Course Content: Essential Sections: 1

MATH 1650 Outline

Notes for MATH1600/1610 are on the last page

(Updated by Rhonda Huettenmueller; August 2012)

The text is Precalculus: Mathematics for Calculus, by Stewart, Redlin, and Watson, 6th edition. Since 2011-2012, we have been using WebAssign for online homework.

Course Content: Essential Sections: 1.10, 1.11, Chapters 2-7, and §12.1-12.3. You can cut back on some of Chapter 3 in favor of some of the recommended additional topics. You should, however, cover oblique asymptotes in 3.6 and complex zeros in 3.5 as these topics might or might not have been covered in 1100. Recommended additional topics: 1.9 (if including graphing calculators as an integral part of your course), 8.1, 8.3 (these sections should be touched on because they reinforce trig., but it is not essential the students master these topics), 12.4 (this reinforces and gives practical application to 12.1- 12.3), 12.5 (this is helpful for math majors, and some calculus instructors like students to have seen induction), 8.4 (time permitting, a brief introduction to vectors is helpful to science students). You will most likely not have time to do all of the above.

Exams: You should give 4 exams plus a comprehensive final. The second exam should be graded and returned before the last day a student can drop with a "W."

Homework: We will use an online homework system, WebAssign, from which you will make assignments. The student has to pay fee, so we must inform students in our syllabus. The homework should account for 10-15% of the grade. If you choose to have students “Self-Enroll” in WebAssign, put the Class Key on your syllabus.

Grades: If you teach a recitation section, you should have about 10-15% of the grade come from recitation (quizzes, projects, etc.). The final exam should count 20%-25% and the rest coming from in-class tests.

Gateway Test: Some instructors have found it useful to give a "gateway" test which tests the students' basic computational ability at things like log rules, trigonometric identities, etc. The idea is that this is like a "driver's" test and students should be able to take the test several times until they develop the required proficiency. Once they have developed their proficiency, their grade should not depend on how many times they had to take the test to acquire that proficiency. See William Cherry if you think you might want to incorporate gateway testing into your course.

Calculators: The suggested calculator policy is that students be required to have a graphing calculator, and that the TI-83 or TI-84 (or equivalent) be recommended, but not required. You should also not permit graphing calculators with CAS, computer algebra system. One some quizzes and/or tests, do not permit calculators. Algebra: Most students struggle with algebra in 1710, so you should include some work (homework assignment, problems on a quiz and/or exam, etc.) involving algebra skills that they will see in calculus. For example, solving for y' and writing expressions in the form of a constant times the variable to a power.

Course Coordinator: Rhonda Huettenmueller (GAB 480, [email protected])

NOTE: Math 1650 covers a lot of material, even for a five hour course. It cannot all be covered during lecture. Recitation instructors will have to cover some of the material, and students will have to learn some material on their own. Emphasize to students that they are expected to spend roughly TEN hours per week outside class studying and doing homework for Math 1650. Also, remind students that Math 1650 is a course for science majors. All students in Math 1650 should be planning to take calculus. Ask to talk with non-science students to help them make sure they are in the right course.

There are many formulas and identities for trigonometry. You should have the students memorize some of the identities (Pythagorean, reciprocal, even/odd, etc.) and the trigonometric functions at the special angles in the first quadrant. You might consider either providing a list of formulas after their first trigonometry test. Do not provide them with any facts involving the unit circle, the fundamental identities (Pythagorean, reciprocal), the double-angle formulas for sine and cosine, and the sum/difference angles.

Math 1650 Sections to Cover

Chapter/Section Time* Goals/Comments

Chapter 1 2½-3 hours Teach students basic proficiency with graphing calculators, review linear equations and nonlinear inequalities, and introduce modeling.

§1.9 ½-1 hour Demonstrate calculator basics: numerical computation, graphing equations, setting the graphing window, zooming/tracing, and tables. It is helpful to bring a calculator with projector attachment to class. (Optional)

§ 1.10 1 hour Review linear equations. Be sure students develop an intuitive understanding of +/-, small, and large slope.

§ 1.11 ½-1 hour Be sure students understand the meaning of "proportionality." Emphasize that proportional does not mean "equal" nor does it mean simply "increasing."

Pages 130-139 ½ hour This is good to discuss together with §1.10. Don't get bogged down Chapter/Section Time* Goals/Comments

in exactly how linear regression works, but it is good for students to have some practice fitting lines to real data and making predictions.

Chapter 2 9 hours Teach students concepts and vocabulary associated with functions and graphs. Students should be able to describe functions in plain English, with a graph, with a formula, or with a table. They should be able to convert between these different ways of describing functions. Students should understand how to shift/stretch/compose functions.

§ 2.1-2.4 3 hours They will find most of this straightforward, although “average rate of change” tends to confuse them, and they struggle with the algebra of simplifying the difference quotient.

§ 2.5 2-3 hours Shifting and stretching are very important skills that will be used over and over again throughout the course. Even/odd is less important.

§ 2.6 1 hour Composition is especially important. They need to be able to find the composition algebraically. You should also assign problems for students to find two functions whose composition is a given function. This will prepare them for the Chain Rule.

