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1. lee
   Key Master

   • 5.3 Polynomials and Polynomials Functions (Week 7-Wednesday Classes)
   1. Explain what terms are and what polynomials are using examples. Do not emphasize on jargons like monomials, binomials, and so on so forth.
   2. Like Terms
   3. Evaluation $P(a)$ of a polynomial $P(x)$ at $x=a$
   4. Adding and subtracting polynomials by combining like terms
   • 5.4 Multiplying Polynomials (Week 7-Wednesday Classes, Thursday classes, Friday Classes)
   1. Multiplying two polynomials by distributive property and simplify the product. Also discuss FOIL as a special case, multiplying two binomials.
   2. Special Products:
      \begin{align*}
      (a+b)^2&=a^2+2ab+b^2\\
      (a-b)^2&=a^2-2ab+b^2\\
      (a+b)(a-b)&=a^2-b^2
      \end{align*}
   • 5.5 The Greatest Common Factor and Factoring By Grouping (Week 7-Thursday Classes, Friday Classes, Week 8-Wednesday Classes)
   1. Notion of Factoring as the reverse process of multiplying
   2. Finding the GCF (Greatest Common Factor) of polynomials
   3. Factoring polynomials by Grouping
   • 5.6 Factoring Trinomials (Week 7-Thursday Classes, Friday Classes, Week 8-Wednesday classes)
   1. Factoring quadratic polynomials, i.e. trinomials of the form $ax^2+bx+c$. As an application, also go over examples like factoring $16x^2+24xy+9y^2$.
   2. Please do not mention about factoring trinomials of the form $ax^2+bx+c$ by grouping. It is not really recommendable method for students.
   3. Factoring by Substitution
   • 5.7 Factoring by Special Products (Week 9-Wednesday/Thursday/Friday Classes)

   Introduce formulas of special products and show examples of factoring using those formulas.

   1. $a^2+2ab+b^2=(a+b)^2$, $a^2-2ab+b^2=(a-b)^2$
   2. $a^2-b^2=(a+b)(a-b)$
   3. $a^3+b^3=(a+b)(a^2-ab+b^2)$, $a^3-b^3=(a-b)(a^2+ab+b^2)$
   • 5.8 Solving Equations by Factoring and Problem Solving (Week 9-Monday/Wednesday/Friday classes)
   Discuss
   1. Zero-Factor Property
   2. Steps of Solving Polynomial Equations by Factoring (p. 317)

   and go over as many examples as you can.

   • 6.1 Rational Expressions, 6.2 Adding and Subtracting Rational Expressions (Week 10-Monday/Wednesday/Friday classes)

   Please cover everything presented in the textbook. Students would understand better if you use analogy between rational numbers and rational expressions.

   Posted 1 year ago #

2. Corwin
   Member

   In 5.6, I disagree with your second point. Although factoring by grouping is not the fastest method method for factoring trinomials, many of our students seem to take well to it. Particularly for those classes which cover 3 sections of material this week, it is much easier to teach this as a natural extension of the methods from 5.5 with our limited time. Then other methods like the "trial and check" method or "factoring by substitution" can be offered as a more advanced technique to strive for.

   Posted 1 year ago #

3. lee
   Key Master

   Corwin,

   I see its merit as you pointed out. I asked instructors not to mention the method (factoring quadratic polynomials by grouping) only because it is not practically the best method. Besides, I was personally thinking that MAT 099 students had better know one method that works the best and keep practicing it.

   I'll leave it to instructors. If instructors feel it is necessary, they are more than welcome to discuss the method in their classes as long as other necessary materials are covered.

   Sorry, I did not mean to make my instructions "too" instructional, but rather suggestive.

   Dr. Lee

   Posted 1 year ago #

4. Lue
   Member

   Dr. Lee
   I have already covered factoring and yes I did use factoring by grouping because it is the easier method for the students to learn. I found that it works for special products as well as any quadratic polynomial of the form ax^2 + bx + c that is not prime. Thus I really don't see why the "Trial and Error" method would be more efficient than grouping technique because we had to discuss gcf by grouping in the previous section so I used it so the students only had to learn that one method. I am willing to show you what I did in class, if necessary.

   Posted 1 year ago #

5. Corwin
   Member

   No apologies needed; just trying to start some discussion. Week nine should be pretty straight forward in 099. I do have a few thoughts on this week's material as well though.

6. For 5.7:
1. Ideally the students will become proficient enough to recognize these special products at a glance. To facilitate this, I normally lead in with a number of examples and let them see the pattern for themselves. For example:

   \begin{align*}
   x^2 + 4x + 4 & = (x+2)^2 \\
   x^2 + 6x + 9 & = (x+3)^2 \\
   x^2 + 8x + 16 & = (x+4)^2 \\
   x^2 + 10x + 25 & = (x+5)^2 \\
   \vdots & \vdots \\
   a^2 + 2ab + b^2 & = (a+b)^2 \\
   \end{align*}

2. I really emphasize this second formula, as it appears frequently, and students will have a difficult time factoring $a^2 - b^2$ without it. Also it seems like a good idea to point out to the students that this doesn't hold for $a^2 + b^2$.
3. I don't have much to add about the sum of cubes formula, although since we're now teaching polynomial division in 101, I'm less concerned with how little this is discussed in 099.
7. For 5.8:
1. For the Zero-Factor Property, I really try to get students to see where this comes from. Normally I ask them to give me two numbers they can multiply to give zero, then watch them sweat for a while before I finally tell them that one of the numbers will have to be zero. Hence, either one or factor or the other must equal zero.
2. For solving by factoring, I always included an example like $(x+2)(x+3)=4$ so that I can teach them to avoid the mistake of setting $x+2=4$ and $x+3=4$. This is a common error I used to see amongst my students, and I also think it helps reinforce the Zero-Factor Property.

Anyway, those are just some of my thoughts on this lesson.