# Mathematics Curriculum Framework

## Grades K-8: Number and Number Sense

**Kindergarten: Number and Number Sense**

Students in kindergarten through grade two have a natural curiosity about their world, which leads them to develop a sense of number. Young children are motivated to count everything around them and begin to develop an understanding of the size of numbers (magnitude), multiple ways of thinking about and representing numbers, strategies and words to compare numbers, and an understanding of the effects of simple operations on numbers. Building on their own intuitive mathematical knowledge, they also display a natural need to organize things by sorting, comparing, ordering, and labeling objects in a variety of collections.

**View Grade Level Standards:**

**Grade 1: Number and Number Sense**

Students in kindergarten through grade two have a natural curiosity about their world, which leads them to develop a sense of number. Young children are motivated to count everything around them and begin to develop an understanding of the size of numbers (magnitude), multiple ways of thinking about and representing numbers, strategies and words to compare numbers, and an understanding of the effects of simple operations on numbers. Building on their own intuitive mathematical knowledge, they also display a natural need to organize things by sorting, comparing, ordering, and labeling objects in a variety of collections.

**View Grade Level Standards:**

**Grade 2: Number and Number Sense**

Students in kindergarten through grade two have a natural curiosity about their world, which leads them to develop a sense of number. Young children are motivated to count everything around them and begin to develop an understanding of the size of numbers (magnitude), multiple ways of thinking about and representing numbers, strategies and words to compare numbers, and an understanding of the effects of simple operations on numbers. Building on their own intuitive mathematical knowledge, they also display a natural need to organize things by sorting, comparing, ordering, and labeling objects in a variety of collections.

**View Grade Level Standards:**

**Grade 3: Number and Number Sense**

Mathematics instruction in grades three through five should continue to foster the development of number sense, with greater emphasis on decimals and fractions. Students with good number sense understand the meaning of numbers, develop multiple relationships and representations among numbers, and recognize the relative magnitude of numbers. They should learn the relative effect of operating on whole numbers, fractions, and decimals and learn how to use mathematical symbols and language to represent problem situations. Number and operation sense continues to be the cornerstone of the curriculum.

**View Grade Level Standards:**

**Grade 4: Number and Number Sense**

Mathematics instruction in grades three through five should continue to foster the development of number sense, with greater emphasis on decimals and fractions. Students with good number sense understand the meaning of numbers, develop multiple relationships and representations among numbers, and recognize the relative magnitude of numbers. They should learn the relative effect of operating on whole numbers, fractions, and decimals and learn how to use mathematical symbols and language to represent problem situations. Number and operation sense continues to be the cornerstone of the curriculum.

**View Grade Level Standards:**

**Grade 5: Number and Number Sense**

Mathematics instruction in grades three through five should continue to foster the development of number sense, with greater emphasis on decimals and fractions. Students with good number sense understand the meaning of numbers, develop multiple relationships and representations among numbers, and recognize the relative magnitude of numbers. They should learn the relative effect of operating on whole numbers, fractions, and decimals and learn how to use mathematical symbols and language to represent problem situations. Number and operation sense continues to be the cornerstone of the curriculum.

**View Grade Level Standards:**

**Grade 6: Number and Number Sense**

Mathematics instruction in grades six through eight continues to focus on the development of number sense, with emphasis on rational and real numbers. Rational numbers play a critical role in the development of proportional reasoning and advanced mathematical thinking. The study of rational numbers builds on the understanding of whole numbers, fractions, and decimals developed by students in the elementary grades. Proportional reasoning is the key to making connections to many middle school mathematics topics.

**View Grade Level Standards:**

**Grade 7: Number and Number Sense**

Mathematics instruction in grades six through eight continues to focus on the development of number sense, with emphasis on rational and real numbers. Rational numbers play a critical role in the development of proportional reasoning and advanced mathematical thinking. The study of rational numbers builds on the understanding of whole numbers, fractions, and decimals developed by students in the elementary grades. Proportional reasoning is the key to making connections to many middle school mathematics topics.

**View Grade Level Standards:**

**Grade 8: Number and Number Sense**

Mathematics instruction in grades six through eight continues to focus on the development of number sense, with emphasis on rational and real numbers. Rational numbers play a critical role in the development of proportional reasoning and advanced mathematical thinking. The study of rational numbers builds on the understanding of whole numbers, fractions, and decimals developed by students in the elementary grades. Proportional reasoning is the key to making connections to many middle school mathematics topics.

