Math Made Easy: Why Do I Care About..?
One of the biggest problems in teaching math is helping the student relate it to the real world. It’s just simply not done for a lot of students throughout their career, so they are disconnected from it, and don’t really see the use of this wonderful tool. I frequently get asked by my students “when am I ever going to use this in the real world?” So, I’m going to try to address that for several different areas of mathematics – just where do we use them? How can these tools be helpful for you?
I’m going to skip addition and subtraction of the four basic operations, as I really hope it’s evident to anyone just where they’re using those on a daily basis (hint: in virtually everything you do). So on to multiplication and division, which I won’t spend too terribly much time on for similar reasons.
Multiplication: Need to figure out how much square footage in your house? Multiplication. How many people can be seated in a room with several rows of chairs? Multiply rows times chairs per row. No need to count. How much is this dinner going to cost if each of us get something worth so much? Multiply. How much are these tickets going to cost for all of us? Same thing. You get the idea, I hope.
Division: You’ll need to use division when you need to know how many things to get when each thing carries so many of something else, and you have so much of that something else (ie, how many buses needed to transport a group of students when each bus has a certain number of seats, or boxes to carry books when each box can fit so many).
Let’s move on from the more obvious to the least.
Fractions: Oh, do my students complain about fractions. Almost to a person, they come to me hating them. They’ve had them thrown at them and been forced to manipulate them, but never really shown why, or how they can actually make their lives easier. And that’s exactly what fractions are for: making our lives easier. There is so much math we can do with them, that is much easier using them than longer more basic methods. Every division problem can be turned into a fraction problem, and the fractions are ALWAYS easier to do – you can simplify them before you do the division, and, frequently, will find you don’t even have to do long division anymore. You can use them to turn division problems into multiplication problems, and it seem almost everyone is better at multiplying than dividing, so you end up taking a difficult problem and turning it into something quite easy. Whenever you split something up, you’ve created a fraction – when you’re talking 3 people in 4, you’re talking fractions. Probabilities (like in poker, blackjack, the roll of a die, or even the chance of rain) can all be written, and more easily figured, as fractions. That ratio on the label of your car tires telling you how the width of the tread relates to the height of the tire- that can be written as a fraction. Slices of pie or cake? Fractions. Fractions can even help you balance your checkbook a little more easily, and do other aspects of your finances. They’re everywhere, and they’re quite useful. I’ll do an article on just how to take advantage of fractions, and make them your friends, later on.
Exponents (Squares and square roots) : Here we’re starting to get into the stuff that, too non-mathematicians and non-scientists, starts to seem more esoteric. As you go higher and higher into math, the operations and functions start seeming to be more and more divorced from the real world, when exactly the opposite is true. We get ALL of these operations and functions *from* the real world. They’re tools we’ve developed to help us figure out how the world works, and how to do things in it. Squares and roots really do exist in the real world, and we need them. The easiest of the roots, the square root, can be seen in your daily world when you’re looking for housing. See that add talking about square feet? If you take the square root of that, you can get an idea of how long by how wide approximately the place is: 100 square feet would be 10 feet by 10 feet, about. (And, vice versa, something that is 10 feet to a side has an area of 100 square feet). 4000 square feet would be just over 63 feet by 63 feet. Realtors like to list properties by square footage because, while it’s honest, a surprisingly large number of people really have no concept of what those numbers mean, so they can make the house or apartment seem bigger than it really is. Any time you’re talking area, you’re talking a square. Factoring: Remember those exercises you did in school where you had to factor numbers, finding all the possible numbers that go into it? Did you miss the reason you did all that? Well, it’s actually pretty simple, and factoring is extremely useful in the real world. It helps you do division, helps you take roots, and helps you simplify fractions. So anytime you do division – like when you’re trying to figure out how many buses you need for a certain number of students – factoring first can make the problem easier, and possibly even eliminate the need for long division. When you’ve got the area of a building, and want to figure out the length and width of it, factoring can help you get to it when you don’t have a calculator, and can’t figure out the square root on your own. When you’re dealing with a large fraction of something (say you’re trying to make sense of the amount of your money you just spent on something), you can use factoring to simplify the fraction and make it look both like something you can make sense of, and make it a little less intimidating.
