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[Algebra 2] [Unit 4] [Radical Functions]
Enduring understanding (Big Idea): Students will understand how radical expressions and radical equations are used
in real – world situations. Students will compare the domain of a radical function with the domain of other functions.
Essential Questions: How do I solve a radical equation? What is a complex number? How are complex numbers used to
solve equations? How do you add, subtract, or multiply complex numbers? Do all solutions found for an equation make
sense in terms of the problem? How does the domain of a radical function compare with domains of other functions? When
are radical expressions used in real-world situations?
BY THE END OF THIS UNIT:
Students will know…
 The Fundamental Theorem of Algebra
Vocabulary: Radicals, roots, Imaginary number,
complex number, complex conjugates, complex roots
Unit Resources
Suggested time for this unit is10 days
Students will be able to…
 Computationally manipulate radical and complex numbers
 Solve equations for ALL possible solutions
 Determine the validity of solutions
 Determine whether solutions of equations involve real or
complex numbers
 Determine extraneous solutions
 Recognize the relationships between functions and their
roots as factors
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving.
2. Reason abstractly.
3. Construct viable arguments.
4. Model with Mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
1 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Algebra
Standard: A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y= g(x)
intersect at the solutions of the equation f(x)=g(x); find the approximate solutions, e.g. , using technology to graph the
functions, make tables of values, or find successive approximations. Include cases where f(x) and /or g(x) are linear,
polynomial, rational, absolute value, exponential, and logarithmic functions. * Include combinations of radical functions.
Concepts and Skills to Master
 Graph equations to find solutions of radical equations, including problems with radicals on both sides of the equation
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Graphing equations
 Domain and Range of functions
 Finding solutions through tables, charts and graphs
 Translations and transformations of graphs
Academic Vocabulary
Square root function, radical function
Suggested Instructional Strategies
Resources
 Remember that most of Section 6-8 had been
 Textbook Correlation:
Section 6-8 Graphing Radical Functions (p. 414, example 4)
previously addressed in Unit 1
Pearson Success Net Interactive Digital Path for 6-8
 Help students recognize how to adjust the calculator
(must log in to access)
to accommodate domain restrictions and excluded
 TI – Graphing Activity for Radical equations
values
 Discovery Ed – Step by step explanation for graphing radical equations
 Use tables and calculator features to find
approximate solutions.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
2 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
Sample Formative Assessment Tasks
Skill-based task
Solve the following using your calculator:
Problem Task: Use text book problems #1 and # 24 on page 418
and # 37 and # 53on page 419
4 x  36  2x  3
Give approximate solutions as needed.
Teacher Created Argumentation Tasks (W1-MP3&6)
a) How many real numbers can you raise to the second power to get an answer of 16?
b) How many real numbers can you raise to the second power to get an answer of 25?
c) Comparing your answers to a and b with what we learned about the square root function, how SHOULD the square root function look? Why do
you think it does NOT look that way?
d) Argue, stating your reasoning, why the square root function should OR should not change based on your answers to questions a and b.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
3 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Algebra
Standard: A-RE1.2 Solve simple radical equations in one variable and give examples of how extraneous solutions may arise.
Concepts and Skills to Master
 Solving radical equations algebraically
 Excluding non – real and extraneous solutions
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Multiplying polynomials
 Solving equations
 Using reciprocal powers to eliminate radicals
 Adding , subtracting, multiplying and simplifying radicals
Academic Vocabulary
Square root equation, radical equation, extraneous roots, rational exponents
Suggested Instructional Strategies
Resources
 Note: This section does not include imaginary
 Textbook Correlation:
Section 6-5 Solving Square Root and Other Radical Equations.
solutions. Be mindful to select problems that have real
(p.390)
solutions or instruct students to label solutions as non –
Section 6 - 4 Rational exponents
real.
 Discovery Ed videos
 You will need to revisit Section 6-4 for rational
Nth Roots: Radical Expressions, Rational Exponents
Nth Roots: Radical Expressions, Rational Exponent: Surface Area
exponents (8th grade math)
 Emphasize to students that they must check solutions to
see if they are extraneous
 Relate to graphing for solutions
 HONORS students should be encouraged to solve
problems like Example Problem 5 algebraically
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
4 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
Sample Formative Assessment Tasks
Skill-based task:
Solve the following. Be sure to identify any extraneous
solutions.
