Different Forms and Different Parameters
Lesson 7 of 21
Objective: SWBAT investigate how different but equivalent forms of the same quadratic function reveal and explain different properties of the function.
At the start of class, I tell students that they have the first ten minutes to work on a brief Delta Math assignment about the graphs of quadratic functions. The first slide of the lesson notes defines the task for students. I share an overview of the task and describe my goals in this narrative video.
For the most part, students work independently for the first ten minutes of class to visually identify the key features of a parabola, "which," I tell students, "is the name for the graph of a quadratic function." Those features are:
- roots and x-intercepts
- axis of symmetry
As they work, I circulate to answer questions, offer encouragement, and to help students notice when they demonstrate that they know the meaning of each of these words.
In the next part of the lesson, I'll want students to be able to use these vocabulary words to describe what they see, and to begin to learn how each of these features can be found in the different algebraic representations of a quadratic function.
When the first ten minutes are up, I put up the second slide of the lesson notes. We have a brief whole-class discussion in which I ask students to identify each of the features of that parabola. Then, I tell everyone to take out their work from yesterday. If you're unfamiliar with the Quadratic Functions in Three Forms assignment, please take a look at yesterday's lesson to see what students have done so far.
Today, students will use this interactive graph on Desmos to check their work and to learn more about the relationship between the graph of a parabola and the three different algebraic forms of a quadratic function that were covered on last night's homework.
I encourage you to play with the applet on your own, and you can also check out this screencast for an overview of how the interactive graph works. I created this activity specifically to address standard F-IF.8a:
- F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
You'll notice that the key words that students tackled in the opener are all part of this standard. The purpose of yesterday's assignment was to practice both processes - factoring and completing the square. Today's interactive graphing activity will now give students the chance to see how the different parameters in each form of a quadratic function affect the behavior of the graph.
Work Time: Two Possible Scenarios
I like using Desmos in this manner, because anyone can pick it up and play with it. The sliders encourage kids to tinker just to see what happens. Students used sliders previously to explore the behavior of linear and exponential functions, and many of them remember that a slider can be animated by pressing the "play" button beside each one. When they start by doing that, I allow them to play for a few minutes, and I'll ask them to describe what they see.
The task at hand is for students to use the interactive graph to check their answers on Quadratic Functions in Three Forms. I project the graph on the front screen and hold the assignment up in my hand. "For each function, use these sliders," I point to the screen, "to make each form of the function match what you've written on the handout. If you've done a problem correctly, you should see that all three of these graphs match," I explain. I dig this assignment because self-checking is built right into it, and there's some sport here: it's satisfying to watch all three parabolas move until they end up in the same place!
That's one scenario, but how does this lesson for work for students who haven't done yesterday's work? I've found that this activity works just as well. Even if a student is unable to factor an expression or complete the square, he can adjust the parameters of one graph to match the function given on the handout, and then adjust the other two functions until they match the first. When students achieve the goal of overlapping all three parabolas, they can record the results on their handout.
The truth is that most kids fall into both camps. Before today's lesson, each student made their own amount of progress toward completing the assignment, but most are not quite done. Maybe they didn't get to the end of the assignment, or maybe they were able to factor an expression to fill in the middle column, but were unable to complete the square. Students can use the online graph to check the work they've done and to fill in what they haven't.
As they work, I enjoy my role moving around the classroom, answering questions, noticing what students notice, and modeling the idea that joy and mathematics truly can happen at the same time. You can see examples of what this looks like in the video that accompanies this lesson.
With a little less than five minutes left in class, I distribute today's exit slip. There is one quadratic function represented in all of the three forms we've studied today. Students are asked to sketch a graph of this function and to label the roots, the vertex, and the y-intercept. I do not provide a grid for graphing, I tell students, because I want them to be able to sketch a rough graph without using that tool. I do want students to pay enough attention to detail that they give a reasonable sketch of where each of they key points lie, but I'm not so concerned about the points in-between.
This exit slip serves two purposes. First of all, I want to get a quick snapshot of whether or not each student can accurately sketch the graph and demonstrate an understanding of the vocabulary by labeling each point. Secondly, I want to be able to see how many students use each representation of the function, and how. It might not be clear how every student produces the graph they do, but if I can pick up a few insights on some of the student work, then that's helpful.
Of course, students are still sitting at computers, and they can just use Desmos to see what this function looks like. I tell them that it's fine to use the computer, but I challenge them to try not to. Whether or not they do, the computer doesn't help in the task of identifying key features. The limited amount of time means that I'll receive the work that kids could do quickly. All of this will help me plan some next steps.