High School Math SOL Standards

Statistics

Functions

Equations and Inequalities

The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Write a linear equation or inequality in one variable to represent a contextual situation.

Solve multistep linear equations in one variable, including those in contextual situations, by applying the properties of real numbers and/or properties of equality.

Solve multistep linear inequalities in one variable algebraically and graph the solution set on a number line, including those in contextual situations, by applying the properties of real numbers and/or properties of inequality.

Rearrange a formula or literal equation to solve for a specified variable by applying the properties of equality.

Determine if a linear equation in one variable has one solution, no solution, or an infinite number of solutions.

Verify possible solution(s) to multistep linear equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables.

Create a system of two linear equations in two variables to represent a contextual situation.

Apply the properties of real numbers and/or properties of equality to solve a system of two linear equations in two variables, algebraically and graphically.

Determine whether a system of two linear equations has one solution, no solution, or an infinite number of solutions.

Create a linear inequality in two variables to represent a contextual situation.

Represent the solution of a linear inequality in two variables graphically on a coordinate plane.

Create a system of two linear inequalities in two variables to represent a contextual situation.

Represent the solution set of a system of two linear inequalities in two variables, graphically on a coordinate plane.

Verify possible solution(s) to a system of two linear equations, a linear inequality in two variable, or a system of two linear inequalities algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

The student will represent, solve, and interpret the solution to a quadratic equation in one variable.

Solve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems.

Determine and justify if a quadratic equation in one variable has no real solutions, one real solution, or two real solutions.

Verify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Expressions and Operations

The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Translate between verbal quantitative situations and algebraic expressions, including contextual situations.

Evaluate algebraic expressions which include absolute value, square roots, and cube roots for given replacement values to include rational numbers, without rationalizing the denominator.

The student will perform operations on and factor polynomial expressions in one variable.

Determine sums and differences of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models.

Determine the product of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models, the application of the distributive property, and the use of area models. The factors should be limited to five or fewer terms (e.g., (4x + 2)(3x + 5) represents four terms and (x + 1)(2x 2 + x + 3) represents five terms).

Factor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors.

Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.

Represent and demonstrate equality of quadratic expressions in different forms (e.g., concrete, verbal, symbolic, and graphical).

The student will derive and apply the laws of exponents.

Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

Simplify and determine equivalent radical expressions involving the square root of a whole number in simplest form.

Simplify and determine equivalent radical expressions involving the cube root of an integer.

Add, subtract, and multiply radicals, limited to numeric square and cube root expressions.

Generate equivalent numerical expressions and justify their equivalency for radicals using rational exponents, limited to rational exponents of 1 2 and 1 3 (e.g., √5 = 5 1 2; √8 3 = 8 1 3 = (2 3 ) 1 3 = 2).

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

Investigate and explain how transformations to the parent function y = x affects the rate of change (slope) and the y-intercept of a linear function.

Write equivalent algebraic forms of linear functions, including slope-intercept form, standard form, and point-slope form, and analyze and interpret the information revealed by each form

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations:

given the graph of a line;

given two points on the line whose coordinates are integers;

given the slope and a point on the line whose coordinates are integers;

vertical lines as x = a; and

horizontal lines as y = c.

Write the equation of a line parallel or perpendicular to a given line through a given point.

Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

For any value, x, in the domain of f, determine f(x), and determine x given any value f(x) in the range of f, given an algebraic or graphical representation of a linear function.

Compare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations.

The student will investigate, analyze, and compare characteristics of functions, including quadratic, and exponential functions, and model quadratic and exponential relationships.

Determine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function; for relations that are functions, determine the domain and range.

Given an equation or graph, determine key characteristics of a quadratic function including xintercepts (zeros), y-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable.

Graph a quadratic function, f(x), in two variables using a variety of strategies, including transformations f(x) + k and kf(x), where k is limited to rational values.

Make connections between the algebraic (standard and factored forms) and graphical representation of a quadratic function.

Given an equation or graph of an exponential function in the form y = abx (where b is limited to a natural number), interpret key characteristics, including y-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable.

Graph an exponential function, f(x), in two variables using a variety of strategies, including transformations f(x) + k and kf(x), where k is limited to rational values.

For any value, x, in the domain of f, determine f(x) of a quadratic or exponential function. Determine x given any value f(x) in the range of f of a quadratic function. Explain the meaning of x and f(x) in context.

Compare and contrast the key characteristics of linear functions (f(x) = x), quadratic functions (f(x) = x 2 ), and exponential functions (f(x) = bx ) using tables and graphs.

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear and quadratic functions.

Formulate investigative questions that require the collection or acquisition of bivariate data.

Determine what variables could be used to explain a given contextual problem or situation or answer investigative questions.

Determine an appropriate method to collect a representative sample, which could include a simple random sample, to answer an investigative question.

Given a table of ordered pairs or a scatterplot representing no more than 30 data points, use available technology to determine whether a linear or quadratic function would represent the relationship, and if so, determine the equation of the curve of best fit.

Use linear and quadratic regression methods available through technology to write a linear or quadratic function that represents the data where appropriate and describe the strengths and weaknesses of the model.

Use a linear model to predict outcomes and evaluate the strength and validity of these predictions, including through the use of technology.

Investigate and explain the meaning of the rate of change (slope) and y-intercept (constant term) of a linear model in context.

Analyze relationships between two quantitative variables revealed in a scatterplot.

Make conclusions based on the analysis of a set of bivariate data and communicate the results.

Statistics

Functions

Equations and Inequalities

Expressions and Operations

The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

Create an absolute value equation in one variable to model a contextual situation.

Solve an absolute value equation in one variable algebraically and verify the solution graphically.

Create an absolute value inequality in one variable to model a contextual situation.

Solve an absolute value inequality in one variable and represent the solution set using set notation, interval notation, and using a number line.

Verify possible solution(s) to absolute value equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

The student will represent, solve, and interpret the solution to quadratic equations in one variable over the set of complex numbers and solve quadratic inequalities in one variable.

Create a quadratic equation or inequality in one variable to model a contextual situation.

Solve a quadratic equation in one variable over the set of complex numbers algebraically.

Determine the solution to a quadratic inequality in one variable over the set of real numbers algebraically.

Verify possible solution(s) to quadratic equations or inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

The student will solve a system of equations in two variables containing a quadratic expression.

Create a linear-quadratic or quadratic-quadratic system of equations to model a contextual situation.

Determine the number of solutions to a linear-quadratic and quadratic-quadratic system of equations in two variables.

Solve a linear-quadratic and quadratic-quadratic system of equations algebraically and graphically, including situations in context.

Verify possible solution(s) to linear-quadratic or quadratic-quadratic system of equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

The student will represent, solve, and interpret the solution to an equation containing rational algebraic expressions.

Create an equation containing a rational expression to model a contextual situation.

Solve rational equations with real solutions containing factorable algebraic expressions algebraically and graphically. Algebraic expressions should be limited to linear and quadratic expressions.

Verify possible solution(s) to rational equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Justify why a possible solution to an equation containing a rational expression might be extraneous.

The student will represent, solve, and interpret the solution to an equation containing a radical expression.

Solve an equation containing no more than one radical expression algebraically and graphically.

Verify possible solution(s) to radical equations algebraically, graphically, and with technology, to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Justify why a possible solution to an equation with a square root might be extraneous.

The student will represent, solve, and interpret the solution to a polynomial equation.

Determine a factored form of a polynomial equation, of degree three or higher, given its zeros or the x-intercepts of the graph of its related function.

Determine the number and type of solutions (real or imaginary) of a polynomial equation of degree three or higher.

Solve a polynomial equation over the set of complex numbers.

Verify possible solution(s) to polynomial equations of degree three or higher algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions in context.

The student will perform operations on and simplify rational expressions.

Add, subtract, multiply, or divide rational algebraic expressions, simplifying the result.

