Algebra 2 with Trigonometry Mathematics: to Hoover City Schools

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1 Jump to Scope and Sequence Map Units of Study Correlation of Standards Special Notes Scope and Sequence Map Conceptual Categories, Domains, Content Clusters, & Standard Numbers NUMBER AND QUANTITY (N) The Complex Number System (CN) Perform arithmetic operations with complex numbers: 1, 2 Use complex numbers in polynomial identities and equations: 3, 4, 5, 6 Vector and Matrix Quantities (VN) Perform Operations on Matrices and use matrices in applications 7,8,9,10,11 2 Units of Study (expected proficiency) 1 st nwks 2 nd nwks 3 rd nwks 4 th nwks 3 ALGEBRA (A) Seeing Structure in Expressions (SSE) Interpret the structure of expressions: 12,13 Write expressions in equivalent forms to solve problems: 14 Arithmetic With Polynomials and Rational Expressions (APR) Perform arithmetic operations on polynomials: 15 Understand the relationship between zeros and factors of polynomials: 16,17 Use polynomial identities to solve problems: 18 Rewrite rational expressions: 19 Creating Equations (CED) Create equations that describe numbers or relationships: 20,21,22,23 Reasoning With Equations and Inequalities (REI) Understand solving equations as a process of reasoning, and explain the reasoning: 24 Represent and solve equations and inequalities graphically: 25 Solve Systems of Equations: 26 Represent and Solve Equations and Inequalities Graphically: 27 FUNCTIONS (F) Interpreting Functions (IF) Interpret functions that arise in applications in terms of the context: 29 Analyze functions using different representations: 30,31,32 3,4 5, , 4 5,6,7 1 3,4 5,6 8 Page 1 of 21

2 Conceptual Categories, Domains, Content Clusters, & Standard Numbers Building Functions (BF) Build a function that models a relationship between two quantities: 33 Build new functions from existing functions: 34,35 Units of Study (expected proficiency) 1 st nwks 2 nd nwks 3 rd nwks 4 th nwks 1 5, 6 7 Linear, Quadratic, and Exponential Models (LE) 6 Construct and compare linear, quadratic, and exponential models and solve problems: 36 Trigonometric Functions (TF) Extend the domain of trigonometric functions using the unit circle: 37,38,39 Model periodic phenomena with trigonometric functions: 40 8 Conic Sections ( ) Understand the graphs and equations of conic sections 28 Emphasis on Parabolas and Circles 28 STATISTICS AND PROBABILITY (S) Conditional Probability and the Rules of Probability (CP) Understand independence and conditional probability and use them to interpret data: 43, 44, 45,46 Use the rules of probability to compute probabilities of compound events in a uniform probability model: 47, 48(+), 49(+), 50 Using Probability to Make Decisions (MD) Use probability to evaluate outcomes of decisions: 41, Units of Study Unit 1- Linear Functions and Equations Create equations and inequalities in one variable and Student understands the.and can solve any given.and can create and solve use them to solve problems. Include equations arising concept of linear equations multi-step equation. multiple equations both 20 A-CED1 from linear and quadratic functions, and simple and can solve a two to general and modeling life three step linear equation. situations. rational and exponential functions.* Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* 21 A-CED2 Student understands concepts of linear functions as it relates to domain and range and can graph a Page 2 of 21...and can graph a general linear function using a variety of graphing techniques....and can model linear data and graph using a variety of methods including regression.

