|
Syllabus and Course Guide for Precalculus – Fall ‘04 |
|
Instructor: Geoff Hagopian
Office: Math 12
Office Hours: M-F
Email: ghagopian@collegeofthedesert.edu
Web Page: http://faculty.collegeofthedesert.edu/ghagopian
Telephone:
Main Text: Precalculus, by Stewart, Redlin and Watson
Tutoring:
Math/Science Study Center in Math 4 --computers and tutors.
Technology:
You are required to rent, borrow or buy a graphing calculator of some
sort.
Catalog
Description:
This is a course of study in the theory and applications of the basic functions
used in calculus: polyonomial, rational, root, exponential, logarithmic,
trigonometric and inverse trigonometic functions and their applications to the
physical world. Topics include
definitions, standard notations and theorems involving these functions and
their application in the physical world.
Overview
Precalculus is a course designed to do just what it suggests:
prepare you for a first course in calculus.
This means learning many definitions and properties of basic functions
and methods of solving equations, but it also—perhaps most importantly—means
learning how to solve 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
This method is described in more detail on the next
page.
Polya’s Four
Step Program for Problem Solving
- UNDERSTANDING THE PROBLEM
o Do you have good definitions for all the words in the problem statement?.
o What is the unknown? That is, what does the problem want to be produced? What are the given data? What condition(s) must be satisfied?
o Is it possible to satisfy the conditions? Are the condition(s) sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
o Draw a figure or make a diagram to help conceptualize what is going on. Introduce suitable notation.
o
Could you restate the problem in an equivalent
way that makes more sense to you?
- DEVISING
A
Polya mentions (1957) that there are many reasonable ways to solve problems. The skill at choosing an effective strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
|
|
Often times an algebra problem is best solved using the algebraic method:
i. Introduce a variable to represent the unknown.
ii. Write related quantities in terms of this variable.
iii. Set up an equation using the variable.
iv.
Solve the equation.
- CARRYING
OUT THE
- Carrying
out your plan of the solution, check each step. Can you see
clearly that the step is correct? Can you prove that it is correct? If your plan isn’t working, you may
need to go back to step 2 and devise a better plan.
- Looking Back
- Can you check the result? Can you derive the solution differently?
o Can you use the result, or the method, for some other problem?
o
Can you generalize your solution to a larger
class of problems?
The Importance of Looking Back
Looking back may be the most important part of problem solving and is the
best opportunity to learn from the problem. The phase was identified by Polya
with admonitions to examine the solution by such activities as checking the
result, checking the argument, deriving the result differently, using the
result, or the method, for some other problem, reinterpreting the problem,
interpreting the result, or stating a new problem to solve.
Teachers and researchers
report, however, that developing the disposition to look back is very hard to
accomplish with students. Some researchers have found little evidence of
looking back among students--even when it is stressed by instruction. One
teacher put it succinctly: "In schools, there is no looking back."
This likely stems from a culture of mathematics education that holds “answer
getting” as the paramount objective.
Also, pressure to cover a prescribed course syllabus; the absence of
tests that measure processes and student frustration contribute to the tendency
not to reflect on what a problem means in a larger context..
The importance of looking back
should outweigh these difficulties.. It is often what you learn after
you have solved the problem that really counts.
Prerequisites:
Intermediate Algebra (
- The use of linear models involving a constant rate of change.
- How to compute the slope of a line with sufficient given information.
- Tables, graphs, and/or equations which represent linear relations.
- How to create a linear model in the form of a table, graph, or equation.
- Methods of solving 2x2 and 3x3 systems of linear equations.
- How to graph and find the equation of circles and parabolas given sufficient information.
- Methods of solving quadratic equations including factoring, completing the square, taking square roots and the quadratic formula.
- Methods of solving linear and quadratic inequalities.
- Representing quadratic relations using a table, graph, or standard equations.
- Methods of graphing circles and parabolas by finding key points such as vertices, intercepts, and other symmetric points.
- How to manipulate rational exponents in algebraic expressions and for solving root equations.
- The use of exponential functions in modeling with a constant growth (decay) factor.
- Tables, graphs, and equations which represent exponential relations.
What We’ll Do
The scheduled 5 hours of
weekly class time will be conducted in a lecture/discussion format. We will all
come prepared to lead and/or follow a discussion on the topic for each
scheduled meeting (tentatively scheduled in the calendar that follows). To be successful, you’ll want to have test
scores whose weighted average exceeds 70% (C), 80% (B) or 90% (A).