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### N Number and Quantity

- N.RN The Real Number System
- Use properties of rational numbers and irrational numbers.
- N.RN.2 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
- N.RN.3 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

- Use properties of rational numbers and irrational numbers.
- N.Q Quantities
- Reason quantitatively and use units to solve problems.
- N.Q.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.
- N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
- N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

- Reason quantitatively and use units to solve problems.
- N.CN The Complex Number System
- Perform arithmetic operations with complex numbers.
- N.CN.1 Know there is a complex number i such that
*i*² = –1, and every complex number has the form*a*+ b*i*with a and b real. - N.CN.2 Use the relation
*i*² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - N.CN.3 Find the conjugate of a complex number.

- N.CN.1 Know there is a complex number i such that
- Use complex numbers in polynomial identities and equations.
- N.CN.8 Solve quadratic equations with real coefficients that have complex solutions.

- Perform arithmetic operations with complex numbers.
- N.VM Vector and Matrix Quantities
- Perform operations on matrices and use matrices in applications.
- N.VM.6 Use matrices to represent and manipulate data, (e.g., to represent payoffs or incidence relationships in a network).
- N.VM.7 Multiply matrices by scalars to produce new matrices, (e.g., as when all of the payoffs in a game are doubled).
- N.VM.8 Add, subtract, and multiply matrices of appropriate dimensions; find determinants of 2×2 matrices.

- Perform operations on matrices and use matrices in applications.

### A Algebra

- A.SSE Seeing Structure in Expressions
- Interpret the structure of expressions.
- A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
- A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.
- A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
*P*(1 +*r*)^{n}as the product of P and (1 +*r*)^{n}.

- A.SSE.2 Use the structure of an expression to identify ways to rewrite it.

- A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
- Write expressions in equivalent forms to solve problems.
- A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
- A.SSE.3c Use the properties of exponents to transform expressions for exponential functions.

- A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

- Interpret the structure of expressions.
- A.APR Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials.
- A.APR.2 Factor higher degree polynomials; identifying that some polynomials are prime.
- A.APR.3 Know and apply the Remainder Theorem: For a polynomial
*p*(*x*) and a number*c*, the remainder on division by (*x*–*c*) is*p*(*c*), so*p*(*c*) = 0 if and only if (*x*–*c*) is a factor of*p*(*x*).

- Use polynomial identities to solve problems.
- A.APR.4 Generate polynomial identities from a pattern.

- Perform arithmetic operations on polynomials.
- A.CED Creating Equations
- Create equations that describe numbers or relationships.
- A.CED.1 Apply and extend previous understanding to create equations and inequalities in one variable and use them to solve problems.
- A.CED.2 Apply and extend previous understanding to create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A.CED.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.
- A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

- Create equations that describe numbers or relationships.
- A.REI Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning.
- A.REI.1 Explain each step in solving a simple 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.

- Solve equations and inequalities in one variable.
- A.REI.2 Apply and extend previous understanding to solve equations, inequalities, and compound inequalities in one variable, including literal equations and inequalities.
- A.REI.3 Solve equations in one variable and give examples showing how extraneous solutions may arise.
- A.REI.3a Solve rational, absolute value and square root equations.
- A.REI.3b Solve exponential and logarithmic equations.

- A.REI.4 Solve radical and rational exponent equations and inequalities in one variable, and give examples showing how extraneous solutions may arise.
- A.REI.5 Solve quadratic equations and inequalities.
- A.REI.5b Solve quadratic equations with complex solutions written in the form
*a*± b*i*for real numbers*a*and b. - A.REI.5c Use the method of completing the square to transform and solve any quadratic equation in
*x*into an equation of the form (*x*–*p*)² =*q*that has the same solutions. - A.REI.5d Solve quadratic inequalities and identify the domain.

- A.REI.5b Solve quadratic equations with complex solutions written in the form

- Solve systems of equations.
- A.REI.7 Represent a system of linear equations as a single matrix equation and solve (incorporating technology) for matrices of dimension 3 × 3 or greater.