3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.* Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* Write a function that describes a relationship between two quantities.* a) Combine standard function types using arithmetic operations. 22 A-CED3 23 A-CED4 29 F-IF5 33 F-BF1 Instructional Recommendations / Resources: Unit 2- Matrices Unit 1- Linear Functions and Equations general linear function in one variable in slopeintercept form. Student can explain the types of solutions of a system of linear equations and find the intersection graphically and algebraically. Student can solve a given formula for a variable. Student can identify the domain and range of a function given a table of values. Student can write a linear equation using slopeintercept form. Page 3 of 21.and finds the intersection graphically, algebraically, with technology, and by linear programming. and use to solve a problem with given criteria....and explain the relationship that maps a domain value to a range value. and model linear functions with point slope form and given two points..and constructs and models life situations involving systems of linear equations and linear programming. and create problems that model life situations. and analyze the function graphically and algebraically to choose domain values that will help predict future outcomes.create functions using linear regression both graphically and by hand to make predictions for life situations. N-VM6 Students can write given 7 data in a matrix format.. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. (+) Add, subtract, and multiply matrices of appropriate dimensions. 8 N-VM7 9 N-VM8 Students can add, subtract and multiply matrices. and manipulate the data to represent numeric relationships Students can multiply a matrix by a constant scalar value and describe when operations on matrices is and collect and represent data in a matrix format and with technology..and describe a scalar situation with words and technology. describe matrix operations from real life

4 Unit 2- Matrices appropriate considering properties of Algebra data and with technology. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of square matrix is nonzero if and only if the matrix has a multiplicative inverse. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). Instructional Recommendations / Resources: 10 N-VM9 11 VM10N 26 A-REI9 Students can identify identity and zero matrices. Students can solve a 2X2 system with row operations. Given matrices, students can show that properties of algebra are true or untrue...and find the inverse of a matrix and solve a 3X3 system with row operations and can perform operations in the appropriate order in the context of the problem. (ie. Solving matrix equations.). use inverse matrices to solve matrix equations and use technology to set up and solve the same systems for 2X2 and 3X3 matrices Unit 3- Quadratics Functions Know there is a complex number i such that = -1, Student can recognize that and express it in standard and every complex number has the form a + bi with a the 1 is described as the form. and b real. 1 N-CN1 complex number and (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Use the relation = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 3 N-CN3 2 N-CN2 that =-1. Student can find a conjugate. Student can perform operations on complex numbers..and find the quotient of a complex number using the conjugate. and use the conjugate to express the expression in standard form. and explain algebraically and graphically the concept of the imaginary number in relation to the roots of a polynomial..find the moduli using complex numbers and model polynomial functions given complex roots. Page 4 of 21

5 Unit 3- Quadratics Functions Solve quadratic equations with real coefficients that have complex solutions 4 N-CN7 (+) Extend polynomial identities to the complex numbers. 5 N-CN8 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 6 N-CN9 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.* Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* 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,(quadratic) polynomial, rational, absolute value, exponential, and logarithmic functions.* Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 20 A-CED1 21 A-CED2 27 A- REI11 31 F-IF8 Student can solve quadratic equations with complex solutions using square roots and the quadratic formula. Student can identify the relationship between complex numbers and the polynomial identities. Student will know and understand the fundamental theorem of algebra (FTA). Student can recognize a quadratic equation in the standard and vertex form. Student can graph a quadratic equation in the standard and vertex form. Student can describe the types and number of solutions that come from a quadratic system. Student understands that there are different equivalent ways to express quadratic functions. Page 5 of 21 and by completing the square and expressing their solution in standard form. and devise factors that will yield a quadratic with real coefficients. and can apply the FTA by solving a quadratic equation and finding all complex roots....and create a general model of a quadratic equation in the vertex form and factored form. and create a general model of a quadratic equation in the vertex form and factored form. and calculate exact and approximate solutions algebraically and graphically with and without technology. and can transform a given quadratic expression into another form by completing the square, multiplying with the distributive property, or factoring (vertex form, factored form and standard and create a quadratic equation that will yield complex roots based on the value of the discriminant. and use a modeled quadratic to predict maximums and minimums. and solve and model life problems that have complex roots. and use quadratic regression to model data in standard form. and use quadratic regression to model data in standard form....and explain and use the solutions when f(x) = g(x). and model a life situation in those three distinct forms.

