# Mathematics Course Curriculum Standards

SOUTH CAROLINA

ACADEMIC STANDARDS

FOR

MATHEMATICS

**South Carolina Department of Education**

** **

2007

# HIGH SCHOOL

# CORE-AREA

# STANDARDS

Algebra 1

The academic standards for the elementary algebra core area establish the process skills and core content for Algebra 1, Mathematics for the Technologies 1, and Mathematics for the Technologies 2, which should provide students with the mathematics skills and conceptual understanding necessary for them to further their mathematical education or to pursue mathematics-related technical careers. These standards will be the basis for the development of the items on the state-required end-of-course examination for Algebra 1 and Mathematics for the Technologies 2.

The content of the elementary algebra standards encompasses the real number system; operations involving exponents, matrices, and algebraic expressions; relations and functions; writing and solving linear equations; graphs and characteristics of linear equations; and quadratic relationships and functions. Teachers, schools, and districts should use the elementary algebra standards to make decisions concerning the structure and content of Algebra 1, Mathematics for the Technologies 1, and Mathematics for the Technologies 2. Content in these three courses may go beyond the elementary algebra standards.

All courses based on the academic standards for elementary algebra must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or individual classes may go beyond these standards. Such decisions will involve choices regarding additional content, activities, and learning strategies and will depend on the objectives of the particular courses or individual classes.

In all courses based on the elementary algebra standards, hand-held graphing calculators are required for instruction and assessment. Students should learn to use a variety of ways to represent data, to use a variety of mathematical tools such as graph paper, and to use technologies such as graphing calculators to solve problems.

**Note**: The term *including* appears in parenthetical statements in the high school mathematics indicators to introduce a list of specifics that are intended to clarify and focus the teaching and learning of the particular concept. That is, within these parenthetical including statements are specified the components of the indicator that are critical for the particular core area with regard both to the state assessments and to the management of time in the classroom. While instruction must focus on the entire indicator, educators must be certain to cover the components specified in the parenthetical *including* statements.

The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs should be built around these processes.

**Standard EA-1: **The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

**Indicators**

EA-1.1 Communicate a knowledge of algebraic relationships by using mathematical terminology appropriately.

EA-1.2 Connect algebra with other branches of mathematics.

EA-1.3 Apply algebraic methods to solve problems in real-world contexts.

EA-1.4 Judge the reasonableness of mathematical solutions.

EA-1.5 Demonstrate an understanding of algebraic relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic).

EA-1.6 Understand how algebraic relationships can be represented in concrete models, pictorial models, and diagrams.

EA-1.7 Understand how to represent algebraic relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs).

**Standard EA-2:** The student will demonstrate through the mathematical processes an understanding of the real number system and operations involving exponents, matrices, and algebraic expressions.

**Indicators**

EA-2.1 Exemplify elements of the real number system (including integers, rational numbers, and irrational numbers).

EA-2.2 Apply the laws of exponents and roots to solve problems.

EA-2.3 Carry out a procedure to perform operations (including multiplication and division) with numbers written in scientific notation.

EA-2.4 Use dimensional analysis to convert units of measure within a system.

EA-2.5 Carry out a procedure using the properties of real numbers (including commutative, associative, and distributive) to simplify expressions.

EA-2.6 Carry out a procedure to evaluate an expression by substituting a value for the variable.

EA-2.7 Carry out a procedure (including addition, subtraction, multiplication, and division by a monomial) to simplify polynomial expressions.

EA-2.8 Carry out a procedure to factor binomials, trinomials, and polynomials by using various techniques (including the greatest common factor, the difference between two squares, and quadratic trinomials).

EA-2.9 Carry out a procedure to perform operations with matrices (including addition, subtraction, and scalar multiplication).

EA-2.10 Represent applied problems by using matrices.

**Standard EA-3:** The student will demonstrate through the mathematical processes an understanding of relationships and functions.

**Indicators**

EA-3.1 Classify a relationship as being either a function or not a function when given data as a table, set of ordered pairs, or graph.

EA-3.2 Use function notation to represent functional relationships.

EA-3.3 Carry out a procedure to evaluate a function for a given element in the domain.

EA-3.4 Analyze the graph of a continuous function to determine the domain and range of the function.

EA-3.5 Carry out a procedure to graph parent functions

EA-3.6 Classify a variation as either direct or inverse.

EA-3.7 Carry out a procedure to solve literal equations for a specified variable.

EA-3.8 Apply proportional reasoning to solve problems.

**Standard EA-4: ** The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.

**Indicators **

EA-4.1 Carry out a procedure to write an equation of a line with a given slope and a *y*-intercept.

