Core Content Connectors by Common Core State Standards: Mathematics Number and Quantity
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Contents |
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 | N-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 | |||
CCCs linked to N-RN.1 | None | ||
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. | |||
CCCs linked to N-RN.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 N-RN.3 | H.NO.2b1 Explain the pattern for the sum or product for combinations of rational and irrational numbers. |
Quantities | N-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 N-Q.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 N-Q.2 | None |
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. | |
CCCs linked to N-Q.3 | None |
The Complex Number System | N-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 N-CN.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 N-CN.2 | None |
3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. | |
CCCs linked to N-CN.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 N-CN.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 N-CN.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 N-CN.6 | None |
Use complex numbers in polynomial identities and equations. | |
7. Solve quadratic equations with real coefficients that have complex solutions. | |
CCCs linked to N-CN.7 | None |
8. Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i). | |
CCCs linked to N-CN.8 | None |
9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. | |
CCCs linked to N-CN.9 | None |
Vector and Matrix Quantities | N-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 N-VM.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 N-VM.2 | None |
3. Solve problems involving velocity and other quantities that can be represented by vectors. | |
CCCs linked to N-VM.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 N-VM.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 N-VM.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 N-VM.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 N-VM.7 | None |
8. Add, subtract, and multiply matrices of appropriate dimensions. | |
CCCs linked to N-VM.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 N-VM.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 N-VM.10 | 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. | |
CCCs linked to N-VM.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 N-VM.12 | None |