Math Matters: Why Do I Need To Know This?

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1 Math Matters: Why Do I Need To Know This? Bruce Kessler, Department of Mathematics Western Kentucky University Episode One 1 Introduction Hi, I m Bruce Kessler and welcome to Math Matters. This is a bold experiment for this math teacher to work in front of the cameras instead of actual students, but hopefully things will go very well. The idea here is that I get to answer this kind of tough question Why do I need to know this? and hopefully you will find the program useful. Hopefully it will help you in your classes and hopefully, hopefully, it will help you follow up and learn even more mathematics than you set out to learn. It s kind of my intent; I ll be honest with you. The purposes of the show, the stated purpose of the show is to illustrate some of the everyday uses of math that we teach in the general ed and teacher ed math classes. On this campus, that would be math 109, 116, 211, and 212. It s really a show for students in those courses, perhaps for students who had those courses and are curious about some of the useful, some of the mathematics there and how it s used, but in a broader sense, this is a program that s for anybody interested in mathematics, how it applies to the real world, and hopefully I can bring you some great examples of that. The title of the program Why do I need to know this? it comes from this question we get, and it s one of the toughest questions we get as math teachers. I ll be honest with you, sometimes we re a little defensive about this question. Hopefully it s born out of curiosity I was a curious kid and I asked this question, but sometimes that question is born out of a desire to not really learn the material and you re looking for some kind of excuse, and when the teacher can t provide a good application on the spot, you ve got that excuse. So we get a little defensive about this from time to time. The other thing is that a lot of the math we talk about in those general ed courses is there to prepare you for more advanced math that has some extremely cool applications. And that s the kind of math we focus on as math teachers a lot of times, so we may not have that great example of how the simple math, the easier math is applied in the real world. That s my job, that s what I m hear to do, is kind of show you some of the cool applications that pop up. Now I have to warn you, you can always come back with this response Well I don t need to know that because I m going to teach social studies to first graders and these kind of things. It s true that if you aim low enough, you can avoid using mathematics at least in your career. A lot of the menial kind of minimum wage jobs that you come across don t require much of a knowledge of mathematics, but let s get real here, you re in college. If your life s aspiration is to work at a place where they have pictures of the food on the register, you don t need a college degree for that. So I m assuming right off the bat that you re not aiming low, that you re trying to get that high paying job that requires a college degree. And I ll tell you, even if your career doesn t use math, it s nearly impossible to avoid it in everyday life. The things we do just to function involve mathematics. If you want to buy a car or a house, very few people can just walk up and write a check. You have to borrow money to do that and that requires a knowledge of interest, compound interest, amortizations, 1

2 and those kinds of things. If you have money you want to invest, you need to understand the pitfalls of that. They re constantly bombarding us with survey data in the papers so you need to be able to understand those kinds of things. Political polls,..., there s no end to the things that we re hit with that require just a basic understanding of mathematics, and hopefully I can bring a lot of those things to life. So I m assuming we re not aiming low and that we are going to probably focus on some careers that require a basic understanding of mathematics. Most college, uh most careers that would require a college degree will require some knowledge of mathematics. Now there are some things, maybe you re going to be a counselor or something like that and you think well I really don t need much math for that, that may be the case, that s fine. But then we have to deal with just surviving and the things I talk about like borrowing money and those kinds of things. So it goes beyond just the career we re talking about. We re also talking about being able to function in society, about being a productive member of society and a lot of the examples I show will kind of focus on that. This is just something that you need know regardless of your career, those kinds of things. The most important thing though is that you train yourself not to get taken by folks who have a better understanding of mathematics. If you re willing to always take numerical information from someone else and believe it, then you re setting yourself up to be conned, basically. I mean ask MC Hammer where his money is today. Well, his accountant took it and it s gone. These are the kinds of things where if he had been paying a little bit of attention, perhaps it could have been stopped. Just being able to spot a bad answer... I can give you a great example of this. One of the houses that we sold, we went to the office to sign all the paperwork to get our money and these kinds of things, and I had already kind of run these numbers, and so when the bottom line was laid out and they had the check all cut laying in front of us, I thought That number doesn t sound right to me. And we had them go back and check, and it turns out that when they had called for the pay-off amount from the bank, they had scribbled down the number and the person couldn t read his own handwriting, so he had put down the wrong digit in the thousands position. Well, that wrong digit was one thousand less than I was supposed to receive. That would have cost me a thousand dollars. We can hope they would have been on the ball and realized at some point their mistake and corrected it, but there s no guarantee of that. I caught the mistake because I had a general kind of knowledge of how much, the way things were working and how much money I was supposed to receive. So, I was my best defender in that case. The big thing is that this program is there for you. It s intended to help you and for that reason I m gonna ask you to give me input on this. Tell me things we have done that helped, tell me things that would make the program better. If you come across some math in of your courses that you don t see a good application for, ask me the tough question. Say Why on earth would I need to know this kind of stuff? and I ll do my very level best to answer it. I have kind of mapped out things I want to talk about, but if you come up with something better, then I am more than willing to go with what you got. We have an address there and we ve got a webpage that we can be contacted at. Please, please use those. 2

