High School Math GSE Standards

Functional & Graphical Reasoning – rational functions

Geometric & Spatial Reasoning – Trigonometry and the Unit Circle

Patterning & Algebraic Reasoning – linear algebra and matrices

Functional & Graphical Reasoning – polynomial functions

Functional & Graphical Reasoning – radical functions

Functional & Graphical Reasoning – exponential and logarithmic functions

Data & Statistical Reasoning – descriptive and inferential statistics

Mathematical Modeling

Mathematical Practices

Communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn.

When collecting and considering data, critically evaluate ethics, privacy, potential bias, and confounding variables along with their implications for interpretation in answering a statistical investigative question. Implement strategies for organizing and preparing big data sets.

Distinguish between population distributions, sample data distributions, and sampling distributions. Use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language.

Calculate and interpret z-scores as a measure of relative standing and as a method of standardizing units.

Given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology.

Model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population.

Given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability.

Summarize and evaluate reports based on data for appropriateness of study design, analysis methods, and statistical measures used.

Explore and analyze structures and patterns for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena.

Find the inverse of exponential and logarithmic functions using equations, tables, and graphs, limiting the domain of inverses where necessary to maintain functionality, and prove by composition or verify by inspection that one function is the inverse of another.

Analyze, graph, and compare exponential and logarithmic functions.

Use the definition of a logarithm, logarithmic properties, and the inverse relationship between exponential and logarithmic functions to solve problems in context.

Create exponential equations and use logarithms to solve mathematical, applicable problems for which only one variable is unknown.

Create and interpret logarithmic equations in one variable and use them to solve problems.

Create, interpret, and solve exponential equations to represent relationships between quantities and analyze the relationships numerically with tables, algebraically, and graphically.

Create, interpret, and solve logarithmic equations in two or more variables to represent relationships between quantities.

Explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena.

Rewrite radical expressions as expressions with rational exponents. Extend the properties of integer exponents to rational exponents.

Solve radical equations in one variable, and give examples showing how extraneous solutions may arise.

Analyze and graph radical functions.

Create, interpret and solve radical equations with one unknown value and use them to solve problems that model real-world situations.

Create, interpret, and solve radical equations in two or more variables to represent relationships between quantities.

Extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-world phenomena.

Graph and analyze quadratic functions in contextual situations and include analysis of data sets with regressions.

Use the structure of an expression to factor polynomials, including the sum of cubes, the difference of cubes, and higher-order polynomials that may be expressed as a quadratic within a quadratic.

Using all the zeros of a polynomial function, list all the factors and multiply to write a multiple of the polynomial function in standard form.

Define complex numbers i such that i² = –1 and show that every complex number has the form a + bi where a and b are real numbers and that the complex conjugate is a - bi.

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

Use the structure of an expression to factor quadratics.

Write and solve quadratic equations and inequalities with real coefficients and use the solution to explain a mathematical, applicable situation.

Solve systems of quadratic and linear functions to determine points of intersection.

Create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations.

Identify the number of zeros that exist for any polynomial based upon the greatest degree of the polynomial and the end behavior of the polynomial by observing the sign of the leading coefficient.

Identify zeros of polynomial functions using technology or pre-factored polynomials and use the zeros to construct a graph of the function defined by the polynomial function. Analyze identify key features of these polynomial functions.

Analyze the behaviors of rational functions to model applicable, mathematical problems.

Rewrite simple rational expressions in equivalent forms.

Add, subtract, multiply and divide rational expressions, including problems in context and express rational expressions in irreducible form.

Graph rational functions, identifying key characteristics.

Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

Develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle.

Define the three basic trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane.

Apply understanding of the angle measures and coordinates of the unit circle to solve practical, real-life problems involving trigonometric equations.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain applicable, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a mathematical, applicable situation.

Use various mathematical representations and structures to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals. Code Expectation

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications.

Use matrices to represent data, and perform mathematical operations with matrices and scalars, demonstrating that some properties of real numbers hold for matrices, but that others do not.