EA-4.2 Carry out a procedure to write an equation of a line with a given slope passing through a given point.

EA-4.3 Carry out a procedure to write an equation of a line passing through two given points.

EA-4.4 Use a procedure to write an equation of a trend line from a given scatterplot.

EA-4.5 Analyze a scatterplot to make predictions.

EA-4.6 Represent linear equations in multiple forms (including point-slope, slope-intercept, and standard).

EA-4.7 Carry out procedures to solve linear equations for one variable algebraically.

EA-4.8 Carry out procedures to solve linear inequalities for one variable algebraically and then to graph the solution.

EA-4.9 Carry out a procedure to solve systems of two linear equations graphically.

EA-4.10 Carry out a procedure to solve systems of two linear equations algebraically.

**Standard EA-5:** The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.

**Indicators**

EA-5.1 Carry out a procedure to graph a line when given the equation of the line.

EA-5.2 Analyze the effects of changes in the slope, m*,* and the *y*-intercept, b*,* on the graph of y = mx + b.

EA-5.3 Carry out a procedure to graph the line with a given slope and a *y*-intercept.

EA-5.4 Carry out a procedure to graph the line with a given slope passing through a given point.

EA-5.5 Carry out a procedure to determine the *x*-intercept and *y*-intercept of lines from data given tabularly, graphically, symbolically, and verbally.

EA-5.6 Carry out a procedure to determine the slope of a line from data given tabularly, graphically, symbolically, and verbally.

EA-5.7 Apply the concept of slope as a rate of change to solve problems.

EA-5.8 Analyze the equations of two lines to determine whether the lines are perpendicular or parallel.

EA-5.9 Analyze given information to write a linear function that models a given problem situation.

EA-5.10 Analyze given information to determine the domain and range of a linear function in a problem situation.

EA-5.11 Analyze given information to write a system of linear equations that models a given problem situation.

EA-5.12 Analyze given information to write a linear inequality in one variable that models a given problem situation.

**Standard EA-6:** The student will demonstrate through the mathematical processes an understanding of quadratic relationships and functions.

#### Indicators

EA-6.1 Analyze the effects of changing the leading coefficient *a* on the graph of y = ax^{2}.

EA-6.2 Analyze the effects of changing the constant *c* on the graph of y = x^{2} + c.

EA-6.3 Analyze the graph of a quadratic function to determine its equation.

EA-6.4 Carry out a procedure to solve quadratic equations by factoring.

EA-6.5 Carry out a graphic procedure to approximate the solutions of quadratic equations.

EA-6.6 Analyze given information to determine the domain of a quadratic function in a problem situation.

Algebra 2

The academic standards for the intermediate algebra core area establish the process skills and core content for Algebra 2, which should provide students with the mathematics skills and conceptual understanding necessary for them to further their mathematical education or to pursue mathematics-related technical careers.

The content of the intermediate algebra standards encompasses functions; systems of equations; systems of linear inequalities; quadratic equations; complex numbers; algebraic expressions; nonlinear relationships including exponential, logarithmic, radical, polynomial, and rational; conic sections; and sequences and series.** **Teachers, schools, and districts should use the intermediate algebra standards to make decisions concerning the structure and content of Algebra 2. Content in this course may go beyond the intermediate algebra standards.

All courses based on the academic standards for intermediate algebra must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or individual classes may go beyond these standards. Such decisions will involve choices regarding additional content, activities, and learning strategies and will depend on the objectives of the particular courses or individual classes.

In all courses based on the intermediate algebra standards, hand-held graphing calculators are required for instruction and assessment. Students should learn to use a variety of ways to represent data, to use a variety of mathematical tools such as graph paper, and to use technologies such as graphing calculators to solve problems.

**Note**: The term *including* appears in parenthetical statements in the high school mathematics indicators to introduce a list of specifics that are intended to clarify and focus the teaching and learning of the particular concept. That is, within these parenthetical including statements are specified the components of the indicator that are critical for the particular core area with regard both to the state assessments and to the management of time in the classroom. While instruction must focus on the entire indicator, educators must be certain to cover the components specified in the parenthetical *including* statements.

The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs should be built around these processes.

**Standard IA-1: **The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

**Indicators**

IA-1.1 Communicate a knowledge of algebraic relationships by using mathematical terminology appropriately.

IA-1.2 Connect algebra with other branches of mathematics.

IA-1.3 Apply algebraic methods to solve problems in real-world contexts.

IA-1.4 Judge the reasonableness of mathematical solutions.

IA-1.5 Demonstrate an understanding of algebraic relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic).

IA-1.6 Understand how algebraic relationships can be represented in concrete models, pictorial models, and diagrams.

