Solving Quadratic Equations

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Objective

SWBAT Assess the structure of a quadratic equation and then select and carry out a solution strategy (factoring, completing the square, or using the Quadratic Formula).

Big Idea

Its important to be flexible in solving quadratic equations. Some equations lend themselves to factoring or completing the square while others are best tackled with the Quadratic Formula.

Warm-Up

15 minutes

As a warm-up, students complete a set of 4 problems Warm-up: Solving Quadratic Equations. These problems are different from the previous day's work in that we are no longer only interested in what makes the function's output zero, but similar in that we are setting a quadratic expression equal to a constant.  

Students work in their table groups while I check homework with the homework rubric. If students have questions about the homework or about the previous day's stations, I leave some time to answer these.  

Direct Instruction

20 minutes

In the previous day's lesson, we solved quadratic equations in order to find the x-intercepts of a quadratic function.  In this lesson, we generalize the process of solving a quadratic equation and work on developing fluency with this process. In the Aligning to Assessments: Do your Homework! reflection I discuss my planning for this lesson while showing some 2014 sample items released by the PARCC consortium that assess students' understanding of quadratic functions and equations.

I begin this part of the lesson by referring to the final exercise in the warm-up (see f(x)=g(x)), determining the value(s) of x that make the output of the quadratic function f(x)=(x-2)2-6 equal to the output of the linear function g(x)=-x+2. In the warm-up activity they solved this system graphically and I ask if anyone is able to come up with a way to solve it algebraically [MP2].  Students may recognize this as a system of equations, but if not I ask more questions to lead them into this line of thinking.  We will eventually end up setting f(x)=g(x) and therefore (x-2)^2-6=-x+2.  I explain that now we have a quadratic equation in one variable that we have tools to solve (factoring, completing the square, Quadratic Formula).  

We discuss the form this equation must be in so that factoring or the Quadratic Formula can be used and how to get it into that form.  We then solve the equation using one of these methods and discuss how to check the answer.  The final step in this initial exploration is to reconcile the answer we obtained algebraically with the answer we obtained graphically during the warm-up. 

After answering questions, I will provide students with another pair of functions that we can use to complete the process described above.  Although I am still anchoring the discussion of solving quadratic equations to quadratic functions, we are now working with non-linear systems as a bridge between quadratic functions and quadratic equations [MP7].

Finally, I present students with an equation like (x-3)- 6 = 2x + 5. I explain solving this equation is equivalent to asking what values of x would make the output of g(x) = (x-3)- 6 equal to the output of f(x) = 2x + 5.  I introduce equations this way to help students see the connection between functions and equations.   

We do several examples together of solving quadratic equations.  In each case I ask students whether factoring and using the Zero-Product Property, completing the square, or using the Quadratic Formula seems most efficient. Students will not agree on the best method (because there is no one best way!) so I ask for a volunteer to solve the equation one way while I solve it another way.  If we don't get the same answer we work together to figure out where the error is [MP3].

 

 

Practice and Reinforcement

40 minutes

To practice solving quadratic equations, I use Solving Quadratics 3 Ways so that students are expected to use each method.  Many students see the efficiency of using the Quadratic Formula and then begin using it for every quadratic equation but I stress the importance of staying flexible.  

Students work with their table partners to complete this task, asking their peers for help before turning to the answer key or to me [MP1].  I circulate during this time and use the 3-Cup System to let me know when students are getting frustrated.

Exit Ticket and Assignment

15 minutes

To determine if students are able to solve using all three methods, I send Quick Polls- Solving Quadratic Equations as an exit ticket through the Navigator system [MP5].  I ask students to solve the first equation by factoring, the second by completing the square and the third by using the Quadratic Formula.  I make note of which method needs the most reinforcement (likely completing the square) to that I can provide more practice when we get to imaginary numbers, later in the unit. 

For homework, I assign a puzzle.  I like to use puzzles when a specific skill (like solving quadratic equations) requires fluency [MP6].  One good source is a website assembled by a retired math Colorado math teacher, Mr. Plecher.