Lesson 6 | Rational and Radical Functions | 11th Grade Mathematics

Objective

Multiply and divide rational expressions and simplify using equivalent expressions.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.APR.D.6 — Rewrite simple rational expressions in different forms; write ^a(x/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Arithmetic with Polynomials and Rational Expressions

A.APR.D.6 — Rewrite simple rational expressions in different forms; write a(x /b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
A.APR.D.7 — Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Arithmetic with Polynomials and Rational Expressions

A.APR.D.7 — Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Foundational Standards

The foundational standards covered in this lesson

A.APR.A.1

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.

Criteria for Success

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

Describe how multiplying or dividing rational expressions is similar to multiplying or dividing numeric fractions.
Use factoring to identify common factors and simplify the expression.
Identify domain restrictions on rational expressions.
Define the vertical and/or horizontal asymptotes graphically by inspection. Compare to the algebraic representation.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students will likely need to review exponent rules in conjunction with rational exponents. They have not worked extensively with this concept since Algebra 1.
Simplifying with rational functions is a tricky concept. Students will need to understand that the best way to simplify a rational expression is to factor the expression first in order to see pairs of factors that simplify to 1. Students will also need to keep track of the factors that are cancelled; the concept of removable discontinuities will be discussed in Lesson 6.

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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

Simplify the rational expression shown below.

$${x^2+2x-3\over2x^2+2x-12}$$

Guiding Questions

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Problem 2

Find the quotient of the following rational expression:

$${x^2+3x-40\over{x^2+2x-35}}$$ $${\div }$$ $${x^2+2x-48\over{x^2+3x-18}}$$

Guiding Questions

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Target Task

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

Problem 1

Find the product and write the solution with the fewest number of factors. Note any domain restrictions in your solution.

$${x-2\over{x^2+x-2}}$$ $${\cdot}$$ $${x^2-3x+2\over{x+2}}$$

References

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 24 — Exit Ticket

Algebra II > Module 1 > Topic C > Lesson 24 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..

Problem 2

Find the quotient and write the solution with the fewest number of factors. Note any domain restrictions in your solution.

$${\left (x-2\over{x^2+x-2}\right )\over\left(x^2-3x+2\over{x+2}\right)}$$

References

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 24 — Exit Ticket

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.

Mix these problems from Kuta Software-Infinite Algebra 2 into the problem set.
Include problems about domain restrictions as well as radicals and rational exponents to provide consistent review throughout the unit.
Include problems where students start with the vertical asymptotes and other values that are not in the domain, and they create an operation with rational expressions problem.

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 24 — Problem Set

Lesson 5

Lesson 7

Lesson Map

Topic A: Introduction to Rational and Radical Functions and Expressions

Define rational functions. Identify domain restrictions of rational functions.

A.APR.D.6 F.IF.B.5

Identify domain restrictions algebraically for non-invertible functions.

F.BF.B.4 F.IF.C.7.B

Graph and transform square root and cubic root functions.

F.BF.B.3 F.IF.C.7.B

Write rational functions in equivalent radical form and identify domain restrictions of rational and radical functions.

F.IF.B.5 N.RN.A.2

Write radical and rational exponent expressions in equivalent forms.

N.RN.A.2

Multiply and divide rational expressions and simplify using equivalent expressions.

A.APR.D.6 A.APR.D.7

Add and subtract rational expressions.

A.APR.D.6 A.APR.D.7

Topic B: Features of Rational Functions and Graphing Rational Functions

Identify asymptotic discontinuities (also known as infinite discontinuities) and removable discontinuities in a rational function and describe why these discontinuities exist.

A.APR.D.6 F.IF.C.7.D

Identify features of rational functions with equal degrees in the numerator and the denominator. Describe how to calculate features of these types of rational functions algebraically.

A.APR.D.6 F.IF.C.7.D

Identify features of rational functions with a larger degree in the denominator than in the numerator. Describe how to calculate these features algebraically.

A.APR.D.6 F.IF.C.7.D

Identify features of rational functions with a larger degree in the numerator than in the denominator. Describe how to calculate these features algebraically.

A.APR.D.6 F.IF.C.7.D

Analyze the graph and equations of rational functions and identify features. Use features of a rational function to identify and construct appropriate equations and graphs.

A.APR.D.6 F.IF.C.7.D

Describe transformations of rational functions.

F.BF.B.3 F.IF.C.7.D

Topic C: Solve Rational and Radical Equations and Model with Rational Functions

Solve simple radical equations.

A.REI.A.2

Solve radical equations and identify extraneous solutions.

A.REI.A.2

Solve rational equations.

A.APR.D.6 A.REI.A.2 A.REI.D.11

Write and solve rational functions for contextual situations.

A.APR.D.6 A.CED.A.2 N.Q.A.1

Analyze rational and radical functions in context and write rational functions for percent applications.

A.APR.D.6 A.CED.A.2 A.REI.A.2

Rational and Radical Functions

Objective

Common Core Standards

Core Standards

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

Foundational Standards

Arithmetic with Polynomials and Rational Expressions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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