Download Radical Functions and Relations

Algebra II, Quarter 2, Unit 2.1
Radical Functions and Relations
Overview
Number of instruction days:
15 - 17
(1 day = 53 minutes)
Content to Be Learned
Mathematical Practices to Be Integrated

Understand the relationship between rational
exponents and radicals.
2 Reason abstractly and quantitatively

Interpret the components of an exponential
expression.

Use the properties of exponents to rewrite an
exponential expression in an equivalent form.


Create radical equations in one variable.

Solve simple radical equations.

Justify the solution method for a simple
equation.

Graph radical functions and show key features
of the graph.

Use the conceptual understanding of functions
to find the inverse of a simple radical function
if the inverse exists.
Create equations to represent exponential
relationships between two quantities.
4 Model with mathematics.

Create radical equations arising in physics and
other disciplines.
5 Use appropriate tools strategically.

Use the regression features on the graphing
technology to create radical equations.

Use graph paper and technology to graph and
analyze radical functions.
Essential Questions

How do you use the properties of exponents
to rewrite radical expressions?

What real-world situations can be modeled
with radical functions?

How do you solve a radical equation in one
variable?

How can key features of a radical function be
identified given different representations?
Providence Public Schools
D-41
Algebra II, Quarter 2, Unit 2.1
Radical Equations and Relations (15 -17 days)
Version 5

Why are there extraneous solutions when
solving radical equations?

What can you determine from the different
features of the graph of a radical function?

What are the similarities and differences
between the domain and range of an inverse
radical function?
Standards
Common Core State Standards for Mathematical Content
Number and Quantity
The Real Number System
N-RN
Extend the properties of exponents to rational exponents.
N-RN.1
Explain how the definition of the meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for a notation for radicals in terms of
rational exponents. For example, we define 51/3 to be the cube root of 5 because we want
(51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
N-RN.2
Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
Algebra
Reasoning with Equations and Inequalities
A-REI
Understand solving equations as a process of reasoning and explain the reasoning
A-REI.1
Explain each step in solving a simple equation as following from the equality of numbers
asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
A-REI.2
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
Functions
Interpreting Functions
F-IF
Interpret functions that arise in applications in terms of the context [Emphasize selection of
appropriate models]
D-42
Providence Public Schools
Radical Equations and Relations (15 - 17 days)
Algebra II, Quarter 2, Unit 2.1
Version 5
F-IF.4
For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.★
Building Functions
F-BF
Build new functions from existing functions [Include simple radical, rational, and exponential functions;
emphasize common effect of each transformation across function types]
F-BF.4
Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write
an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x  1.
Common Core State Standards for Mathematical Practice
2
Reasoning abstractly and quantitatively
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily
attending to their referents—and the ability to contextualize, to pause as needed during the
manipulation process in order to probe into the referents for the symbols involved. Quantitative
reasoning entails habits of creating a coherent representation of the problem at hand; considering the
units involved; attending to the meaning of quantities, not just how to compute them; and knowing and
flexibly using different properties of operations and objects.
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan
a school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later.
They are able to identify important quantities in a practical situation and map their relationships using
such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical results in
the context of the situation and reflect on whether the results make sense, possibly improving the
model if it has not served its purpose.
Providence Public Schools
D-43
Algebra II, Quarter 2, Unit 2.1
Radical Equations and Relations (15 -17 days)
Version 5
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the insight to be gained and
their limitations. For example, mathematically proficient high school students analyze graphs of
functions and solutions generated using a graphing calculator. They detect possible errors by
strategically using estimation and other mathematical knowledge. When making mathematical models,
they know that technology can enable them to visualize the results of varying assumptions, explore
consequences, and compare predictions with data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to explore
and deepen their understanding of concepts.
Clarifying the Standards
Prior Learning
In sixth grade, students began to investigate and write equations to model relationships, which is where
they first encountered exponents. They also began to reason about and solve one-variable equations
and inequalities. In seventh grade, students solved real-world problems using numerical and algebraic
expressions and equations. In eighth grade, students worked with radicals and integer exponents, and
they used square-root and cube-root symbols to represent solutions to equations of the form x2 = p and
x3 = p, where p is a positive rational number. Students also explored linear and nonlinear relationships in
graphs, tables, and equations. They learned to write and solve linear equations. In Grade 8, students
learned how to apply the properties of integer exponents to generate equivalent numerical expressions.
They also used square root and cube root symbols to represent solutions to equations. Students also
evaluated square and cube roots of small perfect cubes. In Algebra I, students created equations that
described numbers and relationships using linear, quadratic, and exponential models (with integer
inputs only). Students represented constraints and interpreted solutions of linear functions only. Also in
Algebra I, students analyzed functions using multiple representations (linear, exponential, quadratic,
absolute-value, step, and piecewise-defined functions.) Finally, students rewrote expressions involving
radicals and rational exponents using the properties of exponents.
Current Learning
Modeling with functions is a critical area for Algebra II. Students identify appropriate types of functions
for given data, adjust parameters as needed, and compare the quality of models. In this unit, students
develop an understanding of the relationship between rational exponents and radicals. They interpret
the parts of an exponential expression, and they use the properties of exponents to rewrite an
D-44
Providence Public Schools
Radical Equations and Relations (15 - 17 days)
Algebra II, Quarter 2, Unit 2.1
Version 5
exponential expression in an equivalent form. They create, graph, and solve radical equations; represent
constraints; and interpret solutions. They also analyze radical functions and their graphs to identify key
features and how the graphs relate to characteristics of model situations. Finally, students rearrange
formulas involving radicals. The following clusters have been identified as major content by the PARCC
Model Frameworks for Mathematics: extending the properties of exponents to rational exponents,
understanding solving equations as a process of reasoning and explaining the reasoning, and
interpreting functions that arise in applications in terms of the context. Defining trigonometric ratios
and solving problems involving right triangles is major content as defined by the PARCC Model
Frameworks for Mathematics. Building new functions from existing functions is classified as additional
content as defined by the PARCC frameworks.
Future Learning
Mastery of concepts related to radical equations and functions will be required for Precalculus and AP
Calculus. In Calculus, students will need to apply rules for derivatives and integrals of radical functions.
Students will also develop techniques for evaluating integrals involving radical functions. Radical
expressions are also common in Trigonometry.
A variety of careers will require understanding of concepts related to radical equations; for example, the
financial industry uses rational exponents to compute interest, depreciation, and other calculations,
such as inflation of a home’s value. One of the simplest formulas in electrical engineering is for voltage,
V = √PR, where P is the power in watts and R is the resistance in ohms. Radical equations are also used
in navigational systems such as GPS.
Additional Findings
There are no additional findings for this unit.
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
Providence Public Schools
D-45
Algebra II, Quarter 2, Unit 2.1
Radical Equations and Relations (15 -17 days)
Version 5
developed within this unit. The assessment should provide you with credible evidence as to your
students’ attainment of the mathematics within the unit.

