Like Wednesday's lesson (two days ago), this lesson pushes the pace a bit. My goal is to put a lot of tools on the table as the first week of the Quadratic Functions unit comes to a close. From there, I want each student to have space to make as much sense as they can, of the tools and how these ideas fit together, over the next few weeks. I find that the richness of a unit on quadratics lies in the number of connections students are able to see between the different representations and the different features of a quadratic function. As my colleague Hilary Yamtich shows in her 12th Grade Math course, the depth of understanding that I want everyone to achieve happens at a different pace for each student, and it's my job to facilitate that.
With that in mind, today's lesson opens with three quadratic expressions (on the first slide of lesson notes) and the instruction, "Find the discriminant of each function. If it's possible, factor the expression."
I want to get a quick pulse of how comfortable students are with calculating the discriminant, and with using it to help decide if an expression can be factored or not. I know that some of students will need to see the formula for the discriminant again, and that others will need to revisit how to use this tool. That's perfect. The repeated use of - and circling back to - a tool like this over the next few weeks will help everyone to make their own sense of it.
I give students a few minutes to give this opener a try, and then we run through the solutions pretty quickly.
Yesterday's Can You Factor It? assignment is due today, and in some classes I'll collect it and we'll move on to the next part of the lesson. In other classes, students may need some extra time and help to get the work done. Again, it's all about giving kids time to come to their own understandings or factoring, of the nature of roots, and of what the discriminant can tell us. I'm flexible today, and more interested in getting the work done than hurrying to next thing. I'll dedicate as much time as needed to help kids get this done, and some sections of this lesson may get pushed to tomorrow.
In most classes, students have just a few questions, and might need just a few more minutes to put some finishing touches on the work. I circulate and encourage kids to form those questions and to explain what they notice. Some students got the big idea pretty quickly, but it's completely normal for kids to need an extra day to get there. To the extent possible, I really want students to feel like it's their own idea that factorable expressions have perfect square discriminants, and that's why I patiently continue to ask what they notice.
Check out the student work I've included here - in particular, look at how the work on the page, and notes in the margins show how actively these students are working to build an understanding of the content.
When students learn how to factor quadratic expressions for the first time, they are drawing on an array of both procedural and conceptual prior knowledge. There's plenty I want students to understand, but the initial hook is to frame this as a puzzle: what are two numbers that have this sum and that product?
As I've written above, some students can make sense of this task pretty quickly, so that by the end of the first week of Unit 6, they're ready to investigate and begin to use the discriminant on their own terms.
Other students need a little more help developing their understanding of factoring. It would be counterproductive to hammer on the discriminant before these students know enough about factoring to appreciate the usefulness of the discriminant in the first place. Students must really get the idea that some quadratic expressions are factorable and others are not. For anyone who needs help getting to that point, we revisit two related approaches from earlier in the year: guess and check, and making lists.
In one of my sections today, we ended up making a list like this, which students have seen before. By engaging in this sort of list-making, students recognize that there's actually a pretty limited number of possibilities, and therefore a limited number of different products we can make from a pair of positive integers whose sum is 17. I asked students if this list is complete, and how they know. Then I asked if them to consider a pair of numbers whose sum is 17 and whose product is 65. The list helps to make it clear that if such numbers exists, they will not be integers. If the numbers we're looking for are not integers, we recognize, then this problem just got a little more complicated. The moment kids understand that distinction is exciting, for them and for me. And once we get there, well, then we're ready to see what the discriminant can do for us!
Even though slides #5-12 on today's lesson notes aren't new, each student's understanding of what they're seeing here will grow from day to day. As I describe in this video, the work that students do each day will continuously reframe these notes. One goal of any introduction to quadratic functions is to help students develop a strong definition for roots. We'll take a brief look at these slides each day - another way to give kids the chance to make the knowledge their own - and each day, and in every section, where those conversations go will change according to how much students understand so far.
I originally tacked this task on to the end of the Can You Factor It? assignment, so some students have already done it. Other students didn't quite get there, and I want to make sure everyone notices what happens when they square any binomial (x + a). In addition to seeing the common structure in each of these expressions, students are building toward learning how to complete the square in a quadratic expression. This list will be important when we get there, and some classes will get there today.
When I project the task at the front of the room, some students will jump right in, and others will need an example of what "(x+1) squared" means. I'll write out an example on the board. "The key here is to notice the patterns," I say. At this point in the year, students are used to tasks like this - something repetitive that could take some time, but that is pretty quickly done once someone realizes what's going on. Understanding this pattern and why it exists, of course, is part knowing how to complete the square.
There are a lot of parts to today's lesson, and it's possible that a class will be in position to run through each of these sections the way they're laid out here. As I've described above, I'm really going to follow my students today, and it's likely that any one of these sections could turn into a deeper focus of today's lesson. Therefore, it's entirely possible that we will need an additional day to finish some initial practice at finding roots before digging into completing the square.
Whenever we do get to this part of the lesson, I start by revisiting by the learning target:
6.2: I can factor a quadratic expression to reveal the roots of the function it defines.
On slide #13 of the lesson notes, I highlight the second half of that learning target; students have been working to learn how to factor a quadratic expression, and now it's time to see why we factor: it's a way to reveal the roots of a quadratic function. I use slide #14 to "find the roots" of a function f(x), we are actually solving the equation f(x) = 0. I ask students to factor that expression, and then we talk about the zero product property and use it to get the pair of roots. Slide #15 is guided practice, and then students are on their own for slide #16. The last of the four equations on that slide is not factorable, and this is what leads us into completing the square.
If and when we do get to that point, I post slide #17, which I use as the first example of how to find roots by completing the square. This is hard work for most kids, and I tell them not to worry about getting it all at once. "We're going to spend the next few weeks practicing all of this," I say. "You're going to be able to figure this out one piece at a time."
When there are a few minutes left in class, I distribute the homework, which is just a Kuta Infinite Algebra practice worksheet. There are 12 quadratic functions, and students should find the roots of all of these. If we've made it through the full lesson today, I'll expect students to be able to finish this work tonight. Most likely, this will extend into the next class, and these practice problems will give my students and me a chance to see how well we've done.