Everyday Math for Everyday Life, Part 2

A Handbook for When It Just Doesn't Add Up
by Mark Ryan

(Page 2 of 2)

A Note on Calculator Use, Mental
Computation, and Estimation

Calculators are great. I have several, I use them often, and couldn't get along without them. Everyone should know how to use them. They're fast, they never make mistakes - assuming, of course, that you punch the right buttons - and there are many math problems that are either too difficult or too time-consuming to do by hand. Throughout this book, I'll go through the calculator steps needed for solving various problems. But, I think it's unwise to use a calculator for every computation you do. When you use calculators for even the simplest of computations, your math muscles atrophy and you are more likely to view mathematics as a foreign, mysterious language that you will never really understand, an esoteric code that only a calculator can "understand." In contrast, when you work with numbers mentally or with pencil and paper, you strengthen your facility for math, and math will gradually come to make more and more sense to you. You will learn that you can do math with your common sense.

If you have forgotten the times table up to 9 times 9 - perhaps because you have been using a calculator even for problems like 9 times 4 or 8 times 6 - you owe it to yourself to relearn it. You should also be able to multiply numbers like 40 times 25 in your head. Four quarters make a dollar, right? So, of course, 4 times 25 is 100, and since 40 is 10 times as big as 4, the answer is 10 times as big as 100, or 1000. And 30 times 120 should be easy to do mentally: multiply 3 by 12 to get 36 and then add two zeros to get 3600. What about 1.4 of 120? - half of 120 is 60, and half of that is 30. When you take a few seconds to think about how problems like this work rather than just pushing buttons on your calculator, your confidence in your math abilities will grow.

In the final analysis, you should do what's comfortable for you. It won't be the end of the world if you decide to use a calculator even for 2 times 3 or 5 times 4. It certainly is reassuring to know that the calculator can't err. And if you're working on an important project where accuracy is absolutely critical, it may make sense to use the calculator, or at least to use it to check your answers. But your confidence and competence in mathematics will likely increase if, at least for the easy stuff, you do the math. Whether you do a problem in your head, on paper, or with a calculator, it's a good idea to also estimate a rough answer to the problem. Estimating builds confidence in math because when you estimate, you have to rely on your own grasp of a problem rather than just following memorized rules and formulas. Estimating is also a good way to catch errors. You shouldn't just blindly accept the answer you get with pencil and paper or with a calculator. Make sure that the answer agrees with your estimate and that it doesn't fly in the face of common sense. If you don't have a rough idea of what kind of answer to expect, you might push the wrong button on your calculator, get a ridiculous answer, and not realize that it's wrong.

A Few Suggestions to Those Who Never
Liked Math or Whose Math Is Rusty

Whether you have never liked math or feel that you've forgotten most of the math you learned in high school, I'm confident that if you work through part I of this book patiently and thoroughly, you'll come away with a solid grasp of all the mathematics you need for practical math problems. Even if you've always hated math, you'll find that the math presented in part I is manageable. You may find that my explanations bring math down to earth and make it easier to grasp than you experienced in school.

There are many reasons for not liking math - for feeling like a fish out of water whenever you deal with numbers. You may have had a bad experience in school that made you feel that you weren't good at math. You may have found math boring, meaningless, or irrelevant - and it's difficult to learn anything we believe is unimportant to us - or your teachers or parents may not have pushed you to succeed in math. And, for girls and women, although things are getting better all the time, there still remains in our culture a gender bias that expects more of boys than girls when it comes to math, science, and technology. The good news is that once you have a real desire to learn mathematics, none of the above will matter much. Everyone can learn math, we're hardwired for it.

When the right approach is taken, math is something we can grasp with our common sense; it need not seem strange or esoteric. Two strategies that make math easier, for example, are to make math concrete and to learn why things are true. Math concepts make more sense to us when we see the connection between what may seem like a foreign or abstract rule or formula and the concrete reality of the world around us. For example, adding, subtracting, multiplying, and dividing negative numbers confuse some people. This is understandable because negative numbers can seem abstract - we can't have -5 apples, for example. Some people might be confused about how to add -8 and -5. For example, they might mistakenly use the multiplication rule that two negatives make a positive for this addition problem. But this problem needn't involve learning or remembering any abstract rules. You can think of negative numbers like debt. Two negatives add up to a bigger negative in precisely the same way that two debts add up to a bigger debt. That's all there is to it. When you make connections like this between seemingly strange math concepts and the familiar things from your day-to-day life, math gets so much easier.

We can see that the area of the triangle is half of the area of the rectangle because triangle ABF (shaded black) is half of the rectangle on the left (rectangle ADBF), and triangle FBC (shaded gray) is half of the rectangle on the right (rectangle FBEC). Therefore, since the area of a rectangle equals base times height (which means the same thing as length times width), the area of a triangle must be half of that, or half of base times height. In short, the formula for the area of a triangle is based on the simple fact that a triangle takes up half the area of a rectangle. When you learn the logic underlying mathematical ideas like these, math becomes far less intimidating.

Copyright © 2002 by The Math Center, Inc.

About the Author

A graduate of Brown University and the University of Wisconsin Law School and a member of the National Council of Teachers of Mathematics, Mark Ryan has been teaching math for over 12 years. He runs the Math Center in Winnetka, Illinois (www.themathcenter.com) where he teaches high school math courses including a course and a related workshop for parents based on a program he developed, The 10 Habits of Highly Successful Math Students. In high school, he twice scored a perfect 800 on the math portion of the SAT, and he not only knows mathematics, he has a gift for explaining it in plain English.