Math Matters: Why Do I Need To Know This?
Bruce Kessler, Department of Mathematics
Western Kentucky University
Episode One
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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,
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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 email address there
and we’ve got a webpage that we can be contacted at. Please, please use those.
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Summary page, Segment 1
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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
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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
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Figure 2, Segment 2
Figure 3, Segment 2
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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.
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Figure 5, Segment 2
Figure 6, Segment 2
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Summary page 1, Segment 2
Summary page 2, Segment 2
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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
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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 1.04. 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 tth year. That’s probably not the best way to think about it.
You’ve got 1.04 up to the tth 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
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Figure 2, Segment 3
Figure 3, Segment 3
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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, 1.01. 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
tth 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 12 years, not because
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 2.718 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)
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Figure 5, Segment 3
Figure 6, Segment 3
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Figure 7, Segment 3
In Figure 7, I elude to one way of defining Euler’s number e, namely
n
1
e = lim 1 +
.
n→∞
n
Recall from calculus that we may also define e as the value such that
Z e
1
du = 1,
ln e =
1 u
and as the infinite series
∞
X
1
e=
.
n!
n=0
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 ert . (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 17.33 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
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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 e0.70 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.
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Figure 9, Segment 3
Summary page 1, Segment 3
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Summary page 2, Segment 3
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