912.HSN Number and Quantity

912.HSNRN The Real Number System

912.HSNRN.A Extend the properties of exponents to rational exponents.

912.HSNRN.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.

912.HSNRN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

912.HSNRN.B Use properties of rational and irrational numbers.

912.HSNRN.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.

912.HSNQ Quantities

912.HSNQ.A Reason quantitatively and use units to solve problems.

912.HSNQ.A.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

912.HSNQ.A.2 Define appropriate quantities for the purpose of descriptive modeling.

912.HSNQ.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
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.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

912.HSASSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.

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

912.HSASSE.B.3c Use the properties of exponents to transform expressions for exponential functions.

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.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.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.

912.HSAREI.B Solve equations and inequalities in one variable

912.HSAREI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

912.HSAREI.B.4 Solve quadratic equations in one variable.

912.HSAREI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x  p)Â² = q that has the same solutions. Derive the quadratic formula from this form.

912.HSAREI.B.4b Solve quadratic equations by inspection (e.g., for xÂ² = 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.

912.HSAREI.C Solve systems of equations

912.HSAREI.C.5 Prove 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.

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

912.HSAREI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

912.HSAREI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.

912.HSAREI.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 Ã— 3 or greater).

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

912.HSAREI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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.HSAREI.D.12 Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes.
912.HSF Functions

912.HSFIF Interpreting Functions

912.HSFIF.A Understand the concept of a function and use function notation

912.HSFIF.A.1 Understand that a function from one set (called the domain) to another set (called the range) 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).

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

912.HSFIF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

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.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

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

912.HSFBF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

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.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

912.HSFLE.A.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

912.HSFLE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

912.HSFLE.A.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

912.HSFLE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).

912.HSFLE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

912.HSFLE.B Interpret expressions for functions in terms of the situation they model

912.HSFLE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.
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.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

912.HSSID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

912.HSSID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

912.HSSID.B Summarize, represent, and interpret data on two categorical and quantitative variables

912.HSSID.B.5 Summarize categorical data for two categories in twoway frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

912.HSSID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

912.HSSID.B.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

912.HSSID.B.6b Informally assess the fit of a function by plotting and analyzing residuals.

912.HSSID.B.6c Fit a linear function for a scatter plot that suggests a linear association.

912.HSSID.C Interpret linear models

912.HSSID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

912.HSSID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

912.HSSID.C.9 Distinguish between correlation and causation.
Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.