Justify and determine equivalent rational algebraic expressions with monomial and binomial factors. Algebraic expressions should be limited to linear and quadratic expressions.

Recognize a complex algebraic fraction and simplify it as a product or quotient of simple algebraic fractions.

Represent and demonstrate equivalence of rational expressions written in different forms.

The student will perform operations on and simplify radical expressions.

Simplify and determine equivalent radical expressions that include numeric and algebraic radicands.

Add, subtract, multiply, and divide radical expressions that include numeric and algebraic radicands, simplifying the result. Simplification may include rationalizing the denominator.

Convert between radical expressions and expressions containing rational exponents.

The student will perform operations on polynomial expressions and factor polynomial expressions in one and two variables.

Determine sums, differences, and products of polynomials in one and two variables.

Factor polynomials completely in one and two variables with no more than four terms over the set of integers.

Determine the quotient of polynomials in one and two variables, using monomial, binomial, and factorable trinomial divisors.

Represent and demonstrate equality of polynomial expressions written in different forms and verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.

The student will perform operations on complex numbers.

Explain the meaning of i.

Identify equivalent radical expressions containing negative rational numbers and expressions in a + bi form.

Apply properties to add, subtract, and multiply complex numbers.

The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families.

Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation.

Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions.

Determine when two variables are directly proportional, inversely proportional, or neither, given a table of values. Write an equation and create a graph to represent a direct or inverse variation, including situations in context.

Compare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions.

The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.

Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions.

Determine the intervals on which the graph of a function is increasing, decreasing, or constant.

Determine the location and value of absolute (global) maxima and absolute (global) minima of a function.

Determine the location and value of relative (local) maxima or relative (local) minima of a function.

For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable.

Describe the end behavior of a function.

Determine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic).

Determine the inverse of a function algebraically and graphically, given the equation of a linear or quadratic function (linear, quadratic, and square root). Justify and explain why two functions are inverses of each other.

Graph the inverse of a function as a reflection over the line y = x.

Determine the composition of two functions algebraically and graphically.

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on univariate quantitative data represented by a smooth curve, including a normal curve.

Formulate investigative questions that require the collection or acquisition of a large set of univariate quantitative data or summary statistics of a large set of univariate quantitative data and investigate questions using a data cycle.

Collect or acquire univariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires.

Examine the shape of a data set (skewed versus symmetric) that can be represented by a histogram, and sketch a smooth curve to model the distribution.

Identify the properties of a normal distribution.

Describe and interpret a data distribution represented by a smooth curve by analyzing measures of center, measures of spread, and shape of the curve.

Calculate and interpret the z-score for a value in a data set.

Compare two data points from two different distributions using z-scores.

Determine the solution to problems involving the relationship of the mean, standard deviation, and z-score of a data set represented by a smooth or normal curve.

Apply the Empirical Rule to answer investigative questions.

Compare multiple data distributions using measures of center, measures of spread, and shape of the distributions.

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear, quadratic, exponential, or a combination of these functions.

Formulate investigative questions that require the collection or acquisition of bivariate data and investigate questions using a data cycle.

Collect or acquire bivariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires.

Represent bivariate data with a scatterplot using technology.

Determine whether the relationship between two quantitative variables is best approximated by a linear, quadratic, exponential, or a combination of these functions.

Determine the equation(s) of the function(s) that best models the relationship between two variables using technology. Curves of best fit may include a combination of linear, quadratic, or exponential (piecewise-defined) functions.

Use the correlation coefficient to designate the goodness of fit of a linear function using technology.

Make predictions, decisions, and critical judgments using data, scatterplots, or the equation(s) of the mathematical model.

Evaluate the reasonableness of a mathematical model of a contextual situation.

The student will compute and distinguish between permutations and combinations.

Compare and contrast permutations and combinations to count the number of ways that events can occur.

Calculate the number of permutations of n objects taken r at a time.

Calculate the number of combinations of n objects taken r at a time.

Use permutations and combinations as counting techniques to solve contextual problems.

Calculate and verify permutations and combinations using technology.

Data Analysis

Algebra and Functions

The student will investigate, analyze, and compare linear, quadratic, and exponential function families, algebraically and graphically, using transformations.

Identify graphs and equations of parent functions for linear, quadratic, and exponential function families.

Describe the transformation from the parent function given the equation or the graph of the function.

Determine and analyze whether a linear, quadratic, or exponential function best models a given representation, including those in context.

Write the equation of a linear, quadratic, or exponential function, given a graph, using transformations of the parent function.

Use a graphical or algebraic representation of a function to solve problems within a context, graphically and algebraically, when appropriate.

Graph a function given the equation of a function, using transformations of the parent function. Use technology to verify transformations of functions.

Compare and contrast linear, quadratic, and exponential functions using multiple representations (e.g., graphs, tables, equations, verbal descriptions).

The student will investigate and analyze characteristics of the graphs of linear, quadratic, exponential, and piecewise-defined functions.

Determine the domain and range of a function given a graphical representation, including those limited by contexts.

Identify intervals on a graph for which a function is increasing, decreasing, or constant.

Given a graph, identify the location and value of the absolute maximum and absolute minimum of a function over the domain of a function.

Given a graph, determine the zeros and intercepts of a function.

Describe and recognize the connection between points on the graph and the value of a function.

Describe the end behavior of a function given its graph.

Identify horizontal and/or vertical asymptotes from the graph of a function, if they exist.

Describe and relate the characteristics of the graphs of linear, quadratic, exponential, and piecewise-defined functions, including those in contextual situations.

The student will represent and interpret contextual situations with constraints that require optimization using linear programming techniques, including systems of linear equations or inequalities, solving graphically and when appropriate, algebraically.

Represent and interpret contextual problems requiring optimization with systems of linear equations or inequalities.

Solve systems of no more than four equations or inequalities graphically and when appropriate, algebraically.

Identify the feasible region of a system of linear inequalities.

Identify the coordinates of the vertices of a feasible region.

Determine and describe the maximum or minimum value for the function defined over a feasible region.

Interpret the validity of possible solution(s) algebraically, graphically, using technology, and in context and justify the reasonableness of the answer(s) or the solution method in context.

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear, quadratic, and exponential functions.

Formulate investigative questions that require the collection or acquisition of bivariate data, where exactly two of the variables are quantitative.

Collect or acquire bivariate data from a representative sample to answer an investigative question.

Represent bivariate data with a scatterplot using technology and describe how the variables are related in terms of the given context.

Make predictions, decisions, and critical judgments using data, scatterplots, or the equation(s) of the mathematical model.

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on the design and implementation of an experiment and/or observational study.

Formulate questions that can be addressed with data and assess the type of data relevant to the question (e.g., quantitative versus categorical).

Investigate, describe, and determine best sampling techniques, such as simple random sampling, stratified sampling, and cluster sampling.

Plan and conduct an experiment and/or observational study. The experimental design should address control, randomization, and minimization of experimental error.

Collect or acquire data to answer a statistical question.

Recognize that data may contain errors, have missing values, or may be biased, and make decisions about how to account for these issues.

Identify biased sampling methods.

Given a plan for an observational study, identify possible sources of bias, and describe ways to reduce bias.

Select, create, and use appropriate visual representations of data to brainstorm solutions.

Use appropriate statistical methods to analyze data.

Communicate the description of an experiment and/or observational study, the resulting data, analysis, and the validity of the conclusions.

The student will calculate and interpret probabilities, including those in contextual situations.

Analyze, interpret, and make predictions based on theoretical probability.

Calculate conditional probabilities for dependent, independent, and mutually exclusive events.

Represent and calculate probabilities using Venn diagrams, probability trees, organized lists, two-way tables, simulations, or other probability models.

Interpret probabilities from simulations or experiments to make informed decisions and justify the rationale.

