IXL Skill Plan | Maine Learning Results: Algebra 2

Maine Learning Results: Algebra 2

Skills available for Maine Algebra 2 standards

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Quantitative Reasoning
Algebraic Reasoning
Statistical Reasoning

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Quantitative Reasoning
Number and Quantity: Complex Number System
9-12.QR.A.4 Perform arithmetic operations with complex numbers.
9-12.QR.A.4 Perform arithmetic operations with complex numbers.

HSN.CN.A.1: Know there is a complex number i (which is a non-real number) such that i² = -1, and every complex number has the form a + bi with a and b real.		1. Introduction to complex numbers
HSN.CN.A.1: Know there is a complex number i (which is a non-real number) such that i² = -1, and every complex number has the form a + bi with a and b real.
1. Introduction to complex numbers
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
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.QR.A.6 Use complex numbers in polynomial identities and equations.
9-12.QR.A.6 Use complex numbers in polynomial identities and equations.

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
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
HSN.CN.C.8: Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).		1. Solve polynomial equations 2. Complex conjugate theorem 3. Conjugate root theorems
HSN.CN.C.8: Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).
1. Solve polynomial equations 2. Complex conjugate theorem 3. Conjugate root theorems
HSN.CN.C.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.		1. Fundamental Theorem of Algebra
HSN.CN.C.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
1. Fundamental Theorem of Algebra

Algebraic Reasoning
Algebra: Seeing Structure in Expressions
9-12.AR.A.1 Interpret the structure of expressions.
9-12.AR.A.1 Interpret the structure of expressions.

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

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
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
SSE.A.1b: Interpret multi-part expressions by viewing one or more of their parts as a single entity. For example, view P(1+r)ⁿ as the product of P and a factor not depending on P and interpret the parts.		1. Interpret parts of quadratic expressions: word problems
SSE.A.1b: Interpret multi-part expressions by viewing one or more of their parts as a single entity. For example, view P(1+r)ⁿ as the product of P and a factor not depending on P and interpret the parts.
1. Interpret parts of quadratic expressions: word problems
HSA.SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ - y⁴ as (x²)² - (y²)², allowing for it to be recognized as a difference of squares that can be factored as (x² - y²)(x² + y²).		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
HSA.SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ - y⁴ as (x²)² - (y²)², allowing for it to be recognized as a difference of squares that can be factored as (x² - y²)(x² + y²).
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.AR.A.2 Write expressions in equivalent forms to reveal information and to solve problems.
9-12.AR.A.2 Write expressions in equivalent forms to reveal information and to solve problems.

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. For example, Watermilfoil in one Maine lake triples in the number of plants each week during the summer when boat propellers are not cleared when exiting the lake. If the lake has 20 plants at the beginning of the season, how many plants will exist at the end of the 12-week summer season? What is the general formula for Watermilfoil growth for this lake?		1. Find the sum of a finite geometric series
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. For example, Watermilfoil in one Maine lake triples in the number of plants each week during the summer when boat propellers are not cleared when exiting the lake. If the lake has 20 plants at the beginning of the season, how many plants will exist at the end of the 12-week summer season? What is the general formula for Watermilfoil growth for this lake?
1. Find the sum of a finite geometric series
Algebra: Arithmetic with Polynomials & Rational Expressions
9-12.AR.A.3 Perform arithmetic operations on polynomials.
9-12.AR.A.3 Perform arithmetic operations on polynomials.

HSA.APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.
HSA.APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.

HSA.APR.A.1a: Perform operations on polynomial expressions (addition, subtraction, multiplication, and division), and compare the system of polynomials to the system of integers.		1. Add and subtract polynomials 2. Multiply polynomials
HSA.APR.A.1a: Perform operations on polynomial expressions (addition, subtraction, multiplication, and division), and compare the system of polynomials to the system of integers.
1. Add and subtract polynomials 2. Multiply polynomials
HSA.APR.A.1b: Factor and/or expand polynomial expressions, identify and combine like terms, and apply the Distributive Property.
HSA.APR.A.1b: Factor and/or expand polynomial expressions, identify and combine like terms, and apply the Distributive Property.