**View Grade Level Standards:**

## Grades K-8: Computation and Estimation

**Kindergarten: Computation and Estimation**

A variety of contexts and problem types are necessary for children to develop an understanding of the meanings of the operations such as addition and subtraction. These contexts often arise from real-life experiences in which they are simply joining sets, taking away or separating from a set, or comparing sets. These contexts might include conversations, such as "How many books do we have altogether?" or "How many cookies are left if I eat two?" or "I have three more candies than you do." Although young children first compute using objects and manipulatives, they gradually shift to performing computations mentally or using paper and pencil to record their thinking. Therefore, computation and estimation instruction in the early grades revolves around modeling, discussing, and recording a variety of problem situations. This approach helps students transition from the concrete to the representation to the symbolic in order to develop meaning for the operations and how they relate to each other.

**View Grade Level Standards:**

**Grade 1: Computation and Estimation**

A variety of contexts and problem types are necessary for children to develop an understanding of the meanings of the operations such as addition and subtraction. These contexts often arise from practical experiences in which they are simply joining sets, taking away or separating from a set, or comparing sets. These contexts might include conversations, such as "How many books do we have altogether?" or "How many cookies are left if I eat two?" or "I have three more candies than you do." Although young children first compute using objects and manipulatives, they gradually shift to performing computations mentally or using paper and pencil to record their thinking. Therefore, computation and estimation instruction in the early grades revolves around modeling, discussing, and recording a variety of problem situations. This approach helps students transition from the concrete to the representation to the symbolic in order to develop meaning for the operations and how they relate to each other.

**View Grade Level Standards:**

**Grade 2: Computation and Estimation**

A variety of contexts and problem types are necessary for children to develop an understanding of the meanings of the operations such as addition and subtraction. These contexts often arise from real-life experiences in which they are simply joining sets, taking away or separating from a set, or comparing sets. These contexts might include conversations, such as "How many books do we have if Jackie gives us five more?" or "About how many students are at two tables?" or "I have three more candies than you do." Although young children first compute using objects and manipulatives, they gradually shift to performing computations mentally or using paper and pencil to record their thinking. Therefore, computation and estimation instruction in the early grades revolves around modeling, discussing, and recording a variety of problem situations. This approach helps students transition from the concrete to the representation to the symbolic in order to develop meaning for the operations and how they relate to each other.

**View Grade Level Standards:**

**Grade 3: Computation and Estimation**

Computation and estimation in grades three through five should focus on developing fluency in multiplication and division with whole numbers and should begin to extend students' understanding of these operations to work with decimals. Instruction should focus on computation activities that enable students to model, explain, and develop proficiency with basic facts and algorithms. These proficiencies are often developed as a result of investigations and opportunities to develop algorithms. Additionally, opportunities to develop and use visual models, benchmarks, and equivalents, to add and subtract fractions, and to develop computational procedures for the addition and subtraction of decimals are a priority for instruction in these grades. Multiplication and division with decimals will be explored in grade five.

**View Grade Level Standards:**

**Grade 4: Computation and Estimation**

Computation and estimation in grades three through five should focus on developing fluency in multiplication and division with whole numbers and should begin to extend students' understanding of these operations to work with decimals. Instruction should focus on computation activities that enable students to model, explain, and develop proficiency with basic facts and algorithms. These proficiencies are often developed as a result of investigations and opportunities to develop algorithms. Additionally, opportunities to develop and use visual models, benchmarks, and equivalents, to add and subtract fractions, and to develop computational procedures for the addition and subtraction of decimals are a priority for instruction in these grades. Multiplication and division with decimals will be explored in grade five.

**View Grade Level Standards:**

**Grade 5: Computation and Estimation**

Computation and estimation in grades three through five should focus on developing fluency in multiplication and division with whole numbers and should begin to extend students' understanding of these operations to work with decimals. Instruction should focus on computation activities that enable students to model, explain, and develop proficiency with basic facts and algorithms. These proficiencies are often developed as a result of investigations and opportunities to develop algorithms. Additionally, opportunities to develop and use visual models, benchmarks, and equivalents, to add and subtract fractions, and to develop computational procedures for the addition and subtraction of decimals are a priority for instruction in these grades. Multiplication and division with decimals will be explored in grade five.

