Math 5 - Trigonometry Syllabus – Section 3386 - Fall ’12
CATALOG COURSE DESCRIPTION: This course is the first of a two semester sequence preparing students for Calculus. In this course you will study functions with an emphasis on the trigonometric functions along with topics in analytic geometry. Topics will include a review of plane and coordinate geometry, functions, including function notation, transformations and inverses, definitions and graphs of the trigonometric functions, modeling periodic behavior, solving triangle problems with the Laws of Sines and Cosines, the conic sections, including an introduction to parametric and polar equations.
PREREQUISITE: MATH-030 AND MATH-040
TEXTBOOK: Precalculus: Mathematics for Calculus., by Stewart, Redlin, and Watson, ISBN 0840068077.
Also, Schaum's Outlines: Geometry, 4th ed., by Rich.
CALCULATORS: You may be restricted to a scientific calculator or no calculator on some tests.
EXAMS: There will be at least five chapter tests and a cumulative final exam. There may also be several projects, as circumstance demands. The point of these is to provide an opportunity to demonstrate your understanding of the concepts covered. As such, exam problems won’t consist entirely of very familiar homework problems.
Teaching and Learning:
According to George Polya, we can articulate three major principles of learning (which governs teaching):
1. Active Learning. It has been said by many people in many ways that learning should be active, not merely passive or receptive; merely by reading books or listening to lectures or looking at moving pictures without adding some action of your own mind you can hardly learn anything and certainly you cannot learn much.
There is another often expressed (and closely related) opinion: The best way to learn anyting is to discover it by yourself. Lichtenberg (an eighteenth century German physicist, better known a s a writer of aphorisms) adds an interesting point:
"What you have been obliged to discover by yourself leaves a path in your mind which you can use again whn the need arises."
Less colorful but perhaps more widely applicable, is the following statement:
"For efficient learning, the learner should discover by himself as large a fraction of the material to be learned as feasible under the given circumstances."
This is the principle of active learning (Arbeitsprinzip, in German) It is a very old principle: it underlies the idea of "Socratic Method."
2. Best Motivation. Learning should be active, we have said. Yet the learner will not act if he has no motive to act. He must be induced to act by some stimulus, by the hope of some reward, for instance. The interest of the material to be learned should be the best stimulus to learning and the pleasure of intensive mental activity should be the best reward for such activity. Yet, where we cannot obtain the best we should try to get the second best, or the third best, and less intrinsic motives of learning should not be forgotten.
For efficient learning, the learner should be interested in the material to be learned and find pleasure in the activity of learning. Yet, beside these best motives for learning, there are other motives too, some of them desirable. (Punishment for not learning may be the least desirable motive.)
Let us call this statement the principle of best motivation.
3. Consecutive phases. Let us start from an often quoted sentence of Kant: "Thus all human cognition begins with intuitions, proceeds from thence to conceptions, and ends with ideas." The English translation uses the terms "cognition, intuition, idea." I am not able (who is able?) to tell in what exact sense Kant intended to use these terms. Yet I beg your permission to present of Kant's dictum:
Learning begins with action and perception, proceeds from thence to words and concepts, and should end in desirable mental habits.
To begin with, please, take the terms of this sentence in some sense that you can illustrate concretely on the basis of your own experience. (to induce you to think about your personal experience is one of the desired effects.) "Learning" should remind you of a classroom with yourself in it as student or teacher. "Action and perception" should suggest manipulating and seeing concrete things such as pebbles, or apples, or Cuisenaire rods; or ruler and compasses; orinsturments in a laboratory; and so on.
These principles proceed from a certain general outlook, from a certain philosophy, and you may have a different philosophy. Now, in teaching as in several other things, it does not matter much what your philosophy is or is not. It matters more whetner you have a philosophy or not. And it matters very much whether you try to live up t your philosophy or not . The only priniciples of teaching which I thoroughly dislike are those to which people pay only lip service.
Much of the course is centered around applying these definitions and theorem by solving problems. The basic outline for general problem solving devised by Polya is a four step program:
1. Understand the problem
2. Devise a plan for solving the problem
3. Carry out the plan
4. Look back
HOMEWORK: Read the text, keep up with the assigned problems (as a minimum) and prepare questions about what you're learning for participation in class. You can expect to learn far more trigonometry individually or in small groups doing homework than you do in class. If you complete (and thoroughly understand) the homework assignment for each section, you will be well prepared to solve questions on tests and quizzes. At the beginning of each class, I will answer as many questions over the previous night's homework as time allows. Generally, you can expect to study at least 2 hours outside of class for every hour of class time. To get credit for homework you’ll need to use an on-line homework site. We'll start with ILRN and then, after the first chapter on geometry, consider moving on to the WebAssign.
For ILRN, your account has already been created. Sign on as a returning user and then use your mycod email address
as your username and your 7-digit student number as your password.