Trigonometric functions (and the Pythagorean theorem) : Now we’ve gotten to where people really start to zone out on math and throw up their hands to say: “why do we need THIS?” Well, it turns out trigonometry, and the functions of it (sine, cosine, tangent, secant, cotangent, cosecant) are EVERYWHERE. Need to figure out how to cut tiles to fill your floor in the most efficient way? Trigonometry can help you. Have to cut down a tree, have limited space to do it (you don’t want to hit nearby buildings or powerlines) – trigonometry can help you figure out how tall that tree is, and where to lay it down. Those functions exist in electricity and light, in the way light reflects and refracts, and can be used to make sense of how things rotate, and just how much faster the outside of something is rotating than its inside (while they describe the same angle in a certain amount of time, the outside edge covers more distance than the inside in the same amount of time). Want to place your furniture to maximize your floor space? Trigonometry.
Polynomials: Remember those harder algebra problems that involved several different x’s or y’s each with a different exponent, and maybe a coefficient (the number that’s multiplying each x or y) and a constant at the end? Did your teacher ever relate those to the real world for you? Well, those have their place, too. Each polynomial can describe different things in reality, and frequently can be used to describe things like motion, the way a wave undulates, the back and forth swinging of a pendulum or a tire swing, or even the way the light of a star behaves or a particular stock on the stock market. The one you probably saw the most, a quadratic (which looks like the expression below), forms a shape called a parabola, which looks like the figure below. It can be used to describe the position of an accelerating object (your car with your foot on the accelerator), or the path a baseball takes through the air when it’s thrown. Most mathematics is done with polynomials, so, whether they look like nonsense or not, it turns out they’re quite useful, and the stuff of which the world is made.
Calculus (Derivatives and integrals): Now we’ve gotten as far as this article is going to take us (there is even more esoteric math out there, but I’m not going to delve into it, since most people won’t be exposed to it outside mathematicians and scientists). Even calculus can be useful in day-to-day life. The derivative and the integral were developed, simultaneously, by Leibniz and Newton, to help describe laws of motion. It turns out things don’t move in such a way as to be easily described by basic mathematics – motions change in a continuous way of tiny infinitesimal changes that add up over time. The derivative and the integral help us make sense of these changes, and to calculate them without having to painstaking pound our heads over a lot of repetitive drudgery. Many aspects of motion are directly related by their derivative – in the polynomial that describes somethings motion through space (across the yard or down the road is sufficient), it turns out that the thing’s velocity is the first derivative of its position, and the acceleration the second. And, to go the other way, the velocity would be the first integral of acceleration, and the position the second. When we do work with magnetic or electric fields, calculus is involved to figure out how much electricity is flowing, or how strong those fields are (sometimes we need the derivative, sometimes we need the integral – it depends on what we’re doing). The same goes for the flow of water – so need to build a dame, and figure out how big your sluice gate needs to be? Calculus. Building a water tower involves calculus, as it will be integrals and derivatives that help us determine where to put it, how big it needs to be, how big the pipeline needs to be, all to achieve desired water pressures. Even the rate at which a towel soaks up water – that can be figure with calculus.
The math gets a lot more complex than that, but, like I said, I’m not going to get into things like linear algebra, branes, bra-ket notation, or the Lagrangian, as they’re things really only scientists and mathematicians are going to deal with, and I don’t think I need to tell them just how useful math is. But, hopefully, after perusing this and seeing a lot of the examples I gave, any hater of math comes away from this article hating it a little bit less. It really is useful to you, and could be so much more if you master just some of the basics.