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Solving Quadratic Equations by Factoring

Lesson Plan

Solving Quadratic Equations by Factoring

Objectives

Students will use factoring as a method to solve quadratic functions. Students will:

  • factor trinomials of various forms:

  • ax² + bx + c = 0, where a = 1

  • ax² + bx + c = 0, where a >1

  • ax² + bx + c = 0, where a, b, and c have a greatest common factor (GCF)

  • apply the Zero Product Property to solve equations of the form(ax + b)(cx + d) = 0

  • obtain solutions to factorable quadratic equations of the form

  • ax² + bx + c = 0, where a = 1

  • ax² + bx + c = 0, where a >1

  • ax² + bx + c = 0, where a, b, and c have a GCF

Essential Questions

  • How can we show that algebraic properties and processes are extensions of arithmetic properties and processes, and how can we use algebraic properties and processes to solve problems?

Vocabulary

  • Binomial: A polynomial with two terms. [IS.1 - Preparation]

  • Trinomial: A polynomial with three terms.

  • Greatest Common Factor: The largest factor that two or more numbers have in common.

  • Factor: A whole number that divides evenly into another number.

  • Zero of a Function: The value of the argument for which the function is zero. Also x-intercept and root of an equation.

Duration

90–120 minutes [IS.2 - All Students]

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

Formative Assessment

  • View
    • Observations during class lessons, discussions, and activities should focus on specific products that students create, particularly the two binomial factors of the trinomial. Require students to multiply the two binomial factors using FOIL and compare the resulting trinomial to the original prompt.

    • Lesson 2 Student Document (M-A1-1-2_Lesson 2 Student Document.doc) requires students to use the zero property of multiplication and evaluates their level of understanding of the logical necessity of a zero product, if one of the factors is equal to zero.

Suggested Instructional Supports

  • View
    Active Engagement, Modeling, Explicit Instruction
    W:

    This lesson helps students to develop skills in solving quadratic equations by factoring and provides them with useful techniques for factoring and for understanding the rationale that supports finding solutions. The lesson includes recognizing and using trinomials in various forms.

    H:
    The example of  x² + x - 6 shows students the relationship between the zeros of the function, the roots of the equation, and the constant terms of the binomial factors of the trinomial. By illustrating these connections, students can see how general solutions are possible.
    E:

    The Zero Product Property is an elementary concept that is familiar to students. In applying it to binomial factors, they can use the property as a tool in a way that has not previously been represented. Students are able to recognize that the property applies not only to monomials, but also to binomials, and is applicable for all real numbers.

    R:

    The think-pair-share activity presents students with representations of all three types of trinomial factoring. By attempting solutions individually, students gain an immediate sense of how well they understand the techniques. In sharing their solution methods and results with partners, they can expand their understanding by seeing different solutions and correcting their own and their partners’ errors.

    E:

    The Solve by Factoring Worksheet requires students to classify as well as factor the trinomials presented. The classification tasks engage students in reviewing their understanding of the individual characteristics of the three types of trinomials. This activity encourages them to use the specific traits of the trinomial to find the unique binomial factors.

    T:

    Students who find the factoring of trinomials a challenging operation will get some satisfaction in the application of the Zero Product Property. The property is easy to understand and use, and makes the steps to solving quadratic equations by identifying and deconstructing binomials more accessible. Students with the knowledge and skills to factor trinomials of higher difficulty will also appreciate this basic technique.

    O:

    This lesson is organized so that students can build upon prior knowledge of factoring and solving linear equations to solve quadratic equations. Students should be introduced, through teacher instruction, to the concepts and procedures for solving quadratics by factoring. During this time students should be given time to individually practice these processes and for discussion with classmates. Students should receive immediate feedback on their work during the activities so they are on track to be successful with homework assignments. The student document can also be used to help students stay organized during classroom instruction.

     

    IS.1 - Preparation
    Consider word walls and different strategies to ensure that the vocabulary is constantly used during the lesson.  
    IS.2 - All Students
    Consider pre-teaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson, based on the results of formative assessment, consider the pacing of the lesson to be flexible based on the needs of the students. Also consider reteaching and/or review both during and after the lesson as necessary.  
    IS.3 - Struggling Learners
    Consider pre-teaching the Zero-product property and factoring.  Strugglling students may need more direct instruction with learning the  concepts critical to this lesson. Throughout the lesson, based on the results of formative assessment, consider the pacing of the lesson to be flexible based on the needs of the students. Also consider reteaching and/or review both during and after the lesson as necessary.  
    IS.4 - All Students
    Consider modeling and doing think alouds to help students understand the problem solving process.  

Instructional Procedures

  • View

    After this lesson, students will know how to solve quadratic equations using factoring. Students are learning how to solve quadratic equations because there are many real-world situations that can be modeled by quadratic equations. Students should have prior knowledge of factoring trinomials. Students will understand that there are two solutions to a quadratic equation and why this is different from solving linear equations. They will also find that in dealing with real-life scenarios not all solutions make sense. They should be able to recognize the solution(s) that fit. Students will be able to check their work by substituting their solutions into the equation.

    Yesterday we looked at quadratic equations and many types of situations that can be modeled using them. One of the things we discussed was the zeros of quadratic equations, which are the solutions. On the graphs we looked at, we noted that the zeros were located where the graph crossed the x-axis. Let’s take a moment to recall one of these examples.” Display the following for students:

    l2-01example1.PNG

    Now solving this equation is rather simple when you can find the zeros right in the graph, but what if you do not have a graph or the zeros are not easy to calculate from the graph? Today, we are going to discuss an algebraic approach that can be used to solve problems like this, as well as story problems that can be modeled using quadratic equations.”

