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Completing the Square

Lesson Plan

Completing the Square

Objectives

This lesson helps students understand how the vertex form can be more useful than the standard form. Students will:

  • use Algebra TilesTM to complete-the-square or convert from standard form to vertex form.

  • be able to convert from vertex form to standard form.

  • use quadratic equations to solve real-world application problems.

Essential Questions

  • How can we determine if a real-world situation should be represented by a quadratic, polynomial, or exponential function?

  • How do quadratic equations and their graphs and/or tables help us interpret events that occur in the world around us?

Vocabulary

  • Completing the square: [IS.1 - Preparation] A technique for finding the roots of quadratic equations that uses the terms to substitute for x2 + bx + c, resulting in a purely quadratic equation with no linear term (y = x2 +bx + c).

  • Vertex Form: A form for a quadratic function; y = a(x – h)2 + k, where the coordinates of the vertex are (h, k).

  • Minimum: The point(s) with the least y-value on the graph of a function; on a parabola that opens upward, the minimum occurs only at the vertex.

  • Maximum: The point(s) with the greatest y-value on the graph of a function; on a parabola that opens downward, the maximum occurs only at the vertex.

Duration

60–90 minutes [IS.2 - All Students]

Prerequisite Skills

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Materials

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Formative Assessment

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    • Teacher observations during group activities should focus on requiring participation by all students, [IS.16 - All Students] challenging individual responses to each component of the activity, and making immediate corrections and remediation of errors of understanding and execution.

    • By completing the Exit Ticket activity (M-A2-2-2_Lesson 2 Exit Ticket.doc and M-A2-2-2_Lesson 2 Exit Ticket KEY.doc), [IS.17 - All Students] students show their understanding and ability to represent vertex form and translate between vertex form and standard form.

Suggested Instructional Supports

  • View
    Active Engagement, Modeling, Explicit Instruction

    [IS.15 - All Students]

    W:  
    In this lesson, students will understand and be able to represent the utility of completing the square both as a technique for solving quadratic equations and as a way of visualizing the relationship between the behavior of quadratic functions and the form of their graphs.
    H:  
    In this lesson, a geometric and visual representation of the quadratic function y = x2 + 6x + 8 challenges students by requiring them to dissect its components and reassemble them as squares and rectangles. Algebra TilesTM provide the model for exploration and visualization.
    E:  
    In Activity 1 with the Algebra TilesTM, students can observe the dimensions of representations of a variety of quadratic functions with a range of b and c terms and a terms with both positive and negative values.
    R:  
    The paired tasks in Activity 5 gives students the opportunity to make their own examples of quadratic functions, translate them between vertex form and standard form, and exchange their results with a partner. By testing themselves and inviting an evaluation by a peer, they can adjust and refine their understanding of the quadratic functions.
    E:  
    The Lesson 2 Exit Ticket evaluates students’ understanding of the relationship between the a, b, and c terms of the quadratic function, the shape of the parabola, location of the vertex, and the dimensions of the Algebra TilesTM.
    T:  
    For students who have difficulty visualizing how the Algebra Tiles represent the characteristics of the quadratic functions, use the form y = ax2 until they become comfortable with the representation of the x2 term. For more advanced students, ask them to visualize and sketch what the rectangles would look like if the a term were a rational number such as .
    O:  
    This lesson begins with a problem that we introduced to students in the first lesson. The task of putting puzzles together grabs their attention. Students have been working extensively with the vertex form and now they get to see why they needed to learn about it. Students will develop an understanding of the usefulness of the vertex form and how it can be applied in real-world situations.

     

