Lesson 18 | Rational and Radical Functions | 11th Grade Mathematics

Objective

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

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.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.REI.A.2 — Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Reasoning with Equations and Inequalities

A.REI.A.2 — Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Criteria for Success

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

Identify domain in context of the problem.
Manipulate a rational or radical equation to identify the features of the function.
Use the context of a problem to assign variables and determine the relationship between these variables.
Describe the solution, in context.
Vary a model for a problem when the context changes.

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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

Alice and Briana each participate in a 5-kilometer race. Alice’s distance covered, in kilometers, after $$t$$ minutes can be modeled by the equation $$a(t)={t\over4}$$. Briana’s progress is modeled by the equation $$b(t)={\sqrt{2t-1}}$$.

Who gets to the finish line first?

Guiding Questions

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References

Illustrative Mathematics Who wins the race?

Who wins the race?, accessed on Nov. 19, 2017, 6:27 p.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.

Modified by Fishtank Learning, Inc.

Problem 2

Chase and his brother like to play basketball. About a month ago they decided to keep track of how many games they have each won. As of today, Chase has won 18 out of the 30 games against his brother.

How many games would Chase have to win in a row in order to have a 75% winning record? Write a function to model this, and then solve to determine the number of games Chase would have to win in a row to have a 75% winning record.

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Illustrative Mathematics Basketball — Part A

Basketball, accessed on Nov. 19, 2017, 11:15 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.

Target Task

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

A group of girls decided to go on a trip to the beach, and the organizer said that the bus would cost $360 to rent. The organizer also told them that if they got 3 more girls to go on the trip, each girl could pay $6 less (which they ended up doing). How many girls ended up going on the trip?

References

She Loves Math Rational Functions and Equations: Application of Rationals

Rational Functions and Equations: Application of Rationals by Lisa Johnson is made available on She Loves Math. Copyright © 2012-2017 She Loves Math (SheLovesMath.com) All Rights Reserved. Accessed Nov. 19, 2017, 6:29 p.m..

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 need to identify the meaning of the expression that models a situation, including the units.
Include problems where students are asked a variety of conceptual problems related to the domain of the context, similar to the target task.

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 27 — Only use the problems that are related to percents
Illustrative Mathematics Basketball — Parts B-D
Mathematics Vision Project: Secondary Mathematics Three Module 4: Rational Expressions and Functions — Lesson 4.1 The Gift
Illustrative Mathematics Fuel Efficiency — This problem will have to be adapted to limit the language that is presented to students. Focus on students identifying the components of the rational function.

Lesson 17

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

Creating Equations

Reasoning with Equations and Inequalities

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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