Define and give contextual examples of complementary, dependent, independent, and mutually exclusive events.

Given two or more events in a problem setting, determine whether the events are complementary, dependent, independent, and/or mutually exclusive.

Compare and contrast permutations and combinations, including those in contextual situations.

Calculate the number of permutations of n objects taken r at a time, without repetition.

Calculate the number of combinations of n objects taken r at a time, without repetition.

The student will describe and apply the properties of normal distribution, including those in contextual situations.

Identify and describe the properties of a normal distribution.

Determine when the normal distribution is a reasonable representation of the data.

Describe how the mean and the standard deviation affect the graph of the normal distribution.

Calculate and interpret the z-score for a data point, given the mean and the standard deviation.

Compare two sets of normally distributed data using a standard normal distribution and zscores, given the mean and the standard deviation.

Represent probability as the area under the curve of a standard normal distribution.

Determine probabilities associated with areas under the standard normal curve, using technology or a table of Standard Normal Probabilities.

Investigate, represent, and determine relationships between a normally distributed data set and its descriptive statistics.

Evaluation of Programming

Applications of Programming

Components of Programming

Data Representation and Storage

The student will write and implement programs using sequencing, selection, and iteration to perform a specific task or solve a problem, including those arising from mathematical and interdisciplinary contexts.

Determine what components of programming are needed to implement a step-by-step plan to perform a specific task or solve a problem.

Write a computer program that includes sequencing, selection (conditionals), and iteration (loops).

Write and implement computer programs to solve mathematical problems using

formulas and equations;

functions;

probability and statistics; and

data-analysis.

The student will create documentation using written comments to annotate the intended purpose of the components of a user-created program.

Create documentation using written comments to:

describe the overall purpose of a program;

align a previously created step-by-step plan to a written program;

describe pre-conditions and post-conditions; and

improve the readability of a program.

The student will verify how programs access and process variables.

Verify that the variable types are aligned to the purpose of the algorithm.

Verify that global variables are set to constant values before run time.

Differentiate between the scopes of variables (e.g., global scope versus local scope) and verify the intended use.

The student will translate a mathematical expression or statement into computer code.

Declare, initialize, and assign variables to represent mathematical expressions or statements.

Implement order of operations, including logical and relational operators.

Translate a mathematical expression or statement into a programming statement(s).

The student will trace existing code to interpret the intended purpose.

Trace existing code of an algorithm to

identify values at each stage of an algorithm; and

predict return values of functions given specific arguments.

Use tracing to describe the intended purpose of existing code for an algorithm.

The student will design a step-by-step plan to perform a task or solve a problem, including those arising from mathematical or interdisciplinary contexts.

Design a step-by-step plan to perform a task or solve a problem using a flowchart or pseudocode that outlines the subtasks needed.

Define the variables needed to perform a task or solve a problem.

Define the constraints of a task or problem (e.g., pre-conditions, post-conditions) to determine the desired input and output.

The student will construct Boolean expressions and implement conditional statements.

Write and implement Boolean expressions using logical and relational operators (e.g., !, &&, ||, ==, <, >, >=, <=, !=).

Write and implement “if” conditional statements.

Write and implement “if/else” conditional statements.

Write and implement compound conditional statements (e.g., nested conditionals, chained conditional statements).

Determine which parts of an algorithm are executed based on a condition being true or false.

The student will perform iteration with loops.

Write and implement “while” and “for” loops.

Differentiate between loops that run a fixed number of times and loops that run an indefinite number of times (e.g., stopping dependent on variable conditions).

Identify conditions that cause infinite loops.

Determine the outcome of code segments that include loops.

The student will write and implement the output phase of a computer program.

Write and implement the output phase of a computer program, which may include:

formatting output in text-based environments;

displaying output through a graphical user interface; and

sending output to a physical device (e.g., speakers, robots, LED lights).

Write output to a file.

The student will write and implement the input phase of a computer program.

Write and implement input statements to store user given values into a program.

Validate input data using exception coding (e.g., using a “while” loop to control valid input by a user).

Determine what output a program will produce given a specific input.

The student will implement library functions.

Implement library functions to process data.

Implement library functions to perform mathematical operations (e.g., random, absolute value, square root, power).

Implement void library functions and return library functions.

Implement overloaded library functions.

The student will write and implement user-defined functions.

Write and implement a user-defined function to complete a task or sub-task.

Write and implement void functions and return functions.

Write and implement functions that accept parameters.

The student will implement pre-defined algorithms, including search routines and sort routines.

Differentiate between types of search routines.

Differentiate between types of sort routines.

Implement pre-defined algorithms.

Implement a search routine on a one-dimensional list or an array, including sequential search and binary search.

Implement a sort routine on a one-dimensional list or an array (e.g., selection sort, insertion sort, merge sort).

The student will represent data and convert data between different number systems.

Represent data in different number systems, including binary, decimal, and hexadecimal.

Convert data between number systems (e.g., binary to decimal, decimal to hexadecimal).

The student will differentiate between variable data types based upon their characteristics.

Describe the characteristics of different variable data types, including

Boolean;

character;

integer;

decimal (double/float); and

string.

Differentiate between variable data types to determine the data type needed based upon intended use (e.g., character versus string, integer versus double/float).

The student will represent data using appropriate data structures.

Given a specific task or problem, determine the appropriate data structure (e.g., lists, arrays, objects) to represent data.

Perform tasks related to lists or arrays (one-dimensional or two-dimensional), including

declare a list or array (one-dimensional or two-dimensional);

choose an appropriate data type for a list or an array; and

fill the list or array with data.

Access and manipulate a particular element of a list or an array.

Implement predefined objects to consolidate related information of different data types.

The student will test a program to match a sample output, using a set of data.

Produce a given output by entering a data set.

Test a program including boundary cases and inaccurate data types to verify the intended outcomes.

The student will identify errors and debug a program using various techniques.

Differentiate among syntax errors, runtime errors, and logic errors.

Debug a program using various techniques:

interpret syntax and runtime error messages;

place controlled breaks;

output intermediate results;

disable a section of code by converting it into a comment;

trace code to identify logic errors; and

use debugging tools available in the programming environment.

The student will compare and contrast the efficiency of computer programs.

Compare and contrast the efficiency of computer programs in terms of

complexity of algorithms with the same intended outcomes;

memory space used; and

run time.

Data and Computing

Data Modeling

Data and Communication

Data Bias

Data in Context

Data and Computing

Data Modeling

Data Bias

Data in Context

identify specific examples of real-world problems that can be effectively addressed using data science.

The student will identify specific examples of real-world problems that can be effectively addressed using data science.

Identify and explain characteristics that best lend themselves to a data driven approach to problem solving.

Formulate questions based on context.

Understand the type of data relevant to the context of the question at hand.

Define relationships between variables and constant relationships.

Create a hypothesis of interest in terms of measurable data.

Define the stages of the data cycle and how each stage is related to the other.

Identify and explain constraints of the data-driven approach.

summarize and interpret data represented in both conventional and emerging visualizations.

The student will be able to summarize and interpret data represented in both conventional and emerging visualizations.

Apply descriptive statistics to explain measures of central tendency and measures of variability/dispersion to describe center and spread in visualizations of distributions.

Define emerging visualizations and describe summarization of characteristics and relationships based on audience and purpose which may include:

a heat map, which uses color to show changes and magnitude of a third variable to a twodimensional plot; and

a bubble chart, which is a multivariate graph that is both a scatterplot and a proportional area chart. Typically, each plotted point then represents a third variable by the area of its circle.

Interpret various emerging visualizations by describing patterns, trends, and relationships between and among the variables.

select statistical models and use goodness of fit testing to extract actionable knowledge directly from data.

The student will select statistical models and use goodness of fit testing to extract actionable knowledge directly from data.

Calculate the theoretical probability of random events and compare them to the observed frequencies.