    IS.1 - Preparation
    Consider reviewing and/or reteaching (possibly through the use of a graphic organizer (e.g. a Frayer Model, Verbal Visual Word Association, Concept Circles)) all of the vocabulary needed for this lesson.  
    IS.2 - All Students
    Consider preteaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based upon the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for reteaching and/or review both during and after the lesson as necessary.  
    IS.3 - All Students
    Consider modifying the numbers associated with this problem so that algebra tiles can be used to model a solution before using a more abstract strategy.  
    IS.4 - Struggling Learners
    Consider having struggling students preview the topic of completing the square and quadratic functions on www.khanacademy.org. Here you can search for lessons provided by the teacher. You may also consider having the student view the Khan Academy to review for homework.  
    IS.5 - All Students
    Consider providing students the opportunity to work in pairs so that ideas can easily be shared as students learn how to complete the square.  
    IS.6 - All Students
    Consider providing a graphic organizer that shows pictures of the algebra tiles in a completing the square final form and a pictorial representation plus an abstract representation of the same situation. Consider making the connections between all three forms of representation with respect to completing the square and the vertex form of a quadratic function explicit.  
    IS.7 - All Students
    Consider allowing students to work in pairs so that student think alouds can be used.  
    IS.8 - Struggling Learners
    Consider providing struggling students graphical representations or access to a graphing calculator  (with encouragement to use the graphing calculator ) so that students aren’t just relying on the abstract form of the function to determine the vertex. Encourage struggling students to use Algebra tiles and then sequence to pictorial representations before proceeding to the abstract strategies.  
    IS.9 - All Students
    Consider using a graphic organizer which contains standard form, vertex form and graphs of each of the functions along with space to note patterns. Pre-plan the assessing and advancing questions used to support students as they discover the patterns.  
    IS.10 - All Students
    Consider using an I Do, We Do, You Do sequence of instruction for these exercises.  
    IS.11 - All Students
    Consider establishing groups before instruction takes place so that you have control over who works together. The group’s dynamics are a factor in successful collaboration.
    IS.12 - All Students
    Consider pairing students before instruction so that you have control over who works together. This helps increase the likelihood that the pairs will contribute equally to the learning.   
    IS.13 - All Students
    Consider creating a graphic organizer to aid in student notetaking.  
    IS.14 - Struggling Learners
    Consider reviewing or reteaching factoring through the use of Algebra Tiles before struggling students complete this activity.  
    IS.15 - All Students
    Also consider Think-Pair-Share, Random Reporter, Think Alouds, Math Journal, and use of graphic organizers. Information on Think-Pair-Share and  Random Reporter can be found on the SAS website at, https://www.pdesas.org/Main/Instruction  
    IS.16 - All Students
    Prior to teaching this lesson, consider the prior knowledge of struggling students as well as misconceptions that are likely to surface. Have a game plan for how to correct misconceptions and connect current concepts to prior knowledge.  
    IS.17 - All Students
    Consider viewing the publication, Teachers’Desk Reference: Essential Practices for Effective Mathematics Instruction in order to review the sections on formative assessment as well as assessing and advancing questions. This publication can be found at: http://www.pattan.net/category/Resources/PaTTAN%20Publications/Browse/Single/?id=4e1f51d3150ba09c384e0000  

Instructional Procedures

  • View

    After this lesson, students will be able to convert quadratic functions from standard form to vertex form using the “completing-the-square” technique. In the previous lessons, students learned how useful the vertex form can be, but not all quadratic functions are written in vertex form. Students will understand that quadratic relationships happen all around us and that the vertex is important to know, especially if the situation is talking about minimizing costs or maximizing revenue. Note that maximum and minimum value occurs at the vertex. The maximum or minimum value is the y-coordinate.

    “Who likes putting puzzles together? Remember the first day of this unit when I asked you to find the vertex of y = x2 + 6x + 8? Well we can find the vertex by putting together a little puzzle.”

    Hand out a set of Algebra TilesTM to each student. [IS.5 - All Students] Students will be working with their tiles on their desk. You may also use the template provided (M-A2-2-1_Algebra Tiles Template.doc).

    “We will be using Algebra TilesTM to make a square. We are going to say that x2 is represented by a square with side length x. We are now going to say that x is represented by a rectangle with length x and width 1. Then we have a single unit with side length 1.”

    “Each part of the quadratic equation tells us how many tiles to use. We need 1 x2, 6 x rectangles, and 8 single units. Let’s count them out.”

    “The goal now is to make a square with the tiles that we have. Begin by putting the x2 square in the top left corner.”

    “Then how would we split up 6 x rectangles evenly?” Students should say 3 and 3.

    “Line up the rectangles: three rectangles along the right side of the x2 square and three rectangles below the x2 square.”

    “Now begin filling in the bottom right-hand corner of the square with our single unit pieces.”

    The two graphics provided are illustrations of the same equation; the only difference is that the one on the right has the outside dimensions labeled.

    “What are the dimensions of this square we made?” Students should come up with x + 3 by x + 3. Students may also note that they were not able to make a whole square.

    “How can we simplify x + 3 by x + 3?” Students should say (x + 3)2.

    “Now did we make a full square? We are missing one unit in the corner, so the equation can’t be (x + 3)2. What do we need to do to the equation?” Students should say minus/subtract 1.

    “Does anyone recognize (x + 3)2 − 1? What form is the equation in? What is the vertex?” [IS.6 - All Students] The equation is now in vertex form and the vertex is at (−3, −1).

    “So any time we are missing a piece, we need to subtract it. What would the equation be in vertex form if we had y = x2 + 6x + 11?” Students should add 3 more units to their square. “Now the equation in vertex form is going to be y = (x + 3)2 + 2, the same as before, except three was added to the k value: −1 + 3 = 2.”

    The two graphics provided are illustrations of the same equation; the only difference is that the one on the right has the outside dimensions labeled.

    “So if we have too many pieces, we have to add that extra amount to the equation. This technique we use to convert an equation from standard form to vertex form is called completing-the-square.”

    Activity 1: Using Algebra TilesTM

    Place the following equations on the board for students to practice using the Algebra TilesTM. They should use their tiles to find the vertex form of the parabola and to determine the vertex. [IS.7 - All Students] Tell students to do one at a time and that they cannot go on to the next one until you have approved their work.



