IXL Skill Plan | Common Core State Standards: Algebra 2

Common Core State Standards: Algebra 2

Skills available for Michigan Algebra 2 standards

Domain

Number and Quantity
Algebra
Functions
Statistics and Probability

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9-12.HSN	Number and Quantity
9-12.HSN-CN The Complex Number System
9-12.HSN-CN.A Perform arithmetic operations with complex numbers.
9-12.HSN-CN.A Perform arithmetic operations with complex numbers.

9-12.HSN-CN.A.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.		1. Introduction to complex numbers
9-12.HSN-CN.A.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.
1. Introduction to complex numbers
9-12.HSN-CN.A.2: Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.		1. Add and subtract complex numbers 2. Multiply complex numbers 3. Powers of i
9-12.HSN-CN.A.2: Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
1. Add and subtract complex numbers 2. Multiply complex numbers 3. Powers of i

9-12.HSN-CN.C Use complex numbers in polynomial identities and equations.
9-12.HSN-CN.C Use complex numbers in polynomial identities and equations.

9-12.HSN-CN.C.7: Solve quadratic equations with real coefficients that have complex solutions.		1. Solve a quadratic equation using square roots 2. Solve a quadratic equation by completing the square 3. Solve a quadratic equation using the quadratic formula 4. Using the discriminant
9-12.HSN-CN.C.7: Solve quadratic equations with real coefficients that have complex solutions.
1. Solve a quadratic equation using square roots 2. Solve a quadratic equation by completing the square 3. Solve a quadratic equation using the quadratic formula 4. Using the discriminant
9-12.HSN-CN.C.8: Extend polynomial identities to the complex numbers.		1. Solve polynomial equations 2. Complex conjugate theorem 3. Conjugate root theorems
9-12.HSN-CN.C.8: Extend polynomial identities to the complex numbers.
1. Solve polynomial equations 2. Complex conjugate theorem 3. Conjugate root theorems
9-12.HSN-CN.C.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.		1. Fundamental Theorem of Algebra
9-12.HSN-CN.C.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
1. Fundamental Theorem of Algebra

9-12.HSA	Algebra
9-12.HSA-SSE Seeing Structure in Expressions
9-12.HSA-SSE.A Interpret the structure of expressions.
9-12.HSA-SSE.A Interpret the structure of expressions.

9-12.HSA-SSE.A.1: Interpret expressions that represent a quantity in terms of its context.
9-12.HSA-SSE.A.1: Interpret expressions that represent a quantity in terms of its context.

9-12.HSA-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.		1. Polynomial vocabulary 2. Sort factors of single-variable expressions 3. Sort factors of multi-variable expressions 4. Interpret parts of quadratic expressions: word problems
9-12.HSA-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
1. Polynomial vocabulary 2. Sort factors of single-variable expressions 3. Sort factors of multi-variable expressions 4. Interpret parts of quadratic expressions: word problems
9-12.HSA-SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.		1. Interpret parts of quadratic expressions: word problems
9-12.HSA-SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
1. Interpret parts of quadratic expressions: word problems
9-12.HSA-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.		Polynomial expressions 1. Factor using a quadratic pattern 2. Factor by grouping 3. Factor sums and differences of cubes 4. Factor polynomials Binomial Theorem 5. Pascal's triangle and the Binomial Theorem 6. Binomial Theorem I 7. Binomial Theorem II Radical expressions 8. Simplify radical expressions with variables I 9. Simplify radical expressions with variables II 10. Simplify radical expressions using conjugates Rational exponents 11. Convert between rational exponents and radicals 12. Simplify expressions involving rational exponents I 13. Simplify expressions involving rational exponents II Rational expressions 14. Simplify rational expressions
9-12.HSA-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
Polynomial expressions 1. Factor using a quadratic pattern 2. Factor by grouping 3. Factor sums and differences of cubes 4. Factor polynomials Binomial Theorem 5. Pascal's triangle and the Binomial Theorem 6. Binomial Theorem I 7. Binomial Theorem II Radical expressions 8. Simplify radical expressions with variables I 9. Simplify radical expressions with variables II 10. Simplify radical expressions using conjugates Rational exponents 11. Convert between rational exponents and radicals 12. Simplify expressions involving rational exponents I 13. Simplify expressions involving rational exponents II Rational expressions 14. Simplify rational expressions

9-12.HSA-SSE.B Write expressions in equivalent forms to solve problems.
9-12.HSA-SSE.B Write expressions in equivalent forms to solve problems.

