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912.A.CED.3 
12 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.4 
9 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.4 
10 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.4 
11 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.4 
12 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.REI.1 
9 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. 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. 


912.A.REI.1 
10 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. 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. 


912.A.REI.1 
11 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. 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. 


912.A.REI.1 
12 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. 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. 


912.A.REI.2 
9 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 


912.A.REI.2 
10 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 


912.A.REI.2 
11 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 


912.A.REI.2 
12 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 


912.A.REI.3 
9 
Algebra 
Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 


912.A.REI.3 
10 
Algebra 
Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 


912.A.REI.3 
11 
Algebra 
Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 


912.A.REI.3 
12 
Algebra 
Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 


912.A.REI.4 
9 
Algebra 
Solve equations and inequalities in one variable. Solve quadratic equations in one variable. 


912.A.REI.4 
10 
Algebra 
Solve equations and inequalities in one variable. Solve quadratic equations in one variable. 


912.A.REI.4 
11 
Algebra 
Solve equations and inequalities in one variable. Solve quadratic equations in one variable. 


912.A.REI.4 
12 
Algebra 
Solve equations and inequalities in one variable. Solve quadratic equations in one variable. 


912.A.REI.4a 
9 
Algebra 
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. 


912.A.REI.4a 
10 
Algebra 
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. 


912.A.REI.4a 
11 
Algebra 
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. 


912.A.REI.4a 
12 
Algebra 
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. 


912.A.REI.4b 
9 
Algebra 
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 Â± bi for real numbers a and b. 


912.A.REI.4b 
10 
Algebra 
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 Â± bi for real numbers a and b. 


912.A.REI.4b 
11 
Algebra 
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 Â± bi for real numbers a and b. 


912.A.REI.4b 
12 
Algebra 
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 Â± bi for real numbers a and b. 


912.A.REI.5 
9 
Algebra 
Solve systems of equations. Prove 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. 


912.A.REI.5 
10 
Algebra 
Solve systems of equations. Prove 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. 


912.A.REI.5 
11 
Algebra 
Solve systems of equations. Prove 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. 


912.A.REI.5 
12 
Algebra 
Solve systems of equations. Prove 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. 


912.A.REI.6 
9 
Algebra 
Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 


912.A.REI.6 
10 
Algebra 
Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 


912.A.REI.6 
11 
Algebra 
Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 


912.A.REI.6 
12 
Algebra 
Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 


912.A.REI.7 
9 
Algebra 
Solve systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x^2 + y^2 = 3. 


912.A.REI.7 
10 
Algebra 
Solve systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x^2 + y^2 = 3. 


912.A.REI.7 
11 
Algebra 
Solve systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x^2 + y^2 = 3. 


912.A.REI.7 
12 
Algebra 
Solve systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x^2 + y^2 = 3. 


912.A.REI.8 
9 
Algebra 
Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. 


912.A.REI.8 
10 
Algebra 
Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. 


912.A.REI.8 
11 
Algebra 
Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. 


912.A.REI.8 
12 
Algebra 
Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. 


912.A.REI.9 
9 
Algebra 
Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). 


912.A.REI.9 
10 
Algebra 
Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). 


912.A.REI.9 
11 
Algebra 
Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). 


912.A.REI.9 
12 
Algebra 
Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). 


912.A.REI.10 
9 
Algebra 
Represent and solve equations and inequalities graphically. 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). 


912.A.REI.10 
10 
Algebra 
Represent and solve equations and inequalities graphically. 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). 


912.A.REI.10 
11 
Algebra 
Represent and solve equations and inequalities graphically. 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). 


912.A.REI.10 
12 
Algebra 
Represent and solve equations and inequalities graphically. 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). 


912.A.REI.11 
9 
Algebra 
Represent and solve equations and inequalities graphically. Explain why the xcoordinates 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. 


912.A.REI.11 
10 
Algebra 
Represent and solve equations and inequalities graphically. Explain why the xcoordinates 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. 


912.A.REI.11 
11 
Algebra 
Represent and solve equations and inequalities graphically. Explain why the xcoordinates 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. 