6 Unit 3- Quadratics Functions Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 32 F-IF9 Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b. 25 A-REI4B Instructional Recommendations / Resources: Students can recognize the differences between two quadratic functions graphically. Students can recognize and calculate when 2 b 4ac 0 yields complex solutions. form). and can compare the properties of the two functions algebraically and graphically by analyzing the rate of change..and solve quadratic equations with complex solutions and write in a+bi form. and can transform the functions using the information retrieved. can model a quadratic function using complex roots in a+bi form. Unit 4- Polynomial Functions Interpret expressions that represent a quantity in terms of its context.* Student can identify the parts of polynomial and can interpret complicated expressions by and use the roots of a polynomial to model a a) Interpret parts of an expression such as terms, functions including the breaking the polynomial polynomial. i.e. maximizing 12 A-SSE1 degree, leading coefficient, down into factors and using the volume of a box factors, and coefficients. rate of change, and factors. the components to graph constructed from a given b) Interpret complicated expressions by viewing the polynomial. sheet of paper. one or more of their parts as a single entity. Use the structure of an expression to identify ways to rewrite it. 13 A-SSE2 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* 14 A-SSE4 Understand that polynomials form a system analogous to the integers; namely, they are closed under the 15 A-APR1 Student can recognize the standard and factored form of a polynomial function. Student can identify like terms of a polynomial expression. Page 6 of 21 and write the polynomial in factored form if given standard form and vice versa. Student can derive the formula for an infinite geometric series. and perform operations on the polynomial functions. and explain how the structure affects a graph and can predict what the graph looks like based on the structure of the polynomial..and solve. and apply concepts when problems solving.

7 Unit 4- Polynomial Functions operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Know and apply the Remainder Theorem: For a polynomial p( 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). 16 A-APR2 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. 17 A-APR3 Prove polynomial identities and use them to describe numerical relationships. 18 A-APR4 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 functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a) Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. b) Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 27 A- REI11 30 F-IF7 Student can recognize factors of a polynomial by evaluating a polynomial where f(x) = 0. Student can identify the zeros of a polynomial given a graph of the function. Students can identity polynomial identities. Student can identify the types and number of solutions that come from two polynomial equations graphically. Student can graph a polynomial function using defined domain values in a table centered about the origin. Page 7 of 21 and use the factors to find the zeroes of a polynomial. and construct a graph using the zeros of a polynomial. and multiply and factor polynomials according to the polynomial identities. and calculate exact and approximate solutions algebraically and graphically with technology. and graph a polynomial function in the factored form, describe end behavior. and explain the type and number of solutions graphically..analyze the rough graph to determine rate of change, local max and min, and increasing/decreasing. and describe numerical relationships such as the binomial and trinomial theorems. and solve polynomial equations graphically with technology and by hand..and use polynomial identities to factor polynomials and describe the graphs based on components.

8 Unit 4- Polynomial Functions Instructional Recommendations / Resources: Unit 5- Radical Functions Solve simple rational and radical equations in one Student can solve a simple variable, and give examples showing how extraneous radical equation. solutions may arise. 24 A-REI2 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 functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a) Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. b) Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, 27 A- REI11 30 F-IF7 Student can identify the types and number of solutions that come from two radical equations graphically. Student can graph a radical function using defined domain values in a table. and solve multi-step radical equations using quadratic techniques and test for extraneous solutions. and calculate exact and approximate solutions algebraically and graphically with and without technology. and manually graph a radical function in the form of. and model a radical equation in a life situation....and explain and use the solutions when f(x) = g(x)..and transforms radical functions and uses the information to interpret data. Page 8 of 21