9-12.HSA-SSE.B.4: 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.		1. Find the sum of a finite geometric series
9-12.HSA-SSE.B.4: 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.
1. Find the sum of a finite geometric series
9-12.HSA-APR Arithmetic with Polynomials and Rational Expressions
9-12.HSA-APR.A Perform arithmetic operations on polynomials.
9-12.HSA-APR.A Perform arithmetic operations on polynomials.

9-12.HSA-APR.A.1: 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.		1. Add and subtract polynomials 2. Multiply polynomials
9-12.HSA-APR.A.1: 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.
1. Add and subtract polynomials 2. Multiply polynomials

9-12.HSA-APR.B Understand the relationship between zeros and factors of polynomials.
9-12.HSA-APR.B Understand the relationship between zeros and factors of polynomials.

9-12.HSA-APR.B.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).		1. Evaluate polynomials using synthetic division 2. Write a quadratic function from its zeros 3. Find the roots of factored polynomials 4. Write a polynomial from its roots
9-12.HSA-APR.B.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).
1. Evaluate polynomials using synthetic division 2. Write a quadratic function from its zeros 3. Find the roots of factored polynomials 4. Write a polynomial from its roots
9-12.HSA-APR.B.3: 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.		Quadratics 1. Solve a quadratic equation using the zero product property 2. Use zeros to match quadratic functions and graphs (Algebra 1) 3. Use zeros to match quadratic functions and graphs (Algebra 2) Polynomials 4. Find the roots of factored polynomials 5. Match polynomials and graphs using zeros
9-12.HSA-APR.B.3: 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.
Quadratics 1. Solve a quadratic equation using the zero product property 2. Use zeros to match quadratic functions and graphs (Algebra 1) 3. Use zeros to match quadratic functions and graphs (Algebra 2) Polynomials 4. Find the roots of factored polynomials 5. Match polynomials and graphs using zeros

9-12.HSA-APR.C Use polynomial identities to solve problems.
9-12.HSA-APR.C Use polynomial identities to solve problems.

9-12.HSA-APR.C.4: Prove polynomial identities and use them to describe numerical relationships.		1. Factor sums and differences of cubes
9-12.HSA-APR.C.4: Prove polynomial identities and use them to describe numerical relationships.
1. Factor sums and differences of cubes
9-12.HSA-APR.C.5: 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. Pascal's triangle 2. Pascal's triangle and the Binomial Theorem 3. Binomial Theorem I 4. Binomial Theorem II
9-12.HSA-APR.C.5: 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. Pascal's triangle 2. Pascal's triangle and the Binomial Theorem 3. Binomial Theorem I 4. Binomial Theorem II

9-12.HSA-APR.D Rewrite rational expressions.
9-12.HSA-APR.D Rewrite rational expressions.

9-12.HSA-APR.D.6: 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.		1. Divide polynomials using long division 2. Simplify rational expressions
9-12.HSA-APR.D.6: 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.
1. Divide polynomials using long division 2. Simplify rational expressions
9-12.HSA-APR.D.7: 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.		1. Multiply and divide rational expressions 2. Add and subtract rational expressions 3. Simplify complex fractions
9-12.HSA-APR.D.7: 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.
1. Multiply and divide rational expressions 2. Add and subtract rational expressions 3. Simplify complex fractions
9-12.HSA-CED Creating Equations
9-12.HSA-CED.A Create equations that describe numbers or relationships.
9-12.HSA-CED.A Create equations that describe numbers or relationships.

