Advanced Algebra 2 Course Number 215

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1 Curriculum Guide for Advanced Algebra 2 Course Number 215 Author: Jessica Labate Completion Date: May 21, 2012 Department of Mathematics Belvidere High School

2 Department Philosophy The principle reason for studying mathematics is to learn how to apply mathematical thought to problem solving. Students are expected to develop facility in performing the fundamental operations that are associated with each course and to acquire a set of meaningful concepts that they can use effectively to solve problems. The ultimate goal of the department is to provide students with a facility to deal with mathematical functions in daily life and for post-secondary education. Course Overview Students in Algebra 2 build upon their knowledge in Algebra 1 and Geometry. The course starts with the review of the foundations of functions, properties of numbers and its operations. Algebra 2 is the study of the complex number system and functions. Students will algebraically represent, model, analyze, and solve mathematical and real-world problems involving functional patterns and relationships. Topics include Linear Functions, Linear Systems, Matrices, Quadratic Functions, Polynomial Functions, Exponential and Logarithmic Functions, Rational and Radical Functions, Piecewise Functions, Conic Sections which involve Circles, Ellipses, Hyperbolas and Parabolas. Sequences and Series and topics on Probability and Statistics are also included.

3 Course Description & Course Proficiency COURSE TITLE: Advanced Algebra II COURSE # 213 TOTAL CREDIT: 5 COURSE LENGTH: 36 weeks PERIODS PER WK: 5 GRADE LEVEL: 10/ 11 /12 PREREQUISITE(S): Algebra I and Advanced Geometry Pursuant to the High School Graduation Act (NJSA 18A: 7, et. Seq.), expectations for this course of study are outlined below OVERVIEW: This course is designed to review and expand the concepts learned in Algebra I and Geometry as well as introduce new concepts. It will enable the student to apply algebraic techniques to better understand the concepts in the more advanced courses in mathematics. TEXTBOOK: Algebra 2, Holt McDougal, 2011 SUPPLEMENTARY MATERIAL: Resources book, Algebra 2, Holt McDougal, 2011 PROFICIENCIES: Successful completion of this course will require the student: 1. Understand and use set notation 2. Understand and use axioms of equality and inequality 3. Use variables to solve word problems 4. Use matrices to solve systems of equations 5. Simplify and evaluate more complex algebraic expressions 6. Simplify rational expressions 7. Understand and use exponents 8. Graph and solve linear equations and inequalities 9. Understand operations with integers and fractions 10. Understand relations and functions 11. Understand the coordinate plane and its use in solving problems 12. Understand and use complex numbers 13. Understand logarithms 14. Solve quadratic equations; determine the nature of the roots 15. Understand factoring of polynomials, solving polynomial equations and inequalities 16. Understand real numbers and radicals 17. Complete daily homework assignments 18. Keep a notebook of class notes, examples, and assignments STUDENT ACHIEVEMENT: STUDY STRATEGIES: 1. Review your notes 2. Practice any examples that were given in class including homework problems 3. Refer back to the book an notes for any topics you are unsure about HOMEWORK EXPECTATIONS: Each homework assignment will be worth up to 5 points. Homework is to be out on desks when the bell rings or won t be counted. Late work will not be given any credit except in the case of an extended absence. Student must make arrangements with teacher immediately upon return to school in order to receive credit. Parents may be notified by phone or when student does not have assignment. PROCEDURES FOR MAKING UP WORK: The student is responsible for any make up work missed from being out of class. This includes homework, class work, notes, and tests/quizzes. Any work that is not made up following the procedures will result in a grade of zero. Board policy applies.