    1. y = x2 + 8x + 2 [y = (x + 4)2 − 14; vertex (−4, −14)] [IS.8 - Struggling Learners]

    2. y = x2 + 2x + 3 [y = (x + 1)2 + 2; vertex (−1, 2)]

    3. y = x2 + 10x + 12 [y = (x + 5)2 − 13; vertex (−5, −13)]

    4. y = x2 + 4x + 11 [y = (x + 2)2 + 7; vertex (−2, 7)]

    5. y = x2 − 6x + 4 [y = (x − 3)2 − 5; vertex (3, −5)]



    “Has anyone noticed a pattern? [IS.9 - All Students] What would happen if I had an odd number of x tiles? We are not always going to have the tiles available, so we may have to draw a picture. We will practice this scenario in the next activity.”

    Activity 2: Think-Pair-Share

    “With your partner, convert the following standard form equations to vertex form by using the formula. When you are done, get together with another pair and check your work.”

    Standard form: y = ax2 + bx + c

    Formula to convert to vertex form:



    1. y = x2 + 5x + 6 [y = (x + 2.5)2 − .25; vertex (−2.5, −.25)]

    Illustrate item 1 by showing steps to

    [IS.10 - All Students]

     

    2. y = x2 + 7x − 4 [y = (x + 3.5)2 − 16.25; vertex (−3.5, −16.25)]

    3. y = x2 − 9x + 25 [y = (x − 4.5)2 + 4.75; vertex (4.5, 4.75)]



    Some students may not like using the tiles, so show them the formula. Students should copy this formula in their notes. See other references in various texts on completing the square for examples.

    Activity 3: Group Work

    Separate the class into groups of four and distribute the Group Word Problems worksheet (M-A2-2-2_Group Word Problems.doc and M-A2-2-2_Group Word Problems KEY.doc). [IS.11 - All Students] Note that the maximum and minimum occurs at the vertex. The maximum and minimum value is the y-coordinate.

    Activity 4: Think-Pair-Share

    “What do you think we would need to do to convert a vertex form equation back into standard form? With a partner, brainstorm how you would convert the quadratic equation y = (x + 4)2 – 5 into standard form.” Tell students that it does not entail a square or a formula. They already have the tools to change this equation. Tell them first to think about this on their own, partner with another student, and then share their ideas with the class.

    “It is important to use the order of operations. What does it mean to square something? What does it mean to combine like terms and to simplify?”

    Show students how easy it is to convert from vertex to standard form.

    (x + 4)2 − 5 = (x + 4)(x + 4) − 5 = x2 + 4x + 4x + 16 − 5 = x2 + 8x + 11

    Give students the following equations to convert to standard form.

    1. y = (x − 3)2 − 17 [y = x2 − 6x − 8]

    2. y = (x + 7)2 − 40 [y = x2 + 14x + 9]

    3. y = (x + 8)2 − 65 [y = x2 + 16x − 1]

    4. y = (x − 6)2 − 1 [y = x2 − 12x + 35]

    Activity 5: Pair Work

    “Individually, you are going to write two equations in standard form and two equations in vertex form. [IS.12 - All Students] Convert your equations to the other form and keep your answers. You are then going to give the four equations you wrote to your partner. You and your partner will then convert the equations to the other form. When both of you are finished, check each other’s work.”

    Hand out the Lesson 2 Exit Ticket (M-A2-2-2_Lesson 2 Exit Ticket.doc and M-A2-2-2_Lesson 2 Exit Ticket KEY.doc) to evaluate students’ understanding.

    Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.

    Routine: Group and partner work is used throughout so that students can help each other. Emphasis should be placed on communicating mathematical ideas with the specific vocabulary words appropriate to the concepts. The lesson requires accurate note-taking skills to enhance the learning experience while creating a useful resource (notes). [IS.13 - All Students]

    “How do we complete the square if there is a number in front of x2? We can either use the tiles or factoring.”

    Example: y = 2x2 + 4x + 6

    “We will try to make 2 squares since 2 is the number in front of x2.”

    “Since we have 2 squares with dimensions x + 1 by x + 1, we can write that as 2(x + 1)2 and then we have 4 single units. So the equation is y = 2(x + 1)2 + 4. The vertex is at (−1, 4) and it has a stretch factor of 2.”

    “Another way we could have changed the equation from standard to vertex form would have been to use factoring. [IS.14 - Struggling Learners] Factor out the number in front of x2.”

    Example: y = 2x2 + 4x + 6

    y = 2(x2 + 2x + 3)
    “Complete the square using only what’s in parentheses (x2 + 2x + 3).”
    (x + 1)2 + 2

    “Now distribute the 2 that we factored out.”

    2(x + 1)2 + 4

    Extension:

    • Students who need additional opportunity for learning can use the tiles or factoring to solve the following problems:

    1. y = 3x2 + 12x − 18 [y = 3(x + 2)2 – 30]

    2. y = −2x2 − 10x + 8 [y = -2(x + 2.5)2 + 20.5]

    3. y = −x2 + 6x − 3 [y = -(x – 3)2 + 6]

    4. y = 4x2 − 16x − 12 [y = 4(x – 2)2 – 28)]

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