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### N Number and quantity

- N.RN The Real Number System
- N.RN.A Extend the properties of exponents to rational exponents.
- N.RN.A.1 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.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

- N.RN.B Use properties of rational and irrational numbers.
- N.RN.B.3 Explain why the sum or product of two rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational.
- Checkpoint opportunity

- N.RN.A Extend the properties of exponents to rational exponents.
- N.Q Quantities
- N.Q.A Reason quantitatively and use units to solve problems.
- 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.
- N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- Checkpoint opportunity

- N.Q.A Reason quantitatively and use units to solve problems.

### A Algebra

- A.SSE Seeing Structure in Expressions
- A.SSE.A Interpret the structure of linear, quadratic, exponential, polynomial, and rational expressions.
- A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
- A.SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
- A.SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.

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

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

- Checkpoint opportunity

- 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.

- A.SSE.A Interpret the structure of linear, quadratic, exponential, polynomial, and rational expressions.
- A.APR Arithmetic with Polynomials and Rational Expressions
- A.APR.A Perform arithmetic operations on polynomials.
- A.APR.A.1 Demonstrate understanding that polynomials form a system analogous to the integers; namely, they are closed under certain operations.
- A.APR.A.1.a Perform operations on polynomial expressions (addition, subtraction, multiplication, division) and compare the system of polynomials to the system of integers when performing operations.
- A.APR.A.1.b Factor and/or expand polynomial expressions, identify and combine like terms, and apply the distributive property.

- Checkpoint opportunity

- A.APR.A.1 Demonstrate understanding that polynomials form a system analogous to the integers; namely, they are closed under certain operations.

- A.APR.A Perform arithmetic operations on polynomials.
- A.CED Creating Equations
- A.CED.A Create equations that describe numbers or relationships.
- A.CED.A.1 Create one-variable equations and inequalities to solve problems, including linear, quadratic, rational, and exponential functions.
- A.CED.A.2 Interpret the relationship between two or more quantities.
- A.CED.A.2.a Define variables to represent the quantities and write equations to show the relationship.
- A.CED.A.2.b Use graphs to show a visual representation of the relationship while adhering to appropriate labels and scales.

- A.CED.A.3 Represent constraints using equations or inequalities and interpret solutions as viable or non-viable options in a modeling context.
- A.CED.A.4 Represent constraints using systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.
- A.CED.A.5 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
- Checkpoint opportunity

- A.CED.A Create equations that describe numbers or relationships.
- A.REI Reasoning with Equations and Inequalities
- A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
- A.REI.A.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 or refute a solution method.

- A.REI.B Solve equations and inequalities in one variable.
- A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
- A.REI.B.4 Solve quadratic equations in one variable.
- A.REI.B.4.a Use the method of completing the square to transform any quadratic equation in
*x*into an equation of the form (*x*–*p*)^{2}=*q*that has the same solutions. Derive the quadratic formula from this form. - A.REI.B.4.b 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*±*bi*for real numbers*a*and*b*.

- A.REI.B.4.a Use the method of completing the square to transform any quadratic equation in
- Checkpoint opportunity

- A.REI.C Solve systems of equations.
- A.REI.C.5 Verify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

- A.REI.D Represent and solve equations and inequalities graphically.
- A.REI.D.10 Demonstrate understanding that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Show that any point on the graph of an equation in two variables is a solution to the equation.
- 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. Include cases where*f*(*x*) and/or*g*(*x*) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. - A.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- Checkpoint opportunity

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

### F Functions

- F.IF Interpreting Functions
- F.IF.A Understand the concept of a function and use function notation.
- F.IF.A.1 Demonstrate understanding that a function is a correspondence from one set (called the domain) to another set (called the range) that 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.A.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.A.3 Demonstrate that a sequence is a functions, sometimes defined recursively, whose domain is a subset of the integers.