9-12.HSA-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.		1. Write inequalities from graphs 2. Solve quadratic equations: word problems 3. Write exponential functions: word problems
9-12.HSA-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.
1. Write inequalities from graphs 2. Solve quadratic equations: word problems 3. Write exponential functions: word problems
9-12.HSA-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.		Linear equations 1. Graph a linear function 2. Write the equation of a linear function Absolute value equations 3. Graph an absolute value function Quadratic equations 4. Graph a quadratic function 5. Write a quadratic function in vertex form from its vertex and another point 6. Write a quadratic function in factored form from its x-intercepts and another point 7. Write a quadratic function from three points Radical equations 8. Graph square root functions 9. Graph cube root functions Exponential and logarithmic equations 10. Write exponential functions: word problems 11. Graph exponential functions 12. Graph logarithmic functions Direct and inverse variation 13. Write and solve direct variation equations 14. Write and solve inverse variation equations 15. Write joint and combined variation equations I 16. Write joint and combined variation equations II 17. Solve variation equations Trigonometric equations 18. Write equations of sine functions from graphs 19. Write equations of sine functions using properties 20. Graph sine functions 21. Write equations of cosine functions from graphs 22. Write equations of cosine functions using properties 23. Graph cosine functions 24. Write equations of sine and cosine functions from graphs 25. Write equations of sine and cosine functions using properties 26. Graph sine and cosine functions
9-12.HSA-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Linear equations 1. Graph a linear function 2. Write the equation of a linear function Absolute value equations 3. Graph an absolute value function Quadratic equations 4. Graph a quadratic function 5. Write a quadratic function in vertex form from its vertex and another point 6. Write a quadratic function in factored form from its x-intercepts and another point 7. Write a quadratic function from three points Radical equations 8. Graph square root functions 9. Graph cube root functions Exponential and logarithmic equations 10. Write exponential functions: word problems 11. Graph exponential functions 12. Graph logarithmic functions Direct and inverse variation 13. Write and solve direct variation equations 14. Write and solve inverse variation equations 15. Write joint and combined variation equations I 16. Write joint and combined variation equations II 17. Solve variation equations Trigonometric equations 18. Write equations of sine functions from graphs 19. Write equations of sine functions using properties 20. Graph sine functions 21. Write equations of cosine functions from graphs 22. Write equations of cosine functions using properties 23. Graph cosine functions 24. Write equations of sine and cosine functions from graphs 25. Write equations of sine and cosine functions using properties 26. Graph sine and cosine functions
9-12.HSA-CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.		Linear inequalities 1. Write two-variable linear inequalities: word problems Systems of equations 2. Solve a system of equations by graphing: word problems 3. Solve a system of equations using substitution: word problems 4. Solve a system of equations using elimination: word problems 5. Solve a system of equations using any method: word problems
9-12.HSA-CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Linear inequalities 1. Write two-variable linear inequalities: word problems Systems of equations 2. Solve a system of equations by graphing: word problems 3. Solve a system of equations using substitution: word problems 4. Solve a system of equations using elimination: word problems 5. Solve a system of equations using any method: word problems
9-12.HSA-CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.		1. Rearrange multi-variable equations
9-12.HSA-CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
1. Rearrange multi-variable equations
9-12.HSA-REI Reasoning with Equations and Inequalities
9-12.HSA-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
9-12.HSA-REI.A Understand solving equations as a process of reasoning and explain the reasoning.

9-12.HSA-REI.A.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.		1. Solve radical equations 2. Solve rational equations
9-12.HSA-REI.A.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
1. Solve radical equations 2. Solve rational equations

9-12.HSA-REI.D Represent and solve equations and inequalities graphically.
9-12.HSA-REI.D Represent and solve equations and inequalities graphically.

9-12.HSA-REI.D.11: 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.		Tables 1. Find solutions using a table 2. Approximate solutions using a table Graphs 3. Solve a system of equations by graphing 4. Solve a system of equations by graphing: word problems 5. Find the number of solutions to a system of equations
9-12.HSA-REI.D.11: 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.
Tables 1. Find solutions using a table 2. Approximate solutions using a table Graphs 3. Solve a system of equations by graphing 4. Solve a system of equations by graphing: word problems 5. Find the number of solutions to a system of equations

9-12.HSF	Functions
9-12.HSF-IF Interpreting Functions
9-12.HSF-IF.B Interpret functions that arise in applications in terms of the context.
9-12.HSF-IF.B Interpret functions that arise in applications in terms of the context.