3 Summary page, Segment 1 3

4 2 Venn Diagrams Resource management Objective: To show how an abstract concept like Venn diagrams can be used in a real-world application. The example will graphically reinforce the set operations of union, intersection, and complement. One of the first things covered in a general mathematics course typically is a little bit of information on sets. And it s presented as a pretty abstract thing. We eventually get around to some good uses for it. I want to show you an example today of how a knowledge of sets and in particular a little trick called Venn diagrams can be used to help in resource management. I m gonna give you a little situation and show you how sets can be used, a knowledge of sets can be used to kind of solve the problem. Here s a situation for you. Let s say a business has 100 non-clerical workers. These are R&D people, sales people, whatever it is, and they have certain resources at their disposal. We ve got 40 administrative positions that can be spread around to these hundred workers. Perhaps they are going into a new building, they re going to have 24 little labs that folks can use and they re going to make a laptop purchase. So they ve decided that they re going to purchase 60 laptop computers and the questions is if you re, let s say you re the office manager or something or someone in charge of allocating these things, the question is How do allocate these things and kind of spread it around? And certainly, if you add 60 and 40 then you ve got 100 so there s enough to go around. But to complicate things, upper management has some minimum requirements here. They ve determined that those hundred people need all three of these resources and they probably know exactly who they are, so you don t have to decide that, but you have to live with that. They ve determined that 4 need only the administrative assistants and research space. They don t need a laptop computer, they re not mobile, they re okay. They ve identified 10 people who need only research space and a laptop, they don t need a secretary to handle situations except interviews or whatever that is and they ve identified 11 people who need only an administrative assistant and a laptop. Maybe these are sales folks that need help setting up appointments and those kinds of things, but they re not involved in any kind of these R&D kind of things. So you ve got to live with these constraints. These are the minimum constraints on the way that the resources are allocated. (Figure 1) Well, the idea with Venn diagrams is that you use circles to represent sets and I m using this big box to kind of say everything s going to happen inside this big box. So, I could use a circle then to represent folks who receive an administrative assistant, and circle for folks who use research space, and a circle for folks who will receive a laptop computer. And what we do is that we arrange these circles so that they can overlap and interact with each other. So when we look at these minimum requirements that are being handed down from upper management, the 10 folks that will receive all three, well, these are the folks who are located where all three of the circles overlap right here and so we can just fill in that number. And I m going to put a little up arrow right there just to say that that s a minimum, that that number could go up. You know if we have enough resources, we could make that number larger. The four who need an administrative assistant and a research space, well that s where these two overlap, but not with the third one. So that would be right there and again I m going to indicate that that number could go up. The 10 that need research space and a laptop right there, okay, and that could go up. And the 11 who need an administrative assistant and a laptop, that s right 4

5 Figure 1, Segment 2 there, okay? (Figure 2) Now let s deal with the resources that we have available. We ve got 40 administrative assistants we can pass around. That means that the total of numbers inside this circle needs to be 40. So to see how many go right here, I just need to subtract the things I ve already allocated from 40, I think that s 25 allocated, so that leaves us with 15. And that number could go down. If these go up, that would have to go down, right? The 24 folks who can use research space, well I ve already allocated all 24 of those. So that zero there means that nobody can just get research space and that s not just a maximum, that s a minimum. I can t go below zero people. So that pretty well locks in then the other numbers, okay, inside that circle. The 60 who can receive a laptop, well, I ve already allocated 31, so that leaves me with 29. And then the last one, well, you may not have thought of this, the folks who receive nothing are in the block, but outside the circle. So, if I just take a hundred, and I subtract all the different amounts that I ve allocated, I think that leaves me with 21. (Figure 3) So with the current structure, I ve got 21 folks who received none of the three resources, and what you d like to do as an office manager is you d like to minimize that number. So you go back and you say, you give them this answer and they say No, no, no, that s too high, we ve got to get that number lower. Well, you can actually show that this is as low as it can get. Let s say I go back in here and the only number I can really raise here is the 11 so I go up to 12 folks getting both an administrative assistant and a laptop. Now, the effect that has is it lowers, right? If I raise this one by 1, I ve got to lower this one so that they all add up to 40 and I have to lower this one so that they add up to 60. Well, the effect, and everything has to add up to 100, so I ve got, I ve added 1, subtracted 1, subtracted 1, guess 5