9 Unit 5- Radical Functions midline, and amplitude. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values Student can recognize the effects of a, h, and k on the and find the value of k given the graph of f(x). and experiment with cases and illustrate with of k (both positive and negative); find the value of k radical form f(x) by f(x) + technology to predict given the graphs. Experiment with cases and illustrate k, k f(x), f(kx), and f(x effects on graphs using 34 F-BF3 an explanation of the effects on the graph using + k). technology. technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 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. Example: f(x) =2x 3 or f(x) = (x+1)/(x-1) for x and 35a F-BF4a Instructional Recommendations / Resources: Students can find the inverse of a relation. and can find the inverse of a function and verify whether two functions are inverses and show graphically that two functions are inverses about the line y=x Unit 6- Exponential and Logarithmic Functions Create equations and inequalities in one variable and Student can recognize an...and create a general and use exponential use them to solve problems. Include equations arising exponential equation in the model of an exponential equation to model data. 20 A-CED1 from linear and quadratic functions, and simple standard form. equation. rational and exponential functions.* Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* 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 21 A-CED2 27 A- REI11 Student can graph an exponential equation. Student can identify the types and number of solutions that come from exponential and logarithmic functions. Page 9 of 21 and create a general model of an exponential equation. and calculate exact and approximate solutions algebraically and graphically with and without technology. and use exponential equation to model data....and explain and use the solutions when f(x) = g(x).

10 Unit 6- Exponential and Logarithmic Functions values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* 29 F-IF5 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 square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. b) Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 30 F-IF7 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using 34 F-BF3 technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 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. 35 F-BF4 Student can identify the domain and range of a function given a table of values. Student can graph an exponential and logarithmic function showing intercepts. Student can recognize the effects of a, h, and k f(x) by f(x) + k, k f(x), f(kx), and f(x + k). Student can solve an equation of the form ( for a simple function f that has an inverse....and explain the relationship that maps a domain value to a range value. and describe end behavior. and find the value of k given the graph of f(x). and write an expression for the inverse. and analyze the function graphically and algebraically to choose domain values that will help predict future outcomes.and uses technology for more complicated cases. and experiment with cases and illustrate with technology to predict effects on graphs using technology. and use the equation to model data. Page 10 of 21

11 Unit 6- Exponential and Logarithmic Functions For exponential models, express as a logarithm the Student can express a and evaluate the and use other bases for b. solution to = d where a, c, and d are numbers, logarithm as the solution to logarithm using and the base b is 2, 10, or e; evaluate the logarithm 36 F-LE4 where a, c, and technology. using technology.* d are numbers and the base b is 2, 10, or e. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* 14 A-SSE4 Instructional Recommendations / Resources: Student can use the formula to find the sum of a finite geometric series. and derive the formula for the sum of a finite geometric series. and use the formula for life problems. Unit 7- Rational Functions Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where Student can set up the problem correctly to solve and solve the problem. and use a computer algebra system for more a(x), b(x), q(x), and r(x) are polynomials with the by long division or complicated examples. 19 A-APR6 synthetic division. 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. 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.* 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. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) A-CED1 21 A-CED2 24 A-REI2 A- REI11 Student can recognize a rational equation. Student can graph a rational equation. Student can solve a simple rational equation. Student can identify the types and number of solutions that come from Page 11 of 21...and create a general model of a rational equation. and create a general model of a rational equation. and solve multi-step rational equations. and calculate exact and approximate solutions algebraically and and use a rational equation to model data. and use a rational equation to model data. and model a rational equation in a life situation....and explain and use the solutions when f(x) =

12 Unit 7- Rational Functions 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.* rational functions. graphically with and without technology. g(x). Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 34 F-BF3 Instructional Recommendations / Resources: Student can recognize the effects of a, h, and k f(x) by f(x) + k, k f(x), f(kx), and f(x + k). and find the value of k given the graph of f(x). and experiment with cases and illustrate with technology to predict effects on graphs using technology. Unit 8- Trigonometric Functions Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and Student can graph trigonometric functions and describe the periods, midline and amplitude..and uses technology for more complicated cases. using technology for more complicated cases.* showing period, midline and amplitude. a) Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. b) Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and 30 F-IF7 Page 12 of 21