IA-1.7 Understand how to represent algebraic relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs).

**Standard IA-2:** The student will demonstrate through the mathematical processes an understanding of functions, systems of equations, and systems of linear inequalities.

**Indicators**

IA-2.1 Carry out a procedure to solve a system of linear inequalities algebraically.

IA-2.2 Carry out a procedure to solve a system of linear inequalities graphically.

IA-2.3 Analyze a problem situation to determine a system of linear inequalities that models the problem situation.

IA-2.4 Use linear programming to solve contextual problems involving a system of linear inequalities.

IA-2.5 Carry out procedures to perform operations on polynomial functions (including f(x) + g(x), f(x) – g(x), f(x) • g(x), f(x) / g(x)).

IA-2.6 Apply a procedure to write the equation of a composition of given functions.

IA-2.7 Carry out a procedure to graph translations of parent functions.

IA-2.8 Carry out a procedure to graph transformations of parent functions (including y = x, y = x^{2}, and y = |x|).

IA-2.9 Carry out a procedure to graph discontinuous functions (including piecewise and step functions).

IA-2.10 Carry out a procedure to determine the domain and range of discontinuous functions (including piecewise and step functions).

IA-2.11 Carry out a procedure to solve a system of equations (including two linear functions and one linear function with one quadratic function).

**Standard IA-3:** The student will demonstrate through the mathematical processes an understanding of quadratic equations and the complex number system.

**Indicators**

IA-3.1 Carry out a procedure to simplify expressions involving powers of *i*.

IA-3.2 Carry out a procedure to perform operations with complex numbers (including addition, subtraction, multiplication, and division).

IA-3.3 Carry out a procedure to solve quadratic equations algebraically (including factoring, completing the square, and applying the quadratic formula).

IA-3.4 Use the discriminant to determine the number and type of solutions of a quadratic equation.

IA-3.5 Analyze given information (including quadratic models) to solve contextual problems.

IA-3.6 Carry out a procedure to write an equation of a quadratic function when given its roots.

**Standard IA-4:** The student will demonstrate through the mathematical processes an understanding of algebraic expressions and nonlinear functions.

**Indicators**

IA-4.1 Carry out a procedure to perform operations (including multiplication, exponentiation, and division) with polynomial expressions.

IA-4.2 Carry out a procedure to determine specified points (including zeros, maximums, and minimums) of polynomial functions.

IA-4.3 Carry out a procedure to solve polynomial equations (including factoring by grouping, factoring the difference between two squares, factoring the sum of two** **cubes, and factoring the difference between two

**cubes).**

IA-4.4 Analyze given information (including polynomial models) to solve contextual problems.

IA-4.5 Carry out a procedure to simplify algebraic expressions involving rational exponents.

IA-4.6 Carry out a procedure to simplify algebraic expressions involving logarithms.

IA-4.7 Carry out a procedure to perform operations with expressions involving rational exponents (including addition, subtraction, multiplication, division, and exponentiation).

IA-4.8 Carry out a procedure to perform operations with rational expressions (including addition, subtraction, multiplication, and division).

IA-4.9 Carry out a procedure to solve radical equations algebraically.

IA-4.10 Carry out a procedure to solve logarithmic equations algebraically.

IA-4.11 Carry out a procedure to solve logarithmic equations graphically.

IA-4.12 Carry out a procedure to solve rational equations algebraically.

IA-4.13 Carry out a procedure to graph logarithmic functions.

IA-4.14 Carry out a procedure to graph exponential functions.

**Standard IA-5:** The student will demonstrate through the mathematical processes an understanding of conic sections.

### Indicators

IA-5.1 Carry out a procedure to graph the circle whose equation is the standard form.

IA-5.2 Carry out a procedure to write an equation of a circle centered at the origin when given its radius.

IA-5.3 Carry out a procedure to graph the ellipse whose equation is the standard form.

IA-5.4 Carry out a procedure to write an equation of an ellipse centered at the origin when given information from among length of major axis, length of minor axis, and vertices.

IA-5.5 Carry out a procedure to graph the hyperbola whose equation is the standard form.

IA-5.6 Carry out a procedure to write an equation of a hyperbola centered at the origin with specified vertices.

IA-5.7 Match the equation of a conic section with its graph.

**Standard IA-6:** The student will demonstrate through the mathematical processes an understanding of sequences and series.

**Indicators**

IA-6.1 Categorize a sequence as arithmetic, geometric, or neither.

IA-6.2 Carry out a procedure to write a specified term of an arithmetic or geometric sequence when given the nth term of the sequence.

IA-6.3 Carry out a procedure to write a formula for the nth term of an arithmetic or geometric sequence when given at least four consecutive terms of the sequence.