**View Grade Level Standards:**

**Grade 6: Computation and Estimation**

The computation and estimation strand in grades six through eight focuses on developing conceptual and algorithmic understanding of operations with integers and rational numbers through concrete activities and discussions that bring an understanding as to why procedures work and make sense.

**View Grade Level Standards:**

**Grade 7: Computation and Estimation**

The computation and estimation strand in grades six through eight focuses on developing conceptual and algorithmic understanding of operations with integers and rational numbers through concrete activities and discussions that bring an understanding as to why procedures work and make sense.

**View Grade Level Standards:**

**Grade 8: Computation and Estimation**

The computation and estimation strand in grades six through eight focuses on developing conceptual and algorithmic understanding of operations with integers and rational numbers through concrete activities and discussions that bring an understanding as to why procedures work and make sense.

**View Grade Level Standards:**

## Grades K-8: Measurement and Geometry

**Kindergarten: Measurement and Geometry**

The exploration of measurement and geometry in the primary grades allows students to learn more about the world around them. Measurement is important because it helps to quantify the world around us and is useful in so many aspects of everyday life. Students in kindergarten through grade two encounter measurement in their daily lives, from their use of the calendar and science activities that often require students to measure objects or compare them directly, to situations in stories they are reading and to descriptions of how quickly they are growing.

**View Grade Level Standards:**

**Grade 1: Measurement and Geometry**

The exploration of measurement and geometry in the primary grades allows students to learn more about the world around them. Measurement is important because it helps to quantify the world around us and is useful in so many aspects of everyday life. Students in kindergarten through grade two encounter measurement in their daily lives, from their use of the calendar and science activities that often require students to measure objects or compare them directly, to situations in stories they are reading and to descriptions of how quickly they are growing.

**View Grade Level Standards:**

**Grade 2: Measurement and Geometry**

The exploration of measurement and geometry in the primary grades allows students to learn more about the world around them. Measurement is important because it helps to quantify the world around us and is useful in so many aspects of everyday life. Students in kindergarten through grade two encounter measurement in their daily lives, from their use of the calendar and science activities that often require students to measure objects or compare them directly, to situations in stories they are reading and to descriptions of how quickly they are growing.

**View Grade Level Standards:**

**Grade 3: Measurement and Geometry**

Students in grades three through five should be actively involved in measurement activities that require a dynamic interaction among students and their environment. Students can see the usefulness of measurement if classroom experiences focus on measuring objects and estimating measurements. Textbook experiences cannot substitute for activities that utilize measurement to answer questions about real problems.

**View Grade Level Standards:**

**Grade 4: Measurement and Geometry**

Students in grades three through five should be actively involved in measurement activities that require a dynamic interaction between students and their environment. Students can see the usefulness of measurement if classroom experiences focus on measuring objects and estimating measurements. Textbook experiences cannot substitute for activities that utilize measurement to answer questions about real problems.

**View Grade Level Standards:**

**Grade 5: Measurement and Geometry**

Students in grades three through five should be actively involved in measurement activities that require a dynamic interaction among students and their environment. Students can see the usefulness of measurement if classroom experiences focus on measuring objects and estimating measurements. Textbook experiences cannot substitute for activities that utilize measurement to answer questions about real problems.

**View Grade Level Standards:**

**Grade 6: Measurement and Geometry**

Measurement and geometry in the middle grades provide a natural context and connection among many mathematical concepts. Students expand informal experiences with geometry and measurement in the elementary grades and develop a solid foundation for further exploration of these concepts in high school. Spatial reasoning skills are essential to the formal inductive and deductive reasoning skills required in subsequent mathematics learning.

**View Grade Level Standards:**

**Grade 7: Measurement and Geometry**

Measurement and geometry in the middle grades provide a natural context and connection among many mathematical concepts. Students expand informal experiences with geometry and measurement in the elementary grades and develop a solid foundation for further exploration of these concepts in high school. Spatial reasoning skills are essential to the formal inductive and deductive reasoning skills required in subsequent mathematics learning.