13 Unit 8- Trigonometric Functions trigonometric functions, showing period, midline, and amplitude. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 37 F-TF1 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* 38 F-TF F-TF5 Instructional Recommendations / Resources: Student can identify the radian measure of an arc subtended by the angle given a unit circle. Students can use the unit circle to find trigonometric functions and interpret radian measures of angles. Students can define the six trigonometric functions using ratios of the sides of a right triangle. Identify the parent graph of basic trigonometric functions. Page 13 of 21 and find the radian measure. and find trigonometric functions to all real numbers and interpret as radian measures of angles. and define the coordinates on the unit circle. and adjust the parent graph to represent the change in amplitude, frequency and midline. and given the radian measure find the angle measure. and given the trigonometric functions find the radian measures. and explain the reciprocal of other functions. and write equations that model the given function and the inverse of the function. Unit 9- Probability and Statistics Describe events as subsets of a sample space (the set 43 S-CP1 Students can calculate and calculate and predict outcomes for of outcomes), using characteristics (or categories) of basic experimental and probabilities of mutually future simulations. the outcomes, or as unions, intersections, or theoretical probability with exclusive and not mutually and without the complements of other events ( or, and, not ). exclusive events complement. Understand the conditional probability of A given B as P (A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 44 S-CP3 Understand conditional probability and independent events. Calculate probabilities in both cases. Understand conditional probability and independent events. Calculate probabilities in both cases. Understand conditional probability and independent events. Calculate probabilities in both cases.

14 Unit 9- Probability and Statistics Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. 45 S-CP4 Construct and use frequency tables to determine probabilities. Construct and use frequency tables to determine probabilities. Construct and use frequency tables to determine probabilities. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 46 S-CP5 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 model. 47 S-CP6 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. 48 S-CP7 (+) 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 49 S-CP8 50 S-CP9 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 41 S-MD6 Recognize and explain the concepts in generic terms. Find conditional probabilities and explain reasoning using models. Apply the Addition Rule to simple examples using Venn Diagrams. Recognize and explain the concepts in generic terms and use given data to find probabilities. Find conditional probabilities and explain reasoning using models. Apply the Addition Rule to simple examples using Venn Diagrams and with word problems. Apply the Multiplication Rule to various problems and interpret results. Use permutations and combinations to compute probabilities of compound events and solve problems Students use probabilities to make fair decision in various situations. Recognize and explain the concepts in generic terms and use given data to find probabilities and create experiments to find probabilities. Find conditional probabilities and explain reasoning using models. Apply the Addition Rule to simple examples using Venn Diagrams and with word problems. Create unique examples that use the Addition Rule. Apply the Multiplication Rule to various problems and interpret results and create unique problems with little guidance. Solve more advanced problems. Use permutations and combinations to compute probabilities of compound events and solve problems and derive these probabilities with permutations and combinations using three or more events. Students use probabilities to make fair decision in various situations and Page 14 of 21

15 Unit 9- Probability and Statistics (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 42 S-MD7 Instructional Recommendations / Resources: Students use probability to make decisions in various real-life situations. justify their results. Students use probability to make decisions in various real-life situations and explain their reasoning and make further predications based on probabilities. Unit 10- Analytic Geometry Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, Students can graph the four basic conic sections and translate circles and parabolas about the and write a standard form equation from the from second-degree equations. 28 centered at the origin coordinate plane based on general form and graph for the given form parabolas and circles. a) Formulate equations of conic sections from their determining characteristics. Correlation of Standards Standards Key AL COS # CCSS # HCS Unit # NUMBER AND QUANTITY: The Complex Number System Know there is a complex number i such that = -1, and every complex number has the form a + bi with a and b real. 1 N-CN1 3 Use the relation = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 2 N-CN2 3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex 3 N-CN3 3 Solve quadratic equations with real coefficients that have complex solutions 4 N-CN7 2 (+) Extend polynomial identities to the complex numbers. 5 N-CN8 2 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 6 N-CN9 3 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 7 N-VM6 2 Page 15 of 21