912.A.REI.11 
12 
Algebra 
Represent and solve equations and inequalities graphically. Explain why the xcoordinates 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. 


912.A.REI.12 
9 
Algebra 
Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes. 


912.A.REI.12 
10 
Algebra 
Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes. 


912.A.REI.12 
11 
Algebra 
Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes. 


912.A.REI.12 
12 
Algebra 
Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes. 


912.F.IF.1 
9 
Functions 
Understand the concept of a function and use function notation. 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). 


912.F.IF.1 
10 
Functions 
Understand the concept of a function and use function notation. 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). 


912.F.IF.1 
11 
Functions 
Understand the concept of a function and use function notation. 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). 


912.F.IF.1 
12 
Functions 
Understand the concept of a function and use function notation. 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). 


912.F.IF.2 
9 
Functions 
Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 


912.F.IF.2 
10 
Functions 
Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 


912.F.IF.2 
11 
Functions 
Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 


912.F.IF.2 
12 
Functions 
Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 


912.F.IF.3 
9 
Functions 
Understand the concept of a function and use function notation. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ? 1 (n is greater than or equal to 1). 


912.F.IF.3 
10 
Functions 
Understand the concept of a function and use function notation. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ? 1 (n is greater than or equal to 1). 


912.F.IF.3 
11 
Functions 
Understand the concept of a function and use function notation. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ? 1 (n is greater than or equal to 1). 


912.F.IF.3 
12 
Functions 
Understand the concept of a function and use function notation. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ? 1 (n is greater than or equal to 1). 


912.F.IF.4 
9 
Functions 
Interpret functions that arise in applications in terms of the context. 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 maximums and minimums; symmetries; end behavior; and periodicity. 


912.F.IF.4 
10 
Functions 
Interpret functions that arise in applications in terms of the context. 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 maximums and minimums; symmetries; end behavior; and periodicity. 


912.F.IF.4 
11 
Functions 
Interpret functions that arise in applications in terms of the context. 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 maximums and minimums; symmetries; end behavior; and periodicity. 


912.F.IF.4 
12 
Functions 
Interpret functions that arise in applications in terms of the context. 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 maximums and minimums; symmetries; end behavior; and periodicity. 


912.F.IF.5 
9 
Functions 
Interpret functions that arise in applications in terms of the context. 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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 


912.F.IF.5 
10 
Functions 
Interpret functions that arise in applications in terms of the context. 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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 


912.F.IF.5 
11 
Functions 
Interpret functions that arise in applications in terms of the context. 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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 


912.F.IF.5 
12 
Functions 
Interpret functions that arise in applications in terms of the context. 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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 


912.F.IF.6 
9 
Functions 
Interpret functions that arise in applications in terms of the context. 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. 


912.F.IF.6 
10 
Functions 
Interpret functions that arise in applications in terms of the context. 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. 


912.F.IF.6 
11 
Functions 
Interpret functions that arise in applications in terms of the context. 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. 


912.F.IF.6 
12 
Functions 
Interpret functions that arise in applications in terms of the context. 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. 


912.F.IF.7 
9 
Functions 
Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 


912.F.IF.7 
10 
Functions 
Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 


912.F.IF.7 
11 
Functions 
Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 


912.F.IF.7 
12 
Functions 
Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 


912.F.IF.7a 
9 
Functions 
Graph linear and quadratic functions and show intercepts, maxima, and minima. 


912.F.IF.7a 
10 
Functions 
Graph linear and quadratic functions and show intercepts, maxima, and minima. 


912.F.IF.7a 
11 
Functions 
Graph linear and quadratic functions and show intercepts, maxima, and minima. 


912.F.IF.7a 
12 
Functions 
Graph linear and quadratic functions and show intercepts, maxima, and minima. 


912.F.IF.7b 
9 
Functions 
Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. 


912.F.IF.7b 
10 
Functions 
Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. 


912.F.IF.7b 
11 
Functions 
Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. 


912.F.IF.7b 
12 
Functions 
Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. 


912.F.IF.7c 
9 
Functions 
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 


912.F.IF.7c 
10 
Functions 
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 


912.F.IF.7c 
11 
Functions 
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 