Understand the relationship between rational exponents and radicals.

Use the properties of exponents to rewrite an exponential expression in an equivalent form.

Create radical equations in one variable and use them in a contextual situation to solve problems.

Solve simple radical equations in one variable and provide examples of how extraneous solutions
arise.

Graph radical functions and show key features of the graph.

Identify, graph, and write equations for inverses of radical functions.
Instruction
Learning Objectives
Students will be able to:

Rewrite expressions involving rational exponents.

Graph and describe the key characteristics of the square root equations.

Graph and explore the concept of nth roots with expressions involving radicals.

Use the coordinates of a function to find and graph an inverse relation.

Find the inverse of a radical function if an inverse function exists.

Verify that two functions are inverses with and without technology.

Use addition, subtraction, multiplication, and division to simplify expressions involving radicals.

Solve simple equations and inequalities involving root functions.

Reflect on and demonstrate understanding of roots, radical expressions, and functions.
Resources
Algebra 2, Glencoe-McGraw Hill, 2010, Teacher Edition / Student Edition

Section 7-2 (pp. 417- 422)

Section 7-3 (pp. 424 - 430)
D-46
Providence Public Schools
Radical Equations and Relations (15 - 17 days)
Algebra II, Quarter 2, Unit 2.1
Version 5

Section 7-4 (pp. 431- 436)

Sections 7-6 – 7-7 (pp. 446 – 459)

Section 7-7: Solving Radical Equations and Inequalities: TI-Nspire

Chapter 7 Resources Masters (pp. 12-30, 38 - 50)

Interactive Classroom CD (PowerPoint Presentations)
Algebra 1 Glencoe McGraw-Hill Online, http://connected.mcgraw-hill.com/connected/login.do:

CCSS Lesson 8: Rational Exponents [see Supplemental Materials section]