Describe the normal curve determined by the mean and standard deviation of a univariate data set.

Fit nonlinear models to data sets and use these models to predict unobserved data values.

Select pairs of variables that identify meaningful clusters of data.

Select an appropriate statistical distribution and test its goodness of fit based on the context of the data being analyzed. Statistical distributions may include, but are not limited to

Normal;

Binomial; and

Poisson.

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 process and prepare data for analysis.

Utilize tools to format and store the data appropriately to allow for effective analysis.

Utilize technology tools to be able to access data effectively from multiple sources (e.g., tables, column separated values, spreadsheets, documents, databases).

Utilize tools and functions (in tools) to effectively explore the data for issues and errors before beginning to process it.

Define the (tools and technological) process to optimally ingest data and to export data after processing.

Utilize tools and their functions to clean and validate data by:

removing data that are incomplete, incorrect, or duplicated;

removing extraneous data or outliers; and

standardizing data to conform to contextual norms (e.g., privacy, sensitive data).

Utilize tools and their functions to combine and store data by:

merging multiple data sets for efficiency purposes; and

optimizing storage of data based on volume, velocity, and variety.

select and utilize appropriate technological tools and functions within those tools to analyze and communicate data effectively.

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

Select and utilize technology tools to effectively generate conventional and unconventional visualizations of data to explore patterns and/or analyze a large data set.

Utilize specific functions in technology tools to perform descriptive and inferential statistical analysis.

Utilize coding to store and extract data more effectively for data analysis.

Select and apply features of technology tools effectively to organize, summarize and gain insight from data.

Select the appropriate visualization based on context and audience and create it using technology tools to effectively communicate an idea.

formulate a top down plan for data collection and analysis, with quantifiable results, based on the context of a problem.

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.

Design a data project plan, which is aligned with the data science cycle, that includes the following components:

definition of the goal of the project as it pertains to a real-world problem;

identification of the various parameters of the problem and stakeholders;

a timeline for the project with deliverables;

Key Performance Indicators (KPI) for the successful data project deliverables;

resource needs and tools for the project;

bias considerations for the sampling process of the project; and

limitations of the project.

Given the context and parameters of a problem, choose from among various sampling techniques, which may include

simple random;

systematic;

stratified; and

cluster;

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 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.

Formulate relevant/clarifying questions to identify potential data biases presented in existing analyses/visualizations.

Effectively read data summaries and visualizations and explain/translate into nontechnical terms in proper context.

Identify potential data biases in terms of data presented and discuss the potential effects of such biases in terms of how they could affect data analysis and decision making.

Identify privacy and consumer protection issues that might be a result of how data is presented.

Describe the types of data that business, industry, and government entities collect and possible ways the data is used.

identify data biases in the data collection process, and understand the implications and privacy issues surrounding data collection and processing.

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.

Identify data biases in the data collection process that include, but are not limited to, confirmation, selection, outliers, overfitting / under fitting, and confounding and describe mitigation strategies for these biases.

Provide examples of sampling biases in terms of data collection and the potential effects.

Identify and describe data biases as a producer as well as a consumer/decision maker of data.

Describe how the data collection process should be focused, relevant, and limited to the scope of the data project plan.

Describe privacy considerations in the collection of data as both a consumer and producer.

use storytelling as a strategy to effectively communicate with data.

The student will use storytelling as a strategy to effectively communicate with data.

Define storytelling and explain the importance of storytelling as a strategy to communicate the idea behind and results of a data science project effectively.

Explain the steps involved in data storytelling and how it relates to the data cycle.

Effectively identify a story worth telling based on the data (looking for trends, correlations, outliers) and by asking a question or forming a hypothesis based on insight and audience.

Effectively select visualizations that simplify the information, highlight the most important data, and communicate key points quickly.

Effectively simplify the information presented to make it more concise and focus the audience's attention on the key parameters that support the student’s hypothesis.

Effectively form a narrative based on data available to provide context, insight, and interpretation to make the analysis more relevant to a given audience.

Explain how data storytelling should include complete and accurate information, and consistent visuals for effective communication.

justify the design, use, and effectiveness of different forms of data visualizations.

The student will justify the design, use, and effectiveness of different forms of data visualizations.

Conduct exploratory data analysis using visualization.

Formulate questions from exploration of a data set to consider how data will communicate a story.

Determine the effectiveness of different data visualization choices based on the data context from conventional statistical charts to unconventional/emerging data visualizations to more complex visualizations.

Create a visualization of a data set and summarize the representation using the context of the data.

Compare two or more different representations to ensure the design communicates the features and behavior of data sets.

Justify design choices (based on data set type, size, context, and audience) of data visualizations to highlight important features, trends, and insights.

assess reliability of source data in preparation for mathematical modeling.

The student will be able to assess reliability of source data in preparation for mathematical modeling.

Explain why determining the reliability of big data sources is a key skill that data scientists use to build data trust across an organization.

Describe the difference between reliability of a data source compared to statistical reliability and validity in research analysis. Assess processing source data for reliability based on validity, completeness, and uniqueness.

acquire and prepare big data sets for modeling and analysis.

The student will be able to acquire and prepare big data sets for modeling and analysis.

Explain the pros and cons of collecting data versus acquiring it from existing sources.

Apply matrix operations using algebraic methods (with the support of technology tools) to:

wrangle the data (sort, select, filter, and replace);

clean the data;

format and enrich the data; and

combine and store the data.

Read data from different sources for preparation and analysis.

Identify important parameters about a big data set based on the context of data collected/acquired.

Define and document the process of ingesting, formatting, and cleaning data for future decision making by:

making data more easily understood by a wider audience; and

connecting data with existing contextual data.

select and analyze data models to make predictions, while assessing accuracy and sources of uncertainty.

The student will select and analyze data models to make predictions, while assessing accuracy and sources of uncertainty.

Identify factors that contribute to the overall behavior of a data set (e.g., true values, bias, and noise).

Fit models based on the behavior of the data, (e.g., models of univariate and bivariate data), in order to make predictions.

Distinguish between linear and nonlinear associations between variables using visualizations.

Identify models that are overly complex and therefore fitting to random noise which decreases their predictive accuracy.

Use regression techniques to perform selection of optimal features.

Recognize the potential implications of removing features.

Select the optimal model for a data set from among a large collection of models, using technological tools.

Computational Methods

Graph Theory

Set and Number Theory

Logical Reasoning

The student will describe and apply sorting and searching algorithms used in processing and communicating information.

Select and apply a sorting algorithm, such as a bubble sort, merge sort, or network sort.

Describe the advantages and disadvantages of various sorting algorithms.

Analyze the knapsack and bin-packing problems.

Select and apply search algorithms to analyze problems.

Determine the average, best-case, and worst-case reasoning for different searches.

The student will use recursive processes.

Compare and contrast iterative and recursive processes.

Use recursive processes to model growth and decay.

Use recursive processes to create fractals.

Use recursive processes to generate the Fibonacci sequence.

Determine if a recursive solution is more efficient than an iterative solution.

The student will identify and apply cryptographic methods.

Compare and contrast ciphers and codes.

Describe the evolution of cipher systems.

Identify the Fundamental Theorem of Arithmetic.

Describe how the complexity of prime factorization is used in cryptography.

Describe modular arithmetic in context (e.g., clocks, days of the week, measures of time).

Analyze the relationship between divisibility and modulus.

Determine congruence within modular arithmetic.

Perform operations within modular arithmetic.

Apply modular arithmetic to problems in context (e.g., cryptography, International Standard Book Number (ISBN), International Bank Account Number (IBAN)).

The student will analyze the limitations of algorithms and their contextual relationships in computing.

Describe maximum complexity of an algorithm using Big O notation.

Describe Turing machines and how they are used to test the limits of computation.

Describe the halting problem and explain how it characterizes the fundamental limitations of computation and undecidability.