9-12.AR.A.4 Understand the relationship between zeros and factors of polynomials.
9-12.AR.A.4 Understand the relationship between zeros and factors of polynomials.

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). For example, consider the polynomial function P(x) = x⁴ – 2x³ + ax² + 8x + 12, where a is an unknown real number. If (x-3) is a factor of this polynomial, what is the value of a?		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
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). For example, consider the polynomial function P(x) = x⁴ – 2x³ + ax² + 8x + 12, where a is an unknown real number. If (x-3) is a factor of this polynomial, what is the value of a?
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
HSA.APR.B.3: Identify zeros of polynomials of degree three or higher when suitable factorizations (in factored form or easily factorable) 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 Polynomials 3. Find the roots of factored polynomials 4. Match polynomials and graphs using zeros
HSA.APR.B.3: Identify zeros of polynomials of degree three or higher when suitable factorizations (in factored form or easily factorable) 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 Polynomials 3. Find the roots of factored polynomials 4. Match polynomials and graphs using zeros

9-12.AR.A.5 Use polynomial identities to solve problems.
9-12.AR.A.5 Use polynomial identities to solve problems.

HSA.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.		1. Factor sums and differences of cubes
HSA.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.
1. Factor sums and differences of cubes
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. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.		1. Pascal's triangle 2. Pascal's triangle and the Binomial Theorem 3. Binomial Theorem I 4. Binomial Theorem II
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. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.
1. Pascal's triangle 2. Pascal's triangle and the Binomial Theorem 3. Binomial Theorem I 4. Binomial Theorem II

9-12.AR.A.6 Rewrite rational expressions.
9-12.AR.A.6 Rewrite rational expressions.

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
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
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
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
Algebra: Creating Equations and/or Inequalities
9-12.AR.A.7 Create equations and/or inequalities that describe numbers or relationships.
9-12.AR.A.7 Create equations and/or inequalities that describe numbers or relationships.

HSA.CED.A.1: 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.		1. Write inequalities from graphs 2. Solve quadratic equations: word problems 3. Write exponential functions: word problems
HSA.CED.A.1: 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.
1. Write inequalities from graphs 2. Solve quadratic equations: word problems 3. Write exponential functions: word problems
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 in standard form from its x-intercepts and another point 8. Write a quadratic function from three points Radical equations 9. Graph square root functions 10. Graph cube root functions Exponential and logarithmic equations 11. Write exponential functions: word problems 12. Graph exponential functions 13. Graph logarithmic functions Direct and inverse variation 14. Write and solve direct variation equations 15. Write and solve inverse variation equations 16. Write joint and combined variation equations I 17. Write joint and combined variation equations II 18. Solve variation equations Trigonometric equations 19. Write equations of sine functions from graphs 20. Write equations of sine functions using properties 21. Graph sine functions 22. Write equations of cosine functions from graphs 23. Write equations of cosine functions using properties 24. Graph cosine functions 25. Write equations of sine and cosine functions from graphs 26. Write equations of sine and cosine functions using properties 27. Graph sine and cosine functions
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 in standard form from its x-intercepts and another point 8. Write a quadratic function from three points Radical equations 9. Graph square root functions 10. Graph cube root functions Exponential and logarithmic equations 11. Write exponential functions: word problems 12. Graph exponential functions 13. Graph logarithmic functions Direct and inverse variation 14. Write and solve direct variation equations 15. Write and solve inverse variation equations 16. Write joint and combined variation equations I 17. Write joint and combined variation equations II 18. Solve variation equations Trigonometric equations 19. Write equations of sine functions from graphs 20. Write equations of sine functions using properties 21. Graph sine functions 22. Write equations of cosine functions from graphs 23. Write equations of cosine functions using properties 24. Graph cosine functions 25. Write equations of sine and cosine functions from graphs 26. Write equations of sine and cosine functions using properties 27. Graph sine and cosine functions
HSA.CED.A.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods such as lobsters, blueberries, and potatoes.		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
HSA.CED.A.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods such as lobsters, blueberries, and potatoes.
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
HSA.CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.		1. Rearrange multi-variable equations
HSA.CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
1. Rearrange multi-variable equations
Algebra: Reasoning with Equations & Inequalities
9-12.AR.A.8 Understand solving equations as a process of reasoning and explain the reasoning.
9-12.AR.A.8 Understand solving equations as a process of reasoning and explain the reasoning.

