Area Models for Multiplying Polynomials and Factoring Quadratic Expressions
Lesson 3 of 21
Objective: SWBAT use area models to find the product polynomials and to factor quadratic expressions.
Today's opener is on the first slide of the lesson notes. It consists of three related number riddles that ask students to find a pair of numbers with a given sum and product. On Monday, this unit began with a similar number riddle. The primary purpose of this opener is for students to get accustomed to finding pairs of numbers with a given sum and product, because that's really what they're doing when they factor a quadratic expression.
As we move into the multiplying and factoring drills that will take up most of today's lesson, I would like students to notice how this problem is connected to the work of the preceding two lessons. First, students should notice - as many of them have already - that "sum and product" problems like these are basically the same thing as the "area and perimeter" problems of the last two days. Furthermore, the solutions to some such problems consist of two different numbers, others consist of two of the same number, and some problems are unsolvable. That's the case with these three problems (as it is with all such problems in which the sum is twice a certain square root, and the products are equal to, one more, and one less than the perfect square).
I give students a few minutes to solve these riddles, and I circulate to try to get them to think about the ideas I've outlined above. I'm going to try push the pace today and try to squeeze a lot in, so after at most five minutes, we take a quick look at the solutions and move on.
The first Student Learning Target of Unit 6 is about multiplying polynomials:
6.1: I can find the product of two polynomials.
I introduce the target by projecting the second slide of the lesson notes (U6 L3 Lesson Notes). I ask students to name the key words in the learning target. They note the word product, and I ask everyone what this word means. "Ok, so this learning target is about multiplication," I say. Then we get the word polynomials. I tell everyone not to worry too much about this big, complicated-sounding word. "A polynomial is really just an algebraic expression like x, or 7x + 3, or x squared minus x," I say. "What's most important is that you can add, subtract, multiply and divide polynomials just like you can with numbers."
From here, the lesson will move quickly through two mini-lessons and two practice sessions. For the first mini-lesson, I use slides #3 through #10 of the lesson notes. Take a look at these slides, each of which prompts students to find the area of a given rectangle. On the first three, the dimensions are simple integers, and students should feel comfortable with the multiplication.
On slide #6, the dimensions are x+2 and x. Many students got a preview of using the distributive property with variables during yesterday's Delta Math practice opener. I run through this to make sure everyone is comfortable using the distributive property, and to review the idea that x*x is x^2. Slide #7 reinforces the same idea.
Slide #8 says, "Here's the BIG IDEA" and the dimensions are x+5 and x+7. I say, "This is what I really want you to see today, and this is what SLT 6.1 is all about. How can we find the product of x+5 and x+7?" A big advantage of starting this unit with rectangles is that now I can transition seamlessly to area models. As you can see in this photo of my notes, I turn the rectangle into an area model by splitting the length into two parts, x and 7, then doing the same for the width. I say, "To find the area of each of these four smaller rectangles, I just have to multiply what's on the outside. I basically have a two-by-two multiplication table with four little multiplication problems to do."
I love using this model because it's instantly approachable by all students. It's clear to see what happens and why in each part of the model. After we find the four products (the area of each smaller rectangle) I write out the expression and we combine like terms. Then we run through one more (very similar) example and move on to some practice time.
Sharp students will ask if we can "solve" for what x will have to be, or find a numerical value for the area of these rectangles. That's a great question. I tell them that if I give length and width in terms of a variable like x, and I tell them the area, then that's a problem we can solve. "We will get there," I say, "but not quite yet. For now we're just practicing how to multiply these polynomials."
Timed Practice: SLT 6.1
I give students 10 minutes to practice using area models to multiply binomials. This Area Models Practice assignment should serve as a confidence builder for students. I've kept the numbers simple because I want students to feel comfortable using the tool, and it's the kind of work that should be done very quickly.
Some students will have questions about the first two problems, because they're not exactly like what they just saw, and I'll remind students that they did see an example like this a few minutes ago, and to check their notes. We may discuss the idea that sometimes it's the "easier" problems - the special cases - that can give us problems, because they're easier than we think.
Some students will ask if they have to draw the rectangle for every problem. I ask if they've done this before. If a student has experience multiplying binomials, I won't require them to use this model. I tell them that as long as they're multiplying correctly, I won't worry too much about the area model. Whenever a student is having a hard time because they're trying to avoid sketching the diagram, I will insist that they use what I've shown them, however.
- Problems #5-7 introduce the idea of squaring binomials. When students have questions about what to do on #6, I say that it means the same thing as #5.
- Problems #9-13 have an important relationship to the Rectangular Gardens work of the last two days. Students will notice that, and I'll say, "Isn't that cool?" Then I'll ask whether #8 is also related.
- Problems #14 and #15 generalize things a bit. Problem #15 introduces a negative number for the first time on this assignment, as well as generalizing the difference of squares. I love to hear the observations students can make about these problems.
Many students can finish the area models assignment in about 10 minutes. Anyone who is not done with the assignment should finish it for homework.
I ask for everyone's attention and I introduce Learning Target 6.2:
I can factor a quadratic expression to reveal the zeroes of the function it defines.
Then I say that factoring is the opposite of what we were just doing, multiplying polynomials. On the board, I write "addition" and say, "What's the opposite?" Kids are quick to shout that it's subtraction, so I write "vs. subtraction" on the board. We repeat for "multiplication vs. division" and "squaring vs. square roots". Below that, I write "multiplying polynomials vs. factoring". To further illustrate what I mean, I say, "If you start with an expression, add 10 then subtract 10, you're back to where you started, right? If square a number then take the square root of the result, you'll be back to the original number. Right now you'll see that if you multiply two binomials, you can back to where you started by factoring."
Before I introduce the first example, I briefly review an important idea from yesterday's lesson on Slides #13-15: parabolas can have two roots, equal roots (one root), or no roots. I show the three graphs, and we might also note the connection of these three graphs to today's opener. We will spend a lot more time with this idea - and there's a bit more to it than I'm showing kids here - but I want students to be grounded in the main reason for factoring right away. "You will soon see that some quadratic expressions are factorable, and some are not," I say.
From here, I show students how to use the Area Model, backwards, with each example (slide #16-18). For the first example, I write x^2 in the upper left corner and 36 in the lower right of my empty area model. "Now we need to think about that 15," I say. "I need two numbers that multiply to make 36 and add up to 15," which brings us around to that number riddle. Some students are quick to find the two numbers, but I've chosen 12 and 3 because they're usually not the first pair that come to kids' minds. I ask students to help me make a list of factor pairs for 36. For each that we come up with, we check to see if the sum is 15 before we keep going.
The next two examples introduce negative numbers, and we'll work through each as time permits. If this doesn't happen today, there will be time to dig deeper into this tomorrow.
However far we get today is perfect. My goal is to put both of these skills on the table so kids can start moving toward mastery right away. I know that each of my students will need a different amount of practice to get there, and that there are all sorts of different cases we'll need to get to over the course of the unit. This lesson gives everyone a place to start.
With three minutes left in class I say that I have a challenge for everyone. "Take out a sheet of scrap paper or a split a sheet of loose-leaf with your neighbor," I say. "Write your name at the top, because I'm going to collect this at the bell." I put up the final slide of the lesson notes, which consists of four different quadratic expressions. "Three of these are factorable, and one is not," I say. "Figure out which one is impossible to factor, and explain why."
Most students just try to factor each one, and if that happens it's great: they're getting the practice with that skill. This also sets the stage for tomorrow's lesson, when we'll go a little deeper into factorability, continue to practice, and see the discriminant for the first time.