Rewrite a system of linear equations using a matrix representation.

Use the inverse of an invertible matrix to solve systems of linear equations.

Utilize linear programming to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as solutions or non-solutions under the established constraints in real-world problems.

Geometric & Spatial Reasoning

Patterning & Algebraic Reasoning

Functional & Graphical Reasoning

Data & Statistical Reasoning

Mathematical Modeling

Mathematical Practices

Communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn.

When collecting and considering data, critically evaluate ethics, privacy, potential bias, and confounding variables along with their implications for interpretation in answering a statistical investigative question. Implement strategies for organizing and preparing big data sets.

istinguish between population distributions, sample data distributions, and sampling distributions. Use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language.

Calculate and interpret z-scores as a measure of relative standing and as a method of standardizing units.

Given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology.

Model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population.

Given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability.

Summarize and evaluate reports based on data for appropriateness of study design, analysis methods, and statistical measures used.

Explore and analyze structures and patterns for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena.

Find the inverse of exponential and logarithmic functions using equations, tables, and graphs, limiting the domain of inverses where necessary to maintain functionality, and prove by composition or verify by inspection that one function is the inverse of another.

Analyze, graph, and compare exponential and logarithmic functions.

Use the definition of a logarithm, logarithmic properties, and the inverse relationship between exponential and logarithmic functions to solve problems in context.

Create exponential equations and use logarithms to solve mathematical, applicable problems for which only one variable is unknown.

Create and interpret logarithmic equations in one variable and use them to solve problems.

Create, interpret, and solve exponential equations to represent relationships between quantities and analyze the relationships numerically with tables, algebraically, and graphically.

Create, interpret, and solve logarithmic equations in two or more variables to represent relationships between quantities.

Explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena.

Rewrite radical expressions as expressions with rational exponents. Extend the properties of integer exponents to rational exponents.

Solve radical equations in one variable, and give examples showing how extraneous solutions may arise.

Analyze and graph radical functions.

Create, interpret and solve radical equations with one unknown value and use them to solve problems that model real-world situations.

Create, interpret, and solve radical equations in two or more variables to represent relationships between quantities.

Extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-world phenomena.

Graph and analyze quadratic functions in contextual situations and include analysis of data sets with regressions.

Use the structure of an expression to factor polynomials, including the sum of cubes, the difference of cubes, and higher-order polynomials that may be expressed as a quadratic within a quadratic.

Using all the zeros of a polynomial function, list all the factors and multiply to write a multiple of the polynomial function in standard form.

Define complex numbers i such that i^2 = –1 and show that every complex number has the form a + bi where a and b are real numbers and that the complex conjugate is a − bi.

Use the relation i^2 = – 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use the structure of an expression to factor quadratics.

Write and solve quadratic equations and inequalities with real coefficients and use the solution to explain a mathematical, applicable situation.

Solve systems of quadratic and linear functions to determine points of intersection.

Create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations.

Identify the number of zeros that exist for any polynomial based upon the greatest degree of the polynomial and the end behavior of the polynomial by observing the sign of the leading coefficient.

Identify zeros of polynomial functions using technology or pre-factored polynomials and use the zeros to construct a graph of the function defined by the polynomial function. Analyze identify key features of these polynomial functions.

Analyze the behaviors of rational functions to model applicable, mathematical problems.

Rewrite simple rational expressions in equivalent forms.

Add, subtract, multiply and divide rational expressions, including problems in context and express rational expressions in irreducible form.

Graph rational functions, identifying key characteristics.

Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

Develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle.

Define the three basic trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane.

Apply understanding of the angle measures and coordinates of the unit circle to solve practical, real-life problems involving trigonometric equations.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain applicable, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a mathematical, applicable situation.

Use various mathematical representations and structures to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications.

Use matrices to represent data, and perform mathematical operations with matrices and scalars, demonstrating that some properties of real numbers hold for matrices, but that others do not.