4 MAJOR PROJECTS TO EXPECT: Tests are given at the end of each chapter/unit. At the end of the year, an extra credit assignment is available for those who wish to complete it. MEASUREMENTS OF STUDENT ACHIEVEMENT: 1. Opening Work- Will be occasionally counted as an unannounced 10 point quiz. Keep all on one sheet of paper for the week. If I decide to collect it, it will be done Friday s right after Friday opening work. If you are absent for a day, write absent by the date. 2. Notebooks- Everyone is required to take notes daily and to keep all papers in date order in a notebook. Occasionally on quiz days, notebooks will be collected and checked; up to 3 points can be received as a quiz bonus. 3. Homework- SEE HOMEWORK EXPECTATIONS 4. Mini Quizzes Each mini quiz will count as 1-20 points. Mini quizzes may or may not be announced. They will occur periodically to check immediate understanding of a skill or topic. 5. Quizzes Each quiz will count as points. There will be at least two quizzes given per chapter/unit. These quizzes will be announced at least the day before. Except in the case of an extended absence, if a student is absent on the day a quiz is given, the quiz must be taken on the day he returns. Additionally, partial credit will be given for work done on quizzes except for multiple choice questions, which will be either right or wrong. 6. Tests Each test will count as 100 points. Tests will be given at the end of each chapter/unit and will be announced at least three (3) days in advance. There will be a review prior to each test. Except in the case of an extended absence, if a student is absent on the day a test is given, the test must be taken on the day he returns. Additionally, partial credit will be given for work done on tests except for multiple choice questions, which will be either right or wrong. PURPOSE AND METHODS OF ASSESSMENT: Assessments are given to accurately measure student knowledge and understanding of the topics. Assessment includes class work, homework, quizzes, and tests. CAREER OBJECTIVE: Emphasis on mathematics achievement to prepare the student for everyday problems as well as more advanced mathematics courses. Applications will represent several different professions. DISCIPLINE PLAN/CLASS RULES: 1. Be in class on time. 2. Have all appropriate materials and supplies at your desk and be seated when the bell rings. 3. No cell phones or ipods are allowed to be used during class time. (Make sure you have a calculator.) 4. Follow directions the first time they are given. 5. Respect the people, equipment, and furnishings in the room. 6. Observe all rules in the Student Handbook. If you choose to break a rule: 1st offense: warning 2nd offense: Phone call home and/or teacher detention. 3rd offense: Referral to office and phone call home. Severe disruptions: Immediate referral and phone call home. Student Signature Teacher Signature Parent/Guardian Signature Parent/Guardian This will allow me to provide timely feedback as needed. Updated 05/07/2012

5 Advanced Algebra 2 Unit I. Foundations for Functions- Properties and Operations High School- Number and Quantity High School- Algebra Using sets of numbers and their properties to solve problems. Simplifying algebraic expressions, expressions with radicals, and expressions with exponents. Provide background and introductory information to speed the progress of units that follow. The Real Number System N-RN Creating Equations A-CED Reasoning with Equations and Inequalities A-REI N-RN 1 N-RN 3 A-CED 1 A-REI 2 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. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Why is number sense and properties important? Why is it important students master numerical operations to manipulate algebraic expressions? Number sense is an intuitive feel for numbers and a common sense approach to using them. Numerical operations are an essential part of the mathematics curriculum. As students progress, they are expected to comprehend more difficult expressions. 1-1 & 1-2 Classify, order, simplify, and use properties of real numbers 1-3 Simplify radical expressions 1-4 Simplify and evaluate algebraic expressions 1-5 Simplify expressions involving exponents Individual White Boards & Markers

6 Unit II. Foundations for Functions- Introduction to Functions High School- Functions Identifying and using functions and their graphs to represent situations. Graphing and transforming parent functions. Provide background and introductory information to speed the progress of units that follow. Interpreting Functions F-IF F-IF 1 F-IF 2 F-IF 5 F-IF 7 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Why is it important students master function notation and graphs? Functions presented as expressions can model many important phenomena. 1-6 & 1-7 Identify the domain and range of relations and functions 1-6 & 1-7 Evaluating functions for inputs 1-8 & 1-9 Identifying appropriate domains for a function and its graph 1-8 & 1-9 Identify the parent functions and their graphs. Additionally, identify how the parent functions transform. Graph the functions. Individual White Boards & Markers