- Represent and solve equations and inequalities graphically.
- A.REI.8 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
- A.REI.9 Solve an equation
*f*(*x*) =*g*(*x*) by graphing*y*=*f*(*x*) and*y*=*g*(*x*) and finding the*x*-value of the intersection point. Include cases where*f*(*x*) and/or*g*(*x*) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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

### F Functions

- F.IF Interpreting Functions
- Understand the concept of a function and use function notation.
- F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and
*x*is an element of its domain, then*f*(*x*) denotes the output of*f*corresponding to the input*x*. The graph of*f*is the graph of the equation*y*=*f*(*x*). - F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F.IF.3 Recognize patterns in order to write functions whose domain is a subset of the integers.

- F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and
- Interpret functions that arise in applications in terms of the context.
- F.IF.4 For a function that models a relationship between two quantities, interpret key features of expressions, graphs and tables in terms of the quantities, and sketch graphs showing key features given a description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
- F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
- F.IF.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.

- Analyze functions using different representations.
- F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
- F.IF.7b Graph square root, cube root, and exponential functions.
- F.IF.7c Graph logarithmic functions, emphasizing the inverse relationship with exponentials and showing intercepts and end behavior.
- F.IF.7d Graph piecewise-defined functions, including step functions.
- F.IF.7e Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
- F.IF.7f Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
- F.IF.7g Graph trigonometric functions, showing period, midline, and amplitude.

- F.IF.8 Write a function in different but equivalent forms to reveal and explain different properties of the function.
- F.IF.8b 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.
- F.IF.8c Use the properties of exponents to interpret expressions for exponential functions.

- F.IF.9 Compare properties of two functions using a variety of representations (algebraically, graphically, numerically in tables, or by verbal descriptions).

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

- Understand the concept of a function and use function notation.
- F.BF Building Functions
- Build a function that models a relationship between two quantities.
- F.BF.1 Use functions to model real-world relationships.
- F.BF.1b Determine an explicit expression, a recursive function, or steps for calculation from a context.
- F.BF.1c Compose functions.

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

- F.BF.1 Use functions to model real-world relationships.
- Build new functions from existing functions.
- F.BF.3 Transform parent functions (
*f*(*x*)) by replacing*f*(*x*) with*f*(*x*) +*k*,*k**f*(*x*),*f*(*k**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. - F.BF.4 Find inverse functions.
- F.BF.4a Write an expression for the inverse of a function.
- F.BF.4b Read values of an inverse function from a graph or a table, given that the function has an inverse.
- F.BF.4c Verify by composition that one function is the inverse of another.
- F.BF.4d Produce an invertible function from a non-invertible function by restricting the domain.

- F.BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

- F.BF.3 Transform parent functions (

- Build a function that models a relationship between two quantities.
- F.LQE Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems.
- F.LQE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
- F.LQE.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
- F.LQE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- F.LQE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

- F.LQE.2 Construct exponential functions, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

- F.LQE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

- Construct and compare linear, quadratic, and exponential models and solve problems.

### G Geometry

- G.GPE Expressing Geometric Properties with Equations
- Translate between the geometric description and the equation for a conic section.
- G.GPE.2 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; graph the circle in the coordinate plane;
- G.GPE.3 Complete the square to find the center and radius of a circle given by an equation.
- G.GPE.4 Derive the equation of a parabola given a focus and directrix; graph the parabola in the coordinate plane.
- G.GPE.5 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant; graph the ellipse or hyperbola in the coordinate plane.

- Translate between the geometric description and the equation for a conic section.

### S Statistics and Probability

- S.ID Interpreting Categorical and Quantitative Data
- Summarize, represent, and interpret data on a single count or measurement variable.
- S.ID.3 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.

- Interpret linear models.
- S.ID.7 Compute (using technology) and interpret the correlation coefficient of a linear fit.
- S.ID.8 Distinguish between correlation and causation.

- Summarize, represent, and interpret data on a single count or measurement variable.
- S.CP Conditional Probability and the Rules of Probability
- Understand independent and conditional probability and use them to interpret data.
- S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
- S.CP.2 Understand that two events
*A*and*B*are independent if the probability of*A*and*B*occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - S.CP.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*. - S.CP.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.
- S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

- Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- S.CP.6 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. - S.CP.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. - S.CP.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. - S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

- S.CP.6 Find the conditional probability of

- Understand independent and conditional probability and use them to interpret data.
- S.MD Using Probability to Make Decisions
- Use probability to evaluate outcomes of decisions.
- S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

- Use probability to evaluate outcomes of decisions.