This site updated April 28, 2015

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

The final exam is Thursday, May 7, at 10:00am in the same room as all the other exams.

You may bring a half-sheet (one side) of notes in your own writing (no photocopies).

The trig identities at the top of the trig section on all old exams will again be there. However, the "Formulas" at the top of page 1 of some old exams (e.g. Spring 2013) will not.

There is one long review session Wednesday, 8:00-noon, in Wilson 1-138. Come for one question or come for an hour or come for a long time. Leave anytime. Come back later if you want. We are providing four hours of help. Come prepared to show you have seriously tried the problems you ask about. Asking questions on Wednesday is not sufficient preparation for a Thursday exam. Most of your preparation should have been done long before Tuesday!

If you have a conflict, contact me well before the exam.

This time you may have half a sheet of paper, one side, as notes in your own writing (no photocopies--prepare it yourself).

Warning: We spend more than two weeks on the three sections of Chapter 7. A lot of the final exam will be on Chapter 7, which it is full of new ideas that you must learn. Please continue to work on Precalculus all the way to the end of the semester. If you don't study Chapter 7 a poor final exam score will result and that will hurt your grade a lot. On the other hand, a good final exam score can help a lot. Take Chapter 7 seriously! (Here is what you need to know from Chapter 7.)

The final exam has slightly less than half its points on trig. Most of the trig is from Chapter 7 (Section 7.3 is on the exam, but not 7.4). Often there is

a multisided figure to solve as in Section 6.4

a derivation from Section 7.1

a derivation from a unit-circle figure as in Section 7.2 (resembling B17-B22)

a derivation from a right-triangle figure as in Section 7.2 (resembling B31-B44)

parts to find in a figure much like Figure 7.3.3

a derivation of one of the results from 7-3-2 through 7-3-5, given the previous results (like on a handout)

a problem of the sort found in 7.3 B4-48 that requires use of a trig identity. (The only trig identities you have to memorize are the Pythagorean Identity and the one for tangent. Any others you might need will be given.)

and more.

[We will have gone over these derivations in class several times. There will have been handouts covering these derivations. Memorize how the derivations go and then you can simply reproduce them on the exam.]

The final exam has slightly more than half its points on the algebra from Chapters 1 through 5. Often there are

problems resembling some of the unit-exam problems that many students missed (Be sure you can do the previous exams from this semester!)

This exam has several word problems where you are to SET UP the equation that could be solved to answer the question. Then you are instructed to not actually solve them. The setup is all that is required.

an expression to evaluate (like Section 1.2, B2-16)

a graph to read

lines in point-slope form

distance in a word problem

graphical factoring

percents in a word problem

word problems

an exponential equation to solve

and more

Old exams are here. We strongly recommend you not look at the solutions until after you have worked on the exam for an hour and 50 minutes. Use the solutions to confirm your answers, not to create answers. Anyone can read the solutions and agree that's how its done. But reading and learning are much different. You learn much more if you try seriously first.

April 15

The results of Exam 3, comments on letter grades, the importance of the final exam, and the possiblity of dropping by Friday, April 17.

The final exam will be in the same room, Thursday May 7 at 10:00 - 11:50 am. It is comprehensive, with about half on trig and most of that half on Chapter 7.

About Chapter 7: Chapter 7 is different. Yes, we are nearing the end of the course, but there is still time for you to learn something new and different. In Chapter 7 you will learn how to derive trig results from simple pictures. You must learn the connections between simple results you know and other, less-familiar, results that can be derived from the things you know. Chapter 7 emphasizes reasoning. You must learn why the results follow from one another and how they connect. There will be a great deal of reasoning required on the final, so learn how it is done.

Long ago we posted links to on-line trig activities. If you did not do them before, you will find them interesting and valuable for the current material and we strongly recommend you do them now.

Thoughts about Precalculus and Mathematics. In mathematics, there are some things you simply must memorize. For example, you must memorize the definition of sine. Other things it is better if you understand them. For example, how are sine and cosine related? Both a unit-circle picture and a trig identity should leap to mind.

Chapter 7 requires memorization of the definition of the new term radian and some other things, but mostly emphasizes relations between trig functions. Most of these formulas (e.g. area of a sector) and identities (e.g. cos(θ + π/2) = - sin(θ)) are best learned by understanding them. Pictures help.

