Prior to this lesson students have learned 1) factoring by grouping and 2) multiplying the sum and difference of two numbers.
Factoring by grouping involves chunking binomials : ax + by + ay + bx is factored into (ax + bx) + (ay + by), and then (a + b)x + (a + b)y, which factor into (a + b)(x + y. Students learn to think of a binomial like (a + b) as a single variable like T that can be factored out; e.g., xT + yT = T(x+y). The Warm-Ups below contain a factoring by grouping problem.
Students have also learned a short-cut for multiplying the sum and difference of two quantities. Specifically, (x + y)(x – y) = x² - y² When multiplying it out using the method they have learned for multiplying polynomials they will get the following:
The middle terms drop out since they are opposites. Today’s lesson has students work in reverse. Given the difference of two squares, they factor it into two binomials that are the sum and difference of the respective square roots of the two terms of the binomial. Some students have a hard time remembering that the middle terms drop out, and will continually multiply everything out. They are spending more time on a problem than need be, but more importantly, they may find it difficult to recognize the reverse of the operation—i.e., factoring the difference of two squares, which is today’s lesson. Therefore, it is important that students recognize both patterns, which comes about through repetition and practice of problems. Supplemental practice sets for students having difficulty should also include various types of multiplication to help with their ability to recognize that pattern.
Warm-Ups.
Problem 1 is factoring by grouping. The (x – 5) binomial is what is factored out. Problem 3 is challenging to students because of the large exponents. The rules are the same; when multiplying powers of the same base, the exponents are added. Possible prompts: “What do we do with the exponents for the x variables?” Problem 4 will be challenging. Possible prompts: “How much time did he spend on the coastal access road?” “How can you figure out the speed?” “What was his speed on the highway?” “How can you figure out the time on the highway?”
Problem 5 is what they have learned in the previous lesson and serves as a reminder that coefficients of the variables are squared as well as the variables themselves.
Preliminaries. Students have completed a lesson in which they multiply the sum and difference of two terms as discussed above. “I’d like you to multiply these binomials in less than 10 seconds and show the answer on your mini-whiteboards,” I say and write on the board:
Most students will get this immediately, though the few who haven’t learned to drop out the middle term automatically as mentioned above will not do the problem in the allotted time.
“Now I want you to do this one in the same time—10 seconds or less. And no calculators.”
Very few if any will get it in the allotted time. It’s x²-225 which I reveal to them and leave on the board.
“I imagine that this was harder because you don’t know the squares of the numbers 11 through 20 as well as you do the squares of 1 through 10. This is something we will be working on because identifying perfect squares is important when we are doing factoring—which is what we will be doing today.”
What I do to encourage memorization of the squares of 11-20 is to have an extra-credit quiz at some point. The quiz consists of problems that have the squares of these numbers, such as 15² - 14² + 1. If you know the squares of 14 and 15, then it is easy to write the problem as 225-196 + 1 which is 30. There are ten problems and a five minute limit. The quiz is worth 5 extra points that are added to their score on the next quiz or test.
Factoring the Difference of Two Squares. “Yesterday we learned that the product of the sum and difference of two numbers is the difference of their squares. In other words (a – b)(a + b) = a² - b².”
“So if I wanted to factor the expression m² - n² I would just work backwards. What are the two binomials this factors into?”
I have them write the answer on their mini-whiteboards and I will see (m-n)(m+n).
“We are just working backwards. Let’s try a few more.”
This was one of their Warm-Up problems, so one would expect it would be familiar, but I’ve ceased being surprised that they find this one challenging.
“Here’s a hint: Take the square root of each term. What’s the square root of 9 first of all?”
They know this is 3 so we move on. “How about the square root of x² ?” There are usually very few hands, so I need to make this familiar. “Remember, the square root is the number which when multiplied by itself equals what we start with. Remember, x² is a compact way of writing the product x∙x. So what is the square root of 9x² ?” Armed with the answer, they now tell me: 3x².
“OK, how about 4y² ?” Most will now get it: 2y
“And what is 1²?” I’ll ask. There is usually some hesitance because the question seems too easy. But someone will volunteer that it is 1. I bring this up because students frequently fail to see that x² - 1 is the difference of two squares, since 1² equals 1. And the expression x² - 1 factors as (x - 1)(x+1).
Examples:
Other Forms. “When we factor the difference of two squares, one of the terms has to be positive. What happens if both are negative? Like this.”
Students will try various things, usually saying (-9 + x)(-9 – x). I point out that -9 squared is positive 81 not negative 81.
“We cannot take the square root of a negative number. Why? We just saw that -9 squared is positive 81. There is no real number that satisfies this; it does not exist. Later you will learn about complex numbers in which we can represent square roots of negative numbers as an “imaginary” number. But in this course, there is no solution for square roots of negative numbers. So what is on the board cannot be factored.”
This is all new information to them, and it will be repeated over the rest of the course on occasion.
“But can I factor this?”
“Remember that subtraction is adding the opposite. So we can reverse the terms here and write it as x² + (-81) which is the same as x² - 81. Can we factor it now?”
They can and they do. “Try this: -1 + x².” They do it, some with hesitation but ultimately they get it done: (x -1)(x + 1)
“OK, somebody tell me what the square root of a²b² is.” There might be some discussion but ultimately more than a few students get this: ab.
“So factor this one, please.”
Homework. The homework problems should be primarily difference of two squares, but I include problem on a worksheet that are a mix of different factoring types. The goal is for students to be able to recognize when there is a difference of two squares, and be able to factor it—and to be able to distinguish these from others, such as y²- y³ which is not the difference of two squares, but may be factored as y²(1 – y).
Below are examples of a mix of different types of problems which will be challenging, and will be discussed the next day.