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### A2.N Number and Quantity

- A2.N-RN The Real Number System
- A2.N-RN.A Extend the properties of exponents to rational exponents.
- A2.N-RN.A.1 Explain how the definition of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
- A2.N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

- A2.N-RN.A Extend the properties of exponents to rational exponents.
- A2.N-Q Quantities
- A2-N-Q.A Reason quantitatively and use units to solve problems.
- A2-N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
- A2.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
- A2.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

- A2-N-Q.A Reason quantitatively and use units to solve problems.
- A2.N-CN The Complex Number System
- A2.N-CN.A Perform arithmetic operations with complex numbers.
- A2.N-CN.A.1 Apply the relation
*i*² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Write complex numbers in the form (*a*+ b*i*) with*a*and*b*real.

- A2.N-CN.A.1 Apply the relation
- A2.N-CN.C Use complex numbers in polynomial identities and equations.
- A2.N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

- A2.N-CN.A Perform arithmetic operations with complex numbers.

### A2.A Algebra

- A2.A-SSE Seeing Structure in Expressions
- A2.A-SSE.A Interpret the structure of expressions.
- A2.A-SSE.A.2 Use structure to identify ways to rewrite polynomial and rational expressions. Focus on polynomial operations and factoring patterns.

- A2-A-SSE.B Write expressions in equivalent forms to solve problems.
- A2.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Include problem-solving opportunities utilizing real-world context and focus on expressions with rational exponents.
- A2.A-SSE.B.3.c Use the properties of exponents to transform expressions for exponential functions.

- A2.A-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.

- A2.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Include problem-solving opportunities utilizing real-world context and focus on expressions with rational exponents.

- A2.A-SSE.A Interpret the structure of expressions.
- A2.A-APR Arithmetic with Polynomials and Rational Expressions
- A2.A-APR.B Understand the relationship between zeros and factors of polynomials.
- A2.A-APR.B.2 Know and apply the Remainder and Factor 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*). - A2.A-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. Focus on quadratic, cubic, and quartic polynomials including polynomials for which factors are not provided.

- A2.A-APR.B.2 Know and apply the Remainder and Factor Theorem: For a polynomial
- A2.A-APR.C Use polynomial identities to solve problems.
- A2.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

- A2.A-APR.D Rewrite rational expressions.
- A2.A-APR.D.6 Rewrite 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.

- A2.A-APR.D.6 Rewrite rational expressions in different forms; write

- A2.A-APR.B Understand the relationship between zeros and factors of polynomials.
- A2.A-CED Creating Equations
- A2.A-CED.A Create equations that describe numbers or relationships.
- A2.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities arising from linear, quadratic, rational, and exponential functions.

- A2.A-CED.A Create equations that describe numbers or relationships.
- A2.A-REI Reasoning with Equations and Inequalities
- A2.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
- A2.A-REI.A.1 Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Extend from quadratic equations to rational and radical equations.
- A2.A-REI.A.2 Solve rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

- A2-A-REI.B Solve equations and inequalities in one variable.
- A2.A-REI.B.4 Fluently solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for
*x*² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as*a*± b*i*for real numbers*a*and b.

- A2.A-REI.B.4 Fluently solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for
- A2-A-REI.C Solve systems of equations.
- A2.A-REI.C.7 Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

- A2-A-REI.D Represent and solve equations and inequalities graphically.
- A2.A-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 problems in real-world context. Extend from linear, quadratic, and exponential functions to cases where*f*(*x*) and/or*g*(*x*) are polynomial, rational, exponential, and logarithmic functions.

- A2.A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations

- A2.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.

### A2.F Functions

- A2.F-IF Interpreting Functions
- A2.F-IF.B Interpret functions that arise in applications in terms of the context.
- A2.F-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. Include problem-solving opportunities utilizing a real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Extend from linear, quadratic and exponential to include polynomial, radical, logarithmic, rational, sine, cosine, tangent, exponential, and piecewise-defined functions.
- A2.F-IF.B.6 Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, tangent, exponential, and piecewise-defined functions.

- A2.F-IF.C Analyze functions using different representations.
- A2.F-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. Extend from linear, quadratic and exponential functions to include square root, cube root, polynomial, exponential, logarithmic, sine, cosine, tangent and piecewise-defined functions.
- A2.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- A2.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions and classify those functions as exponential growth or decay.

- A2.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.). Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, trigonometric, exponential, and piecewise-defined functions.

- A2.F-IF.B Interpret functions that arise in applications in terms of the context.
- A2.F-BF Building Functions
- A2.F-BF.A Build a function that models a relationship between two quantities.
- A2.F-BF.A.1 Write a function that describes a relationship between two quantities. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, exponential, and piecewise-defined functions. Include problem-solving opportunities utilizing real-world context.
- A2.F-BF.A.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
- A2.F-BF.A.1.b Combine function types using arithmetic operations and function composition.