**View Grade Level Standards:**

**Grade 8: Measurement and Geometry**

Measurement and geometry in the middle grades provide a natural context and connection among many mathematical concepts. Students expand informal experiences with geometry and measurement in the elementary grades and develop a solid foundation for further exploration of these concepts in high school. Spatial reasoning skills are essential to the formal inductive and deductive reasoning skills required in subsequent mathematics learning.

**View Grade Level Standards:**

## Grades K-8: Probability and Statistics

**Kindergarten: Probability and Statistics**

Students in the primary grades have a natural curiosity about their world, which leads to questions about how things fit together or connect. They display their natural need to organize things by sorting and counting objects in a collection according to similarities and differences with respect to given criteria.

**View Grade Level Standards:**

**Grade 1: Probability and Statistics**

Students in the primary grades have a natural curiosity about their world, which leads to questions about how things fit together or connect. They display their natural need to organize things by sorting and counting objects in a collection according to similarities and differences with respect to given criteria.

**View Grade Level Standards:**

**Grade 2: Probability and Statistics**

Students in the primary grades have a natural curiosity about their world, which leads to questions about how things fit together or connect. They display their natural need to organize things by sorting and counting objects in a collection according to similarities and differences with respect to given criteria.

**View Grade Level Standards:**

**Grade 3: Probability and Statistics**

Students entering grades three through five have begun to explore the concept of the measurement of chance and are able to determine possible outcomes of given events. Students have utilized a variety of random generator tools, including random number generators (number cubes), spinners, and two-sided counters. In game situations, students have had initial experiences in predicting whether a game is fair or not fair. Furthermore, students are able to identify events as likely or unlikely to happen.

**View Grade Level Standards:**

**Grade 4: Probability and Statistics**

Students entering grades three through five have begun to explore the concept of the measurement of chance and are able to determine possible outcomes of given events. Students have utilized a variety of random generator tools, including random number generators (number cubes), spinners, and two-sided counters. In game situations, students have had initial experiences in predicting whether a game is fair or not fair. Furthermore, students are able to identify events as likely or unlikely to happen.

**View Grade Level Standards:**

**Grade 5: Probability and Statistics**

Students entering grades three through five have begun to explore the concept of the measurement of chance and are able to determine possible outcomes of given events. Students have utilized a variety of random generator tools, including random number generators (number cubes), spinners, and two-sided counters. In game situations, students have had initial experiences in predicting whether a game is fair or not fair. Furthermore, students are able to identify events as likely or unlikely to happen.

**View Grade Level Standards:**

## Grades K-8: Patterns, Functions, and Algebra

**Kindergarten: Patterns, Functions, and Algebra**

**Grade 1: Patterns, Functions, and Algebra**

Stimulated by the exploration of their environment, children begin to develop concepts related to patterns, functions, and algebra before beginning school. Recognition of patterns and comparisons are important components of children's mathematical development.

**View Grade Level Standards:**

**Grade 2: Patterns, Functions, and Algebra**

**Grade 3: Patterns, Functions, and Algebra**

Students entering grades three through five have had opportunities to identify patterns within the context of the school curriculum and in their daily lives, and they can make predictions about them. They have had opportunities to use informal language to describe the changes within a pattern and to compare two patterns. Students have also begun to work with the concept of a variable by describing mathematical relationships within a pattern.

**View Grade Level Standards:**

**Grade 4: Patterns, Functions, and Algebra**

Students entering grades three through five have had opportunities to identify patterns within the context of the school curriculum and in their daily lives, and they can make predictions about them. They have had opportunities to use informal language to describe the changes within a pattern and to compare two patterns. Students have also begun to work with the concept of a variable by describing mathematical relationships within a pattern.

**View Grade Level Standards:**

**Grade 5: Patterns, Functions, and Algebra**

Students entering grades three through five have had opportunities to identify patterns within the context of the school curriculum and in their daily lives, and they can make predictions about them. They have had opportunities to use informal language to describe the changes within a pattern and to compare two patterns. Students have also begun to work with the concept of a variable by describing mathematical relationships within a pattern.

