Precalculus, Math, M-151, Fall 2014
This site updated August 26, 2014
Recent updates will usually be above the first horizontal line.
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
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.

 Pearls before swine
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 -- 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?


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 frequency. 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.

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. 

             [to be continued]

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! 

Reading takes a lot of effort!  But, you will be learning an extremely valuable skill.
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.
        "The huge finding is, the more media people use the worse they are at using any media. We were totally shocked."
        "What's 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.

Course supervisor:  Prof. Warren Esty, Department of Mathematical Sciences,
Wilson Hall 2-238. (406) 994-5354. westy  AT  (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+√(B2-4*A*C))/(2*A) -> P
(-B-√(B2-4*A*C))/(2*A) -> M       
Disp P, M
______________ Here is how to program it:
            Follow each line here with ENTER. Comments you do not type are in green.
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)
Prompt A, B, C
        To find the Prompt command, while writing the program, hit PRGM (again) which brings up a menu.
        Arrow 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]      
        The "->" command is for STOre (it appears as an arrow), on a key near the bottom left.
        It stores numbers in memory We use "P" for "plus" and "M" for "minus".
(-B-√(B2-4*A*C))/(2*A) -> M        ENTER
Disp P, M
         The 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)

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

arrow to 
QUAD and hit ENTER and ENTER again.

Try to 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]
     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:
Your MSU e-mail address is
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 a look.
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.
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:

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.

The famous essayist and "MacArthur Genius Grant" recipient, David Foster Wallace said, in an interview,
    "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.

Calendar: The homework assignments and exam dates on the calendar

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 all courses at MSU, see here.

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, 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.