- A2.F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

- A2.F-BF.A.1 Write a function that describes a relationship between two quantities. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, exponential, and piecewise-defined functions. Include problem-solving opportunities utilizing real-world context.
- A2.F-BF.B Build new functions from existing functions.
- A2.F-BF.B.3 Identify the effect on the graph of replacing
*f*(*x*) by*f*(*x*) +*k*,*k**f*(*x*), 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. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, and exponential functions, and piecewise-defined functions. - A2.F-BF.B.4 Find inverse functions.
- A2.F-BF.B.4.a Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, recognizing that functions
*f*and*g*are inverse functions if and only if*f*(*x*) =*y*and*g*(*y*) =*x*for all values of*x*in the domain of*f*and all values of*y*in the domain of*g*. - A2.F-BF.B.4.b Understand that if a function contains a point (
*a*, b), then the graph of the inverse relation of the function contains the point (b,*a*). - A2.F-BF.B.4.c Interpret the meaning of and relationship between a function and its inverse utilizing real-world context.

- A2.F-BF.B.4.a Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, recognizing that functions

- A2.F-BF.B.3 Identify the effect on the graph of replacing

- A2.F-BF.A Build a function that models a relationship between two quantities.
- A2.F-LE Linear, Quadratic, and Exponential Models
- A2.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
- A2.F-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 logarithms that are not readily found by hand or observation using technology.

- A2.F-LE.A.4 For exponential models, express as a logarithm the solution to
- A2.F-LE.B Interpret expressions for functions in terms of the situation they model.
- A2.F-LE.B.5 Interpret the parameters in an exponential function with rational exponents utilizing real-world context.

- A2.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
- A2.F-TF Trigonometric Functions
- A2.F-TF.A Extend the domain of trigonometric functions using the unit circle.
- A2.F-TF.A.1 Understand radian measure of an angle as the length of the arc on any circle subtended by the angle, measured in units of the circle’s radius.
- A2.F-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of sine and cosine functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

- A2.F-TF.B Model periodic phenomena with trigonometric functions.
- A2.F-TF.B.5 Create and interpret trigonometric functions that model periodic phenomena with specified amplitude, frequency, and midline.

- A2.F-TF.C Apply trigonometric identities.
- A2.F-TF.C.8 Use 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.

- A2.F-TF.A Extend the domain of trigonometric functions using the unit circle.

### A2.S Statistics and Probability

- A2.S-ID Interpreting Categorical and Quantitative Data
- A2.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable.
- A2.S-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal curve, and use properties of the normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, or tables to estimate areas under the normal curve.

- A2.S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
- A2.S-ID.B.6 Represent data of two quantitative variables on a scatter plot, and describe how the quantities are related. Extend to polynomial and exponential models.
- A2.S-ID.B.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.

- A2.S-ID.B.6 Represent data of two quantitative variables on a scatter plot, and describe how the quantities are related. Extend to polynomial and exponential models.
- A2.S-ID.C Interpret models.
- A2.S-ID.C.10 Interpret parameters of exponential models.

- A2.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable.
- A2.S-IC Making Inferences and Justifying Conclusions
- A2.S-IC.A Understand and evaluate random processes underlying statistical experiments.
- A2.S-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
- A2.S-IC.A.2 Explain whether a specified model is consistent with results from a given data-generating process.

- A2.S-IC.B Make inferences and justify conclusions from experiments, and observational studies.
- A2.S-IC.B.3 Recognize the purposes of and differences between designed experiments, sample surveys and observational studies.
- A2.S-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; recognize that estimates are unlikely to be correct and the estimates will be more precise with larger sample sizes.

- A2.S-IC.A Understand and evaluate random processes underlying statistical experiments.
- A2.S-CP Conditional Probability and the Rules of Probability
- A2.S-CP.A Understand independence and conditional probability and use them to interpret data.
- A2.S-CP.A.3 Understand the conditional probability of
*A*given*B*as*P*(*A**a**n**d**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*. - A2.S-CP.A.4 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.
- A2.S-CP.A.5 Recognize and explain the concepts of conditional probability and independence utilizing real-world context.

- A2.S-CP.A.3 Understand the conditional probability of
- A2.S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- A2.S-CP.B.6 Use Bayes Rule to find the conditional probability of
*A*given*B*as the fraction of*B*‘s outcomes that also belong to*A*, and interpret the answer in terms of the model. - A2.S-CP.B.7 Apply the Addition Rule,
*P*(*A**o**r**B*) =*P*(*A*) +*P*(*B*) –*P*(*A**a**n**d**B*), and interpret the answer in terms of the model. - A2.S-CP.B.8 Apply the general Multiplication Rule in a uniform probability model,
*P*(*A**a**n**d**B*) =*P*(*A*)*P*(*B*|*A*) =*P*(*B*)*P*(*A*|*B*), and interpret the answer in terms of the model.

- A2.S-CP.B.6 Use Bayes Rule to find the conditional probability of

- A2.S-CP.A Understand independence and conditional probability and use them to interpret data.