Precalculus, Math, M-151, Fall
This site updated August 26, 2014
updates will usually be above the first horizontal line.
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The will to win is grossly
The will to prepare is far more important.
-- Bobby Knight, basketball coach
students: Read this (long) page!
The beginning: Homework is due every day. Here is the calendar of class work,
homework, and exams.
You will get a copy on colored paper in class the first day.
Follow these links to see how this web site can help
The course requirements, prerequisites, policies,
grading and administration.
Make sure you have met the prerequisite by testing into this class.
Short outline of what you are supposed to
learn in Chapter 1.
We offer free tutoring all day long and
your instructor has office hours. We want to
help. Take advantage of
the opportunities to learn!
This course is different. You will learn what is expected of you as you
go along, but reading the course philosophy
might help you understand why we do what we do.
This page has an index.
This page has links to thoughts about learning.
You will be learning for many years. Learn to learn efficiently. Here
is some of what is known about how to learn. Links about math and jobs.
Exams dates are given on the calendar and you may study from numerous previous exams here.
If you have an exam conflict or a disability, see here.
Calculator programs for TI calculators.
Precalculus requires you to change.
Major topics of each section.
So true! Read your text! Learning to read mathematics with
comprehension is a goal of this
Pearls before Swine is a comic strip you can read here:
of this Course. Read this!
The world is
changing rapidly. Everyday things like iPads, smartphones, apps,
Facebook, Twitter, and massive multi-player on-line games didn't even
exist 10 years ago. What you need to learn about mathematics is
changing rapidly too. Most of the math you learned in high school can
be done by machines. Most of what students used to learn in a typical
precalculus course or even a calculus course can be done by machines
(for free!) So, if you want to learn something with real value, you
need to think about mathematics differently.
This course requires you to change your thinking
This course is different. It is intended to teach
you things that machines can not do for you. It is designed to teach
you what you need to know
about mathematics at the precalculus level that will make you capable
Software (like wolframalpha.com -- try it!)
can compute answers to symbolic algebra and calculus problems, if you know how to communicate with
it. So, learning to communicate mathematics to machines is one step.
But the real world does not pose problems in symbols, so another
important step is to learn how to take real-world problems and convert
them to mathematical notation (The term "indirect" in Section 1.1 refers to this.)
In the past math courses emphasized
1) computational skills (because, not many years
ago, computations were done by hand) and
But now if you want to solve an equation, there is
software that can
"do" that type of math problem. Now, everyone knows you can look up
facts on the web. If you can pose a math question well, you can
probably find an app or web page to answer it, if you can read
and write mathematics well enough. Nevertheless, there is still a lot
of math to learn.You need to learn how to communicate with your
software, interpret graphs, write and
read mathematics to understand how to formulate questions and
interpret answers. Also, you need to have the right question come
to mind, which is a non-trivial skill. We want you to have the
prior knowledge required to think of the right questions and
In summary, if all you know are the facts and
computational skills that math classes previously emphasized (in your
school!), you have little value to anyone. The best person at
multiplying three-digit numbers can be replaced by a $5 calculator. The
best person at solving symbolic algebra problems can be replaced by
free software on a website! We want to add value to you, not
teach you valueless skills.
That is why this course is different--even from
what it used to be a few years ago. The world is changing and what
teach and what you need to learn are changing too.
1) This course encourages you to use machines to do problems that
are already given in symbols,
2) but, you will not get a lot of credit for being able to do
symbolic problems like you learned to do in high school.
3) The intention is to help you develop essential concepts,
abilities, and interpretation skills required to apply math to the real
world and do things machines won't do for you.
4) You will learn to have the right things "come to mind". (It is
one level of learning to know an answer when prompted, another to have
it come to mind without prompting.)
5) Exams will focus on concepts and skills you need to develop,
not on solving problems that machines can solve for free.
6) The point is to add value
to you (knowing how to solve for x in problems posed in symbols
is no longer a valuable skill -- sorry!)
7) Reading and writing math forces you to focus on essential,
valuable, mathematical concepts.
8) Expect exams to be quite different from math exams you have
taken before (even different from previous exams given in this class!)