IA-6.4 Carry out a procedure to write a formula for the nth term of an arithmetic or geometric sequence when given at least four terms of the sequence.

IA-6.5 Represent an arithmetic or geometric series by using sigma notation.

IA-6.6 Carry out a procedure to calculate the sum of an arithmetic or geometric series written in sigma notation.

IA-6.7 Carry out a procedure to determine consecutive terms of a sequence that is defined recursively.

IA-6.8 Carry out a procedure to define a sequence recursively when given four or more consecutive terms of the sequence.

IA-6.9 Translate between the explicit form and the recursive form of sequences.

Geometry

The academic standards for the geometry core area establish the process skills and core content for Geometry and Mathematics for the Technologies 3, which should provide students with the mathematics skills and conceptual understanding necessary for them to further their mathematical education or to pursue mathematics-related technical careers.

The content of the geometry standards encompasses** **properties of basic geometric figures; properties of triangles; properties of quadrilaterals and other polygons; properties of circles, lines, and special segments intersecting circles; transformations; coordinate geometry; vectors; surface area and volume of three-dimensional objects; and proofs.** **Teachers, schools, and districts should use the geometry standards to make decisions concerning the structure and content of Geometry and Mathematics for the Technologies 3. Content in these two courses may go beyond the geometry standards.

All courses based on the academic standards for geometry must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or individual classes may go beyond these standards. Such decisions will involve choices regarding additional content, activities, and learning strategies and will depend on the objectives of the particular courses or individual classes.

In all courses based on the geometry standards, appropriate tools and technologies are required for instruction and assessment. Students should learn to use a variety of ways to represent data, to use a variety of mathematical tools such as graph paper, and to use technologies such as graphing calculators to solve problems.

**Note**: The term *including* appears in parenthetical statements in the high school mathematics indicators to introduce a list of specifics that are intended to clarify and focus the teaching and learning of the particular concept. That is, within these parenthetical including statements are specified the components of the indicator that are critical for the particular core area with regard both to the state assessments and to the management of time in the classroom. While instruction must focus on the entire indicator, educators must be certain to cover the components specified in the parenthetical *including* statements.

The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs should be built around these processes.

**Standard G-1:** The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

**Indicators**

G-1.1 Demonstrate an understanding of the axiomatic structure of geometry by using undefined terms, definitions, postulates, theorems, and corollaries.

G-1.2 Communicate knowledge of geometric relationships by using mathematical terminology appropriately.

G-1.3 Apply basic rules of logic to determine the validity of the converse, inverse, and contrapositive of a conditional statement.

G-1.4 Formulate and test conjectures by using a variety of tools such as concrete models, graphing calculators, spreadsheets, and dynamic geometry software.

G-1.5 Use inductive reasoning to formulate conjectures.

G-1.6 Use deductive reasoning to validate conjectures with formal and informal proofs, and give counterexamples to disprove a statement.

G-1.7 Understand the historical development of geometry.

G-1.8 Connect geometry with other branches of mathematics.

G-1.9 Demonstrate an understanding of how geometry applies to in real-world contexts (including architecture, construction, farming, and astronomy).

G-1.10 Demonstrate an understanding of geometric relationships (including constructions through investigations by using a variety of tools such as straightedge, compass, Patty Paper, dynamic geometry software, and handheld computing devices).

**Standard G-2:** The student will demonstrate through the mathematical processes an understanding of the properties of basic geometric figures and the relationships between and among them.

**Indicators**

G-2.1 Infer missing elements of visual or numerical geometric patterns (including triangular and rectangular numbers and the number of diagonals in polygons).

G-2.2 Apply properties of parallel lines, intersecting lines, and parallel lines cut by a transversal to solve problems.

G-2.3 Use the congruence of line segments and angles to solve problems.

G-2.4 Use direct measurement to determine the length of a segment, degree of an angle, and distance from a point to a line.

G-2.5 Carry out a procedure to create geometric constructions (including the midpoint of a line segment, the angle bisector, the perpendicular bisector of a line segment, the line through a given point that is parallel to a given line, and the line through a given point that is perpendicular to a given line).

G-2.6 Use scale factors to solve problems involving scale drawings and models.

G-2.7 Use geometric probability to solve problems.

**Standard G-3:** The student will demonstrate through the mathematical processes an understanding of the properties and special segments of triangles and the relationships between and among triangles.

**Indicators**

G-3.1 Carry out a procedure to compute the perimeter of a triangle.

G-3.2 Carry out a procedure to compute the area of a triangle.

G-3.3 Analyze how changes in dimensions affect the perimeter or area of triangles.

