HS.N-RN.A The Real Number System: 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.
3 Generalize the properties of integer exponents to rational exponents, and apply these properties to a wider variety of situations, such as rewriting the formula for the volume of a sphere of radius r, V = 4/3πr³ to express the radius in terms of the volume, r = (3/4 × V/π)¹/³.
HS.N-RN.B The Real Number System: Use properties of rational and irrational numbers.
HS.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.
HS.N-Q.A Quantities: 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.
2 Create a coherent representation of problems by considering the units involved, attending to the meaning of quantities and not just how to compute them, and flexibly using properties of operations and objects.
HS.N-CN.B The Complex Number System: 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-VM.A Vector & Matrix Quantities: 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).
HS.N-VM.B.4.c Understand vector subtraction ?? – ?? as ?? + (–??), where –?? is the additive inverse of ??, with the same magnitude as ?? 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.b Compute the magnitude of a scalar multiple c?? using ?c??? = |c|??. Compute the direction of c?? knowing that when |c|?? ≠ 0, the direction of c?? is either along ?? (for c > 0) or against ?? (for c < 0).
HS.N-VM.C.9 Understand that, unlike the 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 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.
1 Understand how matrices can represent systems of equations and are useful when systems contain too many variables to efficiently operate on without technology. (Professional Skills: Information Literacy)
3 Notice, for example, the regularity in the way terms combine to make zero when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1), and how it might lead to the general formula for the sum of a finite geometric series.
HS.A-APR.A Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.
HS.A-APR.A.1 Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
HS.A-APR.B Arithmetic with Polynomials & Rational Expressions: 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). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.)
HS.A-APR.C.5 Know and apply the Binomial Theorem for the expansion of 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
HS.A-APR.D Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.
HS.A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
HS.A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expressions; add, subtract, multiply, and divide rational expressions.
1 Reason with rational expressions like (x² + 5x + 6)/(x + 2) not as a sum divided by a sum, but as a yet-to-be-factored numerator where one of the factors, (x + 2), will make 1 when divided by the denominator.
2 Determine when it is appropriate to use a computer algebra system or calculator instead of paper and pencil to rewrite rational expressions.
3 Understand how types of numbers and operations form a closed system.
HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.
HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
2 Model and solve problems arising in everyday life, society, and the workplace. Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
HS.A-REI.A Reasoning with Equations & Inequalities: 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 a solution method.
2 Describe a logical flow of mathematics, using stated assumptions, definitions, and previously established results in constructing arguments, and explain solving equations as a process of reasoning that demystifies "extraneous" solutions that can arise under certain solution procedures.
HS.A-REI.B.4 Solve quadratic equations in one variable.
HS.A-REI.B.4.a 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.4.b Solve quadratic equations (e.g., for x² = 49) by inspection, 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.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 e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
HS.A-REI.D.12 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.
3 Use graphing calculators and/or computer technology to reason about and solve systems of equations and inequalities.
4 Specify units of measure, label axes to clarify the correspondence with quantities in a problem, calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context.
5 Use the characteristics and structures of function families to understand and generalize about solutions to equations and inequalities.
HS.F-IF.A Interpreting Functions: Understand the concept of a function and use function notation.
HS.F-IF.A.1 Explain 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).
3 Understand a function as a correspondence where each element of the domain is assigned to exactly one element of the range; this structure does not "turn inputs into outputs"; rather, it describes the relationship between elements in two sets.
HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.
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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
2 Graph functions and interpret key features of the graphs or use key features to construct a graph; use technology as a tool to visualize and understand how various functions behave in different representations.
3 Make structural comparisons between linear, exponential, quadratic and higher order polynomial, rational, radical and trigonometric functions to describe commonalities, consistencies, and differences.
HS.F-IF.C Interpreting Functions: Analyze functions using different representations.
HS.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
HS.F-IF.C.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
HS.F-BF.B Building Functions: Build new functions from existing functions.
HS.F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k both positive and negative; find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
HS.F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
2 Both decontextualize-abstract a given situation and representing it symbolically and manipulate the representing symbols without necessarily attending to their referents-and contextualize-pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
3 Use mathematics to model, interpret, and reason about real-world contexts.
HS.F-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.