Lesson 8 Practice [see Supplemental Materials section]
Exam View Assessment Suite
TI-Nspire Teacher Software
www.khanacademy.org
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery section for specific recommendations.
Materials
Graphing calculators, gridded chart paper, string, markers, Wikki Stix.
Instructional Considerations
Key Vocabulary
conjugate
extraneous solution
nth root
radical equation
radical function
inverse function
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: like terms, square root and cube root function.
Providence Public Schools
D-47
Algebra II, Quarter 2, Unit 2.1
Radical Equations and Relations (15 -17 days)
Version 5
In this unit students will synthesize and generalize what they have learned about a variety of function
families and extend their knowledge to include radical functions. The rules of exponents used in a
previous unit are the same as those for rational exponents. Have students use a Venn diagram to
compare the two exponential concepts and identify similarities and differences.
The supplementary resources for Algebra 1 CCSS Lesson 8: Rational Exponents and the Lesson 8 Practice
are accessible online on the Algebra 1 Glencoe McGraw-Hill Math Algebra Student and Teacher
textbooks. First, select the CCSS icon on the homepage of the online textbook, select the CCSS
Supplement tab, and then select the respective lab or lesson. The lessons are also located in the
supplemental section of the curriculum binder.
Since students will have had different experiences with radical expressions, a quick review using a game
format, such as Radical Expressions Jeopardy Review Game accessible on www.superteachertools.com/
jeopardy/usergames/Apr201015/game1271186792.php will activate prior knowledge. The emphasis in
this activity is on developing ease with simplifying an expression with rational exponents to an
equivalent expression with radicals, and vice versa.
A quick review of the characteristics of the quadratic function (domain, range, symmetry, table of
values, graph) using Example 2b on page 419 to visualize the inverse, the square root relation—should
provide a graphical connection. The symmetry of the quadratic function implies that the inverse
relationship will not be a function, which the graph verifies. This should lead into the idea that the
domain of the square root function must be restricted. This concept is new, so using a concrete model
(the Wikki Stix) and actually cutting the parabola in half accentuates the restriction and its effect on the
range. Technology also supports this learning with a graphical representation.
While studying radical functions, interpreting key features of graphs and creating equations, students
describe and explain graphs of radical functions as transformations of related functions, and they adjust
parameters to choose a model that fits.
Consider scaffolding activities to support struggling students. Eventually, students will use radical
functions to solve equations and inequalities. “Reasoned solving” plays a role in Algebra II because the
equations students encounter can have extraneous solutions. Students will need to be able to recognize
the rational exponent/radical form in an equation or inequality in order to know what solution
strategies are necessary to solve the problem. Also, students will have more experience when working
with exponential functions to evaluate expressions with rational exponents.
As students solve problems, ask scaffolding questions such as the following:

Could you begin with a different first step?

Can you find another solution method for the equation?

What do we want to find?

How could you check to see if your answer makes sense?
D-48
Providence Public Schools
Radical Equations and Relations (15 - 17 days)
Algebra II, Quarter 2, Unit 2.1
Version 5

What does it mean to have 6 times twice the amount?

Does it make sense to divide before you multiply?
Be sure to summarize different strategies for solving equations. Journal writing is an appropriate place
for using a summarizing and note taking strategy. You can also ask students to respond to the Essential
Questions at appropriate times throughout the lessons of this unit.
In this unit, students find the inverse of radical functions. Care should be taken in assigning items from
Section 7-2 which align to the CCSS and the focus of the unit.
A nonlinguistic (kinesthetic) representation of inverse functions can be drawn using large grid paper
and string. Put students in groups of 2 or 3. Let students draw the line of reflection y = x on the grid
paper. Give each group a different function to represent with their string. Then have the groups reflect
the function over the line y = x. Have the groups determine if the reflection is a function or not. Have the
groups find the equations of the function or relation using the graph. Have the groups exchange graphs
and determine if the inverse function is correct. Have students share out their findings with the whole
group.
Use word walls, foldables, or graphic organizers to support the learning of vocabulary. Vocabulary
support is essential to student success in Algebra 2. The habit of using precise language in mathematics
is not only a mechanism for effective communication but also a tool for understanding and solving
problems. Describing a concept precisely helps students understand the idea in new ways.
A foldable study organizer, such as the one illustrated on page 408, can also be used to assist students
with note taking skills. Students can create and use a foldable to take notes, define terms, capture key
concepts and ideas, and write illustrated examples. In this unit, there are numerous opportunities to
differentiate instruction and scaffold support in the Study Notebook.
There are abundant online resources available to support the mathematical concepts presented in this
unit as well as future units in Algebra 2. For example, the Khan Academy (www.khanacademy.org) is a
not-for-profit education organization with the goal of “changing education for the better by providing a
free world-class education for anyone anywhere". The website has a comprehensive collection of videos
supporting many concepts in core mathematics classes.
Providence Public Schools
D-49
Algebra II, Quarter 2, Unit 2.1
Radical Equations and Relations (15 -17 days)
Version 5
Notes
D-50
Providence Public Schools