Today's opener is on the first slide of the lesson notes, and it's just like the opener from two days ago, with one important difference. The process of defining variables for width and length, writing an equation for each constraint, and then using substitution to come up with the equation
L^2 + 4L = 23
is all the same. I circulate and coach kids a little bit to make sure they can get that far before I start any whole-class discussions. I want students to have time to play with this problem on their own, and to see for themselves why it's just a bit more difficult than the last one.
What is different is that the solutions to this problem are not whole numbers. As a result, it's hard to solve this problem by guess and check, which means that the algebraic solution will gain some traction. Furthermore, once we commit to an algebraic solution, we'll need to use a method other than factoring to solve the resulting quadratic equation.
Mini-Lesson: Completing the Square
Like I did on Tuesday, I ask everyone's permission to use the variable x in place of l for length, because it feels a little easier. Then we set the right side of the equation equal to zero. Now we've got:
x^2 + 4x - 23 = 0
I scan the room to see if everyone is with me, and then I post the second slide of the lesson notes, with the function
f(x) = x^2 + 4x - 23
"Let's take a closer look at this quadratic function," I say. I proceed to model the method for completing the square to find roots, and the resulting notes look like this. The mini-lesson hits all students at a different point in their learning. Some have very little practice completing the square, and others are getting pretty good at it. My goal is to give everyone a straightforward example to keep in their notes, and then to be able to decide whether or not the strategy yields a reasonable solution.
I ask, "Who can give an approximate value for the square root of 27?" We recognize that it has to be in between 5 and 6, and probably a little closer to 5 than 6. "Can we suppose that it's approximately 5.2?" I ask, usually at the urging of a few confident students. With that decision made, we can see that the roots of the function are ~3.2 and ~-7.2, with the positive number the one that we'll use for length, which, we remember is what x represents. "So if the length is about 3.2, then what's the width?" I ask. Returning to the original problem, we see that the width must be four longer than the length, so that would be about 7.2. Finally, I ask what the area of this rectangle would then have to be, and kids are quick to find that the product of 3.2 and 7.2 is 23.04. "Is that close enough to the area we were trying to get?" I ask. It's satisfying to see a few skeptical students admit that, indeed, this does make sense. Together, we recognize that it feels good when the results of using a new tool make sense.
I should also note that sometimes I'll squeeze in a little mention of simplest radical form. I certainly don't try to pile this on if I see that a class is struggling, but if I'm prompted by students who have come across this idea in their own work, I'll take a moment to show that root 27 can also be written as 3 root 3.
Today, the tone of this Quadratic Functions unit shifts from "mid-unit" where we're seeing new concepts, to "end of unit" where everyone pursues mastery of as much as they can in a short amount of time.
I frame the work of the next few days by providing a quick review of where we stand:
As part of this framing, I return any completed and graded work I've got from the last few days:
Some students still have to finish up on some of this work. The third slide of the lesson notes lists options for what kids should get done, and I say that everyone can work on whatever they'd like. As the name of this lesson suggests, however, most students will end up working on completing the square. That's my focus for this lesson write-up, but the reality may be that there's a great variety of what's going on.
The thing about completing the square is that there's just so much to say about it! There are interesting connections to be made, important skills, and curious revelations to be had about how quadratic functions work. The purpose of Mastery Assignment for SLT 6.3 is to give everyone a chance to practice and to see as much of the interesting stuff as possible. I provide an overview of the assignment and my expectations in this narrative video.
As kids get to work, I'll revisit the opener and make some distinctions between completing the square to rewrite an expression in vertex form and completing the square to solve an equation and reveal the roots. I note in the video that as students continue with the assignment, I may start them at different points depending on which entry point feels more approachable to each kid.
As with many of the assignments that I've shared in last few lessons, I expect this one to take more than a day. I want to provide students with a space where anything can happen, and I try to help kids recognize their role in figuring out what they should do each day. If it turns out that we focus on solving equations today and then graphing tomorrow, that's great. As with the opener, we might talk about simplest radical form, but again, if that's an overwhelming addition right now, it can wait.
Snapshot of a Cool Conversation
Sometimes, a student's misconception about what they're doing when they complete the square can reveal a really neat understanding and provide a place to dig deeper. As she worked on #16 from this assignment, one student got to the point of having root 5 on the right hand side of her solution, and then she got stuck. She asked, "If root 5 is one of the roots, then what's the other?" I was excited to get to clarify for her that that's not the root - it's the distance from the axis of symmetry to each root. That was all I had to say for her to know what to do next, and her satisfaction stemmed from being able to think not only about solving this equation but also in considering the whole function that it represents.
With a little more than ten minutes left in class, I interrupt the workshop to say we have a check in quiz about the graphs of quadratic functions. I tell everyone that they'll pick up right where they're leaving off tomorrow, and I encourage them to see what else they can get done for homework tonight.
The quiz is different from what we've done recently, because it provides graphs of parabolas and asks students to write equations for each. I tell students that as they work, they should refer to some of the work that was returned earlier today. The Gallery Walk and Five Point Graphs were opportunities to sketch graphs from equations, and this is the reverse of that. It's a bit oversimplified, because the leading coefficient is 1 for all of these functions, but I want the focus to be on how well kids can use the features of these graphs - roots, vertex, axis of symmetry, and y-intercept - more than on how quickly each parabola grows.
For kids, this quiz is that it's such a confidence booster! They realize that right now, just as they're getting into some deep, difficult work, that they've learned a lot. What would have been mysterious a few weeks ago makes a lot of sense now, as kids have a pretty good grasp on the structures revealed by the graphs and function rules of quadratics. That's exciting stuff, and it encourages everyone to keep learning more.
For me, this quiz will produce a treasure trove of information about what each student knows. Here are some examples of what I see:
We will kick off tomorrow's class by debriefing on this quiz before diving back into the work. As kids work, I'll pull them aside individually or in small groups to address what I saw on their quizzes.