Previous homework has often emphasized doing calculations. Now, Chapter 7 asks you to understand trig identities in the same way that all higher math is understood--with well-known things connected together in a way that allows you to create (derive) and understand new things. We build from what is known to what is not yet known. This procedure is to pervasive in mathematics that much of the goal of Chapter 7 is to get you to be able to string together simple known things in order to derive other things (for example, trig identities).

Sections 7.1, 7.2, and 7.3 give many important examples. Study how they fit and how those same methods can be used to create trig identities on demand, even if you do not memorize them.

Your Job. In Chapter 7, your job is to learn how to

7.1 Learn that radian measure is defined by s = rθ. Learn how to convert from radians to degrees and vice versa. Learn how to derive the area of a sector. How to derive the arc length when the central angle is given in degrees.

7.2 Learn that angles are regarded as rotations on a unit circle which begin with initial side along the positive x-axis. Two angles are "coterminal" if they terminate at the same point. Therefore, they have the same trig-function values. An angle and its reference angle have the same trig function values, except possibly for the sign (they might have a negative sign). Learn how derive trig identities by representing an angle θ in the first quadrant (not near 45 degrees) and locating the angle in question, say θ+π, and reading the picture for the trig function values of the new angle and how they relate to the trig function values of θ.

Learn how right triangles can be used to illustrate angles with various trig-function values (given with letters), and can be read, with the help of the Pythagorean Theorem, to yield other trig functions values. So, for example, if sin θ is x/2, we can derive tan θ just by looking at a well-labeled picture.

7.3 Learn about derivation and how complicated formulas can be derived by a sequence of simple steps. In particular, learn how the sum-of-angles trig formulas can be derived from a well-labeled picture (learn Figure 7.3.3), and how the other important trig identities follow step-by-step from them (that is, learn the entire sequence from 7.3.1 through 7.3.5).

(7.4 is interesting, but will not be on the final exam.)

---------------------

Here is the curve for Exam 3, Spring 2015:

85-87 A-; 88-100 A;

75-77 B-; 78-81 B; 82-84 B+ ;

60-62 C-; 63-71 C; 72-74 C+ ;

50-52 D-; 53-56 D; 57-59 D+;

below 50 F

How are you doing? Add your three exam scores (Don't average them). If your quizzes are comparable and your homework has been handed in regularly, if the total is

at least 245 of 300, you are on track for at least an A-

at least 215, you are on track for at least a B-

at least 170, you are on track for at least a C-

at least 140, you are on track for at least a D-

139 or below, you are on track for an F.

If you have handed in almost all the homework on time, you may get a notch higher. Often a good homework number raises, say, a C to C+.

There are 200 points on the final, and they can make a big difference, but it is not common for students to do much better on the final than on the first three exams. However, it does happen and some students "get it together" and do much better on the final, in which case they are rewarded with a better grade. On the other hand, some students fail to retain what they learned for earlier exams and do worse on the final, which is comprehensive. If you do poorly on the final we are justified in thinking you do not remember much, in which case a poor course letter grade is appropriate. You are expected to remember everything! So, the final can help a lot or hurt a lot.

The last day to drop is Friday, April 17. Are you going to continue, or not? Make up your mind quickly. If you do not want a letter grade, fill out a "Drop form," get two signatures, including your adviser's signature, and submit it to Montana Hall by the end of Friday, in which case you will get a "W" (withdrawn) instead of a letter grade. A "W" on your transcript does not affect your GPA, but gives you no credit for the course. The "instructor's signature" can be satisfied in three ways--your actual instructor can sign it (go to class to get it), or Dr. Esty can sign it, or it may be signed by an administrative assistant in the Main Math Office (Wilson Hall 2-214). However, you also need your adviser's signature, so allow some time to find that person. If you stay and get a low grade, it will count in your GPA, but it can be replaced if you take the course again and do better. Low grades are replaced and erased by taking the course again and doing better.

March 30

Read all the way down to the first horizontal line.

Exam 3 is Tuesday April 14.

It will be in the same exam room as last time.

New! You may create and bring to the exam half a sheet of paper, one side, as notes in your own writing (no photocopies--prepare it yourself). We will do that for the final exam, too.

If you have a time conflict, see here.

The exam will resemble old exams.