If you have read your text closely, written symbolic math daily on
homework, and assimilated the concepts covered in the text, you will do
well. However, if all you can do is compute numerical answers to
symbolic problems, you will not do well (because the value you added to
yourself would be close to zero.)
Take a look at this brilliant artwork created from words
on this page. How much do you suppose it cost to have this complex work
It cost nothing. It was free on a
web site! I just copied and dropped
in the text
of this page and it arranged the words with size corresponding to word
This (previously) $1000 artwork was free! The point is, machines can do
a lot of what we used to train people to
do (and pay them well for--but not anymore!). This course is aware of
that and will seem
different to you because it teaches useful and valuable skills you were
not taught in high school.
There are big changes from a few years ago. You need a tool
that can do what a graphing calculator can do, but it need not be the
Texas Instruments graphing calculator we will use in class (a TI-83 or
84). If you find a great app, tell your instructor and your
friends! I like "Free Graphing Calculator" (by William Jockusch)
and we have had the app "Graphing Calculator HD" recommended, but there
are many that would work fine. There will be some changes to exams too.
There will be fewer questions asking for calculations that machines can
do. Calculators can help you learn concepts, so we want you to use
calculators a lot, but your understanding of the concepts you develop
can be tested without using calculators.
When we do graphing-calculator activities in class,
you may use your laptop or iPad or Smartphone or calculator to
participate. However, using technology in class for non-math activities
is prohibited and a severe breach of etiquette.
job is to
1.1. Learn what it means for a problem to
be indirect and to become comfortable with working indirectly, that is,
writing about operations you don't actually do.
1.2. Learn how order is expressed in written mathematics and on your
personal calculator so well that you can rapidly evaluate complicated
expressions correctly. This requires learning how to insert parentheses
that are not in the usual written mathematics. Practice until you can
get five problems from B2-B16 correct in a row.
1.3. Learn how functional notation is used to express sequences
of operations. Learn to distinguish the function from the notation used
to express it, and how to apply that function to expressions other than
"x". (This is also discussed in Section 2.2.)
1.4. Learn how to read definitions and theorems (They express
mathematical methods). Learn how to write mathematical methods in
symbolic notation. (This is a course-long project, begun in Section 1.1
and treated as the focus of Section 1.4. Reading and writing are
essential to word problems, so learning to read and write symbolically
1.5. Learn to read graphs (that is, extract information they
contain). Learn how to graph with your graphing calculator and obtain a
"representative" graph. Learn to select and modify windows, and use the
calculator's capabilities to determine key points on graphs. (This is
continued in Section 2.1.)
1.6. Recall the usual algebraic ways to solve equations. Learn to
rapidly classify equations according to which way to solve them, chosen
from the "four ways to solve an equation." (This is a section on making
good algebraic decisions.)
2.1. Learn how to choose windows that make graphs
or have a particular look. Learn how changes in the window will change
the appearance of graphs. (This section continues Section 1.5)
2.2. Learn how notation expresses functional composition. Learn
how given graphs are affected by composition with addition,
subtraction, multiplication, division, and attaching a negative sign.
2.3. Learn that solutions to "f(x) = c"
may not be unique if the
function is not "one-to-one." Learn how to recognize when a function is
not one-to-one and how to deal with the complications that occur when
your calculator has an inverse function but the original function is
[to be continued]
you to change!
It is not just more of the
same--it is not just more math methods.
You must change the way
you think about math symbols to focus on operations and order.
You must change how you
learn math to also learning by reading the text outside of class. (This
is not high school where your teacher has enough time to cover it all.)
You must change your study habits
to devote hours to learning and practicing outside class (even if it is
Learning is uncomfortable. In sports
you don't get stronger until you lift heavy weights and you get sore
with pain. In math you don't get better by doing the easy, painless,
work. It takes concentration and, yes, discomfort. Accept that fact and
you will do much better.
Thoughts about Learning.
takes a lot of
effort! But, you will be learning an extremely valuable skill.
skip the harder parts. In fact, when the going gets rough you need to
slow down and read it several times until it makes sense. If it remains
that only high points are important
(Don't read only
the bold parts).
the rest of the paragraph because you
want to move
along to the next high point.