QUIZZES: There will be regular quizzes throughout the semester. These measure your attendance and give you feedback on current topics of study.
GRADE: Your grade is a weighted average of homework, quiz, chapter test, & final exam scores:
Attendance: 5 pts.
Homework/quiz: 10 pts.
Chapter tests: 55 pts.
Final Exam: 30 pts.
From the course outline:
Before entering the course you must be able to:
a. Interpret slope as a constant rate of change.
b. Recognize and create linearity in tables, graphs, and/or equations.
c. Solve systems of equations by using methods of elimination, substitution and graphing.
d. Graph and/or find the equation of a circle given sufficient information.
e. Solve quadratic equations by factoring, completing the square, and the quadratic formula.
f. Recognize and create quadratic models for relations involving tables, graphs, and equations.
g. Graph a parabola by finding the vertex, intercepts, and other symmetric points.
h. Demonstrate understanding of definitions for function and its related terms: domain and range.
i. Use appropriate notation for function equations and for describing domain and range.
j. Demonstrate understanding of the exponential function, its scaling and growth factors.
k. Understand how to solve similar triangle problems.
l. Demonstrate understanding of triangle congruency theorems involving SSS, SAS, AAS.
m. Basic knowledge about congruence relations such as the congruence of vertical angles
n. Familiarity with Pythagorean theorem.
o. Demonstrate understanding of deductive reasoning in the construction of a proof.
In this course we will:
- Review plane geometry facts and applications of angles, triangles, quadrilaterals and circles.
- Deductive reasoning and proofs.
- Functions including 4 ways of representing functions, function-notation, terminology, rates of change, and representing word problems with functions.
- Transformations of functions
- The algebra of functions including the composition of functions.
- One to one functions and inverse functions.
- The trigonometric definitions using the right triangle and circular approach.
- Solving triangles including right triangles, oblique triangles, Law of Sines, Law of Cosines, and applications.
- Radian and degree measure including applications.
- Trigonometric functions and their graphs, including phase shifts, amplitude, frequency, period, an translations.
- Modeling periodic behavior using trigonometric functions.
- Pythagorean Identities and using identities to deduce other identities.
- Conic Sections including the basic characteristics of parabolas, circles, ellipses and hyperbolas, graphing and applications.
- Parametric Equations including graphing a parametrically defined curve.
- Vectors including analytic and geometric representations and applications.
Course Objectives: Upon completion of this course, students will be able to:
- Apply facts about angles, parallel lines and triangles to deduce further results about a geometric
- Prove when two triangles are congruent or similar.
- Justify the lengths of sides in an isosceles right triangle and in a 30-60-90 triangle.
- Deduce the lengths of sides in quadrilaterals such as trapezoids and rectangles using basic definitions,
- Pythagorean Theorem, perimeter and/or area.
- Calculate the measure of a central angle in a circle using the measure of the intercepted arc and calculate the areas of geometric figures involving circles.
- Apply facts about plane geometric figures to deduce the surface area and volume of three dimensional geometric figures.
- Demonstrate an understanding of the concept of a function by identifying and describing a function graphically, numerically and algebraically.
- Calculate the domain and range for a function expressed as a graph or an equation. From a graph, estimate the intervals where a function is increasing, decreasing and/or has a maximum or minimum value.
- Use and interpret function notation to find “inputs” and “outputs” from the graph, table and/or an equation describing a function.
- From an equation, graph or table, calculate average rates of change by using a difference quotient or by using slopes of secant lines. Analyze average rates of change to determine the concavity of a graph.
- Demonstrate an understanding of the six basic transformations of functions by graphing translated functions including the quadratic functions.
- Represent a word problem (especially a geometric problem) with a function.
- Determine when a function has an inverse (one to one functions) and find the inverse function graphically or algebraically.
- Form new functions through addition, subtraction, multiplication, division and composition.
- Recognize classical and analytic definitions of the trigonometric functions.
- Solve triangles using right triangle trigonometry, the law of sines and the law of cosines.
- Convert from radian to degree measure and vice-versa.
- Graph the 6 trigonometric functions and demonstrate the ability to change parameters and predict corresponding graphic behavior.
- Use trigonometric functions to model periodic behavior.
- Use the basic Pythagorean identities to deduce further identities.
- Recognize the basic features of the graphs of the conic sections (including parabolas, ellipses, circles and hyperbolas) and use those features to graph shifted conics.
- Graph a function defined by parametric equations.
- Represent quantities such as velocity and force with vectors using both a geometric and analytic description.
- Apply vectors and the properties of vectors to solving problems involving force and navigation.
- Analyze independently and set up application problems, thus applying problem solving technique to new situations. Demonstrate the ability to anticipate and check their proposed solutions.
- Communicate effectively with the instructor and mathematical community using proper terminology verbally as well as proper written notation.