§ 2.7 1-1½ hours Have students learn to find inverses of functions defined by formulas, by graphs, by ordinary words, and by tables.

Pages 213-222 1 hour This material is on mathematical modeling and will prepare them to solve applied problems in calculus.

Chapter 3 6-7 hours Teach students about polynomial and rational functions, their graphs, and roots.

§ 3.1 1-1½ hours Emphasize how standard form for a quadratic function is shifting and stretching the graph of y=x^2. Try to get students to understand the process of completing the square, rather than memorize the formula. Sketch some graphs of parabolas and have the students find the formulas!

§ 3.2 1½ hours Having students graph something like y=7-x^2+(x/20)^4 is a good way to teach them that their calculator alone is not so helpful. To really understand the function, they may need theory in addition to their calculator. Give the students some graphs and have them find possible formulas.

§ 3.3 ½ hour Long division is probably best covered in recitation section. Assume students have seen this stuff before.

§ 3.4 1-2 hours I tend to skip Descarte's Rule of Signs and concentrate on the upper and lower bound theorems. This section is not all that important and can be treated lightly if you feel you will need more time for Chapter/Section Time* Goals/Comments

trigonometry and exponentials.

§ 3.5, 3.6 1-2 hours Students should learn basic arithmetic with complex numbers, and the fundamental theorem of algebra.

§ 3.7 2½ hours Students learn a lot by working through these examples, but it takes them a long time to do one. Turn the tables on them by giving them some graphs of rational functions and having the students try to come up with possible formulas for the graphs. Understanding asymptotes will help them with limits in calculus.

Chapter 4 10-12 hours Teach students about exponential functions and the types of phenomena they can be used to model. Teach students the rules for working with logarithms. Try to spend several hours on "modeling" with exponential functions, as in section 4.6. You might mention exponential and logarithmic regression.

§ 4.1 and 4.2 2 hours Compound interest is a good introduction to exponential functions.

§ 4.3 2 hours Connect this with §2.7, emphasizing the inverse function relationship between logarithm and exponent functions

§ 4.4 1 hour Students should be comfortable using multiple logarithm properties to expand/condense log expressions. This will help them later with the chain rule in calculus.

§ 4.5-4.6 3-4 hours Try to do much of this in the context of word problems and tie things together with earlier skills. For example, give them the population values at two points in time, have them find an exponential function describing the population, and then have them find the time the population reaches a certain value.

Chapter 5 5-7 hours Teach students about the unit circle and the trigonometric functions. Students should become very familiar with the graphs of sine and cosine, they should understand period and amplitude, and they should know how to shift and stretch the sine and cosine graphs to model various periodic phenomena.

§ 5.1 ½ hour Get the main idea across. Don't get too bogged down in terminology like “reference number.” They should be able to quickly determine which quadrant contains a given angle.

§ 5.2 1½ hours Students should have a firm grasp of this material so that they are prepared for the rest of the course.

§ 5.3 1 hour Students should be able to sketch the graphs of sine and cosine and their transformations. They should be able to find the period, phase shift, and amplitude, both graphically and algebraically. It is important that they see that the sine and cosine curves are derived from the unit circle. This will also help them to understand the Chapter/Section Time* Goals/Comments

periodic nature of the trigonometric functions.

§ 5.4 1 hour Students should be able to evaluate inverse trig functions without using a calculator. You should also include some problems on reference triangles.

§ 5.5 ½ hour If you are short on time, you can skip this section. In any case, I would not expect students to memorize these. They should at least be aware of the asymptotes and periods of these functions, the local extrema for secant and cosecant, and the x -intercepts for tangent and cotangent.

§ 5.6 1½ hours Students should be able to use sine and cosine to model cyclic behavior to see the relationship between frequency and period.

§ 6.1-6.6 4-5 hours Teach students about angles and right angle trigonometry.

§ 6.1 1 hour Students might have trouble with circular motion, so do plenty of examples.

§ 6.2 1-2 hours Students should be able to solve a right triangle.

§ 6.3 ½ hour Some of this is a review of material covered in Chapter 5.

§ 6.5 and 6.6 2-3 hours Teach students about solving triangles using the Law of Sines and Law of Cosines. Although this material is not essential for calculus, some engineering students will need this material in engineering courses (such as Statics). If you are in a hurry, this can be covered in one lecture.

Chapter 7 7-8 hours Teach students about trigonometric identities, trigonometric equations, and inverse trigonometric functions.

§ 7.1-7.3 3 hours Students need lots of practice with these. Emphasize the fact that they will be using their knowledge of trig identities along with algebra skills to solve problems.

§ 7.4 and § 7.5 2 hours Solving equations. They will use what they learned in the earlier part of the chapter to solve these problems. Students who are weak in their understanding of the unit circle will struggle with certain types of problems.

Chapter 8 0-4 hours Topics like polar coordinates, De Moivre's formula, and vectors reinforce tringonometry and are useful things to cover if you have the time. If you can only do one of these sections, make it §8.1.