Read it all!
Advice, designed for this course,
about how to learn math. (Read this!
It has some helpful, and perhaps surprising, ideas. Here is a copy in Word.)
previous students about how to do well in
this course. Believe it!
Links to articles on learning:
making us stupid?" Read about it here:
is short, so read it!)
original article in The
Atlantic magazine, "Is Google making us stupid," is not short,
but fascinating and worthy of contemplation (However, I don't expect
you to read it):
It has provoked quite
a buzz, so search on
the title will get many hits.
Wikipedia has a summary
and discussion of responses, pro and con, to the original
Learning while multitasking. Recently
the news has had quite a bit about research on multitasking. I
summarized some of it here, and provide links.
huge finding is, the more media people use the worse they are at using
any media. We were totally shocked."
new is that even if you can learn while distracted, it changes how you
learn"--making the learning "less efficient and useful."
A summary of new research on multitasking
says it has a negative effect on learning. You will be better at what
you do if you do one thing at a time. (For example, don't switch
attention to texting [at all!] or Facebook while you are studying.)
And, you will get as much or more done. Don't kid yourself that
multitasking is somehow efficient. It is not.
supervisor: Prof. Warren Esty,
Department of Mathematical Sciences, Wilson Hall 2-238.
AT math.montana.edu (If you want to arrange something, I
prefer phone calls. My office hours are here.)
text: Precalculus, 6th edition,
(The 5th or 4th editions will serve just fine, but correct their typos).
Required graphing-calculator capability:
Calculators play a large
role, and you must have access to graphing-calculator functionality,
but this semester you do not need to buy a stand-alone
calculator if you have an iPad or Smartphone or laptop with an
equivalent calculator app or software.(The iPad app "Graphing
Calculator HD" will serve. I have had the app "Algeo calculator" highly
recommended and it looks good and is free. Probably many other app
would work fine too. If you get an app, you must take the time to learn
to use it!)
prerequisite to stay in Math
M-151. Have you satisfied it?
students who imagine they have actually have not. Check it!)
(which includes sections about these and other topics):
calculus) in High
School, so I have satisfied the prerequisite, right?" No! You
must test into the course.
What you took in high school
is does not
count. What you know
counts. Here are the rules
(you must satisfy the prerequisite!), Work, Calculators, Exams and Grading, Course Goals
In this course, calculators
learning tool, not just a calculating tool. Calculators help in
two main ways. By making lower-level work less time-consuming, we can Calculators.
In this course you are supposed to
develop essential algebraic concepts. Graphing calculators or graphing
apps can help and are required. We will use a stand-alone TI calculator
in class, however, you may use any technology which is more or less
equivalent, including smart apps or any internet-based graphing
program. If you already own an iPad, why pay $100 for a calculator when
you effectively already have one?
learning tool, not just a calculating tool. Calculators help in
two main ways. By making lower-level work less time-consuming, we can
1) Concentrate attention on
essential points, and
2) Increase the rate at which
students gather experience with the subject.
information that you will want to know. For example, copies of previous
exams are available on reserve in the Library. They are also on-line here.
We have free
tutoring! The Math Learning Center (1-110 Wilson) has free
tutoring 8:30am -9:00 pm M-Th and 8:30-2:00
Fridays. Click here
for more about its hours and when you can find a Precalculus instructor
are common-hour exams given at 6:00 pm. The dates are on our calendar.
Mark your personal calendar with these dates and times.
Be there! If you have an unavoidable
academic conflict, or a disability, see here.
If you are taking other common-hour exam
may have a conflict. Look up their exam times
see. If you have an academic conflict, you may be able
to resolve it by signing up (with Dr. Esty in 2-238 Wilson Hall) for
our alternative exam time (probably 4:45 the same day for common-hour
exams). However, you must sign up well in advance. Signing up
the last day is not an option.
We use calculators a great deal. Instructors will use the TI-83 or
TI-84, but you may use other models or iPad or SmartPhone apps. Learn
to use technology. If you use a TI calculator, one program you will
need many times is given next.
