What Does "a" Do?
Lesson 15 of 21
Objective: SWBAT to analyze the role that the leading coefficient plays in a quadratic function.
Today's opener is on the first slide of the lesson notes, and it's a modified take on the rectangle problems that opened this unit. I give students a few minutes to try the problem, and I circulate to take a peek at how everyone approaches this problem.
Given the area and a simple relationship between the dimensions of a rectangle, many students will notice that guess and check is a useful strategy for solving this problem, and I'm not surprised when I see that this is a common strategy. My goal is to show students an algebraic solution to this problem. As students have seen previously, if they've got a solution by another method, then it may be easier to make sense of the algebra.
I call the class to attention to provide these notes. We define the variables w and l for width and length, and then we write an equation for each constraint given in the problem. "What have we got here?" I then ask, hamming it up a little as I point to the pair of equations. It usually works well enough to get at least a few students to see that this is a system of equations. This conversation is important, because kids need to see that just because "Unit 5" is over and we're now nearing the end of "Unit 6," it doesn't mean that topics we've studied will disappear. "What is the best way to solve this system?" I ask, and soon we settle on substitution.
The resulting equation is written in this image, and you'll note that I change the variable from l to x, just to minimize confusion. I explain to students that we can change variables however we'd like, as long as we don't lose track of the changes we make.
Finally, we solve the equation by factoring. Of the two solutions, one is negative and one is positive. Anyone who solved the problem will note that the dimensions are 9 and 12, so this seems about right. I remind students that x represents the length of the rectangle. "Does it matter which of these numbers is the length, and which is the width?" Of course it does! The length is the greater dimension, and must be 12. So we can see that a length of -9 doesn't really make sense in this case. (Although, if you really want to press the issue, -9 is three more than -12, and those two numbers do have a product of 108 if we plug 'em into the area formula, so maybe that does work after all...but for now, I don't lead the whole class on that sort of journey.)
There's a soft transition from yesterday's work session to today's exploration. I post the second slide of the lesson notes, which provides an outline of the work to be done. Some students have already completed and submitted the Five Point Graphs assignment, while others might need a little more time to finish up. Either way is fine. Students are getting what they need and learning as much as they can, at their own speed, as this rigorous study of quadratic function continues.
It's satisfying for everyone involved to review a few ideas if everyone is still finishing up:
- Students will want to confirm that it's ok for a y-intercept and a root to be the same point. When does this happen? Is there a general form for such functions? What does it do to our five point graphs?
- They'll want to make sure that it's ok to have a non-integer for a the axis of symmetry. Of course we can, and this is often the case!
- Then, on #7 and 8, there are no roots. How does the vertex tell us about that. How does the discriminant tell us about that? What does it mean for our graph? Can we generalize?
The better a student's grasp on the learning targets, the more engaging it will be to discuss each of these questions. This unit is unique to the experience my students have this year, because we're really emphasizing math for its own sake. There's not too much context in this unit, just a lot of connections, surprises, and indeed, beauty. Most kids will embrace that. I don't think I could get away with teaching an entire year of 9th grade like this, but there's enough that is inherently satisfying in this unit that kids are happy to play along for a five-week unit.
What Does a Do?
When each student is ready, I provide a copy of the new assignment, What Does a Do?. Now that we've spent a lot of time getting familiar with the features of quadratic functions, it's finally time to consider what happens when the lead coefficient changes to anything other than 1. The structure of this assignment is familiar to students - fill in a table of values and then create a graph from that table - so it's pretty easy for everyone to get started.
Filling in the Table
This is a great opportunity to talk about order of operations, in context. Students often make errors when they first evaluate
2(5)^2 to be 100
-(5)^2 to be 25
but if that happens, then we can review the idea that exponents are evaluated before multiplication, or that if the negative sign is outside of the parentheses, then it's not "attached to" to the 5. As I move around today, I'll watch closely to see if these errors arise. Often, students will have their own arguments about the right way to go about evaluating each expression, and I'll applaud them when they do.
Another approach that helps students make sense of these columns is to point out that every value in the second column will be double the values in the first column. "We've already got x squared in the first column," I tell students. "So g(x) really says two times x squared. How can you use that to quickly fill in this table?" The same goes for the fourth column: m(x) is "negative x squared, so if we already know the values of x squared, what can we write in this column?"
Graphing the Six Functions
A few things are going on as students get started on their graphs. We revisit the importance of carefully scaling the axes, as we have many times this year. If I want to see all of these parabolas on the same graph, then how should I scale the axes? The handout instructs students to create an x-axis that goes from -10 to 10 and a y-axis from -50 to 100, but not how to count in between. Here is another opportunity for students to discuss the work with their colleagues, and to come to consensus on what works best.
I keep a big box of colored pencils available to students (it's probably the best $35 that I spend on my classroom each year), and when they're ready, they choose the six colors they'll use to color-code this graph. When there's an attractive product to be made, students care about it. They naturally pay attention to craftsmanship and start conversations about the merits of each other's work. They help each other out, and learn more from helping each other than they would from listening to me talk.
When the conversation turns to craftsmanship, we must address our attention to detail. Take a look at this student's work, for example. It's a reasonable graph, but look closely at the origin. According to the table, each of these six graphs must pass through the origin. Do they? Does it matter?
Once students have these colored graphs to reference, I push them to make some generalizations. What happens when we put a negative sign in front of x^2? How about a coefficient greater than 1? How about a coefficient less than 1, but still positive? Students will continue to address these questions in the coming days. I don't rush to define anything; I want kids to have as many of their own ideas as possible.
Maybe Today, Maybe Soon
I noted above that we make a "soft transition" from yesterday's work to today's. In general, we're also transitioning to the "end of the year" - in which I show kids that I expect them to take initiative and get stuff done. That's a big deal. Success in this class is all about pushing yourself to do as much as you can, providing evidence of what you know, and seeking out opportunities to learn more about what you don't.
The second part of the What Does a Do? involves sketch a Five Point Graph for each of six functions with different leading coefficients. Very few students will get to this today, but many will approach these functions in the coming days. My favorite scenario is to be able to return their work on Five Point Graphs as students get to the back, where they're repeating that structure with a more challenging twist.
These six functions will provide fertile ground for all sorts of conversations over the next week or so. We'll talk about the features of each graph, and whether or not you can factor an expression like this, and we'll use the same functions to practice applying the quadratic formula. Again, this is unlikely to happen today: that quadratic formula photo was taken a week after initially starting on this assignment. In the coming days, work will be more and more individualized for each student, and everyone will do as much as their able by the end of the unit.