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Click on any standard to search for aligned resources. This data may be subject to copyright. You may download a CSV of the South Carolina Learning Standards for Mathematics if your intention constitutes fair use.

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Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.

Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

Solve an equation of the form ?(?) = ?(?) graphically by identifying the ?coordinate(s) of the point(s) of intersection of the graphs of ? = ?(?) and ? = ?(?). (Limit to linear; quadratic; exponential.) A1

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

Solve mathematical and real-world problems involving quadratic equations in one variable. (Note: A1.AREI.4a and 4b are not Graduation Standards.) a. Use the method of completing the square to transform any quadratic equation in ? into an equation of the form (? )2 = ? that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations 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 ? +?? for real numbers ? and ?. (Limit to noncomplex roots.)

Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation.

Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables. (Note: A1.AREI.6a and 6b are not Graduation Standards.) a. Solve systems of linear equations using the substitution method. b. Solve systems of linear equations using linear combination.

Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.)

Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Find the zeros of a quadratic function by rewriting it in equivalent factored form and explain the connection between the zeros of the function, its linear factors, the x-intercepts of its graph, and the solutions to the corresponding quadratic equation.

Describe the effect of the transformations ??(?), ?(?)+?, ?(? +?), and combinations of such transformations on the graph of ? = ?(?) for any real number ?. Find the value of ? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.)

Extend previous knowledge of a function to apply to general behavior and features of a function. a. 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. b. Represent a function using function notation and explain that ?(?) denotes the output of function ? that corresponds to the input ?. c. Understand that the graph of a function labeled as ? is the set of all ordered pairs (?,?) that satisfy the equation ? = ?(?).

Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.

Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.)

Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)

Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.)

Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ? = ?? +?.)

Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Limit to linear; quadratic; exponential.) (Note: A1.FIF.8a is not a Graduation Standard.) a. 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.

Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.)

Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. (Note: A1.FLQE.1a is not a Graduation Standard.) a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.)

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function.

Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.)

Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays.

Label and define appropriate quantities in descriptive modeling contexts.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in context.

Rewrite expressions involving simple radicals and rational exponents in different forms.

Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms.

Explain why the sum or product of 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.

Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data.

Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem.

Using technology, compute and interpret the correlation coefficient of a linear fit.

Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations.

Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph. (Limit to polynomials with degrees 3 or less.)

Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable.

Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales.

Use systems of equations and inequalities to represent constraints arising in realworld situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.)

Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.

Solve an equation of the form ?(?) = ?(?) graphically by identifying the ?coordinate(s) of the point(s) of intersection of the graphs of ? = ?(?) and ? = ?(?).

Solve simple rational and radical equations in one variable and understand how extraneous solutions may arise.

Solve mathematical and real-world problems involving quadratic equations in one variable. b. Solve quadratic equations 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 ? +?? for real numbers ? and ?. (Note: A2.AREI.4b is not a Graduation Standard.)

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.)

Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.

Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Note: A2.ASE.3b and 3c are not Graduation Standards.) b. Determine the maximum or minimum value of a quadratic function by completing the square. c. Use the properties of exponents to transform expressions for exponential functions.

Write a function that describes a relationship between two quantities. (Note: IA.FBF.1a is not a Graduation Standard.) a. Write a function that models a relationship between two quantities using both explicit expressions and a recursive process and by combining standard forms using addition, subtraction, multiplication and division to build new functions. b. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.

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

Describe the effect of the transformations ??(?), ?(?)+?, ?(? +?), and combinations of such transformations on the graph of ? = ?(?) for any real number ?. Find the value of ? given the graphs and write the equation of a transformed parent function given its graph.

Define functions recursively and recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.

Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.

Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.

Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.

Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Note: A2.FIF.8b is not a Graduation Standard.) b. Interpret expressions for exponential functions by using the properties of exponents.

Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal.

Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. (Note: A2.FLQE.1b is not a Graduation Standard.) b. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. A2

Interpret the parameters in a linear or exponential function in terms of the context.

Know there is a complex number ? such that ?2 = 1, and every complex number has the form ? +?? with ? and ? real.

Solve quadratic equations in one variable that have complex solutions.

Understand the concept of the derivative of a function geometrically, numerically, analytically, and verbally. a. Interpret the value of the derivative of a function as the slope of the corresponding tangent line. b. Interpret the value of the derivative as an instantaneous rate of change in a variety of real-world contexts such as velocity and population growth. c. Approximate the derivative graphically by finding the slope of the tangent line drawn to a curve at a given point and numerically by using the difference quotient. d. Understand and explain graphically and analytically the relationship between differentiability and continuity. e. Explain graphically and analytically the relationship between the average rate of change and the instantaneous rate of change. f. Understand the definition of the derivative and use this definition to determine the derivatives of various functions.