Activities. Chapter 1
your calculator with
the Quadratic Formula. Here is a simple
four-line program for the TI-83 or 84. Here it is:
(-B+√(B2-4*A*C))/(2*A) -> P
(-B-√(B2-4*A*C))/(2*A) -> M
Disp P, M
to program it:
Follow each line here with ENTER. Comments you do not type are in green.
Arrow right to NEW
Enter the name, letter by letter, say, QUAD (the blinking "A" means Alphabetic mode which
refers to the letters in green on your keyboard)
Prompt A, B, C
find the Prompt command, while writing the program, hit PRGM
(again) which brings up a menu.
right to I/O (for Input/Output) and down to Prompt. There
is a comma key above the 7 key.
For "A", type ALPHA A, (then ALPHA B, ALPHA C, then ENTER)
(-B+√(B2-4*A*C))/(2*A) -> P [again "B" is ALPHA B]
"->" command is for STOre (it appears as an arrow), on a key
near the bottom left.
stores numbers in memory We use "P" for "plus" and "M" for "minus".
(-B-√(B2-4*A*C))/(2*A) -> M
Disp P, M
Disp command is for Display, which is also under I/O (hit PRGM, arrow over to I/O, and down to Disp ENTER).
At this line, you can QUIT (2nd QUIT)
something goes wrong, don't worry. Just
QUIT (= 2nd
QUIT in yellow) and resume from where
you were by hitting PRGM and,
this time, EDIT (instead
[Now "quit" and try it out on an
example where you know the answer. For example, to run it, hit
arrow to QUAD and hit
ENTER and ENTER again.
solve x2 - 8x + 15 = 0. Did you get 5, 3?
If not, check your keystrokes.] If you want to do a second
example, you need not begin over, just hit ENTER and it will ask you
for the next value of "A".]
Prior to each exam there will be review sessions [dates and times to be announced]
prepared to show you have seriously tried the problems you ask
about. Do not kid yourself that asking questions on Monday is good
preparation for a Tuesday exam. Most of your preparation should have
been done long before Monday!
We have free tutoring all day long in the "Math
Learning Center". (However, it is closed the hour before the exam.)
Exam 1, like previous exams, comes with instructions. It says, "Show
clear supporting work on problems with several steps. Algebraic problems that display little
or no supporting work will get little or no credit. You do not
need to show work on one-step calculator problems. To solve
numerical problems guess-and-check is legal unless you are requested to
solve them 'algebraically.'”
should look at old versions of Exam 1. View them on-line here.
Be careful when you study the old exams. Not all old
first exams cover the same sections. This semester the first exam
covers Sections 1.1 through 2.2. (Spring 2012 only covered through
2.1.) Each exam says on the top which sections it covers.
You are supposed
to know a lot of algebra already. There are several levels of algebra,
and most of the algebra you learned in school is at a lower level--a
level that will not be
emphasized on the exam. The exam tests you on higher-level skills. It
tests you on material newly
learned in this course.
Be sure you can do the "B" problems. If there is
something you don't know, or don't know how to do, be sure to study
that. Don't be content with the algebra you knew before you signed up
for this course.
You are responsible for
reading and writing mathematics. On the exam we will
state theorems or definitions that you have not seen before and
ask you to read them and use them. This is a skill that cannot be
picked up in an hour or two. You learn to read by reading. We strongly
recommend you learn to read math by reading your text.
On the exam we may ask you to state methods
symbolically (as in Section 1.4).
Be sure to bring your graphing calculator with the
Quadratic Formula programmed into it.
There are many problems you will find hard if you
have not put a lot of time in playing with your calculator and reading
your text. You do not develop reading skills (required on the exam) by
watching your instructor. You do not develop calculator skills without
practicing a lot.
Have your MSU e-mail forwarded
to the e-mail address you actually
Your MSU e-mail address is firstname.lastname@example.org
And you log in from this page:
(which has a link on the MSU home page).
Go there and fix your e-mail address! Faculty will occasionally send
e-mail to you at that address. If you don't have it forwarded, you
won't get it. If you fix it now, you will avoid four years of problems.
Also, give MSU the phone number to actually use.
