Algebra 1 Unit 3: Systems of Equations s2

[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… Students will be able to…  The Fundamental Theorem of Algebra  Computationally manipulate radical and complex numbers  Solve equations for ALL possible solutions  Determine the validity of solutions Vocabulary: Radicals, roots, Imaginary number, complex number, complex conjugates, complex roots  Determine whether solutions of equations involve real or complex numbers  Determine extraneous solutions  Recognize the relationships between functions and their roots as factors Unit Resources Mathematical Practices in Focus: 1. Make sense of problems and persevere in solving. Suggested time for this unit is10 days 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 listed1 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: previously addressed in Unit 1 Section 6-8 Graphing Radical Functions (p. 414, example 4)  Help students recognize how to adjust the calculator Pearson Success Net Interactive Digital Path for 6-8 (must log in to access) to accommodate domain restrictions and excluded values  TI – Graphing Activity for Radical equations  Use tables and calculator features to find  Discovery Ed – Step by step explanation for graphing radical equations

approximate solutions.

Sample Formative Assessment Tasks Skill-based task Problem Task: Use text book problems #1 and # 24 on page 418 Solve the following using your calculator: and # 37 and # 53on page 419 4x+ 36 = 2 x + 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.

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: solutions. Be mindful to select problems that have real Section 6-5 Solving Square Root and Other Radical Equations. solutions or instruct students to label solutions as non – (p.390) 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 exponents (8th grade math) Nth Roots: Radical Expressions, Rational Exponent: Surface Area  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

Sample Formative Assessment Tasks Skill-based task: Problem Task: Solve the following. Be sure to identify any extraneous  Pearson Success Net Performance Task: Chapter 6 - solutions. Task # 2 3 x +3 = 15  Honors: Enrichment 6 - 5 2 (x +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.

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:  Emphasize with students that the number must be in Section 4-8 Complex Numbers (pp. 248 – 249, example 1) complex form prior to proceeding with any operations  Pearson Success Net Interactive Digital Path for 4-8 Ex. -10� = 10 �i 10 = i 10 - 10 i2 10  Discovery Education: Complex Numbers – Electricity A Segment of: Discovering Math: Advanced: Number Concepts Sample Formative Assessment Tasks Skill-based task Problem Task: In your own words, describe a complex number. 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

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 Resources  Relate addition and subtraction to combining like  Textbook Correlation: terms, multiplication to “FOIL” ; box – method; Section 4-8 Complex Numbers(pp. 250 – 251, examples 3 and 4) distribute  Use calculator for computation. Honors students  TI Calculator discovery lesson should be encouraged to discover operational relationships algebraically.

Sample Formative Assessment Tasks Skill-based task Perform addition , subtraction, Problem Task Compare the operations of adding, subtracting and multiplication of the given expressions: multiplying real numbers to those used in simplifying complex 3 + 4i and 5 – 6i numbers.

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 Resources  Students will need to review how to rationalize  Textbook Correlation: denominators with whole number radicals Pg. 251 example # 5  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 Sample Formative Assessment Tasks 4 Problem Task Skill-based task: Simplify the following: 7- 3i Explain the difference between the additive inverse of a complex number and a complex conjugate. Justify your explanation Successive pages contain an unpacking of the standards contained in the unit. Standards are listed8 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]

algebraically. 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 Resources  Students should be encouraged to explore the complex  Textbook Correlation: roots through the use of the calculator Section 4-8 Complex Numbers (p. 252, examples 6 and7)  This is an excellent opportunity to revisit simplifying  Discovery Ed videos -working with imaginary radicals and the discriminant numbers  Honors students should explore Concept Byte 4 – 9 Quadratic Inequalities Sample Formative Assessment Tasks Skill-based task: Find the solutions of the equation 3x2 - x + 2 = 0 . Problem Task: Write a quadratic equation that has imaginary roots. Explain how you wrote your equation and justify your selection. 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 Successive pages contain an unpacking of the standards contained in the unit. Standards are listed9 in alphabetical and numerical order not suggestedthe rootsteaching are order not. Teachersreal?” 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 Resources  Start by solving quadratic equations with real roots and  Textbook Correlation: extend to review writing quadratic equations from given real Section 4-5 Concept Byte pg. 232 Activity 1 and 2 roots. Follow with quadratic equations with imaginary roots ( - extend to include complex numbers as roots. Section 5-5 problems 3 and 4 x2 +4 = 0 ) and relate roots to writing equations  Review multiplication of binomials with and without  Alternate Method for writing equations from roots 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 Problem Task: Write a polynomial with one real root and two roots 3+ 5i and - 2 . 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) Successive pages contain an unpacking of the standards contained in the unit. Standards are listed10 in alphabetical and numerical order notSum suggested of Squares teaching Argumentation order. Teachers mustDiscovery 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:  Review synthetic division Section 5.6 – The Fundamental Theorem of Algebra  Emphasize to students that the number of roots include multiple  TI calculator activity – “Going back to your roots” roots.  Discovery Education video – math explanation –  Review identification of multiplicity from the graph of a polynomial “ Using a given zero to find the remaining zeros” 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 Successiveequation pageshave? contain Explain an unpacking any similarities of the standards or differences contained in thebetween unit. Standards the answers. are listed11 in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.