##
* *Reflection: Problem-based Approaches
Completing the Square of a Quadratic Function - Section 2: Guided Notes

Providing students with an example as an Entry Document instead of a set of problems after the Warm Up increases relevance and interest of the students. A good real world problem that uses completing the square is a Quadratic Function Problem called Throwing Baseballs. This problem comes from the following website:

https://www.illustrativemathematics.org/content-standards/tasks/1279

First, hand the problem to the students to problem only to read, making sure to not include the solutions that are listed on the problem above. Have the students make a know and a need to know list on their paper and then share out with the class.

Some responses of students for the know list are:

- Given two Quadratic Functions
- One function is given as an equation
- One function is given as a graph
- Both boys claim that their baseball went higher than the other

Some responses of students for the need to know list are:

- Find the maximum heights to determine who is right
- How long is each baseball airborne?
- How can we construct a graph of Brett's throw from the equation on the same graph as Andre's throw
- How can we use the graph to confirm our explanations in part a and part b

After students share out with the class and both lists are made, have students generate ideas on Next Steps for a third list.

Some responses for Next Steps include, using the Vertex Formula, Completing the Square to put the function in Vertex Form, Quadratic Formula, and Comparing the Graphs.

Let the students begin working with their table partner using the method of their choice. Provide students with enough time to work on the problem, about 15 or 20 minutes. Then select a few students to share out different methods under the document camera. If none of the students share completing the square, then model this method for them. It is difficult for students to change from Standard Form to Vertex form without several examples and modeling. However, Completing the Square is heavily emphasized in Algebra I, and so is Quadratics.

*Providing Relevance with Problem Based Learning*

*Problem-based Approaches: Providing Relevance with Problem Based Learning*

# Completing the Square of a Quadratic Function

Lesson 6 of 10

## Objective: 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.

*50 minutes*

#### Introduction

*15 min*

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.

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.

*expand content*

#### Guided Notes

*15 min*

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.

#### Resources

*expand content*

#### Independent Practice

*15 min*

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.

*expand content*

#### Exit Slip

*5 min*

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.

*expand content*

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- UNIT 1: Introduction to Functions
- UNIT 2: Expressions, Equations, and Inequalities
- UNIT 3: Linear Functions
- UNIT 4: Systems of Equations
- UNIT 5: Radical Expressions, Equations, and Rational Exponents
- UNIT 6: Exponential Functions
- UNIT 7: Polynomial Operations and Applications
- UNIT 8: Quadratic Functions
- UNIT 9: Statistics

- LESSON 1: Introduction to Quadratic Functions
- LESSON 2: Graphing Quadratic Functions in Standard Form f(x)=ax^2+bx+c.
- LESSON 3: Graphing Quadratic Functions in Vertex Form f(x)=a(x-h)^2 + k.
- LESSON 4: Graphing Quadratic Functions in Intercept Form f(x)= a(x-p)(x-q)
- LESSON 5: Comparing and Graphing Quadratic Functions in Different Forms
- LESSON 6: Completing the Square of a Quadratic Function
- LESSON 7: The Quadratic Formula in Bits and Pieces
- LESSON 8: Solving Quadratic Functions Using the Quadratic Formula
- LESSON 9: Real World Applications of Quadratic Functions
- LESSON 10: Analyzing Polynomial Functions