The Real Number System
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HSN-RN
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Extend the properties of exponents to rational exponents.
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1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define
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CCCs linked to HSN-RN.A.1
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None
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2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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CCCs linked to HSN-RN.A.2
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HS.NO.1a1 Simplify expressions that include exponents.
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HS.NO.1a2 Explain the influence of an exponent on the location of a decimal point in a given number.
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HS.NO1a3 Convert a number expressed in scientific notation.
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H.NO.2c2 Rewrite expressions that include rational exponents.
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Use properties of rational and irrational numbers.
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3. Explain why the sum or product of two 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.
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CCCs linked to HSN-RN.B.3
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H.NO.2b1 Explain the pattern for the sum or product for combinations of rational and irrational numbers.
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The Complex Number System
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HSN-CN
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Perform arithmetic operations with complex numbers.
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1. Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
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CCCs linked to HSN-CN.A.1
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None
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2. Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
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CCCs linked to HSN-CN.A.2
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None
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3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
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CCCs linked to HSN-CN.A.3
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None
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Represent complex numbers and their operations on the complex plane.
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4. 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.
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CCCs linked to HSN-CN.B.4
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None
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5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°.
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CCCs linked to HSN-CN.B.5
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None
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6. 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.
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CCCs linked to HSN-CN.B.6
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None
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Use complex numbers in polynomial identities and equations.
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7. Solve quadratic equations with real coefficients that have complex solutions.
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CCCs linked to HSN-CN.C.7
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None
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8. Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).
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CCCs linked to HSN-CN.C.8
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None
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9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
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CCCs linked to HSN-CN.C.9
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None
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Vector and Matrix Quantities
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HSN-VM
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Represent and model with vector quantities.
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1. 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, \|v\|, \|\|v\|\|, v).
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CCCs linked to HSN-VM.A.1
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None
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2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
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CCCs linked to HSN-VM.A.2
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None
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3. Solve problems involving velocity and other quantities that can be represented by vectors.
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CCCs linked to HSN-VM.A.3
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None
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Perform operations on vectors.
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4. Add and subtract vectors.
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a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
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b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
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c. 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 component-wise.
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CCCs linked to HSN-VM.B.4
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None
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5. Multiply a vector by a scalar.
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a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
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b. Compute the magnitude of a scalar multiple cv using \|\|cv\|\| = \|c\|v. Compute the direction of cv knowing that when \|c\|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
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CCCs linked to HSN-VM.B.5
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None
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Perform operations on matrices and use matrices in applications.
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6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
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CCCs linked to HSN-VM.C.6
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None
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7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
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CCCs linked to HSN-VM.C.7
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None
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8. Add, subtract, and multiply matrices of appropriate dimensions.
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CCCs linked to HSN-VM.C.8
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None
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9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
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CCCs linked to HSN-VM.C.9
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None
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10. 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.
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CCCs linked to HSN-VM.C.10
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None
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11. 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.
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CCCs linked to HSN-VM.C.11
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None
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12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
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CCCs linked to HSN-VM.C.12
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None
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