§ 8.1 1 hour Students should be able to convert the coordinates of points and Chapter/Section Time* Goals/Comments

equations between rectangular form and polar form. This material will help those who later take Multivariable Calculus, where they work with cylindrical and spherical coordinates.

§ 8.3 0-1½ hours When finding the argument of a complex number, be sure students know how to get the angle in the correct quadrant.

Chapter 12 7-11 hours Teach students to work with sequences and series, and the principle of mathematical induction.

§ 12.1 1½ hours Students should be comfortable with sequence and summation notation.

§ 12.2 1 hour Arithmetic sequences

§ 12.3-12.4 2 hours Geometric series are especially important and the practical applications in §12.4 should help motivate this. The book puts too much emphasis on memorizing formulas in §12.4 and not enough emphasis on the connection to geometric series.

Students must understand geometric sequences and when a geometric series converges for Calculus II.

§ 12.5 0-4 hours Students have a very hard time with mathematical induction. Some students will never catch on in the short amount of time we have to treat induction in Math 1650. If you include this material on the final exam, please consider making it bonus.

*Time estimates INCLUDE recitation hours, quizzes, and tests.

You might also considering assigning a few algebra problems from Chapter 1 on occasion. Calculus students often struggle with algebra basics (factoring, working with rational expressions, exponents, radicals, etc.).

The following is a SAMPLE weekly schedule for Math 1650 during a fall semester. Do not feel obligated to follow it. For example, some instructors may want to spend less time on exponential functions and more time on trigonometric modeling. However, if you plan to give William Cherry's "gateway test," then you should be sure to cover the necessary material in time for it. In particular, you should postpone all discussion of complex numbers to the end of the semester. Also, postponing the Law of Sines and Law of Cosines section will be helpful if you plan to give the gateway test.

Week 1

A good part of the first day of class will be taken up discussing your syllabus and course policies. If you are requiring graphing calculators and will be discussing §1.9, this is not a good topic for the first day of class because many students will not have their calculator with them. Postpone §1.9 until late in the week to give students a chance to get their calculators, and tell students to bring their calculators with them to class. Rather than starting in with y  mx  b , one way to set the tone on the first day of class that this is a hard class that will require critical thought is to photocopy some pages from 1.6 and ask the students to find solutions to some of those problems in small groups (for smaller classes), and then discuss the solutions. This also allows the students to get to know some of the other students in the class.

Spend most of the week reviewing linear equations and proportionality (good topics for recitation sections), teaching the students the basics of graphing calculators (also good for recitation section), and variation. The material on modeling is a good introduction to the goals of the course.

Be sure to begin assigning hard homework as early as possible. The students need to realize quickly that they need to set aside a considerable amount of time for this course.

Week 2

Cover sections 2.1-2.4.

Giving a tough, but reasonable, quiz in week 2 is a good way to set the tone for the course.

Week 3

Try to finish Chapter 2. This material should be mostly review so you should try to set a quick pace. Night sections may need to go a bit slower.

Week 4

Cover sections 3.1, 3.2, and 3.3, and most of 3.4 if possible. The mechanics of long and synthetic division is probably best left for recitation section. It is important to do a good job covering 3.1, but you can treat sections 3.2 and 3.3 lightly.

Week 5

Finish Chapter 3. Spend enough time on 3.7. Rational functions may not be the most important class of functions, but what they learn about graphing working with them is worthwhile.

Week 6

Cover the bulk of Chapter 4. Week 7

Spend a good amount of time on the modeling material at the end of Chapter 4, and start Chapter 5.

Week 8

Finish Chapter 5. This is a lot of material and most of your students will not have seen any of it before. Still, much of it can be covered pretty quickly. Spend at least some time on the modeling examples. If the students mastered shifting and stretching, they should be able to do this without too much trouble.

Weeks 9-10

Cover Chapter 6. If you will be giving the gateway test, hand-out the study materials for it around now. Students can memorize trigonometric identities on their own, and the gateway test will help them do this.

Week 11

Cover Chapter 7. If you want to give William Cherry's gateway test, you can do so as soon as the students are comfortable with 7.5, probably in week 12.

Weeks 12-13 (includes Thanksgiving week)

Cover 8.1, and begin Chapter 12. The Wednesday before Thanksgiving is NOT a good day for an exam. However, the Wednesday before Thanksgiving is an excellent day for optional test retakes, for example an optional retake of a gateway style test.

Week 14

Cover 12.2 and 12.3. Give a test if you did not do so before Thanksgiving. Assign homework due pre-finals week. Remember, there is no class on Friday of pre-finals week.

Week 15 (Pre-finals week)

Cover mathematical induction (§12.5), time permitting. The last recitation and last lecture must be review sessions.

For MATH1600 and MATH1610, you should cover more than the above.

For MATH1600 (trig), cover all of the trig chapters including Chapter 8. When you get to the inverse trig functions, you should do a brief lesson on inverse functions in general (one-to-one, horizontal line test, finding inverses, restricting domain so that a function is one-to-one). For MATH1610, cover all 1650 material, except Chapters 5-8. That is, part of Chapter 1, all of Chapter 2, Chapter 3, and Chapter 4 and at least 12.1—12.3.