Solving Quadratic Equations (Delta Math)
Lesson 16 of 21
Objective: SWBAT to solve quadratic equations by factoring, by completing the square, and by using the quadratic formula.
As I describe in this video narrative, the set up for today's lesson is about as simple as they come. I define a task for students, and they set to work. When they need me, I'll teach a mini-lesson or two, either to small groups of students or to the entire class.
As students arrive, I post Student Learning Target 6.5 on the front board, and instruct everyone to log in to their Delta Math accounts. Together, we read the learning target. I ask, "What are the three methods you need to know for solving quadratic equations?" They're written in the learning target, which says, "by any method, including factoring, completing the square, and using the quadratic formula." When students see the assignment on their computer screens I point out that the first two modules require them to use factoring, the next two completing the square, and the last three, to use the quadratic formula. "You'll do as much as you can today," I encourage everyone. "If you're already good at all of this, that's great. If you know how to factor, but not how to complete the square, then you know what you'll need to practice today."
Today's work serves as both a review of what we've done so far and a preview of what's to come. Where the lesson lies on that continuum is different for each student. Some kids are aces at factoring and have already discovered the quadratic formula on their own. Others still lack confidence and proper attention to detail when they're factoring, and they'll need some help setting one side of a quadratic equation equal to zero. This lesson is for both kinds of students. Some will race through the lesson with little help from me, and others will need this opportunity to fill in some gaps. Used thoughtfully, technology makes differentiation like this pretty easy to accomplish.
As I note above, some students won't really need my help today. I'm not only talking about the stellar students. There are always students who surprise me by really rolling when they're just left alone to work with a computer, despite what they show me in the traditional classroom. That's part of why I consider time like this to be essential.
Other students will need plenty of help. For these students, Delta Math helps by giving kids a way to identify what they need from me. Rather than allowing them to just say, "I don't get it!" I can press them to use words from the assignment and from the learning targets to tell what they're missing, and to frame great questions.
Here's an example of a place where I expect some students to need quick help. In the first module, students must solve some equations by factoring. Most kids have limited experience setting one side of the equation equal to zero, however, so I'll need to show them how to do that. When they take a look at these problems and wonder what steps to take, I ask them how this problem is different from what they've seen before. We work together to identify the fact that there are terms on both sides of the equal sign. Once we figure that out, I give a few notes, written directly on a Delta Math-generate example, about what to do before you factor. I might have to give these notes a few times, because students will ask their questions at different times, or because some kids need to see it a few times. That's ok. I'm participating in the assignment with students, doing the same work they're doing, and only jumping in when prompted.
A Novel Method for Factoring
Like many Algebra teachers, I like to frame the task of factoring quadratic expressions as a puzzle: we have to find two numbers that sum to this and add to that. But what happens to this puzzle when the leading coefficient is greater than one?
On the second module students are confronted with this task. For many students, yesterday was the first experience with such expressions, and that in a graphing context. When it comes to factoring, the game is a little different. As much as I love guess and check, there's got to be a better way here, right? Here's a trick I learned from a colleague.
We still want to find two numbers whose sum is the middle coefficient, but whose product is the same as the product of the lead coefficient and the constant term. When we find those numbers, we can fill them into an area model like you see here. We'll complete this area model from the inside out.
Then, fill the other two cells of the area model with the first and last terms. From here, just factor each row and column of the model, and you'll have two binomials right where they've always been in our area models. Beautiful right?
As with the help I provide on the first module, I'll demonstrate this example a few times today. I might invite students to stand with me at the board as I do so. I like to get kids up, thinking in a different posture.
Completing the Square and The Quadratic Formula
Some students have experience with one or both of these methods, others have none. I might be available to help with these skills, but I might be giving most of my attention to struggling students. If that's the case, another huge benefit of Delta Math is that it provides a solution strategy for every exercise. Often, these strategies are different from how I teach, in small and large ways, but I consider that a great thing!
Here's an example of what kids will see if they need help completing the square. Check out the free Delta Math site for yourself to see more examples.