16 Standards Key AL COS # CCSS # HCS Unit # Multiply matrices by scalars to produce new matrices 8 N-VM7 2 (+) Add, subtract, and multiply matrices of appropriate dimensions. 9 N-VM8 2 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative 10 N-VM9 2 operation, but still satisfies the associative and distributive properties. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of square matrix is nonzero if and only if the matrix has a 11 VM10 2 multiplicative inverse. ALGEBRA: Seeing Structure in Expressions Interpret expressions that represent a quantity in terms of its context.* c) Interpret parts of an expression such as terms, factors, and coefficients. 12 A-SSE1 3, 4, 5, 6 d) Interpret complicated expressions by viewing one or more of their parts as a single entity. Use the structure of an expression to identify ways to rewrite it. 13 A-SSE2 3, 4, 5, 6 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* 14 A-SSE4 3, 4, 5, 6 ALGEBRA: Arithmetic With Polynomial and Rational Expressions 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. 15 A-APR1 4, 7 Know and apply the Remainder Theorem: For a polynomial p( 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). 16 A-APR2 4, 7 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. 17 A-APR3 4, 7 Prove polynomial identities and use them to describe numerical relationships. 18 A-APR4 4, 7 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 19 A-APR6 4, 7 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 Hoover A-APR7 4, 7 expressions. ALGEBRA: Creating Equations 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.* 20 A-CED1 1, 3, 6 Create equations in two or more variables to represent relationships between quantities; graph equations on 21 A-CED2 1, 3, 6 Page 16 of 21

17 Standards Key AL COS # CCSS # HCS Unit # coordinate axes with labels and scales.* Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.* 22 A-CED3 1, 3, 6 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.* 23 A-CED4 1, 3, 6 ALGEBRA: Reasoning With Equations and Inequalities Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 24 A-REI2 2,3,4,5,6,7 Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b. A-REI4B 25 2,3,4,5,6,7 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). 26 A-REI9 2,3,4,5,6,7 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, 27 A-REI11 2,3,4,5,6,7 polynomial, rational, absolute value, exponential, and logarithmic functions.* FUNCTIONS: Interpreting Functions Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations b) Formulate equations of conic sections from their determining characteristics. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* 29 F-IF5 1,3,4,5,6,8 Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* d) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. e) Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end 30 F-IF7 1,3,4,5,6,8 behavior. f) 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. 31 F-IF8 1,3,4,5,6,8 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 32 F-IF9 1,3,4,5,6,8 FUNCTIONS: Building Functions Write a function that describes a relationship between two quantities.* a) Combine standard function types using arithmetic operations. 33 F-BF1 1,5,6,7 Page 17 of 21

18 Standards Key AL COS # CCSS # HCS Unit # Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and 34 F-BF3 1,5,6,7 algebraic expressions for them. 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. 35 F-BF4 1,5,6,7 FUNCTIONS: Linear, Quadratic, and Exponential Models For exponential models, express as a logarithm the solution to = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology.* 36 F-LE4 6 FUNCTIONS: Trigonometric Functions Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 37 F-TF1 8 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 38 F-TF2 8 Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* 40 F-TF5 8 STATISTICS AND PROBABILITY: Conditional Probability and the Rules of Probability (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 41 S-MD6 9 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 42 S-MD7 9 Describe events as subsets of a sample space (the set of outcomes), using characteristics (or categories) of the 43 S-CP1 outcomes, or as unions, intersections, or complements of other events ( or, and, not ). 9 Understand the conditional probability of A given B as P (A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability 44 S-CP3 9 of B given A is the same as the probability of B. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate 45 S-CP4 9 conditional probabilities. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 46 S-CP5 9 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 model. 47 S-CP6 9 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. 48 S-CP7 9 (+) 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. 49 S-CP8 9 Page 18 of 21

