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912.N.RN.1 
12 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. 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. For example, we define 5^(1/3) to be the cube root of 5 because we want [5^(1/3)]^3 = 5^[(1/3) x 3] to hold, so [5^(1/3)]^3 must equal 5. 


912.N.RN.2 
9 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 


912.N.RN.2 
10 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 


912.N.RN.2 
11 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 


912.N.RN.2 
12 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 


912.N.RN.3 
9 
High School: Number and Quantity 
Use properties of rational and irrational numbers. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 


912.N.RN.3 
10 
High School: Number and Quantity 
Use properties of rational and irrational numbers. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 


912.N.RN.3 
11 
High School: Number and Quantity 
Use properties of rational and irrational numbers. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 


912.N.RN.3 
12 
High School: Number and Quantity 
Use properties of rational and irrational numbers. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 


912.N.Q.1 
9 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 


912.N.Q.1 
10 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 


912.N.Q.1 
11 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 


912.N.Q.1 
12 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 


912.N.Q.2 
9 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling. 


912.N.Q.2 
10 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling. 


912.N.Q.2 
11 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling. 


912.N.Q.2 
12 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling. 


912.N.Q.3 
9 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 


912.N.Q.3 
10 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 


912.N.Q.3 
11 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 


912.N.Q.3 
12 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 


912.N.CN.1 
9 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Know there is a complex number i such that i^2 = ?1, and every complex number has the form a + bi with a and b real. 


912.N.CN.1 
10 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Know there is a complex number i such that i^2 = ?1, and every complex number has the form a + bi with a and b real. 


912.N.CN.1 
11 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Know there is a complex number i such that i^2 = ?1, and every complex number has the form a + bi with a and b real. 


912.N.CN.1 
12 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Know there is a complex number i such that i^2 = ?1, and every complex number has the form a + bi with a and b real. 


912.N.CN.2 
9 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Use the relation i^2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 


912.N.CN.2 
10 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Use the relation i^2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 


912.N.CN.2 
11 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Use the relation i^2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 


912.N.CN.2 
12 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Use the relation i^2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 


912.N.CN.3 
9 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 


912.N.CN.3 
10 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 


912.N.CN.3 
11 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 


912.N.CN.3 
12 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 


912.N.CN.4 
9 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 


912.N.CN.4 
10 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 


912.N.CN.4 
11 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 


912.N.CN.4 
12 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 


912.N.CN.5 
9 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 + ?3i)^3 = 8 because (1 + ?3i) has modulus 2 and argument 120Â°. 


912.N.CN.5 
10 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 + ?3i)^3 = 8 because (1 + ?3i) has modulus 2 and argument 120Â°. 


912.N.CN.5 
11 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 + ?3i)^3 = 8 because (1 + ?3i) has modulus 2 and argument 120Â°. 


912.N.CN.5 
12 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 + ?3i)^3 = 8 because (1 + ?3i) has modulus 2 and argument 120Â°. 


912.N.CN.6 
9 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. 


912.N.CN.6 
10 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. 


912.N.CN.6 
11 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. 


912.N.CN.6 
12 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. 


912.N.CN.7 
9 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions. 


912.N.CN.7 
10 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions. 


912.N.CN.7 
11 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions. 


912.N.CN.7 
12 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions. 


912.N.CN.8 
9 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x  2i). 


912.N.CN.8 
10 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x  2i). 


912.N.CN.8 
11 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x  2i). 


912.N.CN.8 
12 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x  2i). 


912.N.CN.9 
9 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 


912.N.CN.9 
10 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 


912.N.CN.9 
11 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 


912.N.CN.9 
12 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 


912.N.VM.1 
9 
High School: Number and Quantity 
Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), v, v, v(not bold)). 


912.N.VM.1 
10 
High School: Number and Quantity 
Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), v, v, v(not bold)). 


912.N.VM.1 
11 
High School: Number and Quantity 
Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), v, v, v(not bold)). 


912.N.VM.1 
12 
High School: Number and Quantity 
Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), v, v, v(not bold)). 


912.N.VM.2 
9 
High School: Number and Quantity 
Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 


912.N.VM.2 
10 
High School: Number and Quantity 
Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 


912.N.VM.2 
11 
High School: Number and Quantity 
Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 


912.N.VM.2 
12 
High School: Number and Quantity 
Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 


912.N.VM.3 
9 
High School: Number and Quantity 
Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. 


912.N.VM.3 
10 
High School: Number and Quantity 
Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. 


912.N.VM.3 
11 
High School: Number and Quantity 
Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. 


912.N.VM.3 
12 
High School: Number and Quantity 
Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. 


912.N.VM.4 
9 
High School: Number and Quantity 
Perform operations on vectors. Add and subtract vectors. 


912.N.VM.4 
10 
High School: Number and Quantity 
Perform operations on vectors. Add and subtract vectors. 


912.N.VM.4 
11 
High School: Number and Quantity 
Perform operations on vectors. Add and subtract vectors. 


912.N.VM.4 
12 
High School: Number and Quantity 
Perform operations on vectors. Add and subtract vectors. 


912.N.VM.4a 
9 
High School: Number and Quantity 
Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 


912.N.VM.4a 
10 
High School: Number and Quantity 
Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 


912.N.VM.4a 
11 
High School: Number and Quantity 
Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 


912.N.VM.4a 
12 
High School: Number and Quantity 
Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 


912.N.VM.4b 
9 
High School: Number and Quantity 
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 


912.N.VM.4b 
10 
High School: Number and Quantity 
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 


912.N.VM.4b 
11 
High School: Number and Quantity 
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 


912.N.VM.4b 
12 
High School: Number and Quantity 
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 


912.N.VM.4c 
9 
High School: Number and Quantity 
Understand vector subtraction v  w as v + (w), where (w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. 


912.N.VM.4c 
10 
High School: Number and Quantity 
Understand vector subtraction v  w as v + (w), where (w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. 


912.N.VM.4c 
11 
High School: Number and Quantity 
Understand vector subtraction v  w as v + (w), where (w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. 


912.N.VM.4c 
12 
High School: Number and Quantity 
Understand vector subtraction v  w as v + (w), where (w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. 


912.N.VM.5 
9 
High School: Number and Quantity 
Perform operations on vectors. Multiply a vector by a scalar. 


912.N.VM.5 
10 
High School: Number and Quantity 
Perform operations on vectors. Multiply a vector by a scalar. 


912.N.VM.5 
11 
High School: Number and Quantity 
Perform operations on vectors. Multiply a vector by a scalar. 


912.N.VM.5 
12 
High School: Number and Quantity 
Perform operations on vectors. Multiply a vector by a scalar. 


912.N.VM.5a 
9 
High School: Number and Quantity 
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)). 


912.N.VM.5a 
10 
High School: Number and Quantity 
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)). 


912.N.VM.5a 
11 
High School: Number and Quantity 
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)). 


912.N.VM.5a 
12 
High School: Number and Quantity 
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)). 


912.N.VM.5b 
9 
High School: Number and Quantity 
Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv =? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 


912.N.VM.5b 
10 
High School: Number and Quantity 
Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv =? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 


912.N.VM.5b 
11 
High School: Number and Quantity 
Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv =? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 


912.N.VM.5b 
12 
High School: Number and Quantity 
Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv =? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 


912.N.VM.6 
9 
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.6 
10 
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.6 
11 
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. 