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
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.AR.A.11 Represent and solve equations and inequalities graphically.
9-12.AR.A.11 Represent and solve equations and inequalities graphically.

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
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
Functions: Interpreting Functions
9-12.AR.A.13 Interpret functions that arise in applications in terms of the context.
9-12.AR.A.13 Interpret functions that arise in applications in terms of the context.

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. Key features may include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative and absolute maximums and minimums; symmetries; end behavior; and periodicity.		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
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. Key features may include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative and absolute maximums and minimums; symmetries; end behavior; and periodicity.
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
HSF.IF.B.5: 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.		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
HSF.IF.B.5: 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.
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
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
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

9-12.AR.A.14 Analyze functions using different representations.
9-12.AR.A.14 Analyze functions using different representations.

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

HSF.IF.C.7b: i): Graph square root and piecewise-defined functions, (including step functions and absolute value functions), as well as show and describe key features of the graph.		1. Graph an absolute value function 2. Graph square root functions
HSF.IF.C.7b: i): Graph square root and piecewise-defined functions, (including step functions and absolute value functions), as well as show and describe key features of the graph.
1. Graph an absolute value function 2. Graph square root functions
HSF.IF.C.7b: ii): Graph cube root functions, as well as show and describe key features of the graph.
HSF.IF.C.7b: ii): Graph cube root functions, as well as show and describe key features of the graph.

HSF.IF.C.7c: Graph polynomial functions of degree three or higher, identifying zeros when suitable factorizations (in factored form or easily factorable) are available, and showing end behavior.		1. Graph a quadratic function 2. Match quadratic functions and graphs 3. Match polynomials and graphs
HSF.IF.C.7c: Graph polynomial functions of degree three or higher, identifying zeros when suitable factorizations (in factored form or easily factorable) are available, and showing end behavior.
1. Graph a quadratic function 2. Match quadratic functions and graphs 3. Match polynomials and graphs
HSF.IF.C.7e: i): Graph exponential functions, showing intercepts and end behavior.		1. Match exponential functions and graphs
HSF.IF.C.7e: i): Graph exponential functions, showing intercepts and end behavior.
1. Match exponential functions and graphs
HSF.IF.C.7e: ii): Graph logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline, and amplitude.		Sine functions 1. Find properties of sine functions 2. Graph sine functions 3. Graph translations of sine functions Cosine functions 4. Find properties of cosine functions 5. Graph cosine functions 6. Graph translations of cosine functions Sine and cosine functions 7. Graph sine and cosine functions 8. Graph translations of sine and cosine functions
HSF.IF.C.7e: ii): Graph logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline, and amplitude.
Sine functions 1. Find properties of sine functions 2. Graph sine functions 3. Graph translations of sine functions Cosine functions 4. Find properties of cosine functions 5. Graph cosine functions 6. Graph translations of cosine functions Sine and cosine functions 7. Graph sine and cosine functions 8. Graph translations of sine and cosine functions
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.
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.