G-3.4 Apply properties of isosceles and equilateral triangles to solve problems.

G-3.5 Use interior angles, exterior angles, medians, angle bisectors, altitudes, and perpendicular bisectors to solve problems.

G-3.6 Apply the triangle sum theorem to solve problems.

G-3.7 Apply the triangle inequality theorem to solve problems.

G-3.8 Apply congruence and similarity relationships among triangles to solve problems.

G-3.9 Apply theorems to prove that triangles are either similar or congruent.

G-3.10 Use the Pythagorean theorem and its converse to solve problems.

G-3.11 Use the properties of 45-45-90 and 30-60-90 triangles to solve problems.

G-3.12 Use trigonometric ratios (including sine, cosine, and tangent) to solve problems involving right triangles.

**Standard G-4:** The student will demonstrate through the mathematical processes an understanding of the properties of quadrilaterals and other polygons and the relationships between and among them.

**Indicators**

G-4.1 Carry out a procedure to compute the perimeter of quadrilaterals, regular polygons, and composite figures.

G-4.2 Carry out a procedure to find the area of quadrilaterals, regular polygons, and composite figures.

G-4.3 Apply procedures to compute measures of interior and exterior angles of polygons.

G-4.4 Analyze how changes in dimensions affect the perimeter or area of quadrilaterals and regular polygons.

G-4.5 Apply the properties and attributes of quadrilaterals and regular polygons and their component parts to solve problems.

G-4.6 Apply congruence and similarity relationships among shapes (including quadrilaterals and polygons) to solve problems.

**Standard G-5: ** The student will demonstrate through the mathematical processes an understanding of the properties of circles, the lines that intersect them, and the use of their special segments.

**Indicators**

G-5.1 Carry out a procedure to compute the circumference of circles.

G-5.2 Carry out a procedure to compute the area of circles.

G-5.3 Analyze how a change in the radius affects the circumference or area of a circle.

G-5.4 Carry out a procedure to compute the length of an arc or the area of a sector of a circle.

G-5.5 Apply the properties of the component parts of a circle (including radii, diameters, chords, sectors, arcs, and segments) to solve problems.

G-5.6 Apply the properties of lines that intersect circles (including two secants, two tangents, and a secant and a tangent) to solve problems.

G-5.7 Apply the properties of central angles, inscribed angles, and arcs of circles to solve problems.

**Standard G-6: **The student will demonstrate through the mathematical processes an understanding of transformations, coordinate geometry, and vectors.

**Indicators**

G-6.1 Use the distance formula to solve problems.

G-6.2 Use the midpoint formula to solve problems.

G-6.3 Apply transformations—translation, reflection, rotation, and dilation—to figures in the coordinate plane by using sketches and coordinates.

G-6.4 Apply transformations (including translation and dilation) to figures in the coordinate plane by using matrices.

G-6.5 Carry out a procedure to represent the sum of two vectors geometrically by using the parallelogram method.

G-6.6 Carry out a procedure to determine the magnitude and direction of the resultant of two vectors by using a scale drawing and direct measurement.

G-6.7 Carry out a procedure to compute the magnitude of the resultant of two perpendicular vectors by using the Pythagorean theorem.

G-6.8 Carry out a procedure to determine the direction of the resultant of two perpendicular vectors by using a scale drawing and direct measurement.

**Standard G-7:** The student will demonstrate through the mathematical processes an understanding of the surface area and volume of three-dimensional objects.

**Indicators**

G-7.1 Carry out a procedure to compute the surface area of three-dimensional objects (including cones, cylinders, pyramids, prisms, spheres, and hemispheres).

G-7.2 Carry out a procedure to compute the volume of three-dimensional objects (including cones, cylinders, pyramids, prisms, spheres, hemispheres, and composite objects).

G-7.3 Analyze how changes in dimensions affect the volume of objects (including cylinders, prisms, and spheres).

G-7.4 Apply congruence and similarity relationships among geometric objects to solve problems.

G-7.5 Apply a procedure to draw a top view, front view, and side view of a three-dimensional object.

G-7.6 Apply a procedure to draw an isometric view of a three-dimensional object.

Precalculus

The academic standards for the precalculus core area establish the process skills and core content for Precalculus, which should provide students with the mathematics skills and conceptual understanding necessary for them to further their mathematical education or to pursue mathematics-related technical careers.

The content of the precalculus standards encompasses** **characteristics and behaviors of functions, operations on functions, behaviors of polynomial functions and rational functions, behaviors of exponential and logarithmic functions, behaviors of trigonometric functions, and behaviors of conic sections.** **Teachers, schools, and districts should use the precalculus standards to make decisions concerning the structure and content of Precalculus. Content in this course may go beyond the precalculus standards.

