Flexibility with Quadratic Functions
Lesson 7 of 16
Objective: SWBAT produce a graph of a quadratic equation in any form and highlight important features of the graph.
As I check homework with the homework rubric, students will warm up by individually completing the template Warm-Up Quadratic Functions with information regarding 2 different quadratic functions. Each student will receive a page with the same blank template copied on each side. We used this template in the previous class, starting with a quadratic function in vertex form. Today, I will make this warm-up more challenging by providing the following starting points written on the board:
- a function in standard form (e.g., y = -2x2 + 8x + 24)
- the x-and y-intercepts of a function (e.g., y = -18 and x = -1 and x = 3)
These starting points rely on strong conceptual understanding. Students need to make sense of structure of the given function and then figure out how to translate to new forms [MP2, MP7].
When students have completed their individual work, I ask them to turn to their partner to compare their answers. Together, they should take out the graphing calculator to check their graph. From the graph, they determine if the information they provided about each quadratic function is consistent. [MP5] Finally, I ask for two pairs of students to come to the board to explain their answers using the document camera and projector [MP3].
For most of this period, students will work on a performance task by the Math Assessment Project. Students complete the Forming Quadratics Pretest independently and turn it in to me. There are parts of this assessment that students will not know how to complete but I remind them that the pre-assessment is a tool to gauge what they learn through the activity which follows. I hold on to this pre-test so that I can compare student responses on this and the post-test.
Explore and Reinforce
The Forming Quadratics task from the Mathematics Assessment Project helps students understand that different algebraic forms of a quadratic function reveal different properties of the function's graphical representation [MP2].
The task is designed to reinforce the following concepts:
• The factored form of the function is the best for identifing a quadratic function’s roots.
• The completed square form (or vertex form) of the function reveals a function’s maximum or minimum value.
• The standard form of the function makes it easy to see the function's y-intercept or "starting value."
The group work in this task requires that a set of Forming Quadratics Dominoe cards be prepared in advance. I prefer to copy these dominoes on colored paper and laminate so that they can be used from year to year (see Forming Quadratic Domino Cards for a visual.) The second part of the group work, involves students adding equations to the information provided on the cards. Students use "wet-erase" to write their responses directly on the laminated cards. These can be wiped off and reused with subsequent classes.
Partners for the domino matching activity are assigned based on compatibility and work ethic so that one student will not end up doing more than their share of the work. As students work on the matching activity, I circulate and listen to the discussions. I take note of students' precise use of the many vocabulary terms in this unit and whether they are able to explain that various representations can be used for the same function [MP3, MP6]. I also note what difficulties students have in performing the algebraic manipulations required to switch from one form to another [MP7]. Although I provide some small hints if necessary, I encourage students to use their notes and each other as a resource in completing the activity [MP1].
When groups have completed the activity, I hand them Forming Quadratics Pretest which serves as both the pre- and post-assessment for the Forming Quadratics task. If one or more of the groups needs the entire period to finish the group work, they can complete the post-assessment as homework.