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912.N.VM.6 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 


912.N.VM.7 
9 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 


912.N.VM.7 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 


912.N.VM.7 
11 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 


912.N.VM.7 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 


912.N.VM.8 
9 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate dimensions. 


912.N.VM.8 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate dimensions. 


912.N.VM.8 
11 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate dimensions. 


912.N.VM.8 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate dimensions. 


912.N.VM.9 
9 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 


912.N.VM.9 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 


912.N.VM.9 
11 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 


912.N.VM.9 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 


912.N.VM.10 
9 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 


912.N.VM.10 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 


912.N.VM.10 
11 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 


912.N.VM.10 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 


912.N.VM.11 
9 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 


912.N.VM.11 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 


912.N.VM.11 
11 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 


912.N.VM.11 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 


912.N.VM.12 
9 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. 


912.N.VM.12 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. 


912.N.VM.12 
11 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. 


912.N.VM.12 
12 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. 


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). 


912.A.SSE.2 
11 
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 
12 
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.3 
9 
Algebra 
Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 


912.A.SSE.3 
10 
Algebra 
Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 


912.A.SSE.3 
11 
Algebra 
Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 


912.A.SSE.3 
12 
Algebra 
Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 


912.A.SSE.3a 
9 
Algebra 
Factor a quadratic expression to reveal the zeros of the function it defines. 


912.A.SSE.3a 
10 
Algebra 
Factor a quadratic expression to reveal the zeros of the function it defines. 


912.A.SSE.3a 
11 
Algebra 
Factor a quadratic expression to reveal the zeros of the function it defines. 


912.A.SSE.3a 
12 
Algebra 
Factor a quadratic expression to reveal the zeros of the function it defines. 


912.A.SSE.3b 
9 
Algebra 
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 


912.A.SSE.3b 
10 
Algebra 
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 


912.A.SSE.3b 
11 
Algebra 
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 


912.A.SSE.3b 
12 
Algebra 
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 


912.A.SSE.3c 
9 
Algebra 
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^(12t) ? 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 


912.A.SSE.3c 
10 
Algebra 
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^(12t) ? 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 


912.A.SSE.3c 
11 
Algebra 
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^(12t) ? 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 


912.A.SSE.3c 
12 
Algebra 
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^(12t) ? 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 


912.A.SSE.4 
9 
Algebra 
Write expressions in equivalent forms to solve problems. 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. For example, calculate mortgage payments. 


912.A.SSE.4 
10 
Algebra 
Write expressions in equivalent forms to solve problems. 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. For example, calculate mortgage payments. 


912.A.SSE.4 
11 
Algebra 
Write expressions in equivalent forms to solve problems. 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. For example, calculate mortgage payments. 


912.A.SSE.4 
12 
Algebra 
Write expressions in equivalent forms to solve problems. 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. For example, calculate mortgage payments. 


912.A.APR.1 
9 
Algebra 
Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 


912.A.APR.1 
10 
Algebra 
Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 


912.A.APR.1 
11 
Algebra 
Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 


912.A.APR.1 
12 
Algebra 
Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 


912.A.APR.2 
9 
Algebra 
Understand the relationship between zeros and factors of polynomial. Know and apply the Remainder 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). 


912.A.APR.2 
10 
Algebra 
Understand the relationship between zeros and factors of polynomial. Know and apply the Remainder 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). 


912.A.APR.2 
11 
Algebra 
Understand the relationship between zeros and factors of polynomial. Know and apply the Remainder 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). 


912.A.APR.2 
12 
Algebra 
Understand the relationship between zeros and factors of polynomial. Know and apply the Remainder 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). 


912.A.APR.3 
9 
Algebra 
Understand the relationship between zeros and factors of polynomials. 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. 


912.A.APR.3 
10 
Algebra 
Understand the relationship between zeros and factors of polynomials. 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. 


912.A.APR.3 
11 
Algebra 
Understand the relationship between zeros and factors of polynomials. 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. 


912.A.APR.3 
12 
Algebra 
Understand the relationship between zeros and factors of polynomials. 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. 


912.A.APR.4 
9 
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 
10 
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 
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. 