Other Resources: The Education Portal Academy
has some slick videos
on Precalculus topics. If you want to supplement the lectures, take
The Kahn Academy has Precalculus videos
too, but most of what we are do is in their algebra and trig sections.
Unfortunately, they have very many videos and many seem to develop the
topics slowly and often in a disorganized fashion, so I doubt anyone
would want to wade through them all.
In college your instructor
does not have enough class time to cover all the material. You are responsible for all the material in
the text anyway. You are expected to learn the rest outside
of class by reading the text. The homework and exam questions are all
closely related to things discussed in the text. Read it! Then, if
something is not clear, put in the time and effort to
figure it out.
Make sure you have a Quadratic Formula program in your
the end of the required Precalculus material at this time. Check back
for updates, especially when exams are about to happen and when we
get to Chapter 6 on trig.
quit here. The
rest gives some
interesting links, relevant to education, but not required for
Ourselves to Death"
is a prophetic book that was written in 1985. Here is a cartoon that
illustrates its preface. http://www.recombinantrecords.net/docs/2009-05-Amusing-Ourselves-to-Death.html
extremely interesting. It is amazing the something written then could
still be so relevant (even more relevant) now.
Brain Rules: 12 Principles for
Surviving and Thriving at Work, Home, and School, by John
"The brain is an amazing thing. Most of us have no idea what’s
really going on inside our heads.
"How do we
exactly do sleep and stress do to our brains? Why is multi-tasking a
myth? Why is it so easy to forget—and so important to repeat new
is about what
we know for sure, and what we might do about it."
is the fascinating site. Learn about how to learn. Pay attention to
the "12 rules".
fatty food appears
to take an almost immediate toll on both short-term memory and exercise
performance, according to new research on rats and people. 'We
expected to see changes, but maybe not so dramatic and not in such a
short space of time,' said Andrew Murray, the study’s lead author.’’
a link to a
commercial site on learning (believe it!), and a link to its page
on the role of
sleep in learning.
The famous essayist and
"MacArthur Genius Grant" recipient, David Foster Wallace said, in an
"At a certain point we either gonna have to put away
and discipline ourself about how much time do I spend being passively
entertained? And, how much time do I spend doing stuff that
actually isn't that much fun minute by minute, but that builds certain
muscles in me as a grown up and a human being? And, if we don't do that
... the cultures going to grind to a halt. Because we're gonna get so
interested in entertainment that we're not gonna want to do that work
that generates the income that buys the products that pays for the
advertising that disseminates the entertainment. .... It won't be
anybody else doing it to us, we will have done it to ourselves."
Practice beats talent
when talent doesn't practice
-- unknown author
This site has information about
Calendar: The homework
exam dates on the
Calculator programs (like the
Quadratic Formula) we use frequently.
you think about math, and why you should.
Conflicts: If you
have a time conflict on an
exam, or a
disability, see here.
supervisor contact information.
exams you can use as study guides.
and exam rooms, times.
Conflicts: If you
have a time conflict on an
exam, or a disability, see here.
For final-exam times and common-hour exams times for
all courses at MSU, see here.
Goals: The world is changing rapidly.
(The course philosophy. Use the link to see why this course is somewhat
different than it used to be.)
Changing how you think about math.
Help: Math Learning Center hours (for free
Why not take advantage of it?)
exam dates on the
instructors, e-mails, office hours, rooms
and exam rooms, times.
Advice, designed for this course, about how to learn math.
Advice from previous students
about how to do well in this course.
Outlines of what you need to learn
in Chapters 1&2 (3, 4, 5 [logs and
[trig] will be added later)
Learning Center hours (for free tutoring.
Why not take advantage of it?)
Policies: The course policies.
The course prerequisites.
Topics in a few words.
Textbook: textbook and calculating
technology requirements (a stand-alone calculator is no longer
required--a iPad app or internet-capable phone will suffice. You must
have some technology that allows you to do what a graphing calculator
does. We use a TI-84 in class.)
Click here if (and only
if) your text is the 4th or 5th edition and not the current 6th
and 7 on trigonometry, here
are activities for
and calculator programs for the Law of
webpage is maintained by Warren Esty:
westy at math dot montana dot edu
994-5354. Please report broken links, etc.