**View Grade Level Standards:**

## Algebra I

**Algebra I: Expressions and Operations**

- The student will a) represent verbal quantitative situations algebraically; and b) evaluate algebraic expressions for given replacement values of the variables.
- The student will perform operations on polynomials, including a) applying the laws of exponents to perform operations on expressions; b) adding, subtracting, multiplying, and dividing polynomials; and c) factoring completely first- and second-degree binomials and trinomials in one variable.
- The student will simplify a) square roots of whole numbers and monomial algebraic expressions; b) cube roots of integers; and c) numerical expressions containing square or cube roots.

**View Standards:**

**Algebra I: Equations and Inequalities**

- The student will solve a) multistep linear equations in one variable algebraically; b) quadratic equations in one variable algebraically; c) literal equations for a specified variable; d) systems of two linear equations in two variables algebraically and graphically; and e) practical problems involving equations and systems of equations.
- The student will a) solve multistep linear inequalities in one variable algebraically and represent the solution graphically; b) represent the solution of linear inequalities in two variables graphically; c) solve practical problems involving inequalities; and d) represent the solution to a system of inequalities graphically.
- The student will a) determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; b) write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and c) graph linear equations in two variables.

**View Standards:**

**Algebra I: Functions**

- The student will investigate and analyze linear and quadratic function families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros; d) intercepts; e) values of a function for elements in its domain; and f) connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

**View Standards:**

**Algebra I: Statistics**

- The student, given a data set or practical situation, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.
- The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of linear and quadratic functions.

**View Standards:**

## Geometry

**Geometry: Reasoning, Lines, and Transformations**

- The student will use deductive reasoning to construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; and c) determining the validity of a logical argument.
- The student will use the relationships between angles formed by two lines intersected by a transversal to a) prove two or more lines are parallel; and b) solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal.
- The student will solve problems involving symmetry and transformation. This will include a) investigating and using formulas for determining distance, midpoint, and slope; b) applying slope to verify and determine whether lines are parallel or perpendicular; c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.
- The student will construct and justify the constructions of a) a line segment congruent to a given line segment; b) the perpendicular bisector of a line segment; c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point on the line; e) the bisector of a given angle; f) an angle congruent to a given angle; g) a line parallel to a given line through a point not on the line; and h) an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

**View Standards:**

**Geometry: Triangles**

- The student, given information concerning the lengths of sides and/or measures of angles in triangles, will solve problems, including practical problems. This will include a) ordering the sides by length, given angle measures; b) ordering the angles by degree measure, given side lengths; c) determining whether a triangle exists; and d) determining the range in which the length of the third side must lie.
- The student, given information in the form of a figure or statement, will prove two triangles are congruent.
- The student, given information in the form of a figure or statement, will prove two triangles are similar.
- The student will solve problems, including practical problems, involving right triangles. This will include applying a) the Pythagorean Theorem and its converse; b) properties of special right triangles; and c) trigonometric ratios.

**View Standards:**

**Geometry: Polygons and Circles**

- The student will verify and use properties of quadrilaterals to solve problems, including practical problems.
- The student will solve problems, including practical problems, involving angles of convex polygons. This will include determining the a) sum of the interior and/or exterior angles; b) measure of an interior and/or exterior angle; and c) number of sides of a regular polygon.
- The student will solve problems, including practical problems, by applying properties of circles. This will include determining a) angle measures formed by intersecting chords, secants, and/or tangents; b) lengths of segments formed by intersecting chords, secants, and/or tangents; c) arc length; and d) area of a sector.
- The student will solve problems involving equations of circles.

**View Standards:**

**Geometry: Three-Dimensional Figures**

- The student will use surface area and volume of three-dimensional objects to solve practical problems.
- The student will apply the concepts of similarity to two- or three-dimensional geometric figures. This will include a) comparing ratios between lengths, perimeters, areas, and volumes of similar figures; b) determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; c) determining how changes in area and/or volume of a figure affect one or more dimensions of the figure; and d) solving problems, including practical problems, about similar geometric figures.

