Today's opener is on the first slide of the lesson notes. There are three quadratic expressions to be factored. The first is straightforward enough, but the constant term is 189, so students might have to work a little hard to find two factors of that. Some will get stuck. I tell these students to make a list of pairs of numbers that sum to 30 instead of a list of factors of 189. Guess and check can work in either direction here, and if a change in strategy can make it easier for kids, I want them to see that.
The next two expressions are different cases that will be new to a lot of students. Anyone who made it all the way through the factoring practice that was assigned for homework last night has seen problems like these, and plenty of students will have questions.
The second expression is a difference of two squares. I show students that we can think of the expression x^2 - 81 as x^2 + 0x - 81. In other words, we're looking for two numbers whose product is -81 and whose sum is 0. This hint is usually enough to get kids to factor this. We'll spend more time with the difference of squares later, and I like to allow it to show up consistently in context, so there will be examples like this sprinkled into many of the lessons in the unit.
The third expression has a leading coefficient, but it's possible to start by factoring a 5 out of this expression. I show students how that first step can work. This is another idea that many students need to see just once in order to make it their own. The key point to make here is that it's not always possible to divide all terms by a common factor. When it's possible, like it is here, that's great because it makes our job easier. We'll look at ways to approach problems where it's not possible in an upcoming lesson.
The goal of today's lesson is to get students comfortable assessing the factorability of quadratic expressions, which is a step toward helping them understand the nature of the roots of a quadratic function. Standard A-SSE.3a says kids should be able to "Factor a quadratic expression to reveal the zeroes of the function it defines."
Of course, not all quadratic expressions are factorable, which will be an important idea to understand as we investigate all different sorts of roots. Today, I'm going to introduce students to the discriminant of a quadratic expression, framed as a shortcut for determining factorability. If the discriminant is a perfect square, then the expression is factorable, but I don't tell students about that yet.
For now, I put up slide #2 of lesson notes, which instructs everyone to quickly make a list of the first 20 perfect squares in their notes. Once kids do so, I say, "Now that you have this list, I want you to keep an eye out for the perfect squares as the lesson continues...they're going to play an important role today!"
Knowing one's perfect squares has advantages beyond this - so even though we'll use this list in one context today, it's one that students will revisit for other reasons as the unit continues.
Today will students will learn how to use the discriminant to determine whether or not a quadratic expression can be factored. To frame today's work, I put up the third slide of today's lesson notes, which was yesterday's exit task. I ask students if they think there's a better way to know for sure whether or not such an expression is factorable, and as slide #4 says, "Of course there is!"
But I'm not going to just give it away right away. I put up slide #5 to review the learning target:
6.2: I can factor a quadratic expression to reveal the zeroes of the function it defines.
I say, "Today you'll practice factoring more quadratic expressions, and soon you'll see how factoring can help you find the zeroes - also known as the roots - of a quadratic function." To review the different kinds of roots we might see, I post slides #6, #7, and then #8 to review that a parabola can have two, one, or no zeroes. This is a review of what students have seen in the last two lessons, and it won't be the last time we look at these three images.
With the work framed like that, I distribute the Can You Factor It? handout. I tell the class that some of these expressions can be factored, and some cannot. "Factor every one you can," I say. "If you can't factor an expression, just leave it blank for now." Then, I give students about 10 minutes to do as much as they can.
Students notice the word "discriminant" below each expression, and ask what it means. I say that I'll show them in just a few minutes, and I remind them just to leave a problem blank if they can't factor it.
Introduce the Discriminant
After about ten minutes, I tell everyone that I have some notes I'd like them to see. I introduce the standard form of a quadratic expression, with the coefficients a and b, and the constant c. Then I show the class that the discriminant formula consists of these parameters. This is one of the few times this year that I'm showing kids something without much explanation. They find it a little comforting (after all, this is what I've trained them not to expect!) and a little strange (which is awesome). One student asked, "why is it always -4?" And I had to specifically choose not to get into it.
To practice using this new tool, I project today's classwork on the front board. We come to an agreement that #1 and #3 were pretty easy to factor, but that #2 and #4 were not. Then we practice calculating the discriminant for each. Every time I teach this lesson, I plan to ask, "What do you notice?" but before I even have a chance to get there, at least one student observes with excitement that the discriminant is a perfect square for the factorable expressions, but for the others, it's not. It jumps off the page at everyone, and it's exciting to see kids referring back to their lists of perfect squares to see what we're talking about.
Next I return to yesterday's exit task, and ask everyone to use the discriminant and decide whether or not they buy this conjecture. It works out.
I tell everyone to keep going with the Can You Factor It? assignment. "Factor every expression you can," I say. "If you think that it's impossible to factor one of these, find the discriminant, and see what you find." I'll collect this assignment at the start of tomorrow's class.
Debrief and Extensions
On slide #9 of the lesson notes, there is a table that can be used to summarize the relationship between the nature of the roots and the discriminant of a quadratic function. If we get to this point, I'll lead a whole-class discussion, and we'll fill this in. But I don't push it. If everyone is working hard but not quite done, we'll get to this tomorrow.
At the end of the assignment is the instruction to make another list of perfect squares: for each binomial from (x+1) to (x+12). This will set us up to dig into completing the square next. As with the debrief chart, if students get this far it's great, but there's no pressure - everyone should do as much as they can, and we'll pick up where we need to tomorrow.
No matter how far we get through the lesson, I end with the exit task on slide #10 of the lesson notes as a check in. It's in the same format as yesterday's exit task: there are four quadratic expressions below the question, "Which one is not factorable?"
As students get started, I say, "Ok, this one is a tiny bit of a trick question. There might be more than one that's unfactorable. Just show me what you can figure out."
I'll collect this work as students leave, and it will give a quick snapshot of how well students can use the discriminant as well as how much they understand about how to use it.