| N-RN.A.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 5^1/3 to be the cube root of 5 because we want (5^1/3)^ 3 = 5^(1/3)^3 to hold, so (5^1/3)^ 3 must equal 5. |
Lessons
|
| N-RN.A.2 |
Rewrite expressions involving radicals and rational exponents using the properties of exponents. |
Lessons
|
| N-RN.B.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. |
Lessons
|
| N-Q.A.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 |
Lessons
|
| N-Q.A.2 |
Define appropriate quantities for the purpose of descriptive modeling |
Lessons
|
| N-Q.A.3 |
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities |
Lessons
|
| IA.3 |
Understand, analyze, apply, and evaluate some common voting and analysis methods in addition to majority and plurality, such as runoff, approval, the so-called instant-runoff voting (IRV) method, the Borda method and the Condorcet method |
Lessons
|
| IA.4 |
(+) Describe the role of mathematics in information processing, particularly with respect to the Internet. ( |
Lessons
|
| IA.5 |
(+) Understand and apply elementary set theory and logic as used in simple Internet searches. |
Lessons
|
| IA.6 |
(+) Understand and apply basic number theory, including modular arithmetic, for example, as used in keeping information secure through public-key cryptography. |
Lessons
|
| N-CN.A.1 |
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 |
Lessons
|
| N-CN.A.2 |
Use the relation i 2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. |
Lessons
|
| N-CN.A.3 |
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. |
Lessons
|
| N-CN.B.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 |
Lessons
|
| NCN.B.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° |
Lessons
|
| N-CN.B.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 |
Lessons
|
| N-CN.C.7 |
Solve quadratic equations with real coefficients that have complex solutions |
Lessons
|
| N-CN.C.8 |
(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i) |
Lessons
|
| N-CN.C.9 |
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. |
Lessons
|
| N-VM.A.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). |
Lessons
|
| N-VM.A.2 |
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point |
Lessons
|
| N-VM.A.3 |
(+) Solve problems involving velocity and other quantities that can be represented by vectors |
Lessons
|
| N-VM.B.4.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 |
Lessons
|
| N-VM.B.4.B |
(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. |
Lessons
|
| N-VM.B.4.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. |
Lessons
|
| N-VM.B.5.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) |
Lessons
|
| N-VM.B.5.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) |
Lessons
|
| N-VM.C.6 |
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. |
Lessons
|
| N-VM.C.7 |
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled |
Lessons
|
| N-VM.C.8 |
(+) Add, subtract, and multiply matrices of appropriate dimensions |
Lessons
|
| N-VM.C.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 |
Lessons
|
| N-VM.C.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 |
Lessons
|
| N-VM.C.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 |
Lessons
|
| N-VM.C.12 |
(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area |
Lessons
|
| N-RN.A.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 5^1/3 to be the cube root of 5 because we want (5^1/3)^ 3 = 5^(1/3)^3 to hold, so (5^1/3)^ 3 must equal 5. |
Lessons
|
| N-RN.A.2 |
Rewrite expressions involving radicals and rational exponents using the properties of exponents. |
Lessons
|
| N-RN.B.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. |
Lessons
|
| N-Q.A.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 |
Lessons
|
| N-Q.A.2 |
Define appropriate quantities for the purpose of descriptive modeling |
Lessons
|
| N-Q.A.3 |
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities |
Lessons
|
| IA.3 |
Understand, analyze, apply, and evaluate some common voting and analysis methods in addition to majority and plurality, such as runoff, approval, the so-called instant-runoff voting (IRV) method, the Borda method and the Condorcet method |
Lessons
|
| IA.4 |
(+) Describe the role of mathematics in information processing, particularly with respect to the Internet. ( |
Lessons
|
| IA.5 |
(+) Understand and apply elementary set theory and logic as used in simple Internet searches. |
Lessons
|
| IA.6 |
(+) Understand and apply basic number theory, including modular arithmetic, for example, as used in keeping information secure through public-key cryptography. |
Lessons
|
| N-CN.A.1 |
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 |
Lessons
|
| N-CN.A.2 |
Use the relation i 2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. |
Lessons
|
| N-CN.A.3 |
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. |
Lessons
|
| N-CN.B.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 |
Lessons
|
| NCN.B.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° |
Lessons
|
| N-CN.B.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 |
Lessons
|
| N-CN.C.7 |
Solve quadratic equations with real coefficients that have complex solutions |
Lessons
|
| N-CN.C.8 |
(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i) |
Lessons
|
| N-CN.C.9 |
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. |
Lessons
|
| N-VM.A.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). |
Lessons
|
| N-VM.A.2 |
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point |
Lessons
|
| N-VM.A.3 |
(+) Solve problems involving velocity and other quantities that can be represented by vectors |
Lessons
|
| N-VM.B.4.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 |
Lessons
|
| N-VM.B.4.B |
(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. |
Lessons
|
| N-VM.B.4.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. |
Lessons
|
| N-VM.B.5.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) |
Lessons
|
| N-VM.B.5.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) |
Lessons
|
| N-VM.C.6 |
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. |
Lessons
|
| N-VM.C.7 |
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled |
Lessons
|
| N-VM.C.8 |
(+) Add, subtract, and multiply matrices of appropriate dimensions |
Lessons
|
| N-VM.C.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 |
Lessons
|
| N-VM.C.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 |
Lessons
|
| N-VM.C.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 |
Lessons
|
| N-VM.C.12 |
(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area |
Lessons
|