HSF.IF.C.8a: Use the process of factoring and completing the square in a quadratic function to show zeros, maximum and minimum 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
HSF.IF.C.8a: Use the process of factoring and completing the square in a quadratic function to show zeros, maximum and minimum 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
HSF.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions. For example, apply the properties to financial situations such as identifying appreciation and depreciation rate for the value of a house or car sometime after its initial purchase: V_n = P(1 + r)ⁿ.		1. Evaluate exponential functions 2. Match exponential functions and graphs 3. Exponential functions over unit intervals 4. Solve exponential equations by rewriting the base
HSF.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions. For example, apply the properties to financial situations such as identifying appreciation and depreciation rate for the value of a house or car sometime after its initial purchase: V_n = P(1 + r)ⁿ.
1. Evaluate exponential functions 2. Match exponential functions and graphs 3. Exponential functions over unit intervals 4. Solve exponential equations by rewriting the base
HSF.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
HSF.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Functions: Building Functions
9-12.AR.A.15 Build a function that models a relationship between two quantities.
9-12.AR.A.15 Build a function that models a relationship between two quantities.

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

HSF.BF.A.1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.		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
HSF.BF.A.1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
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.AR.A.16 Build new functions from existing functions.
9-12.AR.A.16 Build new functions from existing functions.

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 20 an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.		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
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 20 an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
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
HSF.BF.B.4: Find inverse functions.
HSF.BF.B.4: Find inverse functions.

HSF.BF.B.4a: Solve an equation of the form f(x) = c (where c represents the output value of the function) for a simple function f that has an inverse and write an expression for the inverse. For example, if f(x) =2 x³, then solving f(x) = c leads to x = (c/2)^1/3, which is the general formula for finding an input from a specific output, c, for this function.		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
HSF.BF.B.4a: Solve an equation of the form f(x) = c (where c represents the output value of the function) for a simple function f that has an inverse and write an expression for the inverse. For example, if f(x) =2 x³, then solving f(x) = c leads to x = (c/2)^1/3, which is the general formula for finding an input from a specific output, c, for this function.
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
Functions: Linear, Quadratic, & Exponential Models
9-12.AR.A.17 Construct and compare linear, quadratic, and exponential models and solve problems.
9-12.AR.A.17 Construct and compare linear, quadratic, and exponential models and solve problems.

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
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
Functions: Trigonometric Functions
9-12.AR.A.19 Extend the domain of trigonometric functions using the unit circle.
9-12.AR.A.19 Extend the domain of trigonometric functions using the unit circle.

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
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
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
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.AR.A.20 Model periodic phenomena with trigonometric functions.
9-12.AR.A.20 Model periodic phenomena with trigonometric functions.

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
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.AR.A.21 Prove and apply trigonometric identities.
9-12.AR.A.21 Prove and apply trigonometric identities.

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
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

Statistical Reasoning
Statistics & Probability: Interpreting Categorical & Quantitative Data
9-12.SR.A.1 Summarize, represent, and interpret data on a single count or measurement variable.
9-12.SR.A.1 Summarize, represent, and interpret data on a single count or measurement variable.

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
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
Statistics & Probability: Making Inferences & Justifying Conclusions
9-12.SR.A.4 Understand and evaluate random processes underlying statistical experiments.
9-12.SR.A.4 Understand and evaluate random processes underlying statistical experiments.

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
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
HSS.IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
HSS.IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?


9-12.SR.A.5 Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
9-12.SR.A.5 Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

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
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
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
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
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
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
HSS.IC.B.6: Evaluate reports based on data. For example, use an article in the local news and interpret the validity of the information presented. Consider animal wildlife reports, medical studies, and/or manufacturer claims.		1. Interpret confidence intervals for population means 2. Analyze the results of an experiment using simulations
HSS.IC.B.6: Evaluate reports based on data. For example, use an article in the local news and interpret the validity of the information presented. Consider animal wildlife reports, medical studies, and/or manufacturer claims.
1. Interpret confidence intervals for population means 2. Analyze the results of an experiment using simulations
Statistics & Probability: Using Probability to Make Decisions
9-12.SR.A.9 Use probability to evaluate outcomes of decisions.
9-12.SR.A.9 Use probability to evaluate outcomes of decisions.

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

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 and replacing the goalie with an extra skater).		1. Choose the better bet
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 and replacing the goalie with an extra skater).
1. Choose the better bet