This site updated August 26, 2014

Recent
updates will usually be above the first horizontal line.

Looking for something? See the page index here.

Looking for something? See the page index here.

The will to win is grossly
overrated.

The will to prepare is far more important.

-- Bobby Knight, basketball coach

The will to prepare is far more important.

-- Bobby Knight, basketball coach

Precalculus 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 you.

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 course.

Pearls before Swine is a comic strip you can read here:

** Philosophy
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
about math.**

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**
and **valuable**.
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

2) facts.

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
techniques.

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
we
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
created?

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.

Section
summaries: Your
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
is critical.)

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
"representative"
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
not one-to-one.

Precalculus requires 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 not "fun").

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.

Read each section. Do not 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 unclear, ask!

Don't just
skim.

Don't expect
that only high points are important
(Don't read only
the bold parts).

Don't skip
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.)

Advice from previous students about how to do well in this course. Believe it!

Links to articles on learning:

"Is
the
internet
making us stupid?" Read about it here:

(Really, it
is short, so read it!)

The
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 article.

Learning while multitasking. Recently the news has had quite a bit about research on multitasking. I summarized some of it here, and provide links.

http://www.math.montana.edu/precalculus/Multitasking2.html

"The huge finding is, the more media people use the worse they are at using any media. We were totally shocked."

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.

****************************************************************************************************

Course supervisor: Prof. Warren Esty, Department of Mathematical Sciences, Wilson Hall 2-238. (406) 994-5354. westy AT math.montana.edu (If you want to arrange something, I prefer phone calls. My office hours are here.)

Required
text: *Precalculus*, 6th edition,
by
Warren Esty.
(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!)

You must satisfy our special prerequisite to stay in Math M-151. Have you satisfied it? (Many incoming students who imagine they have actually have not. Check it!)

Course Policies (which includes sections about these and other topics):

"I
took
precalculus (or
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
about prerequisites.

Prerequisite
(you must satisfy the prerequisite!), Work, Calculators, Exams and Grading, Course Goals
In this course, calculators
are a
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?

In this
course, calculators
are a
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.
Other important
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
there.
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
courses, you
may have a conflict. Look up their exam times
now and
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

Program your calculator with the Quadratic Formula. Here is a simple four-line program for the TI-83 or 84. Here it is:

Prompt A, B, C(-B+√(B

(-B-√(B

Disp P, M ______________ Here is how to program it:

Hit PRGM

Follow each line here with

Arrow right to NEW ENTER

Enter the name, letter by letter, say, QUAD (the blinking "A" means Alphabetic mode which refers to the letters in green on your keyboard)

ENTER

Prompt A, B, C

To find the

Arrow right to I/O (for Input/Output) and down to

For "A", type ALPHA A, (then ALPHA B, ALPHA C, then ENTER)

(-B+√(B

The "->" command is for STO

It stores numbers in memory We use "P" for "plus" and "M" for "minus".

(-B-√(B

Disp P, M

The Disp command is for

At this line, you can QUIT (2nd QUIT)

If 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 of NEW).

[Now "quit" and try it out on an example where you know the answer. For example, to run it, hit

PRGMarrow to QUAD and hit ENTER and ENTER again.

Try to
solve x^{2} - 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]

You must come 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.'”

You 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 read:

https://forge.montana.edu/projects/198/wiki/How_Do_I_Forward_My_Student_Email

Your MSU e-mail address is firstname.lastname@msu.montana.edu

And you log in from this page:

https://cas.montana.edu/idp/Authn/UserPassword

(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.

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 calculator.

************************************************************************************************

This is 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.

You can quit here. The rest gives some interesting links, relevant to education, but not required for Math 151.

"Amusing
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

The book
itself is
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
Medina.

"**The brain is an amazing ****thing****.** Most of us have no idea what’s
really going on inside our heads.

"How do we
learn? What
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
knowledge?

"Brain Rules
is about what
we know for sure, and what we might do about it."

Here
is the fascinating site. Learn about how to learn. Pay attention to
the "12 rules".

"Eating
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.’’
Continued here:

http://well.blogs.nytimes.com/2009/08/13/fatty-foods-affect-memory-and-exercise/?hp

More about learning:

Here
is
a link to a
fascinating
commercial site on learning (believe it!), and a link to its page
on the role of
sleep in learning.

"At a certain point we either gonna have to put away childish things 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

Index:
This site has information about

Calculator programs (like the
Quadratic Formula) we use frequently.

Changing how you think about math, and why you should.

Conflicts: If you have a time conflict on an exam, or a disability, see here. Course supervisor contact information.

Exams: Previous exams you can use as study guides.

instructors, e-mails, rooms 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

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 tutoring. Why not take advantage of it?) Homework assignments and exam dates on the calendar

Instructors: instructors, e-mails, office hours, rooms and exam rooms, times.

Learning: Thoughts about learning.

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 exponentials], 6&7 [trig] will be added later)

Math Learning Center hours (for free tutoring. Why not take advantage of it?) Policies: The course policies. Prerequisites: The course prerequisites.

Office hours

Section 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 edition.

Trigonometry. For Chapters 6 and 7 on trigonometry, here are activities for trigonometry and calculator programs for the Law of Cosines.

This webpage is maintained by Warren Esty: westy at math dot montana dot edu 994-5354. Please report broken links, etc.