Episode 4: February 6, 2010
Real World Math
by Jason Marshall
Today’s article will walk you through the first steps of kicking your calculator dependency. This isn’t something we can accomplish in a single article, so we’ll be revisiting this topic periodically. Today, we’re kicking things off by using a number line, instead of a calculator, to help you keep your signs straight.
Review of Integers
But before we start working to break that calculator habit, let’s briefly review what we talked about last time. The most important thing to take away was the idea that combining negative whole numbers with the natural numbers gives us the very important group of numbers known as integers. These integers can be arranged on a number line with big negative ones extending out indefinitely to the left, zero in the middle, and big positive ones extending out forever to the right. At the end of the article I asked if any integers exist that are neither positive nor negative. What do you think? If you take all the positive and negative integers off the number line, are there any integers left? How about that strange one right in the middle? Yep, that’s the one: zero.
How to Understand the Number Line
Okay, how about the other question? Did you figure out how to put the integers 101, -1, 32, and -2010 in order from smallest to largest? Once you understand the number line and how positive and negative numbers relate to each other, this shouldn’t be too hard. The number -2010 is the smallest since it’s the most negative, then comes -1, then 32, and finally 101 is the largest. But perhaps you’re thinking: How can -2010 be the smallest!? It’s a pretty big number...it has four digits!
Well, here’s a quick and dirty tip to help you keep the relative size of numbers straight. Think about the number line again. Any number to the left of another on the number line must be the smaller of the two. Even though it might be a big number—in that it might have a lot of digits like negative one-trillion—it’s still smaller-than any number to the right of it. Even a seemingly puny one-digit number like zero.
How to Kick Your Calculator Dependency
Okay, with all that covered, let’s talk about our first calculator dependency kicking technique. This one is aimed at making addition of positive and negative integers easier. Let’s say you need to solve a problem like -46+16. Your first instinct might be to go grab your calculator and start punching numbers. Yes, that should give you the right answer, but you run the risk of not understanding the very important question of why it gave you the right answer. And, if you don’t understand “why,” how will you ever know if you’ve made some egregious error, resulting in sharing an embarrassingly ridiculous—and wrong—answer with everybody. Avoid this risk and learn how to do the problem in your head instead. That way you’ll know when something is fishy with a result.
Visualize the Number Line
Let’s start our journey by going back to the number line and thinking about some simple examples. First, imagine you’re standing at the zero marker of the number line with all the negative integers lined up to your left and the positive ones to your right. Really picture this—it’ll help if you have a vivid image in your head. Your number line could be on a beach, a football field, a pasture full of cows—or wherever else makes you happy. I know it seems kinda dorky. It is. But it’ll help.
How to Use the Number Line to Solve Problems
Alright, let’s kick things off with a super simple example: What’s 2+3? I know, I know. I told you it was going to be really simple, so pretend for a minute you don’t already know the answer. It turns out you can use your imaginary number line as a sort-of mental calculator to help solve problems like this.
Here’s how. Start by imagining you’re standing at the zero mark of your number line. Since the first number in the problem 2+3 is positive two, walk two steps in the positive direction (that’s to your right). You’re now standing at the position marked “2.” The second number in 2+3 is positive three, so you next need to walk three additional steps in the positive direction. Now, take a look at the number line and see where you’ve ended up. Of course, you’re at the position marked “5.” So—and I know this isn’t going to be a big surprise to you—you’ve calculated that 2+3=5.
How to Deal With Negative Numbers
Wow, who needs a calculator when you’ve got a number line, huh? Well, okay...I know that was a ridiculously easy problem, and there weren’t even any negative numbers in it. So here’s how to deal with adding positive and negative numbers. Instead of the problem 2+3, let’s say you need to solve the problem -2+3. It’s still pretty simple, but let’s think about how it works with the number line.
Once again, imagine starting at the zero mark on your number line. Since the first number in -2+3 is negative two, you first need to walk two steps in the negative direction (that’s to your left). So you’re now at the position marked “-2.” Now, since the second number in the problem -2+3 is positive 3, you next need to walk three steps in the positive direction. So, starting at “-2,” you take three steps in the positive direction and end up at the position marked “1.” Congratulations! You’ve calculated that -2+3=1.
And that’s the trick for adding positive and negative numbers. Admittedly, it’s not necessarily all that useful for the relatively easy examples we used to demonstrate the method. But it can be extremely helpful when tackling somewhat tougher problems like 3 + (-13) + 14 = ? or perhaps -9 + (-8) + (-3) + 7 = ?. These problem might seem a little intimidating at first, but try mentally walking through them and see if it helps you keep the signs straight. Just take it step by step, walking in the positive or negative direction according to the sign of the number. We’ll go over the answers next time, and we’ll also talk about extending this method to include not just adding negative numbers, but subtracting them too.
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