Classroom Resource

How Does it Change?

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

In this video lesson, students contrast visual patterns that show quadratic relationships with those that show linear and exponential relationships. To analyze the patterns, students generate tables of values, write expressions, and create graphs. They also encounter the term quadratic expression and learn that a quadratic relationship can be written using an expression with a squared term.

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

MA19.8A.4

Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity. [Algebra I with Probability, 4]

Unpacked Content

UP:MA19.8A.4

Vocabulary

  • Expression
  • Terms
  • Coefficient
  • Factors
  • linear expression
  • quadratic expression
  • Exponential expression

Knowledge

Students know:
  • Interpretations of parts of algebraic expressions such as terms, factors, and coefficients.

Skills

Students are able to:
  • Produce mathematical expressions that model given contexts.
  • Provide a context that a given mathematical expression accurately fits.
  • Explain the reasoning for selecting a particular algebraic expression by connecting the quantities in the expression to the physical situation that produced them.

Understanding

Students understand that:
  • Physical situations can be represented by algebraic expressions which combine numbers from the context, variables representing unknown quantities, and operations indicated by the context.
  • Different, but equivalent, algebraic expressions can be formed by approaching the context from a different perspective.
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.A1.4

Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Unpacked Content

UP:MA19.A1.4

Vocabulary

  • Linear expression
  • Quadratic expression
  • Exponential expression
  • Equivalent expressions

Knowledge

Students know:
  • How to recognize the parts of linear, quadratic and exponential expressions and what each part represents.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.
  • That one or more parts of an expression can be viewed as a single entity.

Skills

Students are able to:
  • Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
  • Interpret expressions in terms of a context.
  • View one or more parts of an expression as a single entity and determine the impact parts of the expression have in terms of the context.

Understanding

Students understand that:
  • Making connections among the parts of an expression reveals the roles of important mathematical features of a problem.
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
https://aptv.pbslearningmedia.org/resource/im20-math-ep2-62/how-does-it-change/

CR Resource Type

Audio/Video

Resource Provider

PBS
Accessibility

Accessibility

Video resources: includes closed captioning or subtitles
License

License Type

Custom

Custom License Type (URL)

https://aptv.pbslearningmedia.org/help/terms-of-use/

Comments or Notes

This resource has student practice problems and student task statements.

Approved Date

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