Session 9 (Week 5a)
Chapter 3 Rational Expressions
Section 3.1 Adding and Subtracting Rational Expressions
1. Activities
W. Michael Kelley (2006). The Humongous Book of Calculus Problems.
Complete problems 3.1 - 3.8 [p. 26 - 30]
Wayne W. Dyer (2007). Change Your Thoughts - Change Your Life.
Read Verse 9: Living Humility [p. 44 - 47]
Jack Weiner (2005). The Mathematics Survival Kit. Revised Edition.
Read Getting Started on Survival [p. 1 - 2]
2. What's it all about?
This section provides examples of adding and subtracting rational expressions - expressions involving polynomials in both the numerator and denominator. This is called simplifying the expression, just as 5/6 is a simplification of 1/2 + 1/3.
3. What am I able to do, now that I am finished this section?
4. What will I know?
5. What are the key concepts?
6. What are some relevant Web sites for this material?
Using your favorite search engine, and maybe one or two others, type in a few key words such as "adding rational expressions" or "rational polynomials" .
Here are a few Web sites that I liked:
7. What concept map can I create for this section?
8. What types of Mathematica activities can I create?
Doing the algebra with Mathematica is not very useful, except for a possible check with the answer that one obtained by doing the problem manually. However there is genuine value in obtaining graphs for each of the terms and seeing the result of adding/subtracting them.
Here is my Mathematica Notebook for Week5a.
9. What are some connections to other ideas?
The basic principle for adding/subtracting rational polynomials is the same as that for adding/subtracting rational numbers (i.e. fractions). One first finds the lowest common denominator of the two terms and then performs the required operation. When determining the LCD one should be alert for possible factors which can then be cancelled. Thus one also needs to be very familiar with ways of factoring polynomials. This was the subject of the previous chapter in Kelley.
10. What is my overall reaction to this section?
This chapter builds one's confidence that one can handle any algebraic expression involving polynomials. There is nothing to fear here. That is a great feeling.
11. Comments on Dyer: 9. Living Humility
The ninth verse of the Tao Teh Ching clearly states that one should strive for balance and not over indulge in anything. At one level this applies to learning a topic such as mathematics. There is more to life than mathematics. This is worth remembering while one is engaged in the activity. At a certain point one should cease and switch attention to something else.
Dyer says that this is living humility, but I think a phrase like Living in Balance is to me a more appropriate summary.
11. Comments on Weiner: Getting Started on Survival
I noticed this book while browsing the shelves of the Chapters bookstore yesterday. It appears to be a perfect overlap with the Kelley book that I am using as my primary resource. Thus I am planning on adding it to my regular activities.
Weiner emphasizes that one should begin a session by first reviewing the notes and when one encounters a problem, close the book and try doing the problem yourself. Only when you are completely stumped should you reopen the book, consult whatever sources you can, and figure out what you need to understand. Then try doing the problem again. Repeat this cycle until you have mastered the problem.
This is actually very close to the strategy that I have adopted. Maybe that is why I liked this book when I saw it on the shelf. Weiner's book provides a few more examples of problems to try.
However I have two additional steps in my approach. After I have completed the problems for a section, I then make a set of summary notes (this is one such summary). Second, I redo the problems using Mathematica to see where it provides additional insights and power.