7 Unit III. Linear Functions High School- Algebra High School- Functions Using properties of equality to write and solve linear equations. Applying proportional relationships to rates, similarity, and scale. Writing and graphing linear functions and inequalities. Apply linear relationships to solve real-world problems Creating Equations A-CED Reasoning with Equations and Inequalities A-REI Interpreting Functions F-IF A-CED 2 A-REI 2 A-REI 3 A-REI 10 A-REI 12 F-IF 7 b Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Why are relations and functions represented in multiple ways? How are the properties of functions and functional operations useful? Relations and functions can be represented numerically, graphically, algebraically, and verbally. The properties of functions and function operations are used to model and analyze real world applications and quantitative relationships. 2-1 Solve linear equations and inequalities using a variety of methods 2-2 Apply proportional relationships to rates, similarity, and scale. 2-3 Determine whether a function is linear. Graph a linear function 2-4 Use slope-intercept form and point-slope form to write linear functions. Write linear functions to solve problems. 2-5 Graph linear inequalities on the coordinate plate. Solve problems using inequalities. 2-8 Solve compound inequalities. Solve absolute value equations and inequalities. 2-9 Graph and transform absolute-value functions Individual White Boards & Markers

8 Unit IV. Linear Systems High School- Algebra High School- Functions Graphing systems of linear equations and inequalities. Solving systems of linear equations graphically and algebraically. Graphing and Solving three-dimensional systems. Generalize relationships between two or more variables by using systems. Develop proficiency in solving systems of equations and inequalities. Reasoning with Equations and Inequalities A-REI Interpreting Functions F-IF A-REI 5 A-REI 6 A-REI 10 A-REI 11 A-REI 12 F-IF 7a Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, 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. Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima What methods can be used to solve systems of equations? How are systems of equations useful? A variety of methods are used to solve systems including graphing, substitution, and elimination. Systems are used to model and solve real-world problems. 3-1 Solve systems of equations by using graphs and tables. Classify systems of equations, and determine the number of solutions. 3-1 & 3-2 Solve systems of equations using graphs and tables, substitution, or elimination. 3-2 Solve systems of equations using substitution and elimination. 3-3 Solve systems of linear inequalities and identify points as solutions. 3-5 & 3-6 Graph points and linear equations in three dimensions. Solve systems in three dimensions algebraically. Individual White Boards & Markers

9 Unit V. Quadratic Functions High School- Number and Quantity High School- Algebra High School- Functions Graphing, transforming, and solving quadratic functions. Using and performing operations with imaginary and other complex numbers. Make connections among representations of quadratic functions. Use various methods to solve quadratic equations and apply them to real-world problems. The Complex Number System N-CN Seeing Structure in Expressions A-SSE Reasoning with Equations and Inequalities A-REI Interpreting Functions F-IF N-CN 1 N-CN 2 N-CN 3 N-CN 7 A-SSE 3 A-REI 2 A-REI 4 F-IF 7a F-IF 8a 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. Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Solve quadratic equations with real coefficients that have complex solutions. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. How do quadratic functions model real-world problems and their solutions? Why are complex numbers necessary? How are operations and properties of complex numbers related to those of real numbers? The characteristics of quadratic functions and their representations are useful in solving real-world problems. The domain and range of polynomial functions can be extended to include the set of complex numbers. 5-1 & 5-2 Graph quadratic functions using transformations and standard form. 5-2 Use zeros, symmetry, and maxima or minima in graphing quadratic functions.

10 5-3 Factoring and completing the square to find zeros. 5-3, 5-6 Solve a quadratic equation by graphing, factoring, or using the quadratic formula. 5-5 Solve quadratic equations with complex numbers. 5-5 & 5-9 Define and use imaginary and complex numbers. 5-9 Perform operations with complex numbers.

11 Unit VI. Probability and Statistics High School- Statistics and Probability Find theoretical and experimental probabilities. Apply concepts of probability to solve problems. Interpreting Categorical and Quantitative Data S-ID Making Inferences and Justifying Conclusions S-IC Conditional Probability and Rules of Probability S-CP S-ID 2 S-IC 1 S-IC 3 S-CP 2 S-CP 6 S-CP 7 S-CP 8 S-CP 9 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. (+) Use permutations and combinations to compute probabilities of compound events and solve problems How do probability and statistics model realworld problems and their solutions? Probability and statistics is useful in solving real-world problems. Probability(or), Probability(and), Fundamental Counting Principal, Permutations, Combinations, Factorial, Experimental Probability, Measurements of Central Tendency Individual White Boards & Markers

