Factoring and Completing the Square to Find Zeros
Lesson 7 of 13
Objective: SWBAT factor quadratic expressions to reveal zeros of the function and relate this to graphs and tables. SWBAT complete the square and relate the information to graphs and tables for quadratic functions.
In this particular Entry Ticket: Factoring Quadratics and Completing Square I have students expand upon their previous knowledge about factoring polynomials. This quick entry ticket provides a great way to differentiate. Students that have the concept down can use the entry ticket to further refine their academic conversation skills, while students still struggling with the idea of quadratics are provided with an opportunity to hear how one of their classmates understands and explains the construct. Discussing with a partner gets at the math practice standard MP.3 because students are asked to define their own arguments and also critique the reasoning of their classmates. The last question on the entry ticket (about what the x's look like) is getting at the idea that the x-intercepts will be the solutions to setting each equation to 0. In other words, the prompt is asking students to make the connection between setting y=0 and solving for x with what that solution for x looks like on a graph.
During the Entry Ticket I expect students to be conversing with each other, and I make my way around the class checking in for student understanding.
After students complete the entry ticket I turn to the agenda board and review the learning and language objectives, agenda and homework for the class. I ask students if they have anything else they want to include on the agenda to provide an opportunity to give students increased agency and ownership for their own learning. If, in the case a student brings up an inappropriate idea for the agenda I ask the class for their input on whether or not that should be included – we use the class objectives and essential question as a decision making guide as to whether or not the proposed agenda item is relevant or not.
In this section, I am presenting the Class Notes: Factoring Quadratics and Completing the Square to the class. I ask students to actively take notes on the material, preferably in two-column form. No matter the way students take notes, I am checking in and observing students to be sure I see 100% of pens or pencils moving throughout the lecture.
The presentation then reviews how to apply the idea of factoring polynomials to find the x-intercepts of graphs of quadratic functions. This section of the presentation directly connects to the standard ASSE-3a.
Note that the Powerpoint includes a number of talking points and practice problems for students to stay engaged in the learning process. I really try to help students go back and forth between creating graphs and functions for quadratics. These are also opportunities for the teacher to assess student understanding throughout the lesson and adjust practice accordingly.
The lesson then turns to the how and why of completing the square
In this section of the Class Notes, the class turns to the second main topic of the day: completing the square. The teacher facilitates discussion with the aid of the powerpoint slides and students are taking notes, conversing with each other during think-pair-shares and completing practice problems – all as different means to further student understanding of the concept. This section of class specifically is related to the standard ASSE-3b.
To wrap up class I zoom out and review the main concepts covered thus far in the unit on quadratics with the Reference Sheet: Summary of Forms of Quadratics. I think it is important to take the time to reflect and make connections between unit concepts explicit for students.
During this time, students work in small groups on the Collaborative Work: Factoring and Completing the Square to Interpret Quadratic Functions.
I like to provide students with a Reference Sheet for Completing the Square in which students either fill out the work or I provide the work to them. I encourage students to use the reference sheet as a tool to remind them how to rearrange quadratics and increase their level of independence in solving these types of problems.
The problem set includes both factoring and completing the square problems for students to have to evaluate each problem and see the connection and structure of the two forms of quadratic functions. Having students rewrite quadratic in factored and completed the square form helps show students how each form helps bring different characteristics of quadratics (roots for factored form; vertex for completed the square form) to the surface.
My role during the collaborative work is more of a facilitator. I monitor groups and how each group is interacting and try to value and point out groups that are doing the things that are valued (respectfully sharing and critiquing each other's opinions, staying on the task at hand, etc.).
Exit Ticket + Homework
To close this lesson, students complete the Exit Ticket: Factoring and Completing the Square. Students show that they can factor and complete the square for a quadratic function AND create an equation given critical points on a graph of a quadratic function.
The Homework: Factoring Quadratics and Completing the Square has review problems on factoring to reveal zeros of the function as well as completing the square. Students are also asked to write a 1-2 paragraph response (to complete for homework) to explain the benefits of factoring and completing the square in better understanding quadratic functions for homework.