19 Standards Key AL COS # CCSS # HCS Unit # (+) Use permutations and combinations to compute probabilities of compound events and solve problems 50 S-CP9 9 Special Notes IMPORTANT: A number of CCSS standards for this subject are repeated in the Algebra 1 curriculum, so the specific focus of this level s curriculum is distinguished by highlighting the parts of each applicable standard which will be points of emphasis for Algebra 2 with Trigonometry in yellow. Unlike other math curriculum documents in this cycle, the advanced (i.e. Pre-AP) version of the Algebra 2 with Trigonometry curriculum is actually a separate document. This local curriculum document was developed from the 2010 Alabama Course of Study for Mathematics which was itself based on the newly adopted Common Core State Standards for Mathematics. State COS standards are keyed to CCSS (i.e. Common Core) standards using the lettering and number system employed by the CCSS so that instructional resources which are subsequently designed to support the CCSS can be easily matched back to lessons based on state and local requirements. The symbol (+) is used to designate standards based on mathematics content that students should learn in order to take advanced courses such as calculus. An asterisk (*) is used to designate standards where modeling with mathematics should be stressed because modeling is best interpreted through a relevant context rather than as an isolated topic. The state map of Alabama or the designation (AL) is used to denote additional standards required by the Alabama Board of Education which extend beyond the minimum content defined by the CCSS. The Standards for Mathematical Practice describe the varieties of expertise that mathematics educators at all levels should seek to develop in their students. These standards were developed with input from the National Council of Teachers of Mathematics and the National Research Council, and math teachers should reinforce these process skills when designing daily instructional lessons for students at all grade levels in the Hoover school system: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically Page 19 of 21

20 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning According to the Alabama Quality Teaching Standards (AQTS), teachers of all grade levels and subjects are required to model and reinforce literacy skills for all students. The Alabama Course of Study for Mathematics defines specific college and career readiness anchor standards for grades 6-12 in the areas of reading and writing. Specific grade-appropriate criteria can be found in the state course of study document, but the general anchor standards are defined below: Reading Key Ideas and Details 1. Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text. 2. Determine central ideas or themes of a text and analyze their development; summarize the key supporting details and ideas. 3. Analyze how and why individuals, events, or ideas develop and interact over the course of a text. Craft and Structure 4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. 5. Analyze the structure of texts, including how specific sentences, paragraphs, and larger portions of the text (e.g. a section, chapter, scene, or stanza) relate to each other and the whole. 6. Assess how point of view or purpose shapes the content and style of a text. Integration of Knowledge and Ideas 7. Integrate and evaluate content presented in diverse formats and media, including visually and quantitatively, as well as in words. 8. Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence. 9. Analyze how two or more texts address similar themes or topics in order to build knowledge or to compare the approaches the authors take. Range of Reading and Level of Text Complexity 10. Read and comprehend complex literary and informational texts independently and proficiently. Writing Text Types and Purposes 1. Write arguments to support claims in an analysis of substantive topics or texts using valid reasoning and relevant and sufficient evidence. Page 20 of 21

21 2. Write informative / explanatory texts to examine and convey complex ideas and information clearly and accurate through the effective selection, organization, and analysis of content. 3. Write narratives to develop real or imagined experiences or events using effective technique, well-chosen details, and well-structured event sequences. Production and Distribution of Writing 4. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. 5. Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach. 6. Use technology, including the Internet, to produce and publish writing and to interact and collaborate with others. Research to Build and Present Knowledge 7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the subject under investigation. 8. Gather relevant information from multiple print and digital sources, assess the credibility and accuracy of each source, and integrate the information while avoiding plagiarism. 9. Draw evidence form literary or informational texts to support analysis, reflection, and research. Range of Writing 10. Write routinely over extended time frames (time for research, reflection, and revision) and short time frames (a single sitting or a day or two) for a range of tasks, purposes, and audiences. Page 21 of 21

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