6 Figure 2, Segment 2 Figure 3, Segment 2 6

7 what? I have to add 1 there. So, if I make the only change I can make, it actually makes the number larger. And so you can go back and defend that answer and say Well, that s the best I can do given the constraints you ve given me. (Figure 4) Figure 4, Segment 2 It also enables you, if play with it a little bit, to say Look, if you really want that lower, you ve got to do that, you ve got to change the overlaps you gave me in the minimum requirements. If you go back and you change, I think this was a 4 before, if you lower that to 3, well, that means I can raise this number by one. It means I can raise the number of folks getting research space only to just one. And then I think that number just stays the same. But when you add all those things up, I ve subtracted one, subtracted one, added one, subtracted one. So that number is one lower, so if they re willing to reduce the doubles, the folks getting two of these resources, you can reduce the so-called withouts by one. (Figure 5) If they go through and reduce the number receiving all three of these things, that makes changes in then each of the circles. That number goes up by one, that one goes up by one, that one goes up by one, and that causes then a change of two. The original answer was 21, that comes down to 19 now. So if they reduce the triples, that can reduce the folks without resources by two. (Figure 6) So, if they really want to lower that number then, they can go back and change their minimum requirements for folks receiving both of those things. That s a nice little example of resource management. It comes from a basic knowledge of sets and that s stuff we teach in that general math class. I ll flash some things up here on the screen, then, to kind of summarize what we ve talked about. 7

8 Figure 5, Segment 2 Figure 6, Segment 2 8

9 Summary page 1, Segment 2 Summary page 2, Segment 2 9

10 3 Exponent rules Compound interest and the Rule of Seventy Objective: To show how exponent rules can be used in the course of solving a real-world problem. This segment also introduces the concept of compound interest, and shows how continuously compounded interest can be modeled with a continuous exponential function. There is some foreshadowing of calculus, as the limit definition of Euler s number e is subtly introduced, and an explanation of why the Rule of Seventy is valid. One last thing I wanted to talk to you about today, in the algebra courses right now you re probably going back and reviewing the exponent rules and when we do this, this is very basic stuff. One of the first questions I hear is Why do I need to know exponent rules? It seems so dull, and I m with you, on the surface, it seems kind of dull. I d like to show you an example of where exponent rules are used and it involves understanding investment strategies and eventually I ll leave you with a good little rule of thumb that you can use for investment purposes. I should flash these up. If you take two things with the same base and they re both raised to a power and you multiply them, you can add the exponents. If you divide, you subtract the exponents, and if you take something raised to a power and you raise it to another power, you actually multiply the exponents and that s the kind of thing that you practice in your homework assignments, those kinds of things. (Figure 1) Figure 1, Segment 3 10

11 Well, suppose you have some kind of set amount of money that you want to invest and it could be a compound interest account with a bank, or it could be a mutual fund or some kind of investment like that. Maybe they re offering you something like four percent, which would be a pretty good rate for right now. A good question would be how long will it take to double your money? That s something I would want to know. If I m going to throw this money in the bank and not spend it right now, I d like to know how long I have to wait for it to double. And then they sometimes offer you things that are compounded more often. How important is that to the accumulation of money I m going to get? What if I can get five percent somewhere else? Is that going to make a big difference in how long it s going to take to double? I mean that s a good question. Maybe I get a little risky and I decide to put some money in stocks or bonds or something I can get 10 percent. Will it take half as long to double? That s a good question. Well, let s work this out. This is what I kind of want to work with here and it involves a knowledge of how we calculate interest. A little terminology: the idea of principal, that s the amount you invest and draw interest on and then to calculate interest you take your principal and you multiply it by the rate and time. Now, the rate we convert to a decimal before we do this and the rate and time have to be for the same units. If we re talking about time in terms of years, then we want to use an annual rate. That s the kind of thing I m talking about. And I can write that shorthand as P times r times t P for principal, r for rate, and t for time. To get your new amount, you take your old amount and add the interest. So I can actually write that kind of as a little expression. P is your principal plus P rt. I can factor this is a little algebra trick I can pull the P out of each of these and it leaves me with a 1 + rt. The reason I show you that is I m going to use this formula in just a few minutes here. (Figure 2) This business of compound interest, that means that after certain period of time, I take that interest and I add it to my principal. Then the next time, I don t take it out of the bank and I reinvest it. I take that new amount then as the principal for the next pay period, that s kind of the idea. So you accumulate money. You actually draw interest on your interest that way. So if we re doing four percent, this is the original situation, compounded annually and I m calling the principal 1, 1 what? 1 dollar? I don t know, 1 chunk o money. It doesn t matter, whatever you want to call it. One pot of gold, whatever it is. Well, at the end of that first pay period which is one year, I ve got my principal and then one plus the rate times the time and that s The compounding comes in when you make this amount your new principal. And you do the same calculation again and it goes up a bit. (Figure 3) But here s what s happening, this is where exponent rules come in. The original 1.04 is raised to the first power, you ve got another 1.04 raised to the first power. One plus one is two, so your new exponent is two, okay? That s the first exponent rule we looked at. (Figure 3) Do it the third year, again I raised it so it raises the exponent by one and we keep doing this until the end of the t th year. That s probably not the best way to think about it. You ve got 1.04 up to the t th power. (Figure 3) And I can actually graph that for you, go through at the end of each of these years, let s go back to this, you ve got your dots, that s the amount you have at the end of each year and I ve got a horizontal line here to represent two chunk o money s, whatever that is. So it takes actually 18 years to get above that line. (Figure 4) Now if I go back, here s the other question. What if I compound more often? Well, what 11

