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In Number and Quantity, instructional time should focus on four critical areas: (1) extending the real number system, which includes developing rational exponents and relating rational numbers to irrational numbers through mathematical operations; (2) introducing units as a method of guiding students towards solving multi-step problems; (3) introducing the complex number system, showing how complex numbers can be evaluated using similar operations used with real numbers; (4) introducing the different properties of vectors and matrices and methods of modeling problems using those properties.

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Concepts

N.RN.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 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

7

N.RN.2

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

6

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N.RN.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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N.Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

4

N.Q.2

Define appropriate quantities for the purpose of descriptive modeling.

N.Q.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

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N.CN.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.

3

N.CN.2

Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

3

N.CN.3

(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

2

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N.CN.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.

N.CN.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°.

N.CN.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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N.CN.7

Solve quadratic equations with real coefficients that have complex solutions.

2

N.CN.8

(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).

N.CN.9

(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

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N.VM.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).

2

N.VM.2

(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N.VM.3

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

3

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N.VM.4

(+) Add and subtract vectors.

N.VM.4a

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.

3

N.VM.4b

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

1

N.VM.4c

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.

2

N.VM.5

(+) Multiply a vector by a scalar.

N.VM.5a

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).

2

N.VM.5b

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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N.VM.6

(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

1

N.VM.7

(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

2

N.VM.8

(+) Add, subtract, and multiply matrices of appropriate dimensions.

6

N.VM.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.

2

N.VM.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.

3

N.VM.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.

N.VM.12

(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.