Rewrite a system of linear equations using a matrix representation.

Use the inverse of an invertible matrix to solve systems of linear equations.

Utilize linear programming to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as solutions or non-solutions under the established constraints in real-world problems.

Data & Statistical Reasoning

Geometric & Spatial Reasoning

Patterning & Algebraic Reasoning

Functional & Graphical Reasoning

Numerical Reasoning

Mathematical Modeling

Mathematical Practices

Write, graph, solve, and interpret systems of equations containing one linear and one quadratic equation, given an applicable financial situation.

Collect, analyze, interpret, summarize, and construct displays of data to make predictions within real-world applications.

Interpret measures of central tendency (mean, median, mode) and spread (range, interquartile range, variance, standard deviation) to analyze contextualized data sets.

Construct and interpret common data displays (bar graphs, line graphs, stock bar charts, candlestick charts, box and whisker plots, stem and leaf plots, and circle graphs) to recognize and interpret trends.

Construct and interpret scatterplots to recognize and interpret trends.

Use technology to find, interpret, and graph linear, quadratic, and exponential regression equations to make predictions about the corresponding context.

Use technology to determine the correlation coefficient of linear, quadratic, and exponential regression curves.

Distinguish between causation and correlation for bivariate data.

Create and analyze discrete probability distributions.

Apply the Arithmetic Average Formula to calculate and interpret a d-day simple moving average given a set of n data points, p1 ​ , p2 ​, p3 ​ , ..., pn-1, pn ​ .

Use distributions to identify the key features of the data collected.

Conduct investigative research to solve real-life problems and answer statistical questions involved in business and financial decision-making

Identify a contextual, real-life problem that can be answered using investigative research.

Develop statistical questions that can help solve a real-life problem involved in business and financial decision-making.

Create a statistical study using sound methodology to answer statistical questions and to solve the real-life problem.

Explain how the sample size impacts the precision with which estimates of the population parameters can be made.

Recognize that random selection from a population plays a different role than random assignment in an experiment.

Incorporate random designs in data collection.

Describe ways in which “big data” can be used to make decisions in various business enterprises and in the context of business and financial decision-making.

Interpret results and make connections to the original research question.

Explore and apply functions to model and explain real-life phenomena and to solve complex problems in business and financial contexts.

Examine and identify the key characteristics of functions that model financial situations given the parameters of the context.

Recognize real-world situations where square root, cubic, or rational functions apply.

Create and use inequalities to define domains when creating algebraic expressions and functions.

Solve financial problems given the parameters of the applicable context using a variety of functions.

Describe the meaning of functions and how to determine if a relation is a function or not.

Utilize function notation to represent a functional relation and to evaluate functions.

Create, apply, and interpret linear functions to model real-world financial problems.

Create, apply, and interpret exponential functions of the form y = ab^x and classify them as exponential decay (when 0 < b < 1) or as exponential growth (when b > 1).

Create, apply, and interpret quadratic functions to model real-world financial applications.

Create, apply, and interpret the greatest integer function in real-world financial applications.

Create, apply, and interpret piecewise functions in real-world financial applications.

Apply properties of polygons, circles, and trigonometry to model and explore real-world applications.

Apply concepts of area, volume, and scale factors to a variety of real-world financial applications.

Use factors of dilations to draw to scale in contextual situations.

Use sectors and central angles of a circle to depict proportional categories on a pie chart when given categorical information.

Solve problems using the Pythagorean Theorem and trigonometric functions and their inverses in context.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration, and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Utilize fractions, decimals, percents, and ratios to write and solve a variety of financial problems.

Use fractions, decimals, percents, and ratios to solve problems related to budgets, income tax rates, payroll deductions, pie charts, percent yield, sales tax, percent populations, rent increase, cost savings, debt-to-income ratios, stock splits, floor plans and scale models, trigonometric calculations, banking services, and other business and financial applications.

Convert numerical quantities of one form (fractions, decimals, percents) to another within financial applications.

