Two of the last three lessons have started with openers just like this one. On Tuesday, the purpose of the opener was to show students how to write a system of equations to represent the problem, and then how to use substitution and solve the resulting quadratic equation by factoring. Yesterday, the purpose was to create a similar equation, but one that could not be solved factoring, and to review completing the square.
The process for getting started on today's opener, which is on the first slide of the lesson notes, is the same as it's been previously, and as I circulate, I hope to see that kids feel comfortable defining two variables, writing two equations, and then using substitution to yield the equation
2x^2 - 5x = 228
where x represents the length of the rectangle. Now we've got a lead coefficient greater than one, so students will have to apply some of their more recently acquired skills to solve this one. I should note that, because the solutions to the problem are positive integers, this problem can be solved purely by guess and check, but I also want students to see a solution by factoring.
Of course, guess and check will play a role in factoring this expression. Using this method, which I originally introduced two days ago, requires us to list factor pairs for 456 until we find two whose difference is five, which involves a kind of guessing and checking.
This lesson is an extension of yesterday's, so if you haven't already, please take a look at Moving Toward Mastery: Completing the Square (Day 1). There's just a little review today before kids set back to their work.
I return the Check In Quizzes from the end of yesterday's class, and I field a few questions from students. (See the end of yesterday's lesson for the quiz and an overview of some of the mistakes I'll address with individual students today.)
To help frame today's work - for which there are various options - I post the second slide of the lesson notes, which gives a quadratic function in standard form and prompts students to rewrite it in both factored form and vertex form. I give students a few minutes to work and to discuss what they've got in small groups before eliciting solutions from the class.
After we all agree, I ask what we can learn from each form. I point to the factored expression and ask, "What does this reveal about the function?" Students are able to identify the roots, and we come to the agreement that they are at (-13,0) and (7,0). I repeat the process by pointing to the equation in vertex form, and we identify the axis of symmetry and the vertex.
Next, I ask if anyone knows what the y-intercept might be, and kids are quick to shout out that it's -91, or more specifically, the point (0,-91). "Can you sketch this graph now?" I ask, and I tell students they've got 30 seconds before I show them what the graph will look like. Students are eager to beat me to it. When I flip to slide #3, which to everyone's delight confirms that, indeed, we're getting pretty good at this stuff.
To conclude, I ask the class to observe that all of this is the opposite of what they had to do on the check in quiz. "You'll continue to practice moving between the different representations of quadratic functions," I say. "As you keep working today, pay attention to the features that you can and cannot identify from each form of a quadratic function."
For the rest of class, it's back to work. As I've described in several recent lessons, there's a lot that kids might work on today, and that list is summarized on the last slide of the lesson notes.
Most kids will work on the Mastery Assignment for SLT 6.3: Completing the Square, which I describe in the previous lesson.
My role is of my favorite sort today: to move around the room and make every student feel noticed, and to start conversations with them wherever they are. I might teach an impromptu lesson by putting an example on the front board, or I might just spend a little while at each table. I'll definitely joyously recognize the good work kids are doing. I'll definitely try to identify student experts to whom others can go for help when they're stuck on a particular problem. I'll definitely make it clear to kids that only thing they're not allowed to do is nothing, but beyond that there aren't any rules.