Precalculus main page

The will to win is grossly overrated.
The will to prepare is far more important.
-- Bobby Knight, basketball coach

Practice beats talent
when talent doesn't practice
-- unknown author

Here is the curve for Exam 1, Spring 2013:

85-87 A-, 88-100 A;
75-77 B-, 78-81 B, 82-84 B+ ;
65-67 C-, 68-71 C, 72-74 C+
55-57 D-, 58-61 D, 62-64 D+
below 55 F

Expect your course letter grade to resemble your exam letter grades, possibly modified upward if you almost always do the homework on time and take the required quizzes.
    If you did well, congratulations!
    If you want to do better next time,
1) study two hours outside class for each class hour, on average,
2) read the text (You learn to read by reading, and improving your math reading and writing skills is a major goal of this course)
3) use the Math Learning Center for free tutoring,
4) study efficiently with this advice and this advice from former students

As in any college course, your instructor does not have time to cover all the material during class. But the text covers it all. You are expected to read the text and learn from it outside class.

    The Math Department just had an American instructor of high school math in China give a talk about high school in China. Here is a Chinese school week.

In Chapter 3, your job is to:

3.1 Change how you think about lines. Learn to use the "point-slope" form and the "two-point" form of lines in preference to the famous "y = mx + b" (slope-intercept) form. Do not simplify to slope-intercept form (unless there is a good reason to). Learn how to use point-slope form and two-point form when the points are given functionally or with letters rather than numbers. Learn how linear interpolation works using the "two-point" form.

3.2 Learn how to "complete the square" of quadratics. Learn that when the square is complete the expression exhibits the location of the graph and its axis of symmetry, which is also visible in the Quadratic Formula. Learn how to apply the Quadratic Formula to solve equations in which the unknown is not "x" or the coefficients are not the usual letters (This is related to a good understanding of "placeholders").

3.3 Learn how the formula for distance in the plane is essentially the Pythagorean Theorem. Similarly, equations of circles are given by the distance formula and the Pythagorean Theorem.

3.4 Learn that quadratics which can be written "ax² + bx + c" often may instead be written in factored form "k(x - b)(x - c)" (the "b" and the "c" are not the same in the two forms). We can use graphs to factor and identify quadratics using the "Factor Theorem." If the graph of a quadratic intersects the x-axis at x = c, then "x - c" is a factor. The constant factor, k, is not determined by where the graph crosses the x-axis, however it can be found using one other point.

3.5 Learn that indirect word problems require you to "Build your own formula" which expresses the operations in the particular problem. Cue words indicate operations you express in your formula. Guess-and-check can help you see the operations you need to put into the formula. Usually we build specialized formulas from components that are well-known basic formulas, so write down relevant basic formulas.

3.6. Learn how to approach problems you don't know how to do! (This is what is called "problem solving.") The key is to begin writing the stated information symbolically (even if you don't, at first, see where it is going).

Jan. 31
Exam 1 information is below.

Can Smart Machines do your Job?
http://www.sci-tech-today.com/news/Can-Smart-Machines-Do-Your-Job-/story.xhtml?story_id=13200DRQLE20
January 25 this article came out reinforcing my point about what you are to learn in this course. The article begins by discussing a man who makes $67,000 a year whose job is being replaced by machines. It continues with a broader discussion of the economy and how many good jobs are disappearing.

A very similar article, "New technology is erasing many middle class jobs," was printed in the Bozeman Daily Chronicle Sunday, Jan. 27, 2013: http://tinyurl.com/az5v6ra
(The full URL is here:)
http://www.heraldtribune.com/article/20130123/ARCHIVES/301231035/-1/todayspaper?Title=New-technology-is-erasing-many-middle-class-jobs-By-BERNARD-CONDON-and-PAUL-WISEMAN-AP-Business-Writers

Jan.24
There will be optional review sessions the Sunday and Monday before each exam and the final. They will be in Wilson 1-141 from 6:00 - 8:00 pm. 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!

Exam 1 Room Alert! Exam 1 is Tuesday Feb. 5, at 6:00 pm. It is not in the usual room. The room is:
M&F class hour, section number, instructor,       exam room
8:00 (001) Brandon Smart      Wilson 1-122
9:00 and 10:00 (002 and 003) Jade Schmidt    Reid 105      [Do you know where the building is?]
11:00 (004) Michael Schwager   Wilson 1-132
12:00 (005) Alisia DeHart Wilson 1-121
1:10 and 2:10 (006 and 007) Jocelyn Short   Reid 105
    Exam day there is no class at the regular time.

    If you have an academic time conflict with our 6:00 pm exam (say, you have a class that meets at 6:00 Tuesday evenings), or a documented disability (with a "blue card" from Student Services), please contact the course supervisor Prof. Esty right away: 994-5354 or westy at math.montana.edu. See here for details.

Jan. 15
About Exam 1: Exam problems are mostly like the "B" homework problems. Look at old versions of Exam 1, on-line here. However, not all old first exams cover the same sections. This semester the first exam covers Sections 1.1 through 2.3, and the old first exams only went through 2.1 or 2.2. Last Fall's exam covered through 2.2 (It says on the top of the exam which sections it covers.) So, be sure you can also do problems from Section 2.3 for this semester's exam.
    If there is something you don't know, or don't know how to do, be sure to study that. There are several levels of algebra, and most of the algebra you previously learned is at a lower level--a level that will not be emphasized on the exam. The exam tests you on higher-level skills--on material newly learned in this course. Make sure you can do the "B" problems.
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. 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). This is a reading and writing course and problems on the old exams show you we mean it. If you don't read the text and learn how symbolic mathematics is written, you will have a hard time with the exams. Read in order to learn to read and write.
    There are problems you will find hard if you have not put a lot of time in playing with your calculator. You do not develop calculator skills without practicing a lot.
    Be sure to bring your graphing calculator with the Quadratic Formula programmed into it.

We have free tutoring all day long in the "Math Learning Center". (The linked page lists hours with a Precalculus instructor. However, the MLC 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.'”

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.

Jan. 9, 2013

The 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.
    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 last year. 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.)

There are big changes from previous semesters. 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.

Index: This site has information about
   the first day
    instructors, e-mails, rooms and exam rooms, times.
    the homework assignments and exam dates on the calendar
    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.)
    The world is changing rapidly. (The course philosphy. Use the link to see why this course is somewhat different than it used to be.)
Math Learning Center hours (for free tutoring. Why not take advantage of it?)
    Calculator programs (like the Quadratic Formula) we use frequently.
    The course policies.
    The course prerequisites.
    Previous exams you can use as study guides (But exams this semester will be different).
   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.
    If you have a time conflict on an exam, see here.
    Outlines of what you need to learn in Chapters 1&2 (3, 4, 5 [logs and exponentials], 6&7 [trig] will be added later)
    For Chapters 6 and 7 on trigonometry, here are activities for trigonometry and calculator programs for the Law of Cosines.
    Click here if (and only if) your text is the 4th or 5th edition and not the current 6th edition.
    For final-exam times and common-hour exams times for all courses at MSU, see here.

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").

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.