Calculate and interpret percent of increase and decrease.

Construct, solve, and interpret algebraic ratios and proportions.

Utilize the present and future value of a periodic investment formulas to make calculations regarding long-term investments and retirement planning.

Explore, evaluate, and rearrange formulas applicable to business and financial contexts.

Use and rearrange formulas applicable to real-world contexts.

Utilize the monthly payment formula to assist in calculating the total interest paid (finance charge) when using credit. Compare the total of monthly payments to the original (cash) price.

Interpret and use sigma notation.

Explore and identify how the elements of the present value of a single deposit formula and the periodic deposit investment formula relate to the compound interest formula.

Investigate the impact of changing the value of the different variables in financial formulas to compare the resulting financial outcomes.

Write algebraic formulas for use in spreadsheets and utilize technology to perform both iterate and formulaic calculations.

Use the simple interest formula, I = Prt, and inverse operations to solve for specified variables in banking services applications and other interest problems.

Demonstrate by iteration (both with technology and without) that the compounding process pays “interest on your interest.”

Derive the compound interest formula, A = P(1+ r/t​ )^nt, by using patterns and inductive reasoning, then compute compound interest with and without the formula.

Explore the concept of limits of rational functions in discovering the compound continuous formula. Use technology to investigate and verify what happens as the number of compounds approaches infinity.

Apply the natural base e in the continuous compounding formula, A = Pe^rt.

Use the monthly payment formula to calculate payment amounts in a variety of circumstances.

Write and solve systems of equations and/or inequalities in context of financial applications.

Write, graph, solve, and interpret systems of linear equations given an applicable financial situation.

Write, graph, and interpret systems of equations containing one linear and one exponential equation, given an applicable financial situation.

Write, graph, and interpret systems of a linear and a piecewise function.

Solve linear systems of equations and inequalities to identify points of intersection and define domains in the context of the problem situation.

Geometric & Spatial Reasoning

Functional & Graphical Reasoning

Data & Statistical Reasoning

Probabilistic Reasoning

Patterning & Algebraic Reasoning

Quantitative & Proportional Reasoning

Mathematical Modeling

Mathematical Practices

Conduct investigative research to solve real-life problems and answer statistical investigative questions involved in business and financial decision-making.

Apply statistical methods to design, conduct, and analyze statistical studies. Identify a contextual, real-life problem that can be answered using investigative research.

Build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays. Develop statistical investigative questions that can help solve a real-life problem involved in business and financial decision-making.

Create a statistical study using sound methodology to answer statistical investigative questions and to solve the real-life problem.

Explain how the sample size impacts the precision with which estimates of the population parameters can be made (i.e., the larger the sample size the more precision).

Recognize that random selection from a population plays a different role than random assignment in an experiment.

Incorporate random designs in data collection.

Describe ways in which big data can be used to make decisions in various business enterprises and in the context of business and financial decision-making.

Use distributions to identify the key features of the data collected.

Interpret results and make connections to the original research question.

Use functions to model problem situations in both discrete and continuous relationships.

Determine whether a problem situation involving two quantities is best modeled by a discrete or continuous relationship.

Use linear, exponential, logistic, and piecewise functions to construct a model.

Use functions to model problem situations in both discrete and continuous relationships.

Create and use two-dimensional and three-dimensional representations to model authentic situations.

Solve problems involving inaccessible distances using basic trigonometric principles including extensions of right triangle trigonometry.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use relevant information to create various mathematical representations and structures to solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Use functions to model problem situations in both discrete and continuous relationships.

Represent situations and solve problems using vectors, in areas such as transportation, computer graphics, and the physics of force and motion.

Represent geometric transformations and solve problems using matrices.

Make informed decisions and solve problems with a variety of network models in quantitative situations.

Solve problems represented by a vertex-edge graphs.

Construct, analyze, and interpret flow charts to develop an algorithm to describe processes such as quality control procedures.

Investigate the scