Course Content: Essential Sections: 1

MATH 1650 Outline

(Updated by Rhonda Huettenmueller; August 2011)

The text is Precalculus: Mathematics for Calculus, by Stewart, Redlin, and Watson, 6th edition.

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 a small fee, so we must inform students in our syllabus. The homework should account for 10-15% of the grade.

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. Be aware that many 1710 instructors do not permit calculators.

Course Coordinator: Rhonda Huettenmueller (GAB 480, )

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 (which is what I do after their first trigonometry test) or letting them bring a sheet/card of their own notes with the formulas that occur in the later chapters (Law of Sines, Law of Cosines, half-angle formulas, etc.).

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 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 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 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 -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 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.