Lesson 4 | Quadratic Functions and Solutions | 9th Grade Mathematics

Objective

Factor quadratic expressions using the greatest common factor. Demonstrate equivalence between expressions by multiplying polynomials.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

Seeing Structure in Expressions

A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

Foundational Standards

The foundational standards covered in this lesson

8.EE.A.1

Expressions and Equations

8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Identify a greatest common factor in a quadratic expression and re-write the expression as a product of the greatest common factor and a polynomial.
Multiply a monomial by a polynomial and multiply two binomials.
Determine if two polynomials are equivalent.
Identify quadratic expressions in standard form.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In Unit 6, Lesson 2, students were introduced to some vocabulary of polynomials. In Unit 6, Lesson 3, students multiplied polynomials, including multiplying two binomials, using the properties of exponents. This lesson reengages students in multiplying polynomials that result in quadratic expressions in the interest of identifying when two expressions are equivalent to one another.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

The area of a rectangle, in square units, is represented by $${ 3a^2+3a}$$ for some real number $$a$$.

Find the length and width of the rectangle. How many possible answers are there? List as many as you can find.

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 1 — Example 1

Algebra I > Module 4 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Match each expression on the left with an equivalent expression on the right.

a. $${(x+3)(x-8)}$$	i. $${x^2-5x}$$
b. $${(x-4)(x+6)}$$	ii. $${x^2+2x-24}$$
c. $${x(x-5)}$$	iii. $${x^2-5x}-24$$
d. $${2x(3x+4)}$$	iv. $${6x^2+3x+5}$$
e. $${x(6x+3)+5}$$	v. $${6x^2+8x}$$
f. $${(2x+1)(3x+5)}$$	vi. $${6x^2+13x+5}$$

Guiding Questions

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Problem 3

Consider the algebraic expressions below:

$${(n+2)^2-4}$$ and $${n^2+4n}$$

a. Use the figures below to illustrate why the expressions are equivalent.

b. Find some ways to algebraically verify the same result.

Guiding Questions

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References

Illustrative Mathematics Seeing Dots

Seeing Dots, accessed on July 2, 2018, 10:28 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Rewrite each expression as a product of the greatest common factor and a polynomial.

a. $${12x^2+8x-24}$$

b. $${-9x+81+15x^2}$$

c. $${10x^2-5 }$$

Problem 2

Determine if the three expressions below are equivalent. Show your work to justify your response.

$${(4x-6)(x+2)}$$ $${4x^2+2x-12}$$ $${4(x^2+x-3)}$$

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students identify the greatest common factor in a quadratic expression and factor the monomial out

EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 1 — Problem Set
Kuta Software Free Algebra 1 Worksheets Multiplying Polynomials — (Students may have seen this worksheet in Unit 6)
Illustrative Mathematics Equivalent Expressions

Lesson 3

Lesson 5

Lesson Map

Topic A: Features of Quadratic Functions

Compare quadratic, exponential, and linear functions represented as graphs, tables, and equations.

F.IF.B.4 F.LE.A.2

Identify key features of a quadratic function represented graphically. Graph a quadratic function from a table of values.

F.IF.B.4 F.IF.C.7.A

Calculate and compare the average rate of change for linear, exponential, and quadratic functions.

F.IF.B.4 F.IF.B.6 F.LE.A.3

Topic B: Factoring and Solutions of Quadratic Equations

Factor quadratic expressions using the greatest common factor. Demonstrate equivalence between expressions by multiplying polynomials.

A.APR.A.1 A.SSE.A.2 A.SSE.B.3.A

Identify solutions to quadratic equations using the zero product property (equations written in intercept form).

A.APR.B.3 F.IF.C.8.A

Factor quadratic equations and identify solutions (when leading coefficient is equal to 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor quadratic equations and identify solutions (when leading coefficient does not equal 1).

A.SSE.A.1.A A.SSE.B.3.A

Factor special cases of quadratic equations—difference of two squares.

A.SSE.A.1.A A.SSE.A.2 A.SSE.B.3.A

Factor special cases of quadratic equations—perfect square trinomials.

A.SSE.A.2 A.SSE.B.3.A

Solve quadratic equations by factoring. Compare solutions in different representations (graph, equation, and table).

A.SSE.B.3.A F.IF.C.8.A F.IF.C.9

Solve quadratic equations by taking square roots.

A.REI.B.4.B

Graph quadratic functions using $${x-}$$intercepts and vertex.

A.APR.B.3 F.IF.B.4 F.IF.C.7.A F.IF.C.8.A

Topic C: Interpreting Solutions of Quadratic Functions in Context

Interpret quadratic solutions in context.

A.CED.A.1 F.IF.B.4 F.IF.B.5 F.IF.C.8.A

Quadratic Functions and Solutions

Objective

Common Core Standards

Core Standards

Arithmetic with Polynomials and Rational Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Foundational Standards

Expressions and Equations

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

Problem 1

Problem 2

Additional Practice

Lesson Map

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