All courses based on the academic standards for precalculus must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or individual classes may go beyond these standards. Such decisions will involve choices regarding additional content, activities, and learning strategies and will depend on the objectives of the particular courses or individual classes.

In all courses based on the precalculus standards, hand-held graphing calculators are required for instruction and assessment. Students should learn to use a variety of ways to represent data, to use a variety of mathematical tools such as graph paper, and to use technologies such as graphing calculators to solve problems.

**Note**: The term *including* appears in parenthetical statements in the high school mathematics indicators to introduce a list of specifics that are intended to clarify and focus the teaching and learning of the particular concept. That is, within these parenthetical including statements are specified the components of the indicator that are critical for the particular core area with regard both to the state assessments and to the management of time in the classroom. While instruction must focus on the entire indicator, educators must be certain to cover the components specified in the parenthetical *including* statements.

**Standard PC-1:** The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

**Indicators**

PC-1.1 Communicate knowledge of algebraic and trigonometric relationships by using mathematical terminology appropriately.

PC-1.2 Connect algebra and trigonometry with other branches of mathematics.

PC-1.3 Apply algebraic methods to solve problems in real-world contexts.

PC-1.4 Judge the reasonableness of mathematical solutions.

PC-1.5 Demonstrate an understanding of algebraic and trigonometric relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic).

PC-1.6 Understand how algebraic and trigonometric relationships can be represented in concrete models, pictorial models, and diagrams.

PC-1.7 Understand how to represent algebraic and trigonometric relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs).

**Standard PC-2:** The student will demonstrate through the mathematical processes an understanding of the characteristics and behaviors of functions and the effect of operations on functions.

**Indicators**

PC-2.1 Carry out a procedure to graph parent functions.

PC-2.2 Carry out a procedure to graph transformations (including *–*f(x), a•f(x), f(x) + d, f(x – c), f(-x), f(b • x), |f(x)|, and f(|x|)) of parent functions and combinations of transformations.

PC-2.3 Analyze a graph to describe the transformation (including *–*f(x), a•f(x), f(x) + d, f(x – c), f(-x), f(b • x), |f(x)|, and f(|x|)) of parent functions.

PC-2.4 Carry out procedures to algebraically solve equations involving parent functions or transformations of parent functions.

PC-2.5 Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions.

PC-2.6 Analyze a function or the symmetry of its graph to determine whether the function is even, odd, or neither.

PC-2.7 Recognize and use connections among significant points of a function (including roots, maximum points, and minimum points), the graph of a function, and the algebraic representation of a function.

PC-2.8 Carry out a procedure to determine whether the inverse of a function exists.

PC-2.9 Carry out a procedure to write a rule for the inverse of a function, if it exists.

**Standard PC-3:** The student will demonstrate through the mathematical processes an understanding of the behaviors of polynomial and rational functions.

**Indicators**

PC-3.1 Carry out a procedure to graph quadratic and higher-order polynomial functions by analyzing intercepts and end behavior.

PC-3.2 Apply the rational root theorem to determine a set of possible rational roots of a polynomial equation.

PC-3.3 Carry out a procedure to calculate the zeros of polynomial functions when given a set of possible zeros.

PC-3.4 Carry out procedures to determine characteristics of rational functions (including domain, range, intercepts, asymptotes, and discontinuities).

PC-3.5 Analyze given information to write a polynomial function that models a given problem situation.

PC-3.6 Carry out a procedure to solve polynomial equations algebraically.

PC-3.7 Carry out a procedure to solve polynomial equations graphically.

PC-3.8 Carry out a procedure to solve rational equations algebraically.

PC-3.9 Carry out a procedure to solve rational equations graphically.

PC-3.10 Carry out a procedure to solve polynomial inequalities algebraically.

PC-3.11 Carry out a procedure to solve polynomial inequalities graphically.

**Standard PC-4:** The student will demonstrate through the mathematical processes an understanding of the behaviors of exponential and logarithmic functions.

**Indicators**

PC-4.1 Carry out a procedure to graph exponential functions by analyzing intercepts and end behavior.

PC-4.2 Carry out a procedure to graph logarithmic functions by analyzing intercepts and end behavior.

PC-4.3 Carry out procedures to determine characteristics of exponential functions (including domain, range, intercepts, and asymptotes).

PC-4.4 Carry out procedures to determine characteristics of logarithmic functions (including domain, range, intercepts, and asymptotes).

PC-4.5 Apply the laws of exponents to solve problems involving rational exponents.

PC-4.6 Analyze given information to write an exponential function that models a given problem situation.

