During this unit, there emerge two narratives that have, to this point, played more of a background role in the course. First is the idea of exploring beautiful mathematics just for the sake of it, and that quadratics are fertile, fascinating ground for such an exploration. Students should have the chance to experience wonder and surprise at what they see. Second is an emphasis on rigorous mathematics, and of digging into some advanced algebra.
Today's class begins with a task that will move students toward both goals. The opening task is on the first slide of the lesson notes. The instructions are brief because students are used to filling in charts like this (see this lesson about using Guess and Check, for example). I ask if anyone has questions about how to fill this out, but for the most part, students can see what the chart is asking for.
I give students a few minutes to work as I take a lap or two around the room. I expect to see that students are able to get this done, and I'll answer any questions they've got. Soon, we take a look at the completed chart (on slide #2), everyone checks their work, and then I say that I'd like to focus on the last column - the products - for a little while.
Slide #3 just lists the products, in order, with the question "Do you want to see something beautiful about these numbers?" There is some humor in this question, because it would be easy for a student to say, "no!" -- so I use a little self-deprecation to say that, "as the nerdy math teacher, of course I find this beautiful, and I hope you might too."
Slide #4 then asks the question, "How far is each number from 36?" We figure it out, and I take notes on the board. This is where the beauty begins to appear. Isn't it wonderful to notice that each of these products is "a perfect square away" from another perfect square?
Next, slide #5 summarizes it: "That's called completing the square." I want students to have an idea of what it means, literally, to "complete a square" -- and later, to have a concept for / wonder about the expressions that are factorable, and therefore come out nicely when we solve by completing the square. There's so much cool structural stuff going on here - it's impossible to record it all here, or to expect to show everything to my students - but this idea really has traction, and it gives us a reference point in pure mathematics. (Please see my reflection for this section for some extensions and beautiful ways to explore on your own.)
Moving forward, the question will be: how did we decide that we had to complete the square to 36? Students will often think that it's something special about the number 36, and only 36 here, and this is where it's useful to have another example to look at. One such example follows this exploration, and I'll also refer students back to the list of perfect square binomials from yesterday. We see that what it really depends on is that middle coefficient.
As I wrote in the previous lesson (Finding Roots of All Sorts), my goal is to give all students the time and the space to gradually build toward a rich understanding of quadratic functions: of the different forms of an expression and what each reveals, of the new operations - factoring and completing the square - that allow us to rewrite quadratic expressions in various forms, and of the connections between features of the graph of a quadratic function. All of this takes time, because different students will have different experiences coming to these new understandings.
Up to today, the amount of attention we've been able to place on completing the square really depends on the class. In any given year, I'll have at least one class that has already seen an example and another that hasn't touched it yet, with other classes somewhere on the continuum in between. Either way, students get to see the examples of how to complete the square of (x+6), and then we move on to a few examples.
For the first example, we find the roots by completing the square. The middle coefficient (aka "sum") is still 12, and this time the constant term (the "product") is 30, which wasn't in our list. There's more to solving for roots than for simply re-writing a quadratic expression in vertex form, and I've chosen to go this route because I want kids to have a chance to see an end result and why the solutions make sense. Plenty of my students will need a few more examples before they're comfortable with this, but that's fine.
Everyone will have the opportunity to use this tool today, and the work of today's class and tomorrow's class will provide a chance to work on completing the square. What is required of students on today's assignment is less complicated than finding roots. They will only practice rewriting expressions in vertex form, which involves fewer steps. We will look at another example when we get to the assignment - but as I've been doing, I want to put all this on the table, then work with kids to get it as they can.
Last night's homework was an Infinite Algebra worksheet full of practice exercises for finding roots by factoring. I collect it, and as I do, I'm able to get a quick glimpse at what students can do so far. This is super-quick, informal formative assessment, and all I'm doing is asking myself the question, "does this student know what it means to find roots?" as I collect each piece of work.
As today's class continues we won't be finding roots - simply rewriting quadratic expressions in different forms - but I'll keep in mind what I've seen and take time to check in with kids accordingly.
About the Assignment
Today's assignment is called Quadratic Functions in Three Forms, and I have two objectives when I run this activity.
The first is to give students a chance to apply what they know so far about the first three learning targets for Unit 6, which are listed at the top of the handout. "Maybe you're already great at factoring, and maybe you need some work. Either way, this is a chance to get the practice you need," I say. As students get into the assignment, I show them that in order to move from the first column to the second, they'll have to be able to factor quadratic expressions, which is SLT 6.2. To move from the second column to the first, students must be able to multiply binomials, and that's SLT 6.1. Finally, to move from the first column to third involves completing the square - SLT 6.3 - which we've just seen today.
The other purpose of this assignment is prepare to investigate what happens on a graph during tomorrow's lesson. Each of these representations can reveal different features of a parabola. This is what one of my favorite Common Core Standards is getting at:
F-IF.8: Write a function define by an expression in different but equivalent forms to reveal and explain different properties of the function.
That's a Cadillac standard right there, and it continues in sub-standard F-IF.8a, which is specific to quadratics:
Use the process of factoring and completing the square in a quadratic function to show zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a context.
I like this standard because it pushed me to think about writing the activity you see here.
What Happens in Class
Now, the key to enacting this activity is not to rush into telling students everything at once. Of course, I'm eager for students to see how roots are revealed in one representation of a quadratic expression, while the axis of symmetry and vertex are revealed in another. But when I first distribute the handout, I only show the class how to get started. I don't even name the forms of a quadratic expression. Look at the heading of the third column - I could call this "vertex form" but I don't. In fact, I don't even use the negative sign that we typically see in
y = (x - h)^2 + k
Instead, I leave it as y = (x + h)^2 + k, because I know that we'll get to a full understanding in due time. During tomorrow's lesson, students will check their work by manipulating a graph on Desmos, and as they do, they'll gain first-hand experience seeing just how predictable it is move a parabola that's represented in this form.
The assignment is meant to be pretty self-explanatory - it's another table to fill in, after all! When everyone has the handout, I project it on the board and say, "You're given one representation of a quadratic expression, and you're asked to write two others." We work the first example together. I elicit help from students to factor the expression, and everyone feels comfortable enough with that. Then we have a contextualized chance to complete the square and rewrite the expression in vertex form. I leave the work on the board, and I might fill in a few more solutions as class continues.
One thing I really like about this assignment is that it scaffolds students toward understanding how to complete the square. For some kids, going "backwards" from vertex form to the others really helps to solidify a basic understanding of how this form relates to the other two. If I see that a student is overwhelmed by trying to get this all at once, I'll tell them just to start by filling in the first two columns. If they can get that done, tomorrow's graphing activity will provide an opportunity to match the three forms.
I give students time to work right up until the bell. As time runs out, I tell everyone to finish as much as they can for homework, and that we'll continue to play with this assignment tomorrow.