9-12.HSF-IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.		Linear functions 1. Find the slope of a linear function 2. Graph a linear function Absolute value functions 3. Graph an absolute value function Quadratic functions 4. Characteristics of quadratic functions: graphs 5. Characteristics of quadratic functions: equations 6. Graph a quadratic function 7. Match quadratic functions and graphs Polynomial functions 8. Match polynomials and graphs Radical functions 9. Graph square root functions 10. Graph cube root functions Rational functions 11. Rational functions: asymptotes and excluded values Variation 12. Classify variation 13. Find the constant of variation Exponential and logarithmic functions 14. Graph exponential functions 15. Match exponential functions and graphs 16. Graph logarithmic functions Word problems 17. Identify graphs: word problems
9-12.HSF-IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
Linear functions 1. Find the slope of a linear function 2. Graph a linear function Absolute value functions 3. Graph an absolute value function Quadratic functions 4. Characteristics of quadratic functions: graphs 5. Characteristics of quadratic functions: equations 6. Graph a quadratic function 7. Match quadratic functions and graphs Polynomial functions 8. Match polynomials and graphs Radical functions 9. Graph square root functions 10. Graph cube root functions Rational functions 11. Rational functions: asymptotes and excluded values Variation 12. Classify variation 13. Find the constant of variation Exponential and logarithmic functions 14. Graph exponential functions 15. Match exponential functions and graphs 16. Graph logarithmic functions Word problems 17. Identify graphs: word problems
9-12.HSF-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.		Absolute value functions 1. Domain and range of absolute value functions: graphs 2. Domain and range of absolute value functions: equations Quadratic functions 3. Domain and range of quadratic functions: graphs 4. Domain and range of quadratic functions: equations Polynomial functions 5. Domain and range of polynomials Radical functions 6. Domain and range of radical functions Exponential and logarithmic functions 7. Domain and range of exponential functions 8. Domain and range of logarithmic functions Mixed functions 9. Domain and range
9-12.HSF-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Absolute value functions 1. Domain and range of absolute value functions: graphs 2. Domain and range of absolute value functions: equations Quadratic functions 3. Domain and range of quadratic functions: graphs 4. Domain and range of quadratic functions: equations Polynomial functions 5. Domain and range of polynomials Radical functions 6. Domain and range of radical functions Exponential and logarithmic functions 7. Domain and range of exponential functions 8. Domain and range of logarithmic functions Mixed functions 9. Domain and range
9-12.HSF-IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.		1. Average rate of change 2. Find the slope of a linear function
9-12.HSF-IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
1. Average rate of change 2. Find the slope of a linear function

9-12.HSF-IF.C Analyze functions using different representations.
9-12.HSF-IF.C Analyze functions using different representations.

9-12.HSF-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
9-12.HSF-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

9-12.HSF-IF.C.7b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.		1. Graph an absolute value function 2. Graph piecewise-defined functions 3. Graph square root functions 4. Graph cube root functions
9-12.HSF-IF.C.7b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
1. Graph an absolute value function 2. Graph piecewise-defined functions 3. Graph square root functions 4. Graph cube root functions
9-12.HSF-IF.C.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.		1. Graph a quadratic function 2. Match quadratic functions and graphs 3. Match polynomials and graphs
9-12.HSF-IF.C.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
1. Graph a quadratic function 2. Match quadratic functions and graphs 3. Match polynomials and graphs
9-12.HSF-IF.C.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.		Exponential functions 1. Graph exponential functions 2. Match exponential functions and graphs Logarithmic functions 3. Graph logarithmic functions Sine functions 4. Find properties of sine functions 5. Graph sine functions 6. Graph translations of sine functions Cosine functions 7. Find properties of cosine functions 8. Graph cosine functions 9. Graph translations of cosine functions Trigonometric functions 10. Graph sine and cosine functions 11. Graph translations of sine and cosine functions
9-12.HSF-IF.C.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Exponential functions 1. Graph exponential functions 2. Match exponential functions and graphs Logarithmic functions 3. Graph logarithmic functions Sine functions 4. Find properties of sine functions 5. Graph sine functions 6. Graph translations of sine functions Cosine functions 7. Find properties of cosine functions 8. Graph cosine functions 9. Graph translations of cosine functions Trigonometric functions 10. Graph sine and cosine functions 11. Graph translations of sine and cosine functions
9-12.HSF-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
9-12.HSF-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