There will be review sessions, open to all M151 students. They will be two days before and the day before the exam. For Exam 3 they are Sunday, April 12, at 3:00-5:00 and Monday April 13, 6:00-8:00, both in Wilson room 1-119. You need not stay for the whole session and you may come late. (Wilson is usually locked on Sundays, but you may come to the door on the west side (south end) and get in that way.)

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 spent 1 day on 4.6, 5 days on Chapter 5 on exponents and logs, and 8 days on Chapter 6 (trig). The exam will reflect that. It is not just a trig exam.

The trig problems will go rapidly if you have your calculator programmed with the Law of Cosines, two ways.

There usually are two long problems from 6.4, one of which uses bearing the way we describe it on pages 380ff. If you don't understand bearing, you will lose a significant number of points. Please practice with your calculator a lot so you can do the calculations quickly. We expect you to do calculations with your calculator without writing long Law-of-Cosines equations on the exam page. However, at each stage you must tell us which law you are using.

There will be a word problem with exponential growth or decay.

For more about the Chapter 6 material, read the trig part below.

In Chapter 6, your job is to

6.1 Recognize the geometric cases and learn that the Angle-Side-Side case may yield two different triangles. Know the Pythagorean Theorem and that the sum of the angles of a triangle is 180 degrees. Learn the terms "alternate interior angles" and "vertical angles" discussed at the end of page 343 and beginning of 344.

6.2. Memorize the definitions of sine, cosine, and tangent two ways, both in a right triangle and in a unit circle (You must understand the unit-circle versions. The web activities are very good for that.) Learn how inverse sine and inverse cosine work, especially on the unit circle. Learn that solving "sin x = c" yields two possible answers, not just the one a calculator gives which is sometimes not the one you want. Memorize the values of the trig functions at the three most famous acute angles, 30, 45, and 60 degrees.

6.3. Learn to solve triangles by learning the Law of Sines and Law of Cosines (two ways) and when to use them, described as geometric cases. Program them into your calculator and learn how to use them. In the A-S-S case the Law of Cosines and the Quadratic Formula can be combined to find the third side (and there will often be two answers).

All you need to know to solve trig figures in Chapter 6 is: the angles in a triangle sum to 180 degrees, the Pythagorean Theorem, the definitions of sine, cosine, and tangent, the Law of Sines, and the Law of Cosines.

6.4. Learn to solve multi-sided figures by subdividing them into triangles and then using triangle-trig methods. Learn how "bearing" is expressed. Also, learn to sketch a reasonably good picture of trigonometric figures.

It is not that much, but it does require some memorization.

Most students really like trig and do very well in Chapter 6, triangle-trig. Take this chance to improve your grade and feel good about your math skills by devoting some time to this material.

Information about Trig (for Chapters 6 and 7): Do the on-line trig activities and program your calculator with two versions of the Law of Cosines.

Cool interactive, web-based, activities,

for
learning trigonometry (Chapters 6 and 7)

Learn and have fun at the same
time!

I searched youtube for trig lessons. Most seemed slow and dull. I did
not find an interesting lecture, and most of the lectures are
advertising teasers to get you to buy their whole course. The
activities I recommend (linked above) are the best I have found on the
web. I recommend you read your text and try to
follow it closely.

I strongly recommend you enter two versions of the Law of Cosines as programs into your calculator. The programs are very short and given here:

Here is a TI-83 calculator program to do the Law of Cosines (SAS version to solve
for the opposite side). Program 1:

Hit
PRGM

Follow each line here with
**ENTER**. Comments
you do not type are in black.

Go to NEW

Enter the
name, letter by letter, say, LAWCOS (the
blinking "A" means Alphabetic mode which refers to the
letters in green on your keyboard. Otherwise, to enter "D" hit "ALPHA
(a green key) D")

Name: LAWCOS

Prompt A,B,C

sqrt(A^{2} + B^{2}
- 2*A*B*cos(C)) -> D

Disp D

[In the above, "sqrt" means the square root key, and
"->" is the arrow or "STOre" key.]

In the above program A and B are adjacent sides and C is the included angle. D is the
opposite side (which we would call lower-case "c" if we could).

Keystroke details:

1) Hit "PRGM"

2) go to "NEW" and hit "ENTER"

3) Type in a name, say, "LAWCOS"
and ENTER

4) Find the "Prompt" command by hitting "PRGM"
[again], arrow over to "I/O" (input/output), arrow down to "Prompt" and
hit ENTER.