**View Standards:**

## Algebra, Functions & Data Analysis

**AFDA: Algebra and Functions**

- The student will investigate and analyze linear, quadratic, exponential, and logarithmic function families and their characteristics. Key concepts include a) domain and range; b) intervals on which a function is increasing or decreasing; c) absolute maxima and minima; d) zeros; e) intercepts; f) values of a function for elements in its domain; g) connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; h) end behavior; and i) vertical and horizontal asymptotes
- The student will use knowledge of transformations to write an equation, given the graph of a linear, quadratic, exponential, and logarithmic function.
- The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems using models of linear, quadratic, and exponential functions.
- The student will use multiple representations of functions for analysis, interpretation, and prediction.
- The student will determine optimal values in problem situations by identifying constraints and using linear programming techniques.

**View Standards:**

**AFDA: Data Analysis**

- The student will calculate probabilities. Key concepts include a) conditional probability; b) dependent and independent events; c) mutually exclusive events; d) counting techniques (permutations and combinations); and e) Law of Large Numbers.
- The student will a) identify and describe properties of a normal distribution; b) interpret and compare z-scores for normally distributed data; and c) apply properties of normal distributions to determine probabilities associated with areas under the standard normal curve.
- The student will design and conduct an experiment/survey. Key concepts include a) sample size; b) sampling technique; c) controlling sources of bias and experimental error; d) data collection; and e) data analysis and reporting

**View Standards:**

## Algebra II

**Algebra II: Expressions and Operations**

- The student will a) add, subtract, multiply, divide, and simplify rational algebraic expressions; b) add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and c) factor polynomials completely in one or two variables.
- The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i

**View Standards:**

**Algebra II: Equations and Inequalities**

- The student will solve a) absolute value linear equations and inequalities; b) quadratic equations over the set of complex numbers; c) equations containing rational algebraic expressions; and d) equations containing radical expressions.
- The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically.

**View Standards:**

**Algebra II: Functions**

- The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, determining the
*n*th term, and evaluating summation formulas. - For absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic functions, the student will a) recognize the general shape of function families; and b) use knowledge of transformations to convert between equations and the corresponding graphs of functions.
- The student will investigate and analyze linear, quadratic, absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic function families algebraically and graphically. Key concepts include a) domain, range, and continuity; b) intervals in which a function is increasing or decreasing; c) extrema; d) zeros; e) intercepts; f) values of a function for elements in its domain; g) connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; h) end behavior; i) vertical and horizontal asymptotes; j) inverse of a function; and k) composition of functions, algebraically and graphically.
- The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

**View Standards:**

**Algebra II: Statistics**

- The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions.
- The student will represent and solve problems, including practical problems, involving inverse variation, joint variation, and a combination of direct and inverse variations.
- The student will a) identify and describe properties of a normal distribution; b) interpret and compare z-scores for normally distributed data; and c) apply properties of normal distributions to determine probabilities associated with areas under the standard normal curve.
- The student will compute and distinguish between permutations and combinations.

**View Standards:**

## Trigonometry

**Trigonometry: Triangular and Circular Trigonometric Functions**

- The student, given a point on the terminal side of an angle in standard position, or the value of the trigonometric function of the angle, will determine the sine, cosine, tangent, cotangent, secant, and cosecant of the angle.
- The student will develop and apply the properties of the unit circle in degrees and radians.

**View Standards:**

**Trigonometry: Graphs of Trigonometric Functions**

- The student, given one of the six trigonometric functions in standard form, will a) state the domain and the range of the function; b) determine the amplitude, period, phase shift, vertical shift, and asymptotes; c) sketch the graph of the function by using transformations for at least a two-period interval; and d) investigate the effect of changing the parameters in a trigonometric function on the graph of the function.
- The student will graph the six inverse trigonometric functions.

**View Standards:**

**Trigonometry: Equations and Identities**

- The student will verify basic trigonometric identities and make substitutions, using the basic identities.
- The student will solve trigonometric equations and inequalities.
- The student will determine the value of any trigonometric function and inverse trigonometric function.

**View Standards:**

**Trigonometry: Applications of Trigonometric Functions**

## Computer Mathematics

**Computer Mathematics: Problem Solving**

- The student will design and apply computer programs to solve practical problems in mathematics arising from business and applications in mathematics.

