either case she may either add steps or subtract them. If she’s in a negative mood and adds five steps she’ll be five steps below the landing, at step -5. If her moods doesn’t change, and she subtracts four steps (meaning she’ll walk backwards up the stairs) she’ll end up at step -1. Her mood changes to positive and, she adds four more steps, so she ends up at step 3 (positive three). We started with 0 + -5 = -5. We added a negative five. Then, we said -5 – (-4) = -1. We subtracted a negative four. Finally -1 + 4 = 3. We added a positive 4. (Okay, it seems this one works better when I preen about like a maiden on stairs, but I hope you get the drift.)
Well I find myself out of space, so maybe next time we can look into this a bit further. I’m having fun!
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.