N. Number and Quantity

N-RN.A. Extend the properties of exponents to rational exponents.

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 $5$.

• Evaluating a Special Exponential Expression
• Evaluating Exponential Expressions
• Extending the Definitions of Exponents, Variation 2

N-RN.A.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

• Checking a calculation of a decimal exponent
• Kepler's Third Law of Motion
• Rational or Irrational?

N-RN.B. Use properties of rational and irrational numbers.

• Calculating the square root of 2
• Coordinates of equilateral triangles
• Rational or Irrational?

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.

• Operations with Rational and Irrational Numbers
• Sums of rational and irrational numbers

N-Q.A. Reason quantitatively and use units to solve problems.

• Giving raises
• New Cuyama
• Traffic Jam

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.

• Felicia's Drive
• Fuel Efficiency
• Harvesting the Fields
• How Much is a Penny Worth?
• Ice Cream Van
• Runners' World
• Selling Fuel Oil at a Loss
• Solar Radiation Model
• Weed killer

N-Q.A.2. Define appropriate quantities for the purpose of descriptive modeling.

• Giving raises
• Harvesting the Fields
• Weed killer

N-Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

• Accuracy of Carbon 14 Dating I
• Accuracy of Carbon 14 Dating II
• Bus and Car
• Calories in a sports drink
• Dinosaur Bones
• Felicia's Drive
• Weed killer

N-CN.A. Perform arithmetic operations with complex numbers.

• Complex number patterns
• Computations with Complex Numbers

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.

• Complex number patterns
• Vertex of a parabola with complex roots

N-CN.A.2. Use the relation $i^2 = -1$ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

• Complex Cube and Fourth Roots of 1
• Complex Square Roots
• Powers of a complex number

N-CN.A.3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

• No tasks yet illustrate this standard.

N-CN.B. Represent complex numbers and their operations on the complex plane.

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.

• No tasks yet illustrate this standard.

N-CN.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, $(-1 + \sqrt{3} i)^3 = 8$ because $(-1 + \sqrt3 i)$ has modulus $2$ and argument $120^\circ$.

• Complex Cube and Fourth Roots of 1

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.

• Complex Distance

N-CN.C. Use complex numbers in polynomial identities and equations.

N-CN.C.7. Solve quadratic equations with real coefficients that have complex solutions.

• Completing the square

N-CN.C.8. Extend polynomial identities to the complex numbers. For example, rewrite $x^2 + 4$ as $(x + 2i)(x - 2i)$.

• Complex Cube and Fourth Roots of 1

N-CN.C.9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

• No tasks yet illustrate this standard.

N-VM.A. Represent and model with vector quantities.

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., $\textbf{v}$, $|\textbf{v}|$, $||\textbf{v}||$, $v$).

• No tasks yet illustrate this standard.

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.

• No tasks yet illustrate this standard.

N-VM.A.3. Solve problems involving velocity and other quantities that can be represented by vectors.

• No tasks yet illustrate this standard.

N-VM.B. Perform operations on vectors.

N-VM.B.4. Add and subtract vectors.

• No tasks yet illustrate this standard.

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.

• No tasks yet illustrate this standard.

N-VM.B.4.b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

• No tasks yet illustrate this standard.

N-VM.B.4.c. Understand vector subtraction $\textbf{v} - \textbf{w}$ as $\textbf{v} + (-\textbf{w})$, where $-\textbf{w}$ is the additive inverse of $\textbf{w}$, with the same magnitude as $\textbf{w}$ and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

• No tasks yet illustrate this standard.

N-VM.B.5. Multiply a vector by a scalar.

• No tasks yet illustrate this standard.

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 $c(v_x, v_y) = (cv_x, cv_y)$.

• No tasks yet illustrate this standard.

N-VM.B.5.b. Compute the magnitude of a scalar multiple $c\textbf{v}$ using $||c\textbf{v}|| = |c|v$. Compute the direction of $c\textbf{v}$ knowing that when $|c|{v} \neq 0$, the direction of $c\textbf{v}$ is either along $\textbf{v}$ (for $c > 0$) or against $\textbf{v}$ (for $c < 0$).

• No tasks yet illustrate this standard.

N-VM.C. Perform operations on matrices and use matrices in applications.

N-VM.C.6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

• No tasks yet illustrate this standard.

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.

• No tasks yet illustrate this standard.

N-VM.C.8. Add, subtract, and multiply matrices of appropriate dimensions.

• No tasks yet illustrate this standard.

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.

• No tasks yet illustrate this standard.

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.

• No tasks yet illustrate this standard.

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.

• No tasks yet illustrate this standard.

N-VM.C.12. Work with $2 \times2$ matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

• No tasks yet illustrate this standard.