5) Type in "A,B,C"
[Use
the
alpha
key
to
use
the
alphabet.
There
is
a comma key]

6) Type in the formula (above)

7) "Display" the result (D) by typing PRGM, I/O
and selecting "Disp" and alpha D

8) Quit and run it on a problem where you know
the answer, to check it. (Reproduce one of the examples in
the text.)

Program 2: To use the Law of Cosines to solve for an angle, use this program (short!):

COSANG

Prompt A,B,C

cos^{-1}((C^{2}-A^{2}-B^{2})/(-2*A*B))
-> E

Disp E

This site will be updated when we get closer to Chapter 7.

Be sure to do the trig activities.

March 4

Here is the curve for Exam 2, Spring 2015:

80-82 A-; 83-100 A;

65-67 B-; 68-76 B; 77-79 B+;

50-52 C-; 53-61 C; 62-64 C+ ;

40-42 D-: 43-47 D; 47-49 D+.

below 40 F.

Do not think that exam is "over" and now you no longer have to know how to do the problems. You will see that material again and, in any case, those are the types of things you are supposed to learn how to do in this course.

How are you doing? Add your two exam scores (Don't average them). If your quizzes are comparable and your homework has been handed in regularly, if the total is

at least 160 of 200, you are on track for at least an A-

at least 140, you are on track for at least a B-

at least 110, you are on track for at least a C-

at least 90, you are on track for at least a D-

89 or below, you are on track for an F.

Students who regularly hand in the homework and do okay on the quizzes often get a course letter grade a notch higher. Also, there are many points left. Even if your grade is low now you can try to do much better on the remaining quizzes (quizzes total 100 points, with some dropped), the next 100-point exam, and the 200-point final exam. A strong final can help a lot. Also, a weak final can hurt a lot. You must prove you have retained what you learned by scoring well on the comprehensive final.

If you are not satisfied with your likely grade, you may submit (to the Registrar's office in Montana Hall) a signed drop form (with two signatures, from your instructor and your adviser).

If you want a better grade, study more, or study more effectively. The web has advice here about how to study math. Also, here is advice from previous students.

You are expected to learn by reading the text. If you skipped reading the text and your exam score was poor, it is time to learn from the poor results of your choice. Make a big change and read the text seriously. It takes time and effort. You don't get strong by watching others lift heavy weights! You get strong by lifting heavy weights yourself -- repeatedly!

Many students expect to practice (do homework) outside class, but do not expect to actually learn new things outside class. This is an unfortunate error. There is so much more time outside class than in class! College is (unlike high school) based on the premise that you will actually learn new things outside class. Also, college courses expect you to learn by serious reading. Many students don't read with intent to learn, but college success requires it. If you read in order to "look things up," not much learning results. Finding out how to do something is not the same as learning how to do it! Learning is long-term and "finding out" is short term. If you find out how to do it and then do it a couple of times, that is a very good start which may result in long-term learning. But, too often it does not.

Why not? It is a matter of intent and focus. Those who focus on "getting it done" (those who want someone to show them how to do some problem on some old exam and those who just want to get credit for the homework) need to change their attitude. They need to identify the principles--the common features of all similar problems -- and "learn how to do it." This requires reading the well-selected examples in the text and the explanation of their point, in addition to paying attention to the limited number of examples in the lecture, and identifying the point. The point applies to hundreds of examples. Identify the point and focus on learning it.

Learning takes effort and time.

March 4

If you have a conflict for Exam 2, see here.

Exam 2 is Tuesday after Spring Break, March 17. It covers everything to date but emphasizes 3.1 (lines) through 4.5.

Exam rooms are the same as last time (also listed here).

Please look at previous exams to see what it might be like.

There is usually, but not always:

a word problem with distance

percents

a word problem with average percent change (Examples 24, 25 page 241-242), but the subsection of 4.4 on "annunities" (p. 243) will not be on the exam

finding a polynomial through points (3.4 Example 9, 4.2 Example 11)

a line problem that needs point-slope form (the whole section 3.1)

a quadratic formula problem with unusual letters (page 134)

a problem where a square root must be eliminated (pages 222-224)

a problem requiring the use of properties of power functions (table, page 191)

numerous, mostly shorter, problems on other topics from 3.1 through 4.5.

