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

HS.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.

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

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

HS.N.RN.B.3 Explain why the sum or product of two rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational.

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

HS.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.★

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

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

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

HS.N.CN.A.1 Know there is a complex number 𝑖 such that 𝑖^2=−1, and show that every complex number has the form a+bi where 𝑎 and 𝑏 are real.

HS.N.CN.A.2 Perform arithmetic operations with complex numbers. Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

HS.N.CN.A.3 Use the relation 𝑖^2=−1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

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

HS.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.

HS.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.

HS.N.CN.B.6 (+) Calculate the distance between numbers in the complex plane as the absolute value of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

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

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

HS.N.CN.C.8 (+) Extend polynomial identities to the complex numbers.

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

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

HS.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).

HS.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.

HS.N.VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

HS.N.VM.B. Perform operations on vectors.

HS.N.VM.B.4 (+) Add and subtract vectors.

HS.N.VM.B.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.

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

HS.N.VM.B.4c (+) Demonstrate understanding of 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.

HS.N.VM.B.5 (+) Multiply a vector by a scalar.

HS.N.VM.B.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).

HS.N.VM.B.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).

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

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

HS.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.

HS.N.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

HS.N.VM.C.9 (+) Demonstrate understanding that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

HS.N.VM.C.10 (+) Demonstrate understanding 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.

HS.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.

HS.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.

HS.A.SSE.A Interpret the structure of linear, quadratic, exponential, polynomial, and rational expressions.

HS.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context. ★

HS.A.SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients.

HS.A.SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.

HS.A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.

HS.A.SSE.B Write expressions in equivalent forms to solve problems.

HS.A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

HS.A.SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.

HS.A.SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

HS.A.SSE.B.3c Use the properties of exponents to transform expressions for exponential functions.

HS.A.SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and use the formula to solve problems.★

HS.A.APR.A Perform arithmetic operations on polynomials.

HS.A.APR.A.1 Demonstrate understanding that polynomials form a system analogous to the integers; namely, they are closed under certain operations.

HS.A.APR.A.1a Perform operations on polynomial expressions (addition, subtraction, multiplication, division) and compare the system of polynomials to the system of integers when performing operations.

HS.A.APR.A.1b Factor and/or expand polynomial expressions, identify and combine like terms, and apply the distributive property.

HS.A.APR.B Understand the relationship between zeros and factors of polynomials.

HS.A.APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

HS.A.APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

HS.A.APR.C Use polynomial identities to solve problems.

HS.A.APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

HS.A.APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (𝑥+𝑦)^𝑛 in powers of 𝑥 and 𝑦 for a positive integer 𝑛 , where 𝑥 and 𝑦 are any numbers, with coefficients determined, for example, by Pascal’s Triangle.

HS.A.APR.D Rewrite rational expressions.

HS.A.APR.D.6 Rewrite simple rational expressions in different forms using inspection, long division, or, for the more complicated examples, a computer algebra system.

HS.A.APR.D.7 (+) Demonstrate understanding that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

HS.A.CED.A Create equations that describe numbers or relationships.

HS.A.CED.A.1 Create one-variable equations and inequalities to solve problems, including linear, quadratic, rational, and exponential functions.★

HS.A.CED.A.2 Interpret the relationship between two or more quantities.★

HS.A.CED.A.2a Define variables to represent the quantities and write equations to show the relationship.★

HS.A.CED.A.2b Use graphs to show a visual representation of the relationship while adhering to appropriate labels and scales.★

HS.A.CED.A.3 Represent constraints using equations or inequalities and interpret solutions as viable or non-viable options in a modeling context.★

HS.A.CED.A.4 Represent constraints using systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.★

HS.A.CED.A.5 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.★

HS.A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.

HS.A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify or refute a solution method.

HS.A.REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

HS.A.REI.B Solve equations and inequalities in one variable.

HS.A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

HS.A.REI.B.3a Solve linear equations and inequalities in one variable involving absolute value.

HS.A.REI.B.4 Solve quadratic equations in one variable.

HS.A.REI.B.4a Use the method of completing the square to transform any quadratic equation in 𝑥 into an equation of the form (𝑥–𝑝)^2=𝑞 that has the same solutions. Derive the quadratic formula from this form.

HS.A.REI.B.4b Solve quadratic equations by inspection (e.g., for 𝑥^2=49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝑎±𝑏i for real numbers 𝑎 and 𝑏.

HS.A.REI.C Solve systems of equations.

HS.A.REI.C.5 Verify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

HS.A.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

HS.A.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

HS.A.REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.

HS.A.REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3×3 or greater).

HS.A.REI.D Represent and solve equations and inequalities graphically.

HS.A.REI.D.10 Demonstrate understanding that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Show that any point on the graph of an equation in two variables is a solution to the equation.

HS.A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations 𝑦=𝑓(𝑥) and 𝑦=𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥)=𝑔(𝑥); find the solutions approximately. Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

HS.A.REI.D.12 Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

HS.F.IF.A Understand the concept of a function and use function notation.

HS.F.IF.A.1 Demonstrate understanding that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range: If 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦=𝑓(𝑥).2

HS.F.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

HS.F.IF.A.3 Demonstrate that a sequence is a functions, sometimes defined recursively, whose domain is a subset of the integers.

HS.F.IF.B Interpret functions that arise in applications in terms of the context. Include linear, quadratic, exponential, rational, polynomial, square root and cube root, trigonometric, and logarithmic functions.

M102, M103, M104, M203, M204

HS.F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima; symmetries; end behavior; and periodicity.★