Rearranging and Graphing Quadratics

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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.

Big Idea

This jigsaw lessons lets students dive deep into one connection between quadratic equations and graphs, and then learn from each other in mixed groups!

Entry Ticket

10 minutes

I start class with the Entry Ticket Rearranging and Graphing Quadratics. I have students rearrange and graph one simple quadratic function as a model to the class activity for the day. 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.

Entry Ticket:    Consider the function f(x) = x^2 + 14x + 40

  • What is the concavity of this quadratic function?
  • Factor the function and tell me the x-intercepts of the function
  • Complete the square and tell me the vertex of the function

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.

Jigsaw Activity, Part 1: Expert Groups

10 minutes

This section is run as a jigsaw activity where students are working collaboratively on the Classwork: Jigsaw Activity for Rearranging and Graphing Quadratics. There are two sections to the jigsaw, each taking approximately 20 minutes. In the first sub-section students work in “expert” groups diving deep into one characteristic of quadratic functions and their relationship to the graph. Student then rearrange and form mixed groups, with at least one member from each “expert” group in the mixed group. The lesson then shifts to the mixed groups graphing each of the functions based on the information they gathered about each function. For more information and suggestions on implementing a jigsaw activity the website is an excellent resource on this wonderful technique that can be applied across disciplines.

Expert Groups

In this section, students are assigned to an “expert group” and work on the Classwork: Jigsaw Activity for Rearranging and Graphing Quadratics. One way to arrange students is by their current understanding of the material. I would group students who are struggling with the concavity expert group as that is a relatively straightforward task that they are likely to have success with. More advanced students could be assigned to the vertex expert group as completing the square is often more difficult for students. The beauty of the jigsaw activity is students will all have a chance to teach each other and be exposed and have practice in all of the characteristics and tasks of this assignment. I also thoroughly enjoy running this activity as, on a good day, my role as a teacher shifts dramatically to that of a facilitator and the students take on the role of teaching which is always fantastic to witness.

Jigsaw, Part 2: Mixed Groups and Graphing

20 minutes

Jigsaw: Mixed Groups   

In this section, there is a transition where students go from their expert groups to mixed groups while continuing to make progress on the Classwork: Jigsaw Activity for Rearranging and Graphing Quadratics. I would plan for this transition to take a few minutes, especially if students are not used to the jigsaw format. One way to keep the learning going is to have pre-assigned group names that students can connect to (local sports teams, community hangouts, etc.). When students are in expert groups give them a number so there will be a number 1, 2, 3, …. For each expert group. That way all of the number 1’s can be a mixed group and you can assure each mixed group will have at least one expert from each area.

In the mixed groups students are asked to focus on completing the front of the ­­­­­Classwork: Jigsaw Activity for Rearranging and Graphing Quadratics, namely figuring out the concavity, x-intercepts and vertex for each function. During this time, the teacher is making rounds checking in and providing cues and tips to keep each group on track.  

Once groups are finished I take a few minutes to have the class reflect and engage in some in-process meta-cognitive skills. This can be as easy as asking students what the big picture task is and where we are in the process. This is also a great opportunity to ask students to reflect on what they have done well and what they need to work on to be successful for the remainder of the class.

The mixed group wrap up the Classwork: Jigsaw Activity for Rearranging and Graphing Quadratics by graphing the quadratic functions they have interpreted during the activity.

Creating Equations from Graphs and Matching Graphs

30 minutes

In this section, students work on creating quadratic equations from graphs. The attached set of graphs can be displayed one at a time on a projector or the assignment (Collaborative Work: Creating Equations and Matching Graphs Using Technology)could be printed for students to create an equation and take notes. 

My favorite way to run this activity is through technology. Students work in pairs and each pair has an ipad. Any graphing calculator will do, but the TI N-Spire apps. are excellent tools. 

I project one of the graphs at a time, including the coordinates of the root(s) and vertex. The task is for students to 1. try to match the graph by creating the matching equation and 2. prove their answer is correct by using the calculator app. to identify the root(s) and vertex.


Exit Ticket + Homework

20 minutes

To close this lesson, students complete the Exit Ticket: Rearranging and Graphing Quadratics that asked students to summarize the different forms of quadratics they used to graph and the benefit of each form. I have students complete an Idea Organizer to organizer their thoughts to close the class because the ideas are fresh in their heads from the day’s jigsaw activity. For homework, I ask that students turn in a complete Idea Organizer along with a 1-2 paragraph written response to the prompt. This helps students solidify their thinking about the topic and encourages them to review the class problems as providing examples and evidence of their ideas is an important aspect of writing.