Percents are extremely important. They are mentioned every day in the news and are critical in money matters. If you never need the rest of this course in your life and career, you will still need to grasp percents. Make sure you understand Section 4.4 on percents.

There will be review sessions, open to all M151 students, for each exam. They will be two days before and the day before the exam. For Exam 2 they are Sunday, March 15, at 3:00-5:00 and Monday March 16, 6:00-8:00, both in Wilson room 1-119. You need not stay for the whole session and you may come late. (Wilson is usually locked on Sundays, but you may come to the door on the west side (south end) and get in that way.)

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!

Feb. 9

Section topics Chapters 3 and 4. What you need to know.

Here is the curve for Exam 1, Spring 2015, based on

based on 85/75/60/50.

85-87 A-; 88-100 A;

75-77 B-; 78-81 B; 82-84 B+;

60-62 C-; 63-71 C; 72-74 C+ ;

50-52 D-: 53-56 D; 57-59 D+.

below 50 F.

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) use the Math Learning Center for free tutoring,

3) study efficiently with 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 had an American instructor of high school math in China (a former MSU grad student) give a talk about high school in China. Here is a Chinese school week at his school.

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 "ax

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).

About Chapter 4. Your job is to:

4.1. Learn the shapes of monomials and the approximate possible shapes of polynomials of degrees 1, 2, 3, 4, etc. Learn how many local extrema they can have and how many solutions a polynomial equation of degree n can have.

4.2. Learn how to factor cubics that factor in integers by using the Factor Theorem and a graph. Learn how the Factor Theorem and a graph makes factoring easy if the expression factors in integers.

4.3. Eliminating a square root by squaring may introduce extraneous solutions. Learn to check for extraneous solutions in any problem where you square both sides. Learn that when the relevant theorem is stated with "if..., then..." there is a risk of extraneous solutions, but if the relevant theorem is stated with "iff" there is not. Learn how to solve the equation "x

4.4. [This might be the most important section of all, for percents appear in every newspaper and percents are used when discussing money, credit, investment, and changes of all kinds.] Learn how percents are used to discuss change using multiplication. Successive application of percentage changes is given by successive multiplications. Learn to avoid addition and subtraction in percent problems. Learn how money grows (the Compound Interest Formula, 4-4-6) and how the APY (= effective rate) is computed. Averages are computed using roots, not division (4-4-8).

4.5 Rational Functions, solving rational equations (by reading their graphs), locations of zeros and asymptotes (vertical [p. 254, box] and horizontal) [the most interesting places], end-behavior model (a technical term, it determines horizontal asymptotes, p. 257, box)

Jan. 30, 2015

M&F class hour, section number, instructor, exam room

8:00 (001) KelsiAnne Espinoza Linfield 113 [Do you know where the building is?}

9:00 (002) Kenneth Flagg Gaines 243 [Do you know where the building is?]

10:00 (003) Derek Condor Gaines 043 [Do you know where the building is?]

11:00 (004) Jocelyn Short Linfield 301 [Do you know where the building is?]

1:10 (006) Arthur Huey Wilson 1-121

2:10 (007) Jordan Schupbach Wilson 1-143

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.

Review Sessions: There will be review sessions, open to all M151 students, for each exam. They will be two days before and the day before the exam. For Exam 1 they are Sunday, Feb. 8, at 3:00-5:00 and Monday Feb. 9, 6:00-8:00, both in Wilson room 1-119. You need not stay for the whole session and you may come late. (Wilson is usually locked on Sundays, but you may come to the door on the west side (south end) and get in that way.)

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!

About

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.”

Jan. 12

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.

Instructors and info about them.

Calculator programs for TI calculators that we will use a lot.

Short outline of what you are supposed to learn in Chapter 1.

Section topics, in a few words.

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.

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.

Precalculus requires you to change.

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.

This web site has an index.

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 is
relevant.

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 all Precalculus 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 2.3. (Some semesters only covered through 2.2.) 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

Go there and fix your e-mail address! Faculty will occasionally send e-mail to you at your default MSU e-mail 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 you actually use. Sometimes faculty need to contact students and they don't have their phone number on file.

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.

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".

Rules for studying math, from a newly-published book.

"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

Book (the required textbook)

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.

Disability, see here.

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, 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)

Links to related, but not required, sites.

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.