Explain the P versus NP problem and defend a justification for equality, inequality, or undecidability.

Analyze how the equivalence of P- and NP-class problems might impact society.

The student will represent problems using vertex-edge graphs. The concepts of degree, connectedness, paths, planarity, and directed graphs will be analyzed.

Illustrate the basic terminology of graph theory (e.g., vertex, edge, graph, degree of a vertex).

Use graphs to map situations in which the vertices represent objects, and edges represent a particular relationship between objects.

Identify and describe degree and connectedness.

Determine whether a graph is planar or nonplanar.

Analyze the relationship between faces, edges, and vertices using Euler’s formula (F = E – V + 2).

Use directed graphs (digraphs) to represent situations with restrictions in traversal possibilities.

Determine when graphs are trees.

The student will solve problems through analysis and application of circuits, cycles, Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Optimal solutions will be determined using existing algorithms and student-created algorithms.

Determine whether a graph has an Euler circuit or path, and determine the circuit or path, if it exists.

Determine whether a graph has a Hamilton circuit or path, and determine the circuit or path, if it exists.

Count the number of Hamilton circuits for a complete graph with n vertices.

Use an Euler circuit algorithm to solve optimization problems.

The student will apply graphs to conflict-resolution problems, such as graph coloring, scheduling, matching, and optimization.

Model projects consisting of several subtasks, using a graph.

Use graphs to resolve conflicts that arise in scheduling.

Use graph coloring to determine the chromatic number of a graph.

The student will recognize and apply algorithms to solve configuration, conflict-resolution, and sorting problems.

Recognize algorithms such as nearest neighbor, brute force, and cheapest link as they apply to graphs.

Use Kruskal’s algorithm to determine the shortest spanning tree of a connected graph.

Use Prim’s algorithm to determine the shortest spanning tree of a connected graph.

Use Dijkstra’s algorithm to determine the shortest spanning tree of a connected graph.

The student will use algorithms to schedule tasks to determine a minimum project time.

Specify in a digraph the order in which tests are to be performed.

Identify the critical path to determine the earliest completion time (minimum project time).

Use the list-processing algorithm to determine an optimal schedule.

Create and test scheduling algorithms.

The student will use reasoning to develop and apply logical arguments.

Use Venn diagrams to codify and solve logic problems.

Express logical statements in symbolic form.

Represent a conditional statement as its converse, inverse, and contrapositive.

Describe how symbolic logic can be used to map the processes of computer applications.

Construct a truth table to display all possible input combinations and their outputs.

Identify the rules of inference and model basic logical statements including De Morgan’s Law.

Apply logical reasoning to model contextual situations and make decisions.

The student will apply logic and proof techniques in the construction of a sound argument.

Apply informal logical reasoning to contextual problems (e.g., predicting the behavior of software, solving puzzles).

Outline the basic structure of a proof technique (e.g., direct proof, proof by contradiction, induction).

Deduce the best type of proof for a given problem.

Use the rules of inference to construct direct proofs and proofs by contradiction.

Construct induction proofs involving summations and inequalities.

Use a truth table to prove the logical equivalence of statements.

The student will apply Boolean algebra to represent and analyze the function of logical gates and circuits.

Explain basic properties of Boolean algebra: duality, complements, and standard forms.

Represent verbal statements as Boolean expressions.

Apply Boolean algebra to prove identities and simplify expressions.

Generate truth tables that encode the truth and falsity of two or more statements.

Explain the operation of discrete logic gates.

Describe the relationship between Boolean algebra and electronic circuits.

Analyze a combinational network using Boolean expressions.

Design simple combinational networks that use NAND (AND followed by NOT), NOR (OR followed by NOT), and XOR (exclusive-OR) gates.

The student will use mathematical induction to prove formulas and mathematical statements.

Compare and contrast inductive and deductive reasoning.

Explain the relationship between weak and strong induction.

Construct induction proofs involving a divisibility argument.

Prove the Binomial Theorem through mathematical induction.

The student will identify and use the properties of sets and set operations.

Compare and contrast sets, relations, and functions.

Express relationships between sets using Venn diagrams.

Describe a set using set-builder notation.

Construct new sets using the set operations intersection, union, difference, and complement.

Identify the laws of set theory (e.g., associative, commutative, distributive, De Morgan’s Law).

Use the principle of inclusion and exclusion to determine the size of a set.

Use the properties of set operations to prove set equality.

The student will apply the formulas of combinatorics.

Create a tree diagram to represent relationships between independent events.

Use the Fundamental (Basic) Counting Principle to determine the number of possible outcomes of an event.

Determine the number of combinations possible when subsets of r elements are selected from a set of n elements without regard to order.

Determine the number of permutations possible when r objects selected from n objects are ordered.

Use the pigeonhole principle to solve packing problems to facilitate proofs.

Construct a proof by induction using principles of combinatorics.

The student will use Pascal’s Triangle to analyze numerical patterns and relationships.

Construct Pascal’s Triangle.

Expand binomials having positive integral exponents, using the Binomial Theorem and Pascal’s Triangle.

Compare the binomial coefficient to the calculation of combinations.

Identify the Fibonacci numbers within Pascal’s Triangle.

Two- and Three-Dimensional Figures

Polygons and Circles

Triangles

Reasoning, Lines and Transformations

The student will create models and solve problems, including those in context, involving surface area and volume of rectangular and triangular prisms, cylinders, cones, pyramids, and spheres.

Identify the shape of a two-dimensional cross section of a three-dimensional figure.

Create models and solve problems, including those in context, involving surface area of threedimensional figures, as well as composite three-dimensional figures.

Solve multistep problems, including those in context, involving volume of three-dimensional figures, as well as composite three-dimensional figures.

Determine unknown measurements of three-dimensional figures using information such as length of a side, area of a face, or volume.

The student will determine the effect of changing one or more dimensions of a threedimensional geometric figure and describe the relationship between the original and changed figure.

Describe how changes in one or more dimensions of a figure affect other derived measures (perimeter, area, total surface area, and volume) of the figure.

Describe how changes in surface area and/or volume of a figure affect the measures of one or more dimensions of the figure.

Solve problems, including those in context, involving changing the dimensions or derived measures of a three-dimensional figure.

Compare ratios between side lengths, perimeters, areas, and volumes of similar figures.

Recognize when two- and three-dimensional figures are similar and solve problems, including those in context, involving attributes of similar geometric figures.

The student will prove and justify theorems and properties of quadrilaterals, and verify and use properties of quadrilaterals to solve problems, including the relationships between the sides, angles, and diagonals

Solve problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.

Prove and justify that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the slope formula, the distance formula, and the midpoint formula.

Prove and justify theorems and properties of quadrilaterals using deductive reasoning.

Use congruent segment, congruent angle, angle bisector, perpendicular line, and/or parallel line constructions to verify properties of quadrilaterals.

The student will verify relationships and solve problems involving the number of sides and measures of angles of convex polygons.

Solve problems involving the number of sides of a regular polygon given the measures of the interior and exterior angles of the polygon.

Justify the relationship between the sum of the measures of the interior and exterior angles of a convex polygon and solve problems involving the sum of the measures of the angles.

Justify the relationship between the measure of each interior and exterior angle of a regular polygon and solve problems involving the measures of the angles.

The student will solve problems, including those in context, by applying properties of circles.

Determine the proportional relationship between the arc length or area of a sector and other parts of a circle.

Solve for arc measures and angles in a circle formed by central angles.

Solve for arc measures and angles in a circle involving inscribed angles.

Calculate the length of an arc of a circle.

Calculate the area of a sector of a circle.

Apply arc length or sector area to solve for an unknown measurement of the circle including the radius, diameter, arc measure, central angle, arc length, or sector area.

The student will solve problems in the coordinate plane involving equations of circles.

Derive the equation of a circle of given the center and radius using the Pythagorean Theorem.