PC-4.7 Apply the laws of logarithms to solve problems.

PC-4.8 Carry out a procedure to solve exponential equations algebraically.

PC-4.9 Carry out a procedure to solve exponential equations graphically.

PC-4.10 Carry out a procedure to solve logarithmic equations algebraically.

PC-4.11 Carry out a procedure to solve logarithmic equations graphically.

**Standard PC-5:** The student will demonstrate through the mathematical processes an understanding of the behaviors of trigonometric functions.

**Indicators**

PC-5.1 Understand how angles are measured in either degrees or radians.

PC-5.2 Carry out a procedure to convert between degree and radian measures.

PC-5.3 Carry out a procedure to plot points in the polar coordinate system.

PC-5.4 Carry out a procedure to graph trigonometric functions by analyzing intercepts, periodic behavior, and graphs of reciprocal functions.

PC-5.5 Carry out procedures to determine the characteristics of trigonometric functions (including domain, range, intercepts, and asymptotes).

PC-5.6 Apply a procedure to evaluate trigonometric expressions.

PC-5.7 Analyze given information to write a trigonometric function that models a given problem situation involving periodic phenomena.

PC-5.8 Analyze given information to write a trigonometric equation that models a given problem situation involving right triangles.

PC-5.9 Carry out a procedure to calculate the area of a triangle when given the lengths of two sides and the measure of the included angle.

PC-5.10 Carry out a procedure to solve trigonometric equations algebraically.

PC-5.11 Carry out a procedure to solve trigonometric equations graphically.

PC-5.12 Apply the laws of sines and cosines to solve problems.

PC-5.13 Apply a procedure to graph the inverse functions of sine, cosine, and tangent.

PC-5.14 Apply trigonometric relationships (including reciprocal identities; Pythagorean identities; even and odd identities; addition and subtraction formulas of sine, cosine, and tangent; and double angle formulas) to verify other trigonometric identities.

PC-5.15 Carry out a procedure to compute the slope of a line when given the angle of inclination of the line.

**Standard PC-6:** The student will demonstrate through the mathematical processes an understanding of the behavior of conic sections both geometrically and algebraically.

**Indicators**

PC-6.1 Carry out a procedure to graph the circle whose equation is the standard form.

PC-6.2 Analyze given information about the center and the radius or the center and the diameter to write an equation of a circle.

PC-6.3 Apply a procedure to calculate the coordinates of points where a line intersects a circle.

PC-6.4 Carry out a procedure to graph the ellipse whose equation is the standard form.

PC-6.5 Carry out a procedure to graph the hyperbola whose equation is the standard form.

PC-6.6 Carry out a procedure to graph the parabola whose equation is the standard form.

Probability and Statistics

The academic standards for the data analysis and probability core area establish the process skills and core content for Probability and Statistics and Mathematics for the Technologies 4, which should provide students with the mathematics skills and conceptual understanding necessary for them to further their mathematical education or to pursue mathematics-related technical careers.

The content of the data analysis and probability standards encompasses** **design of a statistical study; collection, organization, display, and interpretation of data; basic statistical methods of analyzing data; and basic concepts of probability. Teachers, schools, and districts should use the data analysis and probability standards to make decisions concerning the structure and content of Probability and Statistics and Mathematics for the Technologies 4. Content in these two courses may go beyond the data analysis and probability standards.

All courses based on the academic standards for data analysis and probability must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or individual classes may go beyond these standards. Such decisions will involve choices regarding additional content, activities, and learning strategies and will depend on the objectives of the particular courses or individual classes.

In all courses based on the data analysis and probability standards, hand-held graphing calculators are required for instruction and assessment. Students should learn to use a variety of ways to represent data, to use a variety of mathematical tools such as graph paper, and to use technologies such as graphing calculators to solve problems.

**Note**: The term *including* appears in parenthetical statements in the high school mathematics indicators to introduce a list of specifics that are intended to clarify and focus the teaching and learning of the particular concept. That is, within these parenthetical including statements are specified the components of the indicator that are critical for the particular core area with regard both to the state assessments and to the management of time in the classroom. While instruction must focus on the entire indicator, educators must be certain to cover the components specified in the parenthetical *including* statements.

**Standard DA-1:** The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

**Indicators**

DA-1.1 Execute procedures to conduct simple probability experiments and collect data by using manipulatives (including spinners, dice, cards, and coins).

DA-1.2 Execute procedures to find measures of probability and statistics by using tools such as handheld computing devices, spreadsheets, and statistical software.

DA-1.3 Execute procedures to conduct a simulation by using random number tables and/or technology (including handheld computing devices and computers).