    The following notes, models, and examples should be shown to students to explain the lesson. Visual and auditory learners will be able to see and/or hear the process that is involved in solving quadratics by factoring.

    Zero-Product Property

    For any a and b, if ab = 0, then either a = 0, b = 0, or a and b equal 0.

    l2-02zeroproduct.PNG

     

    l2-03zeroproduct.PNG

    Solving Equations by Factoring [IS.3 - Struggling Learners]

    Step 1: Make the equation equal to _0_.

    Step 2: ___Factor___ the trinomial.

    Step 3: Apply the ___Zero Product Property__ (set each factor equal to __0_; then ____solve___).

     

     

    l2-04type1eqs.PNG

    Students should have an understanding of factoring trinomials from previous instruction. Depending on the skill level of your students, you may have to vary how much review of factoring trinomials you provide.


    l2-05type2plusfactors.PNG

    Type 3: Equations of the form ax² + bx + c = 0 with a GCF

    Students should be instructed to factor out a GCF before beginning the rest of the solving process as in type 1 and 2.

    Note: At first, many of these equations look as if they are type 2 equations; yet, after factoring a GCF, the problem may reveal a type 1 equation. If the GCF is not factored out of the equation before beginning the factoring process, the solutions will be the same but the factored forms will be different. (This is demonstrated below.)


    l2-06type3.PNG

    Without factoring out the GCF in problem 1: (3x + 6)(x – 5) = 0, factored, but not completely, since 3 can be factored out of 3x + 6. But solving 3x + 6 = 0 gives a solution of −2, the same as in Example 1. This relationship is important because when students are asked to factor something completely, the answer of (3x + 6)(x – 5) would not be correct since it is not completely factored. A similar situation can be shown with Example 2.

     

    Activity 1

    Think-pair-share (interpersonal and verbal intelligences): Place a problem on the board and have students individually work out the problem on paper. After 3 to 5 minutes, have students pair up to discuss their answers. Direct students to discuss any errors and help each other decide on a correct answer. Then have a class discussion on the correct answer and anything students noticed during their discussions such as common errors, arithmetic mistakes, procedural mistakes, etc. You may have a student display the process for the class on the board.

    Sample problems for students:

     

    l2-07activity1.PNG

     

    Activity 2: Real-Life Scenarios [IS.4 - All Students]

    Problem 1: The length of a rectangle is 3 more inches than its width. Find the dimensions of the rectangle with an area of 108 square inches.

    l2-08square.PNG

     

    1. This problem uses a type 1 scenario and also uses the concepts of area of a rectangle and the distributive property.

    2. It is important to explain at this point that in applied situations not all solutions make sense. Have a discussion with students about which answer works and why. (−12 is a solution but does not make sense because a length cannot be negative, thus making 9 the only possible solution to the width).

    Solution: width = 9 in., length = 12 in.

    Problem 2: The length and width of an 8-inch by 12-inch photograph are reduced by the same amount to make a new photograph with an area that is 1/3 of the original. By how many inches will the dimensions of the photograph have to be reduced?

    l2-09rect.PNG

    1. This problem uses a type 1 scenario, the concept of area of a rectangle, and using FOIL (First Outside Inside Last when multiplying two binomials).

    2. For situation 2 there are two possible solutions that are both positive (16 and 4), but discuss with students which one makes sense in the given situation. Since the possible solutions represent the value that is deducted from each side of the photograph, the only answer that would work is 4. An answer of 16 is not reasonable because it is not possible to cut 16 inches off a photograph that only has 12 inches on one side and 8 on the other.

    Solution: Reduce the dimensions of the photograph by 4 inches.

    Give students the following problems to work on independently for about 10 to 15 minutes. Have students label the type of each problem before beginning to work on it. After independent work time, have students pair up to compare and discuss answers. As students are finishing, have some students write the work for each problem on the board and then discuss the problems as a class. Hand out the Solving Quadratics by Factoring Worksheet (M-A1-1-2_Solving Quadratics by Factoring Worksheet.doc), as desired, for students to work on. (This resource is good as a day 2 follow-up lesson.)

     

    l2-10solutions.PNG

     


    Extension:

    • Routine: Use the Lesson 2 Student Document (M-A1-1-2_Lesson 2 Student Document.doc) to give students a structured format for taking notes. Provide this resource to students, as needed, to allow them to keep more organized and structured notes.

    • Have students reflect on factoring trinomials and whether they remember the process (intrapersonal). This should be done prior to going through the examples of solving quadratics by factoring. Display two problems (one at a time) and have students work through the factoring process on a white board (or piece of paper). Have students hold up their work when finished and make corrections and adjust teaching where needed to meet the needs of your students.

    • Alternate Method: For Activity 1, you can do the activity once after presenting all three situations or one situation at a time (after each of the methods), having students change partners for each situation. This approach might allow students more reflection and discussion on each of the methods, if time permits.

    • Visual Learners: For Activity 2, use the Problem Solving Graphic Organizer (M-A1-1-2_Problem Solving Graphic Organizer.docx and M-A1-1-2_Problem Solving Graphic Organizer Blank.docx) to help students organize their word- problem solving techniques more efficiently. This can help many students, especially those who need their work to be more visual and organized. There are two resources: one with sequential steps and ideas already filled in, and another that has a blank flow chart. Use whichever document fits the needs of your students.

    • Assign to students an Internet word-problem activity (see the Related Resources section). This activity will help build students’ understanding and ability to read and evaluate important information from a word problem. This is a great way to give students more practice with word problems.

Related Instructional Videos

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DRAFT 10/21/2010
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