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912.A.APR.4 
11 
Algebra 
Use polynomial identities to solve problems. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2  y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples. 


912.A.APR.4 
12 
Algebra 
Use polynomial identities to solve problems. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2  y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples. 


912.A.APR.5 
9 
Algebra 
Use polynomial identities to solve problems. Know and apply that the Binomial Theorem gives 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.). 


912.A.APR.5 
10 
Algebra 
Use polynomial identities to solve problems. Know and apply that the Binomial Theorem gives 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.). 


912.A.APR.5 
11 
Algebra 
Use polynomial identities to solve problems. Know and apply that the Binomial Theorem gives 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.). 


912.A.APR.5 
12 
Algebra 
Use polynomial identities to solve problems. Know and apply that the Binomial Theorem gives 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.). 


912.A.APR.6 
9 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.6 
10 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.6 
11 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.6 
12 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.7 
9 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.7 
10 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.7 
11 
Algebra 
Rewrite rational expressions. 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. 


912.A.APR.7 
12 
Algebra 
Rewrite rational expressions. 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. 


912.A.CED.1 
9 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.1 
10 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.1 
11 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.1 
12 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.2 
9 
Algebra 
Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 


912.A.CED.2 
10 
Algebra 
Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 


912.A.CED.2 
11 
Algebra 
Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 


912.A.CED.2 
12 
Algebra 
Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 


912.A.CED.3 
9 
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.3 
10 
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.3 
11 
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.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.A.SSE.1 
9 
Algebra 
Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. 


912.A.SSE.1 
10 
Algebra 
Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. 


912.A.SSE.1 
11 
Algebra 
Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. 


912.A.SSE.1 
12 
Algebra 
Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. 


912.A.SSE.1a 
9 
Algebra 
Interpret parts of an expression, such as terms, factors, and coefficients. 


912.A.SSE.1a 
10 
Algebra 
Interpret parts of an expression, such as terms, factors, and coefficients. 


912.A.SSE.1a 
11 
Algebra 
Interpret parts of an expression, such as terms, factors, and coefficients. 


912.A.SSE.1a 
12 
Algebra 
Interpret parts of an expression, such as terms, factors, and coefficients. 


912.A.SSE.1b 
9 
Algebra 
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 a factor not depending on P. 


912.A.SSE.1b 
10 
Algebra 
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 a factor not depending on P. 


912.A.SSE.1b 
11 
Algebra 
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 a factor not depending on P. 


912.A.SSE.1b 
12 
Algebra 
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 a factor not depending on P. 


912.A.SSE.2 
9 
Algebra 
Interpret the structure of expressions. Use the structure of an expression to identify ways to rewrite it. For example, see x^4  y^4 as (x^2)^2  (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2  y^2)(x^2 + y^2). 


912.A.SSE.2 
10 
Algebra 
Interpret the structure of expressions. Use the structure of an expression to identify ways to rewrite it. For example, see x^4  y^4 as (x^2)^2  (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2  y^2)(x^2 + y^2). 