Solve problems in the coordinate plane involving equations of circles:

given a graph or the equation of a circle in standard form, identify the coordinates of the center of the circle;

given the coordinates of the endpoints of a diameter of a circle, determine the coordinates of the center of the circle.

given a graph or the equation of a circle in standard form, identify the length of the radius or diameter of the circle.

given the coordinates of the endpoints of the diameter of a circle, determine the length of the radius or diameter of the circle.

given the coordinates of the center and the coordinates of a point on the circle, determine the length of the radius or diameter of the circle; and

given the coordinates of the center and length of the radius of a circle, identify the coordinates of a point(s) on the circle.

Determine the equation of a circle given:

a graph of a circle with a center with coordinates that are integers;

coordinates of the center and a point on the circle;

coordinates of the center and the length of the radius or diameter; and

coordinates of the endpoints of a diameter.

The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams.

Translate propositional statements and compound statements into symbolic form, including negations (~𝑝, read “not p”), conjunctions (p ∧ 𝑞, read “p and q”), disjunctions (p ∨ 𝑞, read “p or q”), conditionals (p → q, read “if p then q”), and biconditionals (p ↔ q, read “p if and only if q”), including statements representing geometric relationships.

Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement, and recognize the connection between a biconditional statement and a true conditional statement with a true converse, including statements representing geometric relationships.

Use Venn diagrams to represent set relationships, including union, intersection, subset, and negation.

Interpret Venn diagrams, including those representing contextual situations.

The student will analyze, prove, and justify the relationships of parallel lines cut by a transversal.

Prove and justify angle pair relationships formed by two parallel lines and a transversal, including:

corresponding angles;

alternate interior angles;

alternate exterior angles;

same-side (consecutive) interior angles; and

same-side (consecutive) exterior angles.

Prove two or more lines are parallel given angle measurements expressed numerically or algebraically.

Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal.

The student will solve problems, including contextual problems, involving symmetry and transformation.

Locate, count, and draw lines of symmetry given a figure, including figures in context.

Determine whether a figure has point symmetry, line symmetry, both, or neither, including figures in context.

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include:

translations;

reflections over any horizontal or vertical line or the lines y = x or y = -x;

clockwise or counterclockwise rotations of 90°, 180°, 270°, or 360° on a coordinate grid where the center of rotation is limited to the origin; and

dilations, from a fixed point on a coordinate grid.

The student will determine the relationships between the measures of angles and lengths of sides in triangles, including problems in context.

Given the lengths of three segments, determine whether a triangle could be formed.

Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie.

Order the sides of a triangle by their lengths when given information about the measures of the angles.

Order the angles of a triangle by their measures when given information about the lengths of the sides.

Solve for interior and exterior angles of a triangle, when given two angles.

The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.

Use definitions, postulates, and theorems (including Side-Side-Side (SSS); Side-Angle-Side (SAS); Angle-Side-Angle (ASA); Angle-Angle-Side (AAS); and Hypotenuse-Leg (HL)) to prove and justify two triangles are congruent.

Use algebraic methods to prove that two triangles are congruent.

Use coordinate methods, such as the slope formula and the distance formula, to prove two triangles are congruent.

Given a triangle, use congruent segment, congruent angle, and/or perpendicular line constructions to create a congruent triangle (SSS, SAS, ASA, AAS, and HL).

The student will, given information in the form of a figure or statement, prove and justify two triangles are similar using direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar triangles.

Use definitions, postulates, and theorems (including Side-Angle-Side (SAS); Side-Side-Side (SSS); and Angle-Angle (AA)) to prove and justify that triangles are similar.

Use algebraic methods to prove that triangles are similar.

Use coordinate methods, such as the slope formula and the distance formula, to prove two triangles are similar.

Describe a sequence of transformations that can be used to verify similarity of triangles located in the same plane.

Solve problems, including those in context involving attributes of similar triangles.

The student will model and solve problems, including those in context, involving trigonometry in right triangles and applications of the Pythagorean Theorem.

Determine whether a triangle formed with three given lengths is a right triangle.

Find and verify trigonometric ratios using right triangles.

Model and solve problems, including those in context, involving right triangle trigonometry (sine, cosine, and tangent ratios).

Solve problems using the properties of special right triangles.

Solve for missing lengths in geometric figures, using properties of 45°-45°-90° triangles, where rationalizing denominators may be necessary.

Solve for missing lengths in geometric figures, using properties of 30°-60°-90° triangles, where rationalizing denominators may be necessary.

Solve problems, including those in context, involving right triangles using the Pythagorean Theorem and its converse, including recognizing Pythagorean Triples.

Algebra - Functions

Algebra - Equations and Inequalities

Algebra - Expressions and Operations

Identify an equation when provided a verbal description in real world applications.

The description and equation could include addition and subtraction from 0 through 50 or multiplication or division from 0 through 40.

Interpret trends in data, including in real world applications.

Trends could include positive slopes of 1 through 10, negative slopes of -1 through -10, and slopes of 0.

Tell time and measure elapsed time to the minute using analog and digital clocks, including with context.

Times could be in one minute increments in a.m. or p.m. and could include up to +/- 24 hours or multiple days of elapsed time. Contexts will relate the time to an appropriate activity.

Evaluate expressions with one variable in real world applications, including using money.

Expressions could include addition, subtraction, multiplication, or division with solutions 1 through 100.

Use currency for problems involving $100.00 or less.

Identify equivalent expressions and evaluate expressions using powers 1-3.

Expressions: to the first power could result in a number from 1 through 20, to the second power could result in a number from 1 through 25 (squaring 1 through 5), and to the third power could result in a number from 1 through 125 (cubing 1 through 5)

Solve one- and two-step linear equations with one variable and solutions from 0 through 40.

Equations could range from having one step of addition, subtraction, multiplication, or division to having two steps with two different operations.

Find the amount of sales tax and total cost for a purchase.

Problems could include finding the total cost for a purchase when given the cost(s) of 1 to 3 items and the total sales tax or finding the amount of sales tax when given the cost(s) of 1 to 3 items and the total cost of the purchase.

Match the graph on a number line with the correct inequality.

The graph on a number line could represent an inequality with descriptive words or the symbols <, >, ≤, or ≥.

Identify a missing value in input and output tables based on a given function.

The missing value could range from multiples of 1 through 30.

Analytic Geometry

Functional Relationships

Characteristics of Functions

The student will identify and analyze the properties of conic sections and sketch a graph given an equation.

Given a translation or rotation matrix, determine an equation for the transformed function or conic section.

Convert between standard and general forms of conic equations by completing the square.

Graph conic sections from equations written in general or standard form using transformations.

Identify characteristics of conic sections including center, vertices, axes, symmetry, foci, directrix, eccentricity, and asymptotes.

Represent applications of conic sections.

The student will use parametric equations to model and solve problems in context.

Graph and analyze parametric equations and use the graph to determine solutions.

Use parametric equations to model contextual problems, including motion over time.

The student will perform operations with vectors in the coordinate plane.

Use vector notation.

Perform the operations of addition, subtraction, and scalar multiplication, graphically and algebraically on vectors.

Find the dot (inner) product of two vectors and use it to determine the angle between two vectors.

Determine if two vectors are orthogonal.

Express complex numbers in vector notation.

Verify properties of the dot product.

Determine the components of a vector.

Determine the norm (magnitude) of a vector.

Find a unit vector in the same direction of a given vector.

Apply vectors to problems in context.

The student will investigate and identify the characteristics of the graphs of polar equations.

Classify polar equations (rose, cardioid, limaçon, lemniscate, spiral, and circle), given the graph or the equation.

Determine the effects of changes in the parameters of polar equations on the graph, using graphing technology.

Convert between polar and rectangular forms of coordinates.

Convert between complex numbers written in rectangular form and polar form.

Convert equations between polar and rectangular forms.

