Learning Resource Type

Classroom Resource

Rewriting Quadratic Expressions in Factored Form (Part 3): Algebra 1, Episode 17: Unit 7, Lesson 8 | Illustrative Math

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form.

Through repeated reasoning, students are able to generalize the equivalence of these two forms: (x + m)(x – m) and x² – m² (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other.

Students also consider why a difference of two squares (such as x² – 25) can be written in factored form, but a sum of two squares (such as x² + 25) cannot be, even though both are quadratic expressions with no linear term.

    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): 8 - Grade 8 Accelerated

    MA19.8A.11

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

    Unpacked Content

    UP:MA19.8A.11

    Vocabulary

    • quadratic equation
    • Square root
    • Factoring
    • Completing the square
    • quadratic formula
    • Derive
    • Real numbers
    • Imaginary numbers
    • Complex numbers

    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.
    • 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:
    • Take the square root of both sides of an equation.
    • Factor quadratic expressions in the form x2+bx+c where the leading coefficient is one.
    • Use the factored form to find zeros of the function.
    • Complete the square.
    • Use the quadratic formula to find solutions to quadratic equations.
    • Manipulate equations to rewrite them into other forms.

    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 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).
    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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