9-12.HSF-IF.C.8a: 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.		1. Solve a quadratic equation by factoring 2. Solve a quadratic equation by completing the square
9-12.HSF-IF.C.8a: 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.
1. Solve a quadratic equation by factoring 2. Solve a quadratic equation by completing the square
9-12.HSF-IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions.		1. Evaluate exponential functions 2. Match exponential functions and graphs 3. Exponential functions over unit intervals 4. Solve exponential equations by rewriting the base
9-12.HSF-IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions.
1. Evaluate exponential functions 2. Match exponential functions and graphs 3. Exponential functions over unit intervals 4. Solve exponential equations by rewriting the base
9-12.HSF-IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
9-12.HSF-IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

9-12.HSF-BF Building Functions
9-12.HSF-BF.A Build a function that models a relationship between two quantities.
9-12.HSF-BF.A Build a function that models a relationship between two quantities.

9-12.HSF-BF.A.1: Write a function that describes a relationship between two quantities.
9-12.HSF-BF.A.1: Write a function that describes a relationship between two quantities.

9-12.HSF-BF.A.1b: Combine standard function types using arithmetic operations.		Polynomials 1. Add and subtract polynomials 2. Multiply polynomials Rational expressions 3. Multiply and divide rational expressions 4. Add and subtract rational expressions Functions 5. Add and subtract functions 6. Multiply functions 7. Divide functions
9-12.HSF-BF.A.1b: Combine standard function types using arithmetic operations.
Polynomials 1. Add and subtract polynomials 2. Multiply polynomials Rational expressions 3. Multiply and divide rational expressions 4. Add and subtract rational expressions Functions 5. Add and subtract functions 6. Multiply functions 7. Divide functions

9-12.HSF-BF.B Build new functions from existing functions.
9-12.HSF-BF.B Build new functions from existing functions.

9-12.HSF-BF.B.3: 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.		Translations 1. Translations of functions Reflections 2. Reflections of functions Dilations 3. Dilations of functions Mixed transformations 4. Transformations of absolute value functions 5. Function transformation rules 6. Transformations of functions 7. Describe function transformations
9-12.HSF-BF.B.3: 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.
Translations 1. Translations of functions Reflections 2. Reflections of functions Dilations 3. Dilations of functions Mixed transformations 4. Transformations of absolute value functions 5. Function transformation rules 6. Transformations of functions 7. Describe function transformations
9-12.HSF-BF.B.4: Find inverse functions.
9-12.HSF-BF.B.4: Find inverse functions.

9-12.HSF-BF.B.4a: 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.		Find inverse functions 1. Find inverse functions and relations Exponential equations 2. Solve exponential equations using common logarithms 3. Solve exponential equations using natural logarithms Logarithmic equations 4. Solve logarithmic equations I 5. Solve logarithmic equations II
9-12.HSF-BF.B.4a: 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.
Find inverse functions 1. Find inverse functions and relations Exponential equations 2. Solve exponential equations using common logarithms 3. Solve exponential equations using natural logarithms Logarithmic equations 4. Solve logarithmic equations I 5. Solve logarithmic equations II
9-12.HSF-LE Linear, Quadratic, and Exponential Models
9-12.HSF-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
9-12.HSF-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.

9-12.HSF-LE.A.4: For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.		Exponential and logarithmic form 1. Convert between exponential and logarithmic form: rational bases 2. Convert between natural exponential and logarithmic form 3. Convert between exponential and logarithmic form: all bases 4. Evaluate logarithms Solve exponential equations 5. Solve exponential equations using common logarithms 6. Solve exponential equations using natural logarithms
9-12.HSF-LE.A.4: For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Exponential and logarithmic form 1. Convert between exponential and logarithmic form: rational bases 2. Convert between natural exponential and logarithmic form 3. Convert between exponential and logarithmic form: all bases 4. Evaluate logarithms Solve exponential equations 5. Solve exponential equations using common logarithms 6. Solve exponential equations using natural logarithms
9-12.HSF-TF Trigonometric Functions
9-12.HSF-TF.A Extend the domain of trigonometric functions using the unit circle.
9-12.HSF-TF.A Extend the domain of trigonometric functions using the unit circle.