Determine and verify the intersection of the graphs of two polar equations, using graphing technology.

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.

Multiply matrices by a scalar.

Add, subtract, and multiply matrices.

Represent problems with a system of no more than three linear equations.

Express a system of linear equations as a matrix equation.

Solve a system of equations using matrices.

Determine the inverse of a two-by-two or three-by-three matrix using paper and pencil.

Verify two matrices are inverses using matrix multiplication.

Verify the commutative and associative properties for matrix addition and multiplication.

The student will identify and analyze the properties of polynomial, rational, piecewise-defined, absolute value, radical, and step functions and sketch the graphs of the functions.

Use mathematical reasoning to identify polynomial, rational, piecewise-defined, absolute value, radical, and step functions, given an equation or graph.

Given multiple representations of a polynomial, rational, piecewise-defined, absolute value, radical, and step function, analyze:

domain and range;

roots (including complex roots);

intercepts;

symmetry (including even and odd functions);

relative and/or absolute maximum and minimum points.

asymptotes (horizontal, vertical, and oblique/slant;

points of discontinuity;

intervals for which the function is increasing, decreasing or constant;

end behavior; and

Sketch the graph of a polynomial, rational, piecewise-defined, absolute value, radical, and step function.

The student will determine the limit of a function if it exists.

Verify estimates about the limit of a function using graphing technology.

Determine the limit of a function algebraically and verify with graphing technology.

Determine the limit of a function numerically and verify with graphing technology.

Use proper limit notation, including when describing the end behavior of a function.

As the variable approaches a finite number,

determine the limit of a function numerically by direct substitution;

determine the limit of a function using algebraic manipulation;

estimate the limit of a function using a table; and

determine the limit of a function from a given graph.

As the variable approaches positive or negative infinity, analyze the limit of a function to describe the end behavior.

The student will analyze and describe the continuity of functions.

Describe continuity of a function.

Use mathematical notation to communicate and describe the continuity of functions including polynomial, rational, piecewise, absolute value, radical, and step function, using graphical and algebraic methods.

Prove continuity at a point, using the definition.

Classify types of discontinuity based on which condition of continuity is violated.

The student will analyze compositions of functions to determine and verify inverses of functions.

Construct the composition of functions algebraically and graphically.

Determine the domain and range of composite functions algebraically and graphically.

Develop the inverse of a function algebraically and graphically.

Compare the domain and range of the inverse of a function with the original function, both algebraically and graphically.

Use mathematical reasoning to generalize and communicate the criteria for an inverse function to exist.

The student will analyze the characteristics of exponential and logarithmic functions, and sketch the graphs of the functions.

Generalize characteristics of exponential and logarithmic functions from an equation or a graph.

Define e and estimate its value.

Convert between equations written in logarithmic and exponential form.

Use laws of exponents and properties of logarithms to solve equations and simplify expressions.

Represent contextual problems, using exponential and logarithmic functions, to include common and natural logarithms.

Sketch the graph of exponential and logarithmic functions and identify asymptotes, end behavior, intercepts, domain, and range.

The student will analyze sequences and finite series, and model and solve problems in context using sequences and series.

Use and interpret the notation: ∑, n, n th, and an.

Derive the formulas associated with arithmetic and geometric sequences and series.

Determine the nth term, an, for an arithmetic or geometric sequence.

Determine the sum, Sn, if it exists, of an arithmetic or geometric series.

Model and solve problems in context, using sequences and series.

Distinguish between a convergent and divergent series.

Describe convergent series in relation to the concept of a limit.

Inferential Statistics

Probability

Descriptive Statistics

Data in Context

The student will use a statistical cycle to formulate questions, describe types of data, data sources, and constraints within the context of a problem.

Define the stages of the statistical cycle and how each stage relates to the others.

Formulate questions and conclusions based on context.

Understand the type of data relevant to the question at hand (e.g., quantitative versus categorical).

Compare and contrast population and sample, and parameter and statistic.

Identify and explain constraints of the statistical approach.

The student will compare and contrast data collection methods to plan and conduct an observational study.

Investigate and describe sampling techniques (e.g., simple random sampling, stratified sampling, systematic sampling, cluster sampling).

Determine which sampling technique is best, given a particular context.

Investigate and explain biased influences inherent within sampling methods and various forms of response bias.

Use the statistical cycle to plan and conduct an observational study to answer a question or address a problem.

The student will utilize the principles of experimental design to plan and conduct a well-designed experiment.

Describe the principles of experimental design, including:

treatment/control groups;

blinding/placebo effects;

experimental units/subjects; and

blocking/matched pairs and completely randomized designs.

Evaluate the principles of experimental design to address comparison, randomization, replication, and control within the context of the problem.

Compare and contrast controlled experiments and observational studies and the conclusions that may be drawn from each.

Use the statistical cycle to plan and conduct a well-designed experiment to answer a question or address a problem.

Select a data collection method appropriate for a given context.

The student will represent and analyze data visualizations of univariate quantitative data, including dot plots, stemplots, boxplots, cumulative frequency graphs, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers, within the context of a problem.

Create and interpret graphical displays of data, including dot plots, stemplots, boxplots, cumulative frequency graphs, and histograms, using appropriate technology.

Examine the graphs within the context of the problem by analyzing:

shape;

measures of center;

spread; and

unusual features of the data (e.g., outliers, clusters, gaps).

The student will represent and analyze numerical characteristics of univariate quantitative data sets to describe patterns and departures from patterns within the context of a problem

Interpret measures of central tendency: mean, median, and mode.

Interpret measures of spread: range, interquartile range, variance, and standard deviation.

Identify possible outliers, using an algorithm.

Investigate and explain the influence of outliers on a univariate data set.

Investigate and explain ways in which standard deviation addresses variability by examining the formula for standard deviation.

The student will represent, compare, and analyze distributions of two or more univariate quantitative data sets, numerically and graphically.

Create graphical displays of data, including back-to-back stemplots, parallel dot plots, parallel boxplots, and histograms, using appropriate technology.

Compare and contrast two or more univariate data sets, numerically and graphically, within the context of a problem by analyzing:

shape;

measures of center;

measures of spread; and

unusual features of the data (e.g., clusters, gaps, outliers).

The student will represent and analyze categorical data, using two-way tables and other graphical displays, to describe patterns and relationships.

Create and interpret graphical displays of univariate categorical data, including bar graphs within the context of the problem, using appropriate technology.

Create and interpret graphical displays comparing distributions of two or more univariate categorical data sets including segmented and side-by-side bar graphs within the context of the problem, using appropriate technology.

Generate and interpret a two-way table as a summary of the information obtained from two categorical variables.

Calculate and interpret marginal, relative, and conditional frequencies to analyze data in a two-way table within the context of a problem.

The student will represent and analyze quantitative bivariate data with scatterplots to identify and describe the relationship between two variables.

Create scatterplots, using appropriate technology.

Examine and interpret scatterplots in the context of the problem by analyzing:

the form of relationship for linear and nonlinear trends;

the direction of the relationship for positive, negative, or no association;

the strength of the relationship such as strong, moderate, or weak; and

the presence of unusual features within the data (e.g., clusters, gaps, influential points, outliers).

The student will create and interpret a linear model using the least squares regression method to assess the relationship between two quantitative variables

Create the least squares regression model using technology to interpret the contextual meaning of the slope and y-intercept.

Using technology, calculate and interpret the correlation coefficient, r, within the context of a problem.

Using technology, calculate and interpret the coefficient of determination, r 2 , within the context of a problem.

Use regression lines to make predictions, and identify the limitations of the predictions, such as extrapolation.

Calculate and interpret a residual to understand the error of a prediction.

Using technology, calculate and interpret the standard deviation of the residuals, s.

The student will apply properties of sampling distributions and inference procedures to make decisions about population proportions. The student will apply properties of sampling distributions and inference procedures to make decisions about population proportions.