12 Unit VII. Polynomial Functions High School- Algebra High School- Functions Multiplying, dividing, & factoring polynomials Solving polynomial equations Graphing polynomial functions by finding the zeros, y-intercept, and end behavior Solve problems with polynomials Identify characteristics of polynomial functions. Seeing Structure in Expressions A-SSE Arithmetic with Polynomials and Rational Expressions A-APR Interpreting Functions F-IF A-SSE 1 A-SSE 2 A-APR 1 Interpret expressions that represent a quantity in terms of its context. Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). 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. A-APR 2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). A-APR 3 A-APR 5 A-APR 6 F-IF 4 F-IF 7 c Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle.1 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. 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. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. How do polynomial functions model real-world problems and their solutions? The characteristics of polynomial functions and their representations are useful in solving real-world problems. 6-1 Identify parts of a polynomial function 6-1 & 6-2 Polynomials are closed under addition, subtraction, & multiplication. Also adding, subtracting, and multiplying polynomials. 6-2 Apply the Binomial Theorem through Pascal s Triangles to expand polynomials 6-3 Dividing polynomials

13 6-3 & 6-4 Use division of polynomials to apply the remainder theorem and determine if a polynomial is a factor. 6-4 Identify the structure of a polynomial and determine how to factor it. 6-7 Factoring a polynomial function to reveal its zeros and constructing a rough graph of the function. 6-7 Begin to interpret key features of graphs of polynomials 6-7 Graph polynomial functions identifying zeros and end behavior Smart Board, Calculators, Graphing Calculators

14 Unit VIII. Radical Functions High School- Number and Quantity High School- Algebra Simplifying radical expressions. Solving radical equations. Apply algebraic reasoning to solve problems with rational and radical expressions. Make connections among multiple representations of rational and radical functions. The Real Number System- N-RN Reasoning with Equations and Inequalities A-REI N-RN 1 N-RN 2 A-REI 2 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. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. How do radical functions model real-world problems and their solutions? How are expressions involving radical and exponents related? The characteristics of radical functions and their representations are useful in solving real-world problems. 8-6 Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents. 8-8 Solve radical equations. Individual White Boards & Markers

15 Unit IX. Exponential and Logarithmic Functions High School- Algebra High School- Functions High School- Geometry Exponential functions and their graphs. Logarithms, the inverse of exponents, and logarithmic functions. Solving problems involving exponents and logarithms. Communicate the relationship between exponential and logarithmic functions. Solving problems using exponential and logarithmic functions. Seeing Structure in Expressions A-SSE Interpreting Functions F-IF Building Functions F-BF Congruence G-CO A-SSE 3c F-IF 7e F-IF 8b F-BF 4 F-BF 5 G-CO 2 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. 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. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). How do exponential functions model real-world problems and their solutions? How do logarithmic functions model real-world problems and their solutions? How are expressions involving exponents and logarithms related? The characteristics of exponential and logarithmic functions and their representations are useful in solving real-world problems. 7-1 Write, evaluate, and graph exponential functions and determine growth or decay based on the exp. base.

16 7-2 Create and graph inverse functions from a table and a function. 7-3 Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions. 7-4 Use properties to simplify logarithmic expressions. Translate between logarithms in any base. 7-5 Solve exponential and logarithmic equations. 7-6 Use the number e to write and graph exponential functions representing real-world situations. Solve equations and problems involving e or natural logarithms. Individual White Boards & Markers

17 Unit X. Rational Functions High School- Algebra Simplifying rational expressions. Graphing rational functions. Solving rational equations. Apply algebraic reasoning to solve problems with rational and radical expressions. Make connections among multiple representations of rational and radical functions. Arithmetic and Polynomials and Rational Expressions A-APR Reasoning with Equations and Inequalities A-REI A-APR 6 A-APR 7 A-REI 2 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 (+) 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. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. How do rational functions model real-world problems and their solutions? The characteristics of rational functions and their representations are useful in solving real-world problems. 8-1 Solve problems involving direct, inverse, and joint variations. 8-2 Simplify rational expressions. Multiply and divide rational expressions. 8-3 Add and subtract rational expressions. Simplify complex fractions. 8-5 Solve rational equations. Identify extraneous solutions. Individual White Boards & Markers

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

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