912.HSN Number and Quantity

912.HSNCN The Complex Number System

912.HSNCN.A Perform arithmetic operations with complex numbers.

912.HSNCN.A.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.

912.HSNCN.A.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

912.HSNCN.C Use complex numbers in polynomial identities and equations.

912.HSNCN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

912.HSNCN.C.8 Extend polynomial identities to the complex numbers.

912.HSNCN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
912.HSA Algebra

912.HSASSE Seeing Structure in Expressions

912.HSASSE.A Interpret the structure of expressions

912.HSASSE.A.1 Interpret expressions that represent a quantity in terms of its context.

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

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

912.HSASSE.A.2 Use the structure of an expression to identify ways to rewrite it.

912.HSASSE.B Write expressions in equivalent forms to solve problems

912.HSASSE.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.

912.HSAAPR Arithmetic with Polynomials and Rational Expressions

912.HSAAPR.A Perform arithmetic operations on polynomials

912.HSAAPR.A.1 Understand 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.

912.HSAAPR.B Understand the relationship between zeros and factors of polynomials

912.HSAAPR.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).

912.HSAAPR.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.

912.HSAAPR.C Use polynomial identities to solve problems

912.HSAAPR.C.4 Prove polynomial identities and use them to describe numerical relationships.

912.HSAAPR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y) to the n power 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.

912.HSAAPR.D Rewrite rational expressions

912.HSAAPR.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.

912.HSAAPR.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 expression; add, subtract, multiply, and divide rational expressions.

912.HSACED Creating Equations

912.HSACED.A Create equations that describe numbers or relationships

912.HSACED.A.1 Create equations and inequalities in one variable and use them to solve problems.

912.HSACED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

912.HSACED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

912.HSACED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

912.HSAREI Reasoning with Equations and Inequalities

912.HSAREI.A Understand solving equations as a process of reasoning and explain the reasoning

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

912.HSAREI.D Represent and solve equations and inequalities graphically

912.HSAREI.D.11 Explain why the xcoordinates 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.
912.HSF Functions

912.HSFIF Interpreting Functions

912.HSFIF.B Interpret functions that arise in applications in terms of the context

912.HSFIF.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.

912.HSFIF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

912.HSFIF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

912.HSFIF.C Analyze functions using different representations

912.HSFIF.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.

912.HSFIF.C.7b Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.

912.HSFIF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

912.HSFIF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

912.HSFIF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

912.HSFIF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

912.HSFIF.C.8b Use the properties of exponents to interpret expressions for exponential functions.

912.HSFIF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

912.HSFBF Building Functions

912.HSFBF.A Build a function that models a relationship between two quantities

912.HSFBF.A.1 Write a function that describes a relationship between two quantities.

912.HSFBF.A.1b Combine standard function types using arithmetic operations.

912.HSFBF.B Build new functions from existing functions

912.HSFBF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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.

912.HSFBF.B.4 Find inverse functions.

912.HSFBF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

912.HSFLE Linear, Quadratic, and Exponential Models

912.HSFLE.A Construct and compare linear, quadratic, and exponential models and solve problems

912.HSFLE.A.4 For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

912.HSFTF Trigonometric Functions

912.HSFTF.A Extend the domain of trigonometric functions using the unit circle

912.HSFTF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

912.HSFTF.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.

912.HSFTF.B Model periodic phenomena with trigonometric functions

912.HSFTF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

912.HSFTF.C Prove and apply trigonometric identities

912.HSFTF.C.8 Prove the Pythagorean identity sin^{2}(θ) + cos^{2}(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
912.HSS Statistics and Probability

912.HSSID Interpreting Categorical and Quantitative Data

912.HSSID.A Summarize, represent, and interpret data on a single count or measurement variable

912.HSSID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

912.HSSIC Making Inferences and Justifying Conclusions

912.HSSIC.A Understand and evaluate random processes underlying statistical experiments

912.HSSIC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

912.HSSIC.A.2 Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation.

912.HSSIC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies

912.HSSIC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

912.HSSIC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

912.HSSIC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

912.HSSIC.B.6 Evaluate reports based on data.

912.HSSMD Using Probability to Make Decisions

912.HSSMD.B Use probability to evaluate outcomes of decisions

912.HSSMD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

912.HSSMD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.