Learning Resource Type

Classroom Resource

Rewriting Quadratic Expressions in Factored Form (Part 1): Algebra 1, Episode 15: Unit 7, Lesson 6 | Illustrative Math

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

In this video lesson, students begin to rewrite quadratic expressions from standard to factored form.

Students relate the numbers in the factored form to the coefficients of the terms in standard form, looking for a structure that can be used to go in reverse—from standard form to factored form (MP7).

(This lesson only looks at expressions of the form (x + m)(x + n) and (x – m)(x – n) where m and n are positive.)

    Mathematics (2019) Grade(s): 8 - Grade 8 Accelerated

    MA19.8A.6

    Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

    Unpacked Content

    UP:MA19.8A.6

    Vocabulary

    • Function
    • zero of a function
    • Roots
    • parabola
    • vertex form of a quadratic expression
    • Minimum and maximum value
    • Axis of symmetry
    • Completing the square
    • Exponential growth and decay

    Knowledge

    Students know:
    • The vertex form of a quadratic expression asf (x) = a(x
    • h)2 + k, where (h, k) is the vertex of the parabola.
    • Techniques for generating equivalent forms of an algebraic expression including factoring and completing the square for quadratic expressions and using properties of exponents,
    • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

    Skills

    Students are able to:
    • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures,
    • Factor quadratic expressions with leading coefficient of one
    • Complete the square in quadratic expressions.

    Understanding

    Students understand that:
    • An expression may be written in various equivalent forms.
    • Some forms of the expression are more beneficial for revealing key properties of the function.
    Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

    MA19.A1.5

    Use the structure of an expression to identify ways to rewrite it.

    Unpacked Content

    UP:MA19.A1.5

    Vocabulary

    • Terms
    • Linear expressions
    • Equivalent expressions
    • Difference of two squares
    • Factor
    • Difference of squares

    Knowledge

    Students know:
    • Algebraic properties.
    • When one form of an algebraic expression is more useful than an equivalent form of that same expression.

    Skills

    Students are able to:
    • Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.

    Understanding

    Students understand that:
    • Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions.
    Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

    MA19.A1.6

    Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

    Unpacked Content

    UP:MA19.A1.6

    Vocabulary

    • Quadratic expression
    • Zeros
    • Complete the square
    • Roots
    • Zeros
    • Solutions
    • x-intercepts
    • Maximum value
    • Minimum value
    • Factor
    • Roots
    • Exponents
    • Equivalent form
    • Vertex form of a quadratic expression

    Knowledge

    Students know:
    • Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
    • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

    Skills

    Students are able to:
    • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
    • Factor quadratic expressions.
    • Complete the square in quadratic expressions.
    • Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.

    Understanding

    Students understand that:
    • Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.
    Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

    MA19.A1.9

    Select an appropriate method to solve a quadratic equation in one variable.

    Unpacked Content

    UP:MA19.A1.9

    Vocabulary

    • Completing the square
    • Quadratic equations
    • Quadratic formula
    • Inspection
    • Imaginary numbers
    • Binomials
    • Trinomials

    Knowledge

    Students know:
    • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
    • The method for completing the square.
    • Notational methods for expressing complex numbers.
    • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.

    Skills

    Students are able to:
    • Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
    • Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
    • Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
    • Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
    • Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

    Understanding

    Students understand that:
    • Solutions to a quadratic equation must make the original equation true and this should be verified.
    • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
    • Different procedures for solving quadratic equations are necessary under different conditions.
    • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
    • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.11

    Solve quadratic equations with real coefficients that have complex solutions.

    Unpacked Content

    UP:MA19.A2.11

    Vocabulary

    • Complex solution
    • Quadratic equation
    • Real coefficients

    Knowledge

    Students know:
    • strategies for solving quadratic equations

    Skills

    Students are able to:
    • apply the quadratic equation.
    • provide solutions in complex form.

    Understanding

    Students understand that:
    • all quadratic equations have two solutions: real or imaginary.
    • Some contextual situations are better suited to quadratic solutions.
    Link to Resource

    CR Resource Type

    Audio/Video

    Resource Provider

    PBS
    Accessibility

    Accessibility

    Video resources: includes closed captioning or subtitles
    License

    License Type

    Custom
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