**View Standards:**

**Computer Mathematics: Program Design**

- The student will design, write, document, test, and debug a computer program.
- The student will write program specifications that define the constraints of a given problem.
- The student will design an algorithm to solve a given problem.
- The student will divide a given problem into modules by task and implement the solution.
- The student will translate mathematical expressions into programming expressions by declaring variables, writing assignment statements, and using the order of operations.
- The student will select and call library functions to process data, as appropriate.
- The student will implement conditional statements that include "if/then" statements, "if/then/else" statements, case statements, and Boolean logic.
- The student will implement pre-defined algorithms, including sort routines, search routines, and simple animation routines.

**View Standards:**

**Computer Mathematics: Program Implementation**

- The student will design and implement the input phase of a program, which will include designing screen layout, getting information into the program by way of user interaction and/or file input, and validating input.
- The student will design and implement the output phase of a computer program, which will include designing output layout, accessing available output devices, using output statements, and labeling results.
- The student will design and implement computer graphics to enhance output.
- The student will implement various mechanisms for performing iteration with an algorithm.
- The student will select and implement appropriate data structures, including arrays (one- and/or two-dimensional) and objects.

**View Standards:**

**Computer Mathematics: Data Manipulation**

- The student will define and use appropriate variable data types that include integer, real (fixed and scientific notation), character, string, Boolean, and object.
- The student will describe the way the computer stores, accesses, and processes variables, including the following topics: the use of variables versus constants, parameter passing, scope of variables, and local versus global variables.

**View Standards:**

**Computer Mathematics: Program Testing**

- The student will test a program using an appropriate set of data. The test data should include boundary cases and test all branches of a program.
- The student will debug a program using appropriate techniques (e.g., appropriately placed controlled breaks, the printing of intermediate results, and other debugging tools available in the programming environment), and identify the difference among syntax errors, runtime errors, and logic errors.

**View Standards:**

## Probability and Statistics

**Probability and Statistics: Descriptive Statistics**

- The student will analyze graphical displays of univariate data, including dotplots, stemplots, boxplots, cumulative frequency graphs, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers.
- The student will analyze numerical characteristics of univariate data sets to describe patterns and departures from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers.
- The student will compare distributions of two or more univariate data sets, numerically and graphically, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features.
- The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points.
- The student will determine and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plot to assess linearity.
- The student will make logarithmic and power transformations to achieve linearity.
- The student, using two-way tables and other graphical displays, will analyze categorical data to describe patterns and departures from patterns and to determine marginal frequency and relative frequencies, including conditional frequencies.

**View Standards:**

**Probability and Statistics: Data Collection**

- The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting.
- The student will plan and conduct a survey. The plan will address sampling techniques and methods to reduce bias.
- The student will plan and conduct a well-designed experiment. The plan will address control, randomization, replication, blinding, and measurement of experimental error.

**View Standards:**

**Probability and Statistics: Probability**

- The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive.
- The student will determine probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule.
- The student will develop, interpret, and apply the binomial and geometric probability distributions for discrete random variables, including computing the mean and standard deviation for the binomial and geometric variables.
- The student will simulate probability distributions, including binomial and geometric.
- The student will identify random variables as independent or dependent and determine the mean and standard deviations for random variables and sums and differences of independent random variables.
- The student will identify properties of a normal distribution and apply the normal distribution to determine probabilities.

**View Standards:**

**Probability and Statistics: Inferential Statistics**

- The student, given data from a large sample, will determine and interpret appropriate point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, difference between two means (independent and paired), and slope of a least-squares regression line.
- The student will apply and interpret the logic of an appropriate hypothesis-testing procedure. Tests will include large sample test for proportion, mean, difference between two proportions, difference between two means (independent and paired); chi-squared tests for goodness of fit, homogeneity of proportions, and independence; and slope of a least-squares regression line.
- The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means
- The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures.

**View Standards:**

## Discrete Mathematics

**Discrete Mathematics: Graphs**

- The student will model problems, using vertex-edge graphs. The concepts of valence, connectedness, paths, planarity, and directed graphs will be investigated.
- The student will solve problems through investigation and application of circuits, cycles, Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Optimal solutions will be sought using existing algorithms and student-created algorithms.
- The student will apply graphs to conflict-resolution problems, such as map coloring, scheduling, matching, and optimization.
- The student will apply algorithms relating to trees, networks, and paths. Appropriate technology will be used to determine the number of possible solutions and generate solutions when a feasible number exists.

**View Standards:**

**Discrete Mathematics: Election Theory and Fair Division**

- The student will analyze and describe the issue of fair division in discrete and continuous cases.
- The student will investigate and describe weighted voting and the results of various election methods. These may include approval and preference voting as well as plurality, majority, runoff, sequential runoff, Borda count, and Condorcet winners.
- The student will identify apportionment inconsistencies that apply to issues such as salary caps in sports and allocation of representatives to Congress. Historical and current methods will be compared.

**View Standards:**

**Discrete Mathematics: Computer Mathematics**

**Discrete Mathematics: Recursion and Optimization**

- The student will use algorithms to schedule tasks in order to determine a minimum project time. The algorithms will include critical path analysis, the list-processing algorithm, and student-created algorithms.
- The student will solve linear programming problems.
- The student will use the recursive process and difference equations with the aid of appropriate technology to generate a) compound interest; b) sequences and series; c) fractals; d) population growth models; and e) the Fibonacci sequence.
- The student will apply the formulas of combinatorics in the areas of a) the Fundamental (Basic) Counting Principle; b) knapsack and bin-packing problems; c) permutations and combinations; and d) the pigeonhole principle.

**View Standards:**

## Mathematical Analysis

**Mathematical Analysis: Functions**

- The student will investigate and identify the properties of polynomial, rational, piecewise, and step functions and sketch the graphs of the functions.
- The student will investigate and identify the characteristics of exponential and logarithmic functions to graph the function, solve equations, and solve practical problems.
- The student will apply compositions of functions and inverses of functions to practical situations and investigate and verify the domain and range of resulting functions.
- The student will determine the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity.
- The student will investigate and describe the continuity of functions.

**View Standards:**

**Mathematical Analysis: Analytic Geometry**

- The student will investigate, graph, and identify the properties of conic sections from equations in vertex and standard form.
- The student will perform operations with vectors in the coordinate plane and solve practical problems using vectors.
- The student will identify, create, and solve practical problems involving triangles.

**View Standards:**

**Mathematical Analysis: Equations**

- The student will investigate and identify the characteristics of the graphs of polar equations.
- The student will use parametric equations to model and solve practical problems.
- The student will use matrices to organize data and will add and subtract matrices, multiply matrices, multiply matrices by a scalar, and use matrices to solve systems of equations.

**View Standards:**

**Mathematical Analysis: Discrete Mathematics**

## Data Science

**Data Science: Data in Context**

**Data in Context** - Understanding data science facilitates critical examination of questions and supports informed data-driven decision making.

- The student will identify specific examples of real-world problems that can be effectively addressed using data science.
- The student will be able to formulate a top-down plan for data collection and analysis, with quantifiable results, based on the context of a problem.

**View Standards:**

**Data Science: Data Bias**

**Data Bias** - Data bias may result from the types of methods used for data collection, processing, representation, analysis, and use.

- The student will recognize the importance of data literacy and develop an awareness of how the analysis of data can be used in problem solving to effect change and create innovative solutions.
- The student will be able to identify data biases in the data collection process, and understand the implications and privacy issues surrounding data collection and processing.

**View Standards:**

**Data Science: Data and Communication**

**Data and Communication** - Data visualizations are used to communicate insights about complex data sets to support making decisions.

- The student will use storytelling as a strategy to effectively communicate with data.
- The student will justify the design, use and effectiveness of different forms of data visualizations.

**View Standards:**

**Data Science: Data Modeling**

**Data Modeling** - Mathematical models are used to predict future, unobserved data values.

- The student will be able to assess reliability of source data in preparation for mathematical modeling.
- The student will be able to acquire and prepare big data sets for modeling and analysis.
- The student will select and analyze data models to make predictions, while assessing accuracy and sources of uncertainty.
- The student will be able to summarize and interpret data represented in both conventional and emerging visualizations.
- The student will select statistical models and use goodness of fit testing to extract actionable knowledge directly from data.

**View Standards:**

**Data Science: Data and Computing**

**Data and Computing** - Technology is used to effectively prepare, analyze, and communicate with data.

- The student will be able to select and utilize appropriate technological tools and functions within those tools to process and prepare data for analysis.
- The student will be able to select and utilize appropriate technological tools and functions within those tools to analyze and communicate data effectively.

**View Standards:**