Apply the rules of differentiation to functions. a. Know and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions. b. Use the constant multiple, sum, difference, product, quotient, and chain rules to find the derivatives of functions. c. Understand and apply the methods of implicit and logarithmic differentiation.

Apply theorems and rules of differentiation to solve mathematical and real-world problems. a. Explain geometrically and verbally the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem. b. Write an equation of a line tangent to the graph of a function at a point. c. Explain the relationship between the increasing/decreasing behavior of ? and the signs of ?. Use the relationship to generate a graph of ? given the graph of ?, and vice versa, and to identify relative and absolute extrema of ?. d. Explain the relationships among the concavity of the graph of ?, the increasing/decreasing behavior of ? and the signs of ?. Use those relationships to generate graphs of ?, ?, and ? given any one of them and identify the points of inflection of ?. e. Solve a variety of real-world problems involving related rates, optimization, linear approximation, and rates of change.

Understand the concept of the integral of a function geometrically, numerically, analytically, and contextually. a. Explain how the definite integral is used to solve area problems. b. Approximate definite integrals by calculating Riemann sums using left, right, and mid-point evaluations, and using trapezoidal sums. c. Interpret the definite integral as a limit of Riemann sums. d. Explain the relationship between the integral and derivative as expressed in both parts of the Fundamental Theorem of Calculus. Interpret the relationship in terms of rates of change.

Understand the concept of a limit graphically, numerically, analytically, and contextually. a. Estimate and verify limits using tables, graphs of functions, and technology. b. Calculate limits, including one-sided limits, algebraically using direct substitution, simplification, rationalization, and the limit laws for constant multiples, sums, differences, products, and quotients. c. Calculate infinite limits and limits at infinity. Understand that infinite limits and limits at infinity provide information regarding the asymptotes of certain functions, including rational, exponential and logarithmic functions.

Understand the definition and graphical interpretation of continuity of a function. a. Apply the definition of continuity of a function at a point to solve problems. b. Classify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity. c. Understand the Intermediate Value Theorem and apply the theorem to prove the existence of solutions of equations arising in mathematical and realworld problems.

Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.)

Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.)

Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.

Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.

Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

Solve an equation of the form ?(?) = ?(?) graphically by identifying the ?coordinate(s) of the point(s) of intersection of the graphs of ? = ?(?) and ? = ?(?). (Limit to linear; quadratic; exponential.)

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

Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation

Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables. (Note: FA.AREI.6a and 6b are not Graduation Standards.) a. Solve systems of linear equations using the substitution method. b. Solve systems of linear equations using linear combination.

Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.)

Describe the effect of the transformations ??(?), ?(?)+?, ?(? +?), and combinations of such transformations on the graph of ? = ?(?) for any real number ?. Find the value of ? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.)

Extend previous knowledge of a function to apply to general behavior and features of a function. a. 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. b. Represent a function using function notation and explain that ?(?) denotes the output of function ? that corresponds to the input ?. c. Understand that the graph of a function labeled as ? is the set of all ordered pairs (?,?) that satisfy the equation ? = ?(?).

Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.

Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.)

Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)

including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ? = ?? +?.)

Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Limit to linear; quadratic; exponential.) (Note: FA.FIF.8a is not a Graduation Standard.) a. 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.

Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.)

Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. (Note: FA.FLQE.1a is not a Graduation Standard.) a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function

Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.)

Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays. FA

Label and define appropriate quantities in descriptive modeling contexts.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in context.

Rewrite expressions involving simple radicals and rational exponents in different forms.

Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms.

Explain why the sum or product of 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.

Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.

Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data.

Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem.

Using technology, compute and interpret the correlation coefficient of a linear fit.

Use probability to evaluate outcomes of decisions by finding expected values and determine if decisions are fair.

Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions.

Understand statistics and sampling distributions as a process for making inferences about population parameters based on a random sample from that population.

Distinguish between experimental and theoretical probabilities. Collect data on a chance event and use the relative frequency to estimate the theoretical probability of that event. Determine whether a given probability model is consistent with experimental results.

Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems.

Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle.

Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction.

Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems.

Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects.

Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a. opposite sides of a parallelogram are congruent; b. opposite angles of a parallelogram are congruent; c. diagonals of a parallelogram bisect each other; d. rectangles are parallelograms with congruent diagonals; e. a parallelograms is a rhombus if and only if the diagonals are perpendicular.

Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships.

Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions.

Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image.

Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other.

Prove two triangles are congruent by applying the Side-Angle-Side, Angle-SideAngle, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a. vertical angles are congruent; b. when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c. any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d. perpendicular lines form four right angles.

Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a. measures of interior angles of a triangle sum to 180; b. base angles of isosceles triangles are congruent; c. the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d. the medians of a triangle meet at a point. G

Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems.

Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieris principle. G

Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications.

Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems.

Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula.

Use coordinates to prove simple geometric theorems algebraically.

Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope.

Given two points, find the point on the line segment between the two points that divides the segment into a given ratio.

Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates.

Use geometric shapes, their measures, and their properties to describe real-world objects.

Use geometry concepts and methods to model real-world situations and solve problems using a model.

Understand 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. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other.

Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results.

Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a. A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c. The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides.

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

Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.

Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.

Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem

Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.

Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.

Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).

Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations.

Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable.

Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable.

Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.

Solve an equation of the form ?(?) = ?(?) graphically by identifying the ?coordinate(s) of the point(s) of intersection of the graphs of ? = ?(?) and ? = ?(?).

Solve simple rational and radical equations in one variable and understand how extraneous solutions may arise.

Solve mathematical and real-world problems involving quadratic equations in one variable. (Note: IA.AREI.4a and 4b are not Graduation Standards.) a. Use the method of completing the square to transform any quadratic equation in ? into an equation of the form (? )2 = ? that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations 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 ? +?? for real numbers ? and ?.

Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.

Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Note: IA.ASE.3b is not a Graduation Standard.) a. Find the zeros of a quadratic function by rewriting it in equivalent factored form and explain the connection between the zeros of the function, its linear factors, the x-intercepts of its graph, and the solutions to the corresponding quadratic equation. b. Determine the maximum or minimum value of a quadratic function by completing the square.

Write a function that describes a relationship between two quantities. (Note: IA.FBF.1a is not a Graduation Standard.) a. Write a function that models a relationship between two quantities using both explicit expressions and a recursive process and by combining standard forms using addition, subtraction, multiplication and division to build new functions. b. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.

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

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

Define functions recursively and recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.

Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.

Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.

Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.

Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Note: IA.FIF.8b is not a Graduation Standard.) b. Interpret expressions for exponential functions by using the properties of exponents.

Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal.

Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables.

Interpret the parameters in a linear or exponential function in terms of the context.

Know there is a complex number ? such that ?2 = 1, and every complex number has the form ? +?? with ? and ? real.

Solve quadratic equations in one variable that have complex solutions.

Express a logarithm as the solution to the exponential equation, ???? = ? where ?, ?, and ? are numbers and the base ? is 2, 10, or ?; evaluate the logarithm using technology.

Know and apply the Division Theorem and the Remainder Theorem for polynomials.

Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph.

Prove polynomial identities and use them to describe numerical relationships.

Apply the Binomial Theorem to expand powers of binomials, including those with one and with two variables. Use the Binomial Theorem to factor squares, cubes, and fourth powers of binomials.

Apply algebraic techniques to rewrite simple rational expressions in different forms; using inspection, long division, or, for the more complicated examples, a computer algebra system.

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.

Solve an equation of the form ?(?) = ?(?) graphically by identifying the ?coordinate(s) of the point(s) of intersection of the graphs of ? = ?(?) and ? = ?(?).

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions.

Represent a system of linear equations as a single matrix equation in a vector variable.

Using technology for matrices of dimension 3 3 or greater, find the inverse of a matrix if it exists and use it to solve systems of linear equations

Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.

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 including applications to finance.

Write a function that describes a relationship between two quantities. b. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.

Describe the effect of the transformations ??(?), ?(?)+?, ?(? +?), and combinations of such transformations on the graph of ? = ?(?) for any real number ?. Find the value of ? given the graphs and write the equation of a transformed parent function given its graph

Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse functions if and only if f(x) = y and g(y) = x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain. a. Use composition to verify one function is an inverse of another. b. If a function has an inverse, find values of the inverse function from a graph or table.

Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents.

Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.

Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.

Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.

Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Note: PC.FIF.7a d are not Graduation Standards.) a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. b. Graph radical functions over their domain show end behavior. c. Graph exponential and logarithmic functions, showing intercepts and end behavior. d. Graph trigonometric functions, showing period, midline, and amplitude.

Understand that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle.

Define sine and cosine as functions of the radian measure of an angle in terms of the ?- and ?-coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions. a. Define the tangent, cotangent, secant, and cosecant functions as ratios involving sine and cosine. b. Write cotangent, secant, and cosecant functions as the reciprocals of tangent, cosine, and sine, respectively.