9-12.HSF-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.		1. Convert between radians and degrees 2. Radians and arc length
9-12.HSF-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
1. Convert between radians and degrees 2. Radians and arc length
9-12.HSF-TF.A.2: 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.		1. Coterminal angles 2. Reference angles 3. Find trigonometric ratios using the unit circle
9-12.HSF-TF.A.2: 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.
1. Coterminal angles 2. Reference angles 3. Find trigonometric ratios using the unit circle

9-12.HSF-TF.B Model periodic phenomena with trigonometric functions.
9-12.HSF-TF.B Model periodic phenomena with trigonometric functions.

9-12.HSF-TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.		Sine functions 1. Find properties of sine functions 2. Write equations of sine functions from graphs 3. Write equations of sine functions using properties Cosine functions 4. Find properties of cosine functions 5. Write equations of cosine functions from graphs 6. Write equations of cosine functions using properties
9-12.HSF-TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Sine functions 1. Find properties of sine functions 2. Write equations of sine functions from graphs 3. Write equations of sine functions using properties Cosine functions 4. Find properties of cosine functions 5. Write equations of cosine functions from graphs 6. Write equations of cosine functions using properties

9-12.HSF-TF.C Prove and apply trigonometric identities.
9-12.HSF-TF.C Prove and apply trigonometric identities.

9-12.HSF-TF.C.8: Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.		1. Find trigonometric ratios using a Pythagorean or reciprocal identity 2. Find trigonometric ratios using multiple identities
9-12.HSF-TF.C.8: Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
1. Find trigonometric ratios using a Pythagorean or reciprocal identity 2. Find trigonometric ratios using multiple identities

9-12.HSS	Statistics and Probability
9-12.HSS-ID Interpreting Categorical and Quantitative Data
9-12.HSS-ID.A Summarize, represent, and interpret data on a single count or measurement variable.
9-12.HSS-ID.A Summarize, represent, and interpret data on a single count or measurement variable.

9-12.HSS-ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.		1. Find probabilities using the normal distribution I 2. Find z-values 3. Find values of normal variables 4. Distributions of sample means
9-12.HSS-ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
1. Find probabilities using the normal distribution I 2. Find z-values 3. Find values of normal variables 4. Distributions of sample means
9-12.HSS-IC Making Inferences and Justifying Conclusions
9-12.HSS-IC.A Understand and evaluate random processes underlying statistical experiments.
9-12.HSS-IC.A Understand and evaluate random processes underlying statistical experiments.

9-12.HSS-IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.		1. Identify biased samples
9-12.HSS-IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
1. Identify biased samples
9-12.HSS-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
9-12.HSS-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.


9-12.HSS-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
9-12.HSS-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

9-12.HSS-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.		1. Experiment design
9-12.HSS-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
1. Experiment design
9-12.HSS-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.		1. Find confidence intervals for population means 2. Find confidence intervals for population proportions
9-12.HSS-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
1. Find confidence intervals for population means 2. Find confidence intervals for population proportions
9-12.HSS-IC.B.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.		1. Analyze the results of an experiment using simulations
9-12.HSS-IC.B.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
1. Analyze the results of an experiment using simulations
9-12.HSS-IC.B.6: Evaluate reports based on data.		1. Interpret confidence intervals for population means 2. Analyze the results of an experiment using simulations
9-12.HSS-IC.B.6: Evaluate reports based on data.
1. Interpret confidence intervals for population means 2. Analyze the results of an experiment using simulations
9-12.HSS-MD Using Probability to Make Decisions
9-12.HSS-MD.B Use probability to evaluate outcomes of decisions.
9-12.HSS-MD.B Use probability to evaluate outcomes of decisions.

9-12.HSS-MD.B.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
9-12.HSS-MD.B.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

9-12.HSS-MD.B.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).		1. Choose the better bet
9-12.HSS-MD.B.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
1. Choose the better bet