After a few days of investigating the graphs of different quadratic functions, today's opener (on the first slide of the lesson notes) takes us back a few days to finding the roots of a quadratic function by factoring.
It's always instructive to see what my students can do when we circle back to a topic from, say, a week ago. Some kids will be thrilled to return to factoring. Others will still some help to make the skill their own. By reviewing factoring in a new, deeper context, I hope to give students more ways to make sense of it.
While students complete the opener in their notes, I return the exit slips from yesterday's lesson. This gives me the chance to share some brief, conversational feedback with kids on how they did with their quick graph of a parabola, and to say that they're going to learn more about that today. If they haven't yet submitted it, I also collect their work on the Quadratic Functions Gallery Walk.
After a few minutes, we run through the solution. In some classes this will be a quick, confidence building review, while in others students may need to review the definition of roots and to see again what it means to factor an expression. Everyone achieves mastery on their own schedule, and today's lesson is designed to give kids space to advance whatever knowledge they've gained so far.
Today's mini-lesson extends directly from the opener. In this narrative video, I explain how I try to build on the experiences of my students during the gallery walk to try to develop some shortcuts. Once students understand that the vertex is an important point on a parabola - and that finding it can make the work of graphing a quadratic function a lot easier - they'll want to know shortcuts for identifying it. Many students have had the experience of realizing that, "if I could just find the lowest point, the rest of this is pretty easy." I focus on one way to think about that today.
On the second slide of today's lesson notes, I briefly take students back to the second day of school, when I told them what success looks like in this class. With less than eight weeks to go in this course, it's useful to take a moment to orient ourselves within our work. On the third slide is learning target 6.4:
I can graph a quadratic function and show all of the following features: the roots, the vertex, the axis of symmetry, and the y-intercept.
I tell students that they're going to see how to use these features to save themselves some time in graphing quadratic functions.
With that in mind, we take another look at the opener. It's on the fourth slide of the lesson notes, with the question, "What are the key features of a parabola?" I ask students to think about SLT 6.4. We recognize that we already have the roots, and the conversation comes around to finding the vertex.
The hand gestures I use in the video are the same that I use with my students. As I trace a parabola through the air and place my fingers where the roots might be, I ask everyone to visualize a parabola with two roots: "If we know these two roots, then, can you imagine that the vertex is going to be halfway in-between them?" I move one hand to where the yet-to-be-officially-named axis of symmetry would be, and trace that through the air as well. The point of this visualization exercise is to think, rather informally, about the relationship between the roots and the vertex of a parabola.
Take a look at this photo of my notes on the board. To build these notes, I started by eliciting from students what the x-coordinate of the vertex would be, if it's halfway in-between the roots, which are at (-3,0) and (7,0). If necessary, I help students to think about that question by figuring out that the distance from -3 to 7 is 10, and that half of that distance is 5. "So we're looking for a number that's five up from -3 and five below 7," I say. (This is similar to the way I showed students how to think about finding the median during our statistics unit.)
Once we agree that we're looking for x=2, we have to find f(2), and unlike yesterday, this is the only value for which we need to evaluate the function. You'll see that on the left side of my board notes. With two roots and a vertex, we have enough to make a rough sketch of this function. To check our work, I ask students to estimate what the y-intercept might be. On the graph it's clear that it's a little bit above -25, and the value of -21 in the function rule makes beautifully perfect sense as the y-intercept.
From here, it's time to practice.
A Note about Context
It is also important for teachers to note that the relationship I've outlined here shows up in the quadratic formula and when we find roots by completing the square. When half the square root of the discriminant is added to or subtracted from the axis of symmetry, we can see the roots. I encourage kids to explore this on their own when we get that far, but wanted to give you a chance to think about it now. To me, seeing connections like this is what makes a rigorous study of quadratic functions so rewarding.
Now students get 20-25 minutes to practice what they've just seen. This lesson jumped off from yesterday's, in which all students accomplished a different amount of work and at the end, showed me what they knew about making a quick sketch of a parabola.
Today we began to look at some relationships between the features of quadratic functions, and students saw that they can save some time by considering just a few key points on a parabola.
On the fifth slide of today's lesson notes are eight quadratic functions to be graphed. This is more than most students will be able to finish in just 20 minutes, but I want to make sure that there's enough for everyone to do as much as they can. The instructions take students through the same process they've just seen: find the roots, use the roots to find the vertex, look at the function rule to see the y-intercept, and then produce an informal graph that includes these points. I specifically instruct students not to use graph paper, because I really want them to focus on the informal part, and to really focus on just those key points.
The first function serves as a guided example, which I use to show students exactly what I'm looking for. The number line at the bottom of that image is one method for finding the x-coordinate of the vertex.
Note that exercises d and e will have a non-integer axis of symmetry. Students will discover this on their own, wonder aloud whether or not that's ok, and then with a little guidance, be satisfied to see that it makes sense. Note also that f, g, and h share the same axis of symmetry but different roots. Watch to see how students explain this to themselves, and to see whether or not they're able to attribute the role of the middle coefficient to the behavior of the axis of symmetry.
For some classes, it might be overwhelming to be asked to graph eight functions at once. For these groups, I adapt today's activity by giving one function at a time.
On slides 6 and 7 are two extensions to this activity. The first is to find two more points on each function, and the second is to re-write each function in vertex form.
To close today's class, I ask everyone to circle up. When we're ready, I ask everyone to "give a thumbs up if you feel like you got better at graphing parabolas today." Most kids will do so, and then I ask if anyone wants to share what they learned today. We have time for a few kids to share, and it's always instructive to hear about today's lesson in the words of my students.