- F.IF.A.1 Demonstrate understanding that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range: If
- F.IF.B Interpret functions that arise in applications in terms of the context. Include linear, quadratic, exponential, rational, polynomial, square root and cube root, trigonometric, and logarithmic functions.
- 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima; symmetries; end behavior; and periodicity.
- F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
- F.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.
- Checkpoint opportunity

- F.IF.C Analyze functions using different representations.
- F.IF.C.7 Graph functions expressed symbolically and show key features of the graphs, by hand in simple cases and using technology for more complicated cases.
- F.IF.C.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- F.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

- 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.
- F.IF.C.8.a Use the process of factoring and/or completing the square in quadratic and polynomial functions, where appropriate, to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
- F.IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as identifying appreciation and depreciation rate for the value of a house or car sometime after its initial purchase.

- F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
- Checkpoint opportunity

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

- F.IF.A Understand the concept of a function and use function notation.
- F.BF Building Functions
- F.BF.A Build a function that models a relationship between two quantities.
- F.BF.A.1 Write a function that describes a relationship between two quantities. Functions could include linear, exponential, quadratic, simple rational, radical, logarithmic, and trigonometric.
- F.BF.A.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F.BF.A.1.b Combine standard function types using arithmetic operations.

- 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.
- Checkpoint opportunity

- F.BF.A.1 Write a function that describes a relationship between two quantities. Functions could include linear, exponential, quadratic, simple rational, radical, logarithmic, and trigonometric.
- F.BF.B Build new functions from existing functions.
- F.BF.B.3 Identify the effect on the graph of replacing
*f*(*x*) by*f*(*x*) +*k*,*kf*(*x*),*f*(*kx*), and*f*(*x*+*k*) for specific values of*k*(both positive and negative); find the value of*k*given the graphs. Include, linear, quadratic, exponential, absolute value, simple rational and radical, logarithmic, and trigonometric functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph. Include recognizing even and odd functions from their graphs and algebraic expressions for them. - F.BF.B.4 Find inverse functions algebraically and graphically.
- F.BF.B.4.a Solve an equation of the form
*f*(*x*) =*c*for a simple function*f*that has an inverse and write an expression for the inverse. Include linear and simple polynomial, rational, and exponential functions.

- F.BF.B.4.a Solve an equation of the form
- Checkpoint opportunity

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

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

- F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).
- F.LE.A.3 Use graphs and tables to demonstrate that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

- F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
- F.LE.B Interpret expressions for functions in terms of the situation they model.
- F.LE.B.5 Interpret the parameters in a linear or exponential function (of the form
*f*(*x*) =*b*+^{x}*k*) in terms of a context. - Checkpoint opportunity

- F.LE.B.5 Interpret the parameters in a linear or exponential function (of the form

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

### S Statistics and Probability

- S.ID Interpreting Categorical and Quantitative Data
- S.ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.
- S.ID.A.1 Differentiate between count data and measurement variable.
- S.ID.A.2 Represent measurement data with plots on the real number line (dot plots, histograms, and box plots).
- S.ID.A.3 Compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different variables, using statistics appropriate to the shape of the distribution for each measurement variable.
- Checkpoint opportunity

- S.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
- S.ID.B.6 Represent data on two categorical variables on a clustered bar chart and describe how the variables are related. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S.ID.B.7 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
- S.ID.B.7.a Fit a linear function to data where a scatter plot suggests a linear relationship and use the fitted function to solve problems in the context of the data.
- S.ID.B.7.b Use functions fitted to data, focusing on quadratic and exponential models, or choose a function suggested by the context. Utilize technology where appropriate.
- S.ID.B.7.c Informally assess the fit of a function by plotting and analyzing residuals.

- Checkpoint opportunity

- S.ID.C Interpret linear models.
- S.ID.C.8 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
- S.ID.C.9 Compute (using technology) and interpret the linear correlation coefficient.
- S.ID.C.10 Distinguish between (linear) correlation and causation.
- Checkpoint opportunity

- S.ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.