Describe the shape, center, and spread of the sampling distribution of a proportion within the context of a problem.

Given a problem, construct a one sample z confidence interval:

identify the basic conditions for inference: random sample, independence, and normality;

calculate a confidence interval using technology; and

interpret the interval within the context of the problem.

Explain how changes in confidence level and sample size affect width of the confidence interval and margin of error.

Calculate and interpret a point estimate and margin of error of a confidence interval for a proportion within the context of the problem.

Explain how and why the hypothesis testing procedure allows one to reach a statistical decision.

Given a problem, apply the one sample z hypothesis testing procedures:

construct appropriate null and alternate hypotheses;

identify the basic conditions for inference: random sample; independence, and normality;

calculate and interpret the p-value using technology;

determine and justify whether to reject the null hypothesis; and

interpret the results within the context of the problem.

Use the statistical cycle to plan and conduct a statistical study about a proportion to answer a question or address a problem with inference.

The student will apply properties of sampling distributions and inference procedures to make decisions about populations.

Describe the shape, center, and spread of the sampling distribution of a mean within the context of a problem.

Calculate and interpret a point estimate and a margin of error for a confidence interval of a mean within the context of a problem.

Describe the use of the Central Limit Theorem in satisfying the assumptions and conditions for inference about a mean.

Identify the properties of a t distribution.

Given a problem, construct a one sample t confidence interval:

identify the basic conditions for inference: random sample, independence, and approximate normality;

calculate a confidence interval using technology; and

interpret the interval within the context of the problem.

Given a problem, apply the one sample t hypothesis testing procedures:

construct appropriate null and alternate hypotheses;

identify the basic conditions for inference: random sample, independence, and approximate normality;

calculate and interpret the p value using technology;

determine and justify whether to reject the null hypothesis; and

interpret the results within the context of the problem.

The student will organize information and apply probability rules to compute probabilities of events within the context of a problem.

Given two or more events, determine whether the events are complementary, dependent, independent, and/or mutually exclusive, and compute the probability of those events.

Represent and calculate probabilities using Venn diagrams, tree diagrams, and two-way tables.

Apply the addition rule, the multiplication rule, and complementary rule to calculate probabilities.

Calculate conditional probabilities to determine the association or independence of two events.

The student will represent and interpret situations using discrete random distributions, including binomial distributions.

Identify discrete random variables and create a table to represent valid discrete probability distributions within the context of a problem.

Calculate and interpret the mean (expected value) and standard deviation for a discrete random variable within the context of a problem.

Determine if a discrete random variable satisfies the conditions for a binomial distribution.

Design and conduct a simulation of a binomial distribution.

Calculate and interpret probabilities from a binomial distribution within the context of a problem.

Calculate the mean and standard deviation for binomial distributions.

Describe the center, shape, and spread of a discrete random variable within the context of a problem.

The student will represent and interpret situations using normal distributions.

Compare and contrast discrete and continuous distributions.

Represent probability as the area under the curve of a normal distribution using the Empirical Rule and graphing technology.

Describe the center, shape, and spread of normal distributions within the context of a problem.

Compare and contrast two or more sets of normally distributed data using z-scores, percentiles, or probabilities within the context of a problem.

Standardize a data value from a normal distribution and interpret the z-score within the context of a problem.

Calculate and interpret probabilities of a normal distribution using technology within the context of a problem.

Identities and Equations

Graphs of Trigonometric Functions

Circular Trigonometry

Triangle Trigonometry

The student will determine the degree and radian measure of angles; sketch angles in standard position on a coordinate plane; and determine the sine, cosine, tangent, cosecant, secant, and cotangent of an angle, given a point on the terminal side of an angle in standard position or the value of a trigonometric function of the angle.

Define a radian as a unit of angle measure and determine the relationship between the radian measure of an angle and the length of the intercepted arc in a circle.

Determine the degree and radian measure of angles to include both negative and positive rotations in the coordinate plane.

Find both positive and negative coterminal angles for a given angle.

Identify the quadrant or axis in/on which the terminal side of an angle lies.

Draw a reference right triangle when given a point on the terminal side of an angle in standard position.

Draw a reference right triangle when given the value of a trigonometric function of an angle (sine, cosine, tangent, cosecant, secant, and cotangent).

Determine the value of any trigonometric function (sine, cosine, tangent, cosecant, secant, and cotangent) when given a point on the terminal side of an angle in standard position.

Given one trigonometric function value, determine the other five trigonometric function values.

Calculate the length of an arc of a circle in radians.

Calculate the area of a sector of a circle.

The student will develop and apply the properties of the unit circle in degrees and radians.

Convert between radian and degree measure of special angles of the unit circle without the use of technology.

Define the six circular trigonometric functions of an angle in standard position on the unit circle.

Apply knowledge of right triangle trigonometry, special right triangles, and the properties of the unit circle to determine trigonometric functions values of special angles (0°, 30°, 45°, 60°, and 90°) and their related angles in degree and radians without the use of technology.

The student will graph and analyze trigonometric functions and apply trigonometric functions to represent periodic phenomena.

Sketch the graph of the six parent trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for at least a two-period interval.

Determine the domain and range, amplitude, period, and asymptote locations for a trigonometric function, given a graph or an equation.

Describe the effects of changing the parameters (A, B, C, or D in the standard form of a trigonometric equation) on the graph of the function using graphing technology.

Sketch the graph of a transformed sine, cosine, and tangent function written in standard form by using transformations for at least a two-period interval, including both positive and negative values for the domain.

Apply trigonometric functions and their graphs to represent periodic phenomena.

The student will graph the six inverse trigonometric functions.

Determine the domain and range of the inverse trigonometric functions.

Use the restrictions on the domain of an inverse trigonometric function to determine a value of the inverse trigonometric function.

Graph inverse trigonometric functions.

The student will evaluate expressions involving the six trigonometric functions and the inverse sine, cosine, and tangent functions.

Determine the values of trigonometric functions, with and without graphing technology.

Determine angle measures by using the inverse trigonometric functions, with and without a graphing technology.

Evaluate composite functions that involve trigonometric functions and inverse trigonometric functions.

The student will use basic trigonometric identity substitutions to simplify and verify trigonometric identities.

Use trigonometric identities to make algebraic substitutions to simplify and verify trigonometric identities. The basic trigonometric identities include

reciprocal identities;

Pythagorean identities;

sum and difference identities;

double-angle identities; and

half-angle identities.

Apply the sum, difference, and half-angle identities to evaluate trigonometric function values of angles that are not integer multiples of the special angles to solve problems, including contextual situations.

The student will solve trigonometric equations and inequalities.

Solve trigonometric equations with and without restricted domains algebraically and graphically.

Solve trigonometric inequalities algebraically and graphically.

Verify and justify algebraic solutions to trigonometric equations and inequalities, using graphing technology.

The student will determine the sine, cosine, tangent, cotangent, secant, and cosecant of the acute angles in a right triangle and use these ratios to solve for missing sides and angle measures, including application in contextual problems.

Define and represent the six triangular trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in a right triangle.

Describe the relationships between side lengths in special right triangles (30°-60°-90° and 45°-45°-90°).

Use the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines to solve contextual problems.

Represent and solve contextual problems involving right triangles, including problems involving angles of elevation and depression.

The student will find the area of any triangle and solve for the lengths of the sides and measures of the angles in a non-right triangle using the Law of Sines and the Law of Cosines.

Apply the Law of Sines, and the Law of Cosines, as appropriate, to find missing sides and angles in non-right triangles.

Recognize the ambiguous case when applying the Law of Sines and the potential for two triangle solutions in some situations.

Solve problems that integrate the use of the Law of Sines and the Law of Cosines and the triangle area formula (Area = 1 2 absinC, where a and b are triangle sides and C is the included angle) to find the area of any triangle, including those in contextual problems.