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.

Checkpoint opportunity

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.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.

Checkpoint opportunity

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.

Checkpoint opportunity

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.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.B Solve equations and inequalities in one variable

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)² = 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 x² = 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.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Checkpoint opportunity
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.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.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).

Checkpoint opportunity

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.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.

Checkpoint opportunity

912.HSFLE Linear, Quadratic, and Exponential Models

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

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.HSFTF Trigonometric Functions

912.HSFTF.C Prove and apply trigonometric identities

912.HSFTF.C.8 Prove the Pythagorean identity sin²(Θ) + cos²(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.HSG Geometry

912.HSGCO Congruence

912.HSGCO.C Prove geometric theorems

912.HSGCO.C.9 Prove theorems about lines and angles.

912.HSGCO.C.10 Prove theorems about triangles.

912.HSGCO.C.11 Prove theorems about parallelograms.

Checkpoint opportunity

912.HSGSRT Similarity, Right Triangles, and Trigonometry

912.HSGSRT.A Understand similarity in terms of similarity transformations

912.HSGSRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:

912.HSGSRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

912.HSGSRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

912.HSGSRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

912.HSGSRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Checkpoint opportunity

912.HSGSRT.B Prove theorems involving similarity

912.HSGSRT.B.4 Prove theorems about triangles.

912.HSGSRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Checkpoint opportunity

912.HSGSRT.C Define trigonometric ratios and solve problems involving right triangles

912.HSGSRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

912.HSGSRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

912.HSGSRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Checkpoint opportunity

912.HSGC Circles

912.HSGC.A Understand and apply theorems about circles

912.HSGC.A.1 Prove that all circles are similar.

912.HSGC.A.2 Identify and describe relationships among inscribed angles, radii, and chords.

912.HSGC.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

912.HSGC.A.4 Construct a tangent line from a point outside a given circle to the circle.

Checkpoint opportunity

912.HSGC.B Find arc lengths and areas of sectors of circles

912.HSGC.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Checkpoint opportunity

912.HSGGPE Expressing Geometric Properties with Equations

912.HSGGPE.A Translate between the geometric description and the equation for a conic section

912.HSGGPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

912.HSGGPE.A.2 Derive the equation of a parabola given a focus and directrix.

Checkpoint opportunity

912.HSGGPE.B Use coordinates to prove simple geometric theorems algebraically

912.HSGGPE.B.4 Use coordinates to prove simple geometric theorems algebraically.

912.HSGGPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Checkpoint opportunity

912.HSGGMD Geometric Measurement and Dimension

912.HSGGMD.A Explain volume formulas and use them to solve problems

912.HSGGMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

912.HSGGMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Checkpoint opportunity
912.HSS Statistics and Probability

912.HSSCP Conditional Probability and the Rules of Probability

912.HSSCP.A Understand independence and conditional probability and use them to interpret data

912.HSSCP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

912.HSSCP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

912.HSSCP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

912.HSSCP.A.4 Construct and interpret twoway frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities.

912.HSSCP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

912.HSSCP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model

912.HSSCP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

912.HSSCP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) — P(A and B), and interpret the answer in terms of the model.

912.HSSCP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model.

912.HSSCP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

Checkpoint opportunity

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.