3
x  3  15
x
Problem Task:
 Pearson Success Net Performance Task: Chapter 6 Task # 2
 Honors: Enrichment 6 - 5
2
 53  4
Teacher Created Argumentation Tasks (W1-MP3&6)
Explain why taking the even root of a number yields positive AND negative answers while taking the odd root of a number
yields a positive OR negative answer. Give an example of each to support your reasoning.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
5 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Number and Quantity
Standard: N-CN.1 Know there is a complex number i such that i2=-1, and every complex number has the form a+bi with a
and b real.
Concepts and Skills to Master
 Use i to represent the square root of a negative number
 Recognize and identify the parts of a complex number
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Simplifying radicals
Academic Vocabulary
Imaginary number, complex numbers, real numbers
Suggested Instructional Strategies
Resources
 Emphasize with students that i does not equal -1.
 Textbook Correlation:
Section 4-8 Complex Numbers (pp. 248 – 249, example 1)
 Emphasize with students that the number must be in
complex form prior to proceeding with any
 Pearson Success Net Interactive Digital Path for 4-8
operations
Ex.
10  10  i 10  i 10  10i 2  10

Discovery Education: Complex Numbers – Electricity A Segment
of: Discovering Math: Advanced: Number Concepts
Sample Formative Assessment Tasks
Problem Task: In your own words, describe a complex number.
Skill-based task
Given: Perform addition , subtraction, multiplication (and
division) of the given expressions 3 +4i and 5 – 6i
Teacher Created Argumentation Tasks (W1-MP3&6)
Imaginary Numbers Discovery
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
6 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Number and Quantity
Standard N-CN.2 Use the relation i2 =-1 and the commutative, associative, and distributive properties to add, subtract, and
multiply complex numbers.
Concepts and Skills to Master
 Add, subtract , multiply complex numbers
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Simplifying radicals
 Powers of “ i
Academic Vocabulary
Suggested Instructional Strategies
 Relate addition and subtraction to combining like
terms, multiplication to “FOIL” ; box – method;
distribute
 Use calculator for computation. Honors students
should be encouraged to discover operational
relationships algebraically.
Sample Formative Assessment Tasks
Skill-based task Perform addition , subtraction,
multiplication of the given expressions:
3 + 4i and 5 – 6i
Resources
 Textbook Correlation:
Section 4-8 Complex Numbers(pp. 250 – 251, examples 3 and 4)
 TI Calculator discovery lesson
Problem Task Compare the operations of adding, subtracting and
multiplying real numbers to those used in simplifying complex
numbers.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
7 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title:
Standard N – CN.3 (+) Find the conjugate of a complex number, use conjugates to find the moduli and quotients of complex
numbers
Concepts and Skills to Master
 (+)Rationalize denominators containing complex numbers by using the conjugate
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Simplifying radicals
 Rationalizing denominators
 Powers of “ i ”
Academic Vocabulary
conjugates, complex conjugates
Suggested Instructional Strategies




Students will need to review how to rationalize
denominators with whole number radicals
Students should be encouraged to revisit difference of
squares patterns
Standard students should be encouraged to work with the
calculator and reading the solutions from the calculator
Honors students should be encouraged to work with
more difficult conjugates, ie. those that will need to be
simplified after rationlizing
Resources
 Textbook Correlation:
Pg. 251 example # 5
Sample Formative Assessment Tasks
Skill-based task: Simplify the following:
4
7  3i
Problem Task
Explain the difference between the additive inverse of a complex
number and a complex conjugate. Justify your explanation
algebraically.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
8 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Number and Quantity
Standard: N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.