DA-1.4 Design and conduct a statistical research project and produce a report that summarizes the findings.

DA-1.5 Apply the principles of probability and statistics to solve problems in real-world contexts.

DA-1.6 Communicate a knowledge of data analysis and probability by using mathematical terminology appropriately.

DA-1.7 Judge the reasonableness of mathematical solutions on the basis of the source of the data, the design of the study, the way the data are displayed, and the way the data are analyzed.

DA-1.8 Compare data sets by using graphs and summary statistics.

**Standard DA-2:** The student will demonstrate through the mathematical processes an understanding of the design of a statistical study.

**Indicators**

DA-2.1 Classify a data-collection procedure as a survey, an observational study, or a controlled experiment.

DA-2.2 Compare various random sampling techniques (including simple, stratified, cluster, and systematic).

DA-2.3 Analyze a data-collection procedure to classify the technique used as either simple cluster, systematic, or convenience sampling.

DA-2.4 Critique data-collection methods and describe how bias can be controlled.

DA-2.5 Judge which of two or more possible experimental designs will best answer a given research question.

DA-2.6 Generate a research question and design a statistical study to answer a given research question.

**Standard DA-3:** The student will demonstrate through the mathematical processes an understanding of the methodology for collecting, organizing, displaying, and interpreting data.

**Indicators**

DA-3.1 Use manipulatives, random number tables, and technology to collect data and conduct experiments and simulations.

DA-3.2 Organize and interpret data by using pictographs, bar graphs, pie charts, dot plots, histograms, time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots.

DA-3.3 Select appropriate graphic display(s) from among pictographs, bar graphs, pie charts, dot plots, histograms, time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots when given a data set or problem situation.

DA-3.4 Represent frequency distributions by using displays such as categorical frequency distributions/Pareto charts, histograms, frequency polygons, and cumulative frequency distributions/ogives.

DA-3.5 Classify a scatterplot by shape (including linear, quadratic, and exponential).

DA-3.6 Classify graphically and analytically the correlation between two variables as either positive, negative, or zero.

DA-3.7 Carry out a procedure to determine an equation of a trend line for a scatterplot exhibiting a linear pattern by using visual approximation.

DA-3.8 Carry out a procedure using technology to determine a line of best fit for a scatterplot exhibiting a linear pattern.

DA-3.9 Explain the meaning of the correlation coefficient* r*.

DA-3.10 Use interpolation or extrapolation to predict values based on the relationship between two variables.

**Standard DA-4:** The student will demonstrate through the mathematical processes an understanding of basic statistical methods of analyzing data.

**Indicators**

DA-4.1 Classify a variable as either a statistic or a parameter.

DA-4.2 Compare descriptive and inferential statistics.

DA-4.3 Classify a variable as either discrete or continuous and as either categorical or quantitative.

DA-4.4 Use procedures and/or technology to find measures of central tendency (mean, median, and mode) for given data.

DA-4.5 Predict the effect of transformations of data on measures of central tendency, variability, and the shape of the distribution.

DA-4.6 Use procedures and/or technology to find measures of spread (range, variance, standard deviation, and interquartile range) and outliers for given data.

DA-4.7 Use procedures and/or technology to find measures of position (including median, quartiles, percentiles, and standard scores) for given data.

DA-4.8 Classify a distribution as either symmetric, positively skewed, or negatively skewed.

DA-4.9 Explain the significance of the shape of a distribution.

DA-4.10 Use a knowledge of the empirical rule to solve problems involving data that are distributed normally.

DA-4.11 Use control charts to determine whether a process is in control.

**Standard DA-5:** The student will demonstrate through the mathematical processes an understanding of the basic concepts of probability.

**Indicators**

DA-5.1 Construct a sample space for an experiment and represent it as a list, chart, picture, or tree diagram.

DA-5.2 Use counting techniques to determine the number of possible outcomes for an event.

DA-5.3 Classify events as either dependent or independent.

DA-5.4 Categorize two events either as mutually exclusive or as not mutually exclusive of one another.

DA-5.5 Use the concept of complementary sets to compute probabilities.

DA-5.6 Use the binomial probability distribution to solve problems.

DA-5.7 Carry out a procedure to compute simple probabilities and compound probabilities (including conditional probabilities).

DA-5.8 Use a procedure to find geometric probability in real-world contexts.

DA-5.9 Compare theoretical and experimental probabilities.

DA-5.10 Construct and compare theoretical and experimental probability distributions.

DA-5.11 Use procedures to find the expected value of discrete random variables and construct meaning within contexts.

DA-5.12 Understand the law of large numbers.

DA-5.13 Carry out a procedure to compute conditional probability by using two-way tables.