Curriculum  /  Math  /  9th Grade  /  Unit 7: Quadratic Functions and Solutions  /  Lesson 4

Quadratic Functions and Solutions

Lesson 4

Math

Unit 7

9th Grade

Lesson 4 of 13

Objective

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

Common Core Standards

Core Standards

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

Foundational Standards

  • 8.EE.A.1

Criteria for Success

  1. 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.
  2. Multiply a monomial by a polynomial and multiply two binomials.
  3. Determine if two polynomials are equivalent. 
  4. Identify quadratic expressions in standard form.

Tips for Teachers

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

25-30 minutes

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

15-20 minutes

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

Target Task

5-10 minutes

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

Next

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

Lesson 5
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Quadratic Functions

Topic B: Factoring and Solutions of Quadratic Equations

Topic C: Interpreting Solutions of Quadratic Functions in Context

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