12 Figure 2, Segment 3 Figure 3, Segment 3 12

13 Figure 4, Segment 3 has changed in this calculation is that it s still four percent, but now the amount of time is one fourth of a year. So that gives me a different number, At the end of the second quarter I compound this interest again and this 1.01 becomes my new principal. But again, these are exponent rules and so as I do this the next quarter goes up to three. Into the first year, this is now four. I ve compounded four times at the end of that first year. So after the t th year, it is now 1.01 that has changed, and it s now four times t, because I m doing it four times a year. (Figure 5) So, if I go back, and let me put these dots right on top of the others, it turns out they re right on top of the others. It doubles in 17 1 years, not because 2 I ve drawn that much more money, but because now I can get my money out or check on it in a quarter year increments as opposed to whole year increments. But those dots are basically right on top of each other. (Figure 6) Now, here s a trick. This looks like it has nothing to do with what I m doing, but it does. If you take this expression and you think about plugging in bigger and bigger numbers in, when that s one, it s one plus one over one, which is two raised to the first power that s two. And then when I go to ten, it gets bigger, and when I go to 100, it gets bigger. But watch what happens. It s getting bigger all the time, but some of the decimals start to kind of stay the same. There, there, you get really big numbers now. It actually levels out it s a weird idea it kind of levels out to this number blah blah blah. It s an irrational number. It s a non-terminating, non-repeating decimal and it s hard to write down, so we typically just use the letter e for this. (Figure 7) 13

14 Figure 5, Segment 3 Figure 6, Segment 3 14

15 Figure 7, Segment 3 In Figure 7, I elude to one way of defining Euler s number e, namely ( e = lim n. n n) Recall from calculus that we may also define e as the value such that and as the infinite series ln e = e = e 1 1 du = 1, u n=0 1 n!. Well, if I m thinking about this calculation, this r to the n, I can do this kind of deal. I can replace n with rk, and when I do that, the r s cancel and here s your exponent rules again. I can pull this r outside and so this business is the e and it s raised to the r power. And for t years, again you just pull this t out and it s e rt. (Figure 7) So, what I ve got is a nice curve you can graph on your graphing calculator. I don t have to plot these points and I can just draw that curve, and that e is a button on your calculator that you can draw and it crosses two around years. (Figure 8) If I bump this up to five, it does make a sizeable difference in the amount of time it takes to double. And if I go to ten, it doubles in exactly half the time it turns out. (Figure 8) That s a good little rule of thumb. In fact, it s called the Rule of Seventy, and that is that 15

16 whatever interest rate you have and how long it takes to double, you take 70 and divide it by the interest rate, and that s how many years it takes. And to show you that, you know your rate, you always move your decimal two places, that is you divide by 100. Well, where 70 comes from is e 0.70 is roughly equal to two, and so if you think about multiplying by r, this 0.70 divided by r replaces t, and t turns out to be 70 divided by k. (Figure 9) So that s a nice little rule of thumb all the finance people know this rule and if you talk to some kind of financial planner they ll tell you How soon do you want this to double? That ll determine how risky you ll have to be and what kind of interest rate you have to get to make it happen. Figure 8, Segment 3 Okay, and you can flash up the summary stuff on this as well. Closing That s our first episode of Math Matters: Why Do I Need To Know This? I hope you find some of this stuff useful. I realize a lot of these things I went through very quickly, so let me just tell you that all the things you ve seen here today are downloadable from our webpage and they ll flash that up there at the end of the program. Anything that was too quick and you want to go back and check and see it yourself, go to our page and download it and look things over. Also, come back to me with suggestions. I d love to hear from you. And so with that, I m going to wrap up the first episode. See you next week. 16

17 Figure 9, Segment 3 Summary page 1, Segment 3 17

18 Summary page 2, Segment 3 18

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