Posts
All Comments
The question arises, how did Menaechmus come to think of obtaining curves by cutting a cone? On this we have no information whatever. (pg. 110)Lost in the mists of time!
The appropriateness of when to introduce the function concept varies depending on the views of the individual instructor and the student population one is working with. I value the importance of the function concept and notation, but feel that teaching the function notation and the “vertical line test” without follow-up material relating to curve analysis (max, min etc.), composition and inverse, and transformation of functions is of questionable value particularly in the case of a gen. ed. math course that is to be taken by a broad range of students.
I've found that it is possible to discuss many topics related to the behavior of polynomial, rational, exponential and logarithmic functions without the additional two or three days of material required to introduce the function notation to a gen. ed. student population.
Given time, I like to cover functions by introducing the basic notation, followed by curve analysis in which the concepts of increasing and decreasing behavior, as well as maximum and minimum values are discussed. I then like to address function composition and inverse, the transformation topics (which can be somewhat tricky for students) and finish with a unit on the application of functions to a range of optimization problems common in many Calculus courses. I typically have the students use the TI 83/84 calculators to find the maximum and minimum values that will be found algebraically in a Calculus course.
So, I typically spend four to five weeks covering
1) notation
2) increasing, decreasing, max, min
3) composition, inverse
4) transformations
5) applications/modeling
Continuing in the project to blog the major topics in the Pre-Calculus curriculum (as I see them) brings us to finding the rational roots of polynomial functions.
The topics involved in studying the Rational Roots Theorem seem to me to fall into two major categories – polynomial factorization and the analysis of polynomial functions and their roots.
Suppose the polynomial P(x) with integer coefficients satisfies the following conditions:
(A) If P(x) is divided by x^2-4x+3, the remainder is 65x-68
(B) If P(x) is divided by x^2+6x-7, the remainder is -5x+a
Then we know that a={?}.
Let us find the remainder bx+c when P(x) is divided by x^2+4x-21.
Condition (A) implies that {?}b+c={?}.
Condition (B) implies that {?}b+c={?}.
It follows that b={?} and c={?}
I've been using Michael Sullivan's Algebra and Trigonometry textbook for the last few years to teach College Algebra/Pre-Calculus/Trigonometry on the quarter system, but we're switching to John Coburn's Algebra and Trigonometry next year. They're both pretty good textbooks.
In building Pre-Calculus curriculum I've drawn mainly from four textbooks:
One of the things I liked about the Sobel/Lerner textbook was the coverage of rational expressions that several of the other books I had considered didn't have. I like rational expressions because this topic requires a firm grasp of many of the most important concepts from algebra – factoring and multiplying of bi- and trinomials, simplifying complex expressions, and combining like terms in the context of manipulating algebraic fractions. If students understand numerical fractions it helps a lot.
When I teach College Algebra courses at Clatsop CC, I often begin with exercises like (x+7)(2x-3) – (x+1)(x+5) to address these topics and prepare the students for when they see this again in working with rational expressions. Another good example is (x+6)(3x+1) – (x+2)^2, to get them used to seeing the squared binomial. ( I make the squared binomial a regular visitor in most of my algebra classes). These expressions often appear as the numerator of a combined fraction in a problem like (x+7)/(x+1) – (x+5)/(2x-3).
Here is a link to a collection of problems I often assign for this topic.
I recently began to reacquaint myself with the College Board Math II Subject Test which I had taken after taking Pre-Calculus in the spring of 1982. The first question on the sample test I looked at was intriguing, and an understanding of rational expressions is really useful in finding a quick solution:
If 3x+6=(k/4)(x+2), then k=
a) ¼
b) 3
c) 4
d) 12
e) 24
In this problem, dividing through by (x+2) so that 3=k/4 (and 12=k) gives the almost instantaneous answer we need for a timed test. This is an interesting problem because it really gets at the concepts involved in working with factors in an equation.
In the Sobel/Lerner textbook, Sections 1.7, 1.8 and 1.9 cover the algebra review (multiplying polynomials, combining like terms, factoring polynomials and rational expressions) necessary to move on. This is where it is important to illustrate these topics with problems, because when I say “combining like terms,” I don't mean 2x+5x. While this type of problem could be appropriate when first teaching the concept, in the context of review for Pre-Calculus, a problem from the Sobel/Lerner text like (x^3-2x+1)(2x)+(x^2-2)(3x^2-2) is better practice for using these skills together.
This is something that I consider extremely important and that textbooks and assessments often don't include enough of – using the skills together. Learning skills in isolation is useful to grasp each skill individually, but to really DO MATH, a student must be able to make decisions about what to do and when. Something that I like about the Pre-Calculus curriculum is that it lends itself well to the type of problem in which the tools of algebra must be applied in a variety of situations.
The Brown/Robbins text covers complex numbers and the quadratic formula in Chapter 1 (1-4, 1-5) and then the solution of equations involving rational expressions in Section 2-2. Chapter 2 goes on to examine the graphing of quadratic and polynomial curves and finishes with material on finding rational roots.
The Sullivan book covers polynomials and algebra in sections R.4, R.5 and R.7, Coburn covers this in sections R.3, R.4 and R.5.
The Foerster and Dolciani texts don't really cover much algebra review at all, but, as a result, they explore a number of topics the other books don't. I'll probably follow a path similar to the Brown/Robbins book and talk about rational roots next.
I suppose that my experience with Pre-Calculus curriculum began in Steve Patterson’s Pre-Calculus class at Briarcliff High School in the 1981-82 school year. Pre-Calculus always stuck out in my mind because it was the only math course that was completely locally developed. Algebra I, II, and Geometry all had Regents exams and the Calculus course was AP Calculus.I'm hoping Rich will write more posts for us.
I remember studying Conic Sections, Polynomial Long Division and Synthetic Division, the Rational Roots Theorem (and its proof), elementary Discrete Math (permutations, combinations and binomial probability), Polar Coordinate graphing and hand calculating Riemann Sums at the end of the year. I took the College Board Math Achievement Test II (now the SAT Subject Test Math II) after completing the Pre-Calculus course so I recently looked at some current sample questions and saw these same topics – Analytic Geometry, Permutations & Combinations, Synthetic Division, Functions, Sequences & Series.
During the 1999-2000 school year, I taught Pre-Calculus at Maine Central Institute in Pittsfield, Maine. The school was using the Chicago Series text Functions, Statistics and Trigonometry for their Pre-Calculus course. I know that some teachers like the Chicago Series and FST in particular, but I didn’t really get much use out of the textbook, and began to supplement. Standard textbooks can be supplemented quite easily because the order and difficulty level of the topics is often similar. I found that the Chicago Series was very difficult to supplement and just began to create separate materials for the students. I collected these assignments in a binder and showed this to the University of Maine math department when I was interviewing for an adjunct position the following year (yeah - I didn’t stay at MCI very long – they were sticklers for using the approved textbook). I taught as an adjunct at UMaine for two years before beginning their MA program in Math.
