Core Content Connectors by Common Core State Standards: Mathematics Number and Quantity

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Mathematics High School—Number and Quantity Overview

The Real Number System

• Extend the properties of exponents to rational exponents
• Use properties of rational and irrational numbers.

Quantities

• Reason quantitatively and use units to solve problems.

The Complex Number System

• Perform arithmetic operations with complex numbers
• Represent complex numbers and their operations on the complex plane
• Use complex numbers in polynomial identities and equations.

Vector and Matrix Quantities

• Represent and model with vector quantities.
• Perform operations on vectors.
• Perform operations on matrices and use matrices in applications.

The Real Number System	HSN-RN
Extend the properties of exponents to rational exponents.
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 5(⅓) to be the cube root of 5 because we want (5(⅓))³ = 5(⅓)³ to hold, so (5(⅓))³ must equal 5.
CCCs linked to HSN-RN.A.1	None
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
CCCs linked to HSN-RN.A.2	HS.NO.1a1 Simplify expressions that include exponents.
	HS.NO.1a2 Explain the influence of an exponent on the location of a decimal point in a given number.
	HS.NO1a3 Convert a number expressed in scientific notation.
	H.NO.2c2 Rewrite expressions that include rational exponents.
Use properties of rational and irrational numbers.
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.
CCCs linked to HSN-RN.B.3	H.NO.2b1 Explain the pattern for the sum or product for combinations of rational and irrational numbers.

Quantities	HSN-Q
Reason quantitatively and use units to solve problems.
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.
CCCs linked to HSN-Q.A.1	H.ME.1a1 Determine the necessary unit(s) to use to solve real-world problems.
	H.ME.1a2 Solve real-world problems involving units of measurement
2. Define appropriate quantities for the purpose of descriptive modeling.
CCCs linked to HSN-Q.A.2	None
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
CCCs linked to HSN-Q.A.3	None

The Complex Number System	HSN-CN
Perform arithmetic operations with complex numbers.
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.
CCCs linked to HSN-CN.A.1	None
2. Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CCCs linked to HSN-CN.A.2	None
3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
CCCs linked to HSN-CN.A.3	None
Represent complex numbers and their operations on the complex plane.
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.
CCCs linked to HSN-CN.B.4	None
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°.
CCCs linked to HSN-CN.B.5	None
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.
CCCs linked to HSN-CN.B.6	None
Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
CCCs linked to HSN-CN.C.7	None
8. Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).
CCCs linked to HSN-CN.C.8	None
9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
CCCs linked to HSN-CN.C.9	None

Vector and Matrix Quantities	HSN-VM
Represent and model with vector quantities.
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).
CCCs linked to HSN-VM.A.1	None
2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CCCs linked to HSN-VM.A.2	None
3. Solve problems involving velocity and other quantities that can be represented by vectors.
CCCs linked to HSN-VM.A.3	None
Perform operations on vectors.
4. Add and subtract vectors.
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.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
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.
CCCs linked to HSN-VM.B.4	None
5. Multiply a vector by a scalar.
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).
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).
CCCs linked to HSN-VM.B.5	None
Perform operations on matrices and use matrices in applications.
6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
CCCs linked to HSN-VM.C.6	None
7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
CCCs linked to HSN-VM.C.7	None
8. Add, subtract, and multiply matrices of appropriate dimensions.
CCCs linked to HSN-VM.C.8	None
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.
CCCs linked to HSN-VM.C.9	None
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.
CCCs linked to HSN-VM.C.10	None
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.
CCCs linked to HSN-VM.C.11	None
12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
CCCs linked to HSN-VM.C.12	None