Concepts and Skills to Master
 Solving quadratic equations with complex solutions
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Solving quadratic equations with real solutions
 Quadratic Formula
 Simplifying Radicals – from prior lessons
Academic Vocabulary
Complex solutions
Suggested Instructional Strategies
 Students should be encouraged to explore the complex
roots through the use of the calculator
 This is an excellent opportunity to revisit simplifying
radicals and the discriminant
 Honors students should explore
Concept Byte 4 – 9 Quadratic Inequalities
Sample Formative Assessment Tasks
Resources
 Textbook Correlation:
Skill-based task: Find the solutions of the equation 3x 2  x  2  0 .
Problem Task: Write a quadratic equation that has imaginary
roots. Explain how you wrote your equation and justify your
selection.
Section 4-8 Complex Numbers (p. 252, examples 6 and7)
 Discovery Ed videos -working with imaginary
numbers
Teacher Created Argumentation Tasks (W1-MP3&6)
 Quadratics Roots Discovery
After completion, ask students to explain why their discovery for each discriminant makes sense based on their previous math and Algebra
2 knowledge. For example, “Why does a negative discriminant yield two complex roots, and how does the graph of the parabola show that
the roots are not real?”
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
9 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Number and Quantity
Standard: N-CN.8 Extend polynomial identities to the complex numbers. For example, rewrite x2+4 as (x+2i) (x-2i).
Concepts and Skills to Master
 Writing complex number solutions as factors
 Finding a quadratic function from the given roots.
 Write polynomial equations of degree more than 2 given one or more roots
SUPPORTS FOR TEACHERS
Critical Background Knowledge

Roots as factors of a quadratic equation
Multiplication of binomials with and without complex numbers

Academic Vocabulary
Roots, solutions, Complex solutions
Suggested Instructional Strategies
 Start by solving quadratic equations with real roots and
extend to review writing quadratic equations from given real
roots. Follow with quadratic equations with imaginary roots
( x2  4  0 ) and relate roots to writing equations
 Review multiplication of binomials with and without
complex numbers
 Emphasize to students that complex roots always are in
paired form
Sample Formative Assessment Tasks
Skill-based task: Write a polynomial of least degree given the
roots 3  5i and  2 .
Resources
 Textbook Correlation:
Section 4-5 Concept Byte pg. 232 Activity 1 and 2
- extend to include complex numbers as roots.
Section 5-5 problems 3 and 4
 Alternate Method for writing equations from roots
Problem Task: Write a polynomial with one real root and two
complex roots. Write a polynomial with two real roots and two
complex roots. Explain why it is necessary for complex roots to
be in pairs.
Teacher Created Argumentation Tasks (W1-MP3&6)
Sum of Squares Argumentation Discovery
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
10 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
[Algebra 2] [Unit 4] [Radical Functions]
CORE CONTENT
Cluster Title: Number and Quantity
Standard: N-CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials
Concepts and Skills to Master
 To determine the number of solutions based on the degree of the polynomial
 Find all the zeros of a polynomial function
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Quadratic Formula; Synthetic division; Graphing polynomials; Finding solutions of polynomials using the graphing calculator
Academic Vocabulary
Zeros, roots, complex solutions
Suggested Instructional Strategies
Resources
 Review procedures for using the quadratic formula for finding roots
 Textbook Correlation:
Section 5.6 – The Fundamental Theorem of Algebra
 Review synthetic division
 TI calculator activity – “Going back to your roots”
 Emphasize to students that the number of roots include multiple
roots.
 Discovery Education video – math explanation –
“Using a given zero to find the remaining zeros”
 Review identification of multiplicity from the graph of a polynomial
function
 Honors Students should be encouraged to explore the Rational Root
Theorem
Sample Formative Assessment Tasks
Skill-based task – Given a polynomial of degree n , explain how you
Problem Task: Describe when to use synthetic division and
determine the number of zeros of the polynomial.
when to use the Quadratic Formula to determine the linear
factors of the polynomial.
Text book problems pg. 323 #38 - 40
Teacher Created Argumentation Tasks (W1-MP3&6)
Create or research real-world situations that could represent a linear equation, quadratic equation, and cubic equation. How many solutions
SHOULD each equation have, based on the Fundamental Theorem of Algebra? Based on the real-world situations, how many solutions DOES each
equation have? Explain any similarities or differences between the answers.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed
11 in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.