Classroom Resource

Completing the Square (Part 1): Algebra 1, Episode 20: Unit 7, Lesson 12 | Illustrative Math

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

Previously in this video series, students saw that a squared expression of the form (x + n)² is equivalent to x² + 2nx + n². This means that, when written in standard form ax² + bx + c (where a is 1), b is equal to 2n and c is equal to n². Here, students begin to reason the other way around. They recognize that if ax² + bx + c is a perfect square, then the value being squared to get c is half of b, or (b/2)². Students use this insight to build perfect squares, which they then use to solve quadratic equations.

Students learn that if we rearrange and rewrite the expression on one side of a quadratic equation to be a perfect square, that is if we complete the square, we can find the solutions of the equation.

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.A1.5

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).
Link to Resource
https://aptv.pbslearningmedia.org/resource/im20-math-ep20-712/completing-the-sq…

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/#.YD-bCNhKiUk

Approved Date

habradley
ALSDE LOGO

This site may contain links to other websites or content belonging to or originating from third parties or links to websites and features in banners or other advertising. Such external links are not investigated, monitored, or checked for accuracy, adequacy, validity, reliability, availability, or completeness by us. WE DO NOT WARRANT, ENDORSE, GUARANTEE, OR ASSUME RESPONSIBILITY FOR THE ACCURACY OR RELIABILITY OF ANY INFORMATION OFFERED BY THIRD-PARTY WEBSITES LINKED THROUGH THIS SITE.
Alabama Learning Exchange © 2002-2023 by Alabama State Department of Education is licensed under Attribution-NonCommercial-ShareAlike 4.0 International

Design by Adaptive Theme