Completing the Square of a Quadratic Function

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SWBAT Complete the Square of a Quadratic Function to change between Standard Form and Vertex Form, as well as Solve the Quadratic Function,

Big Idea

To use Algebra Tiles to visualize the meaning of Completing the Square and to work through some advanced uses of Completing the Square.


15 minutes

To introduce students to how to Complete the Square of a Quadratic Function, I begin this lesson with an activity involving Algebra Tiles.  I post the following four problems along the top of the board:

1.  x^2 + 2x + _________

2.  x^2 + 4x + _________

3.  x^2 + 8x + _________

4.  x^2+12x + _________

Each pair of students will need a set of Algebra Tiles with one large square, 12 rectangles, and 36 small squares.  I tell the students that x squared is represented by the large square, each rectangle is an x because it is 1 by x, and the small squares are ones because they are one by one.  

I instruct students to set up the first problem with one large square for x squared and two rectangles for two x, and to find the number of small squares it takes to complete a square that is equal on all sides.  Then students proceed working through problems two through four with their table partner.  I have students sketch their answers on their own paper before moving to another problem.

Students should find the value of c above to be 1,4, 16, and 36 for problems one through four.  I have students draw sketches on the board to share with students on how they completed the square.  I have more than one student share if there are different responses.  

In number three, two different students shared the diagrams pictured below.

1. Diagram of Student One

2. Diagram of Student Two

I discuss with students that both diagrams are correct, but that Diagram Two shows a more visual representation of the Algebraic Method to Complete the Square that I will show next, in the Guided Notes.


Guided Notes

15 minutes

After introducing students to Completing the Square using Algebra Tiles, I then show students two uses of Completing the Square in the Guided Notes.  In the Guided Notes, I demonstrate for students how to Solve a Quadratic Equation by Completing the Square, and how to use Completing the Square to change from Standard Form to Vertex Form.  I model some of the examples in the Guided Notes in the video below.

Independent Practice

15 minutes

After working with students on the Guided Notes to Complete the Square, I assign an Independent Practice.  In this Independent Practice, I want students to work the five given problems in two different ways on a Comparison Chart that I copy for them.  I copy the chart front and back to provide enough space to work the five problems from the Independent Practice. 

On the left side of the Comparison Chart, students are to solve the Quadratic Function. On the right side of the chart, students are to change the same problem from Standard Form to Vertex Form.

I provide students about 15 minutes to work on the five problems, as I walk around to monitor their progress.  By using the Comparison Chart, I want students to recognize the difference of Completing the Square on one side of the equation to Change to Vertex Form, and using both sides of the equation to solve.

After about 15 minutes, I have students start checking their work on the Exit Slip.

Exit Slip

5 minutes

I have students work the Exit Slip for this lesson on their own paper with their table partner.  I have students check their work on the Independent Practice by using a graphing calculator.  If students have not completed the Independent Practice, I have them check only the problems that are complete.  

Students are to enter the original problem from the Independent Practice and Compare it to the Equation that the student put into Vertex Form.  These should be equivalent equations if the student did not make a mistake, and they should have an identical graph.  

The student may also check that the zeros are correct that they solved for in the left column and where the graph crosses the x-axis on the graphing calculator.  If students did not make any mistakes, their solutions should match the graph as well.

Students may also compare their work to their table partner's work for any corrections.  Students are to to make as many corrections as possible, and try to identify the reasons for the mistakes. Again, students may only be able to check a few of the problems before they leave class.

The Intervention of checking for mistakes using the calculator should help students be able to finish the assignment successfully.  I assign the problems that are not complete as homework to be handed in the following day.