Today's opener is on the first slide of the lesson notes, and once again, it's about patterns. I give everyone a few minutes to get started, and I hope to observe that kids are pretty confident with this review task. The familiarity students have with tasks like this should open up some space to explore some new ideas. As I describe in this video, students should already feel comfortable moving both forward and backward through a linear or geometric sequence.
In considering the third sequence (which I don't rush to name because I want to leave room for students to construct an understanding of what this is), we find some curious behavior when we try to extend it backwards to the numbers that might precede the first term. Whatever consensus we come to, this sequence gives us a way to talk about the graph of
y = x^2
which is the focus of the mini-lesson that follows this opener.
Today's mini-lesson extends directly from the opener. I ask students to name the first sequence, which typically elicits both responses: it's an arithmetic sequence and it can be represented by a linear function. I everyone if they can imagine graph a linear function, and everyone feels good about that.
I repeat that structure for the second sequence. It's a geometric sequence that can be modeled by an exponential function, and ask students to use their fingers to draw in the air the shape that this graph would make. If anyone is unsure of what to do here, I direct their attention to anyone who is correctly tracing an exponential curve through the air.
"So what about this one?" I ask, pointing to the third sequence. "What should we call this, and what will the graph look like?" Students will have ideas, and some might even recognize already that it can be modeled by a quadratic function whose graph will be a parabola. Most kids will have to see it, so I suggest that we make a table and plot the points.
On the bottom half of the front board, I draw a table of values from 1 to 5, filling it in with the first five terms. Then, picking up where we left off on the opener, I make some blank rows above that, and extend the input column with 0 and the first five negative integers. Then we're able to fill it in by working our way up. With the table made, we can once again see the "pattern within the pattern" as we look at how the outputs change.
At this point, I hope that students can recognize that each output is the square of its input, so we're talking about the function
f(x) = x^2
Whether I can elicit that from a student or not, I write the rule on the board and ask everyone to make sure they know where I'm getting it. Once kids see it, testing a few input and output pairs, they buy it.
"This is a quadratic function," I say, "and today you're going to graph a few of these." I explain that just like y=x gives us a basic line - through the origin with a slope of 1 - this "parent function" will show us the basic graph of all the quadratic functions that follow. I set up a pair of axes on the board, plot the 11 points in our table, and ask that students do the same in their notes. After connecting the dots, I say "This is a parabola." Kids have seen some parabolas over the course of the last few lessons, and I hope that today's opener and this mini-lesson provide a foundation for why this function is shaped like it is. It's an understanding that I'll be looking for each student to demonstrate as we move into today's gallery walk.
Introduce the Task / Explanation of Documents
Now we get started on a gallery walk that will continue into tomorrow's lesson. To set up the lesson, I print each page (except for page one, the cover sheet) of this Quadratic Functions Gallery Walk. I group the functions by color, posting them in four sets around the room. For example, three "blue functions" are posted with the blue instructions on the back wall.
Each student gets a copy of this graphing template. I explain the task: "To complete this gallery walk, you must choose one function from each colored set, make a table of values and to plot the points for each function." So functions "A", "B", and "C" are the "Green Functions" and students must choose one of these before repeating for the "Blue Functions" (D, E, F), the "Purple Functions" (G, H, J), and the "Orange Functions" (K and L). By completing the gallery walk in that order, students are exposed to a new twist in each set. The green functions are pretty easy to see on the domain -5 to 5, but the blue functions are a little more "off-center," which will require students to reconsider the domain they choose. The purple functions are in vertex form, and the orange functions have less than two roots.
As I've described with previous gallery walks, student choice and student movement around the room result in increased buy-in and interesting conversations while the work happens.
Everyone gets started, and there's a lively bustle around the classroom as students choose the functions they'll work on. I circulate with students and answer clarifying questions about how to get started. At some point everyone may need to see an example, which I provide in two ways. On the fourth slide of the lesson notes is an example function. I show students how to make a table by using each input. This example is like the green functions, because there's a clear turning point between -5 and 5. As this activity progresses, students will continue to build an understand of how important that turning point is.
There are also some examples are posted on the wall. For function C, for example, I provide a table of values that shows how each input yields an output, and a graph on the domain -5 to 5. Students are free to use this one if they want, and it's encouraging to see how many kids will take a look at this before trying to do the same for A or B on their own. With that scaffold in place, I provide an extension by putting up sticky notes for other exemplars, and invite students to take a sticky and make the exemplar to post for others to see.
Where this Lesson Fits & Where We're Going Next
Moving forward, We're going to use these functions to talk about the features of a parabola. As students move on to the blue functions, they see that it is important to choose a domain that shows where the function turns around. Over the course of the next few days, we'll give the turning point a name, the vertex, and we'll formalize ways to find that point. For now I want students to notice some of these features on their own, so we can build toward that knowledge. Look at U6 L8 this student's work, for example. After using the domain -5 to 5 on function E, this student realized that she'd get a more complete graph by using some inputs less than -5, and she added those points to the graph. Other students try one domain, reconsider it, and do a second draft before setting up the axes and graphing the function.
In this video narrative, I describe how this activity helps students get the idea that the graph of a function is a set of points. It's important to have this experience to really drive that point home: there's no magic when we look at the graph of a parabola. That graph is a set of points, and those points come from somewhere.
Today's work is likely to be slow going because the arithmetic takes time. This is a structure students have seen before in this class: we'll do something the "long way" before learning and formalizing some shortcuts. The long way is important because it draws on some foundational skills and provides experience that students will reference moving forward. We'll start to explore some shortcuts in tomorrow's lesson.
I buzz around the classroom and encourage students in their work right up until the bell.