The unit on polynomials includes operations on polynomials, which also includes monomials. Students have had some experience with the addition and subtraction of polynomials in seventh grade. These lessons constitute mostly a review. Although I teach the addition and subtraction as separate lessons, I combine them here. In the first lesson, I also include a section on terms that pertain to polynomials. Using correct terminology and vocabulary is important if we want students to articulate explanations of what they are doing when solving problems, rather than hearing “I multiplied this guy by the other and got a bunch of stuff that I combined to get this.”
I focus on the following terms: monomial, binomial, trinomial, polynomial. I also cover the degree of variables, monomials and polynomials.
This unit includes multiplication and division of powers which students saw in an earlier unit—this expands on the concept and the problems become more complex. The unit also covers multiplication of monomials by polynomials, and the multiplication of polynomials which includes the FOIL shortcut for multiplying two binomials (e.g., (x +2)(x – 3). I teach both methods, but find that the FOIL method is useful when students learn factoring of trinomials which is in the next unit.
The topics of multiplying and factoring polynomials is usually when I hear the first of the “When will I ever use this stuff?” I find that once students get the feel of the operations, the complaints diminish considerably, but in answer to the question of when they will ever use it, I tell them that if they go into science, engineering or math, they will be working with them frequently. To the inevitable response from someone who tells me they are going to be a lawyer, I reply that they will need to know it to get a good grade in the class. That generally works.
Warm-Ups.
2. A jet traveling 600 mph makes a trip in 33 hours less time than a train traveling at 50 mph. How long does the train trip take? Answer: Let x = time (hrs) for train; then x – 33= hours for jet. 600(x – 33) = 50x; 600x -19,800 = 50x; 550x = 19,800; x =36 hours.
3. A train left Omaha at 9 AM traveling at 50 mph. At 1 PM a plane also left Omaha and traveled in the same direction at 300 mph. At what time did the plane overtake the train? Answer: The train travels for 4 hours prior to the plane leavin Omaha, and thus travels 50 × 4 miles (200 miles). Speed of jet relative to the train is 250 mph. 250t = 200; t = 200/250 =4/5 hr or 48 minutes.
4. -5(x – 30) = ? Answer: -5x + 150
5. Simplify. (x-30) + (x +30) Answer: 2x
For Problem 1, students have had complex fractions in seventh grade. They may need to be reminded that the larger fraction bar represents division of the numerator by the denominator; i.e., 1÷2/3. Students will have questions on Problem 2. Prompts “What should we have x equal?” “What distances are we saying are equal? How do we represent the distance each travels?” After setting up the equation, students will want to use a calculator which I don’t allow for this problem. Students have worked with distributing a negative number across a binomial in an earlier lesson. A prompt may be “Can we write it as -5(x + (-30) )?” Similarly, students should be able to simplify the expressions in Problem 5, which also acts as a segue to this lesson.
Going Over Homework. There are likely to be questions on the uniform motion problems assigned from the previous lesson. I go over the problems they have questions on, and pick students who did the problems correctly to do them at the board.
Addition of Polynomials. After going through the terms and vocabulary discussed above, I turn to the addition of polynomials. Problem 5 of the Warm-Ups is in fact what the lesson is about, and in my experience, most students know how to do this, though some still have difficulty with the concept of combining like terms. This is an opportunity to clarify what that means, and how to do so.
The problems are more complex than what they’ve been used to, since they will now contain powers of variables, as this example shows:
“We need to arrange terms in order of decreasing or increasing degree for a particular variable. What will this look like?” I have them work in their notebooks and go around to see who has done it right and who is having difficulty. What I want to see is:
This can be simplified to:
What is the degree of this polynomial?” The highest degree is given by the first term. The degree of a monomial is the sum of the exponents. This means that the degree of x²y is 3, which is the degree of the polynomial. Terms should also be written in descending order
Examples.
Some common mistakes that I see are combining variables like y² with y to get 2y or other terms. I therefore emphasize what like terms are and that y² and 2y² are like terms and their sum is 3y². I will also see students adding the exponents so that y² and 2y² are expressed as 3y^4.
Homework: Problems continue in this vein; students are to simplify and identify the degree.
Subtracting Polynomials. This lesson is given the following day. Students have had some experience with distributing a negative number across a binomial which is really the main skill of the lesson.
“I have a question for you: How do we simplify –(2x + 1)?”
Someone may say “distribute the negative” and as I said in a previous section, I don’t lose sleep over people saying it that way, but I also am quick to clarify what that means. “Yes, and what you are doing is distributing –1. The expression –(2x+1) is the same as –1(2x + 1). So someone distribute it for me.”
We get: –2x –1. I have them try others. – (–3x–5): –1(–3x–5) = 3x + 5.
As necessary, I may show that –1(–3x–5)= –1(–3x + (–5) ) to make sure students remember that they are multiplying –1 by –5 and therefore getting positive 5 as the product.
“Yesterday we added polynomials. Today, as you’ve probably guessed, we’ll be subtracting them. And when we subtract polynomials that’s what’s going on. So let’s do this subtraction:”
“And this is the same as:”
“Let’s see if you can do the rest.”
They should get the following:
More examples.
Homework. The homework problems should be a mix of adding and subtracting polynomials.