Use special triangles to determine geometrically the values of sine, cosine, tangent for ? 3 , ? 4 , and ? 6 , and use the unit circle to express the values of sine, cosine, and tangent for ? ?, ? +?, and 2? ? in terms of their values for ?, where ? is any real number.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Define the six inverse trigonometric functions using domain restrictions for regions where the function is always increasing or always decreasing.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sin?, cos?, or tan?, given sin?, cos?, or tan?, and the quadrant of the angle.

Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems.

Derive the formulas for the length of an arc and the area of a sector in a circle, and apply these formulas to solve mathematical and real-world problems.

Use the geometric definition of a parabola to derive its equation given the focus and directrix.

Use the geometric definition of an ellipse and of a hyperbola to derive the equation of each given the foci and points whose sum or difference of distance from the foci are constant.

Prove the Laws of Sines and Cosines and use them to solve problems.

Use the Law of Sines and the Law of Cosines to solve for unknown measures of sides and angles of triangles that arise in mathematical and real-world problems.

Derive the formula ? = 1 2 ??sin? for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Find the conjugate of a complex number in rectangular and polar forms and use conjugates to find moduli and quotients of complex numbers.

Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Determine the modulus of a complex number by multiplying by its conjugate and determine the distance between two complex numbers by calculating the modulus of their difference.

Solve quadratic equations in one variable that have complex solutions.

Extend polynomial identities to the complex numbers and use DeMoivres Theorem to calculate a power of a complex number.

Know the Fundamental Theorem of Algebra and explain why complex roots of polynomials with real coefficients must occur in conjugate pairs.

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.

Multiply a vector by a matrix of appropriate dimension to produce another vector. Work with matrices as transformations of vectors.

Apply 22 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Represent and model with vector quantities. Use the coordinates of an initial point and of a terminal point to find the components of a vector.

Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors. a. Add and subtract vectors using components of the vectors and graphically. b. Given the magnitude and direction of two vectors, determine the magnitude of their sum and of their difference.

Multiply a vector by a scalar, representing the multiplication graphically and computing the magnitude of the scalar multiple.

Use matrices to represent and manipulate data. (Note: This Graduation Standard is covered in 8.)

Perform operations with matrices of appropriate dimensions including addition, subtraction, and scalar multiplication.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

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.

Describe events as subsets of a sample space and a. Use Venn diagrams to represent intersections, unions, and complements. b. Relate intersections, unions, and complements to the words and, or, and not. c. Represent sample spaces for compound events using Venn diagrams.

Use the multiplication rule to calculate probabilities for independent and dependent events. 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.

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.

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

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Calculate the conditional probability of an event A given event B as the fraction of Bs outcomes that also belong to A, and interpret the answer in terms of the model.

Apply the Addition Rule and the Multiplication Rule to determine probabilities, including conditional probabilities, and interpret the results in terms of the probability model.

Use permutations and combinations to solve mathematical and real-world problems, including determining probabilities of compound events. Justify the results.

Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.

Create residual plots and analyze those plots to compare the fit of linear, quadratic, and exponential models to a given data set. Select the appropriate model and use it for interpolation.

Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.

Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).

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.

Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.

Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.

Find linear models using median fit and regression methods to make predictions. Interpret the slope and intercept of a linear model in the context of the data.

Compute using technology and interpret the correlation coefficient of a linear fit.

Differentiate between correlation and causation when describing the relationship between two variables. Identify potential lurking variables which may explain an association between two variables.

Develop the probability distribution for a random variable defined for a sample space in which a theoretical probability can be calculated and graph the distribution.

Calculate the expected value of a random variable as the mean of its probability distribution. Find expected values by assigning probabilities to payoff values. Use expected values to evaluate and compare strategies in real-world scenarios.

Construct and compare theoretical and experimental probability distributions and use those distributions to find expected values.

Use probability to evaluate outcomes of decisions by finding expected values and determine if decisions are fair.

Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions.

Understand statistics and sampling distributions as a process for making inferences about population parameters based on a random sample from that population.

Distinguish between experimental and theoretical probabilities. Collect data on a chance event and use the relative frequency to estimate the theoretical probability of that event. Determine whether a given probability model is consistent with experimental results.

Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods to reduce bias.

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.

Distinguish between experiments and observational studies. Determine which of two or more possible experimental designs will best answer a given research question and justify the choice based on statistical significance.

Evaluate claims and conclusions in published reports or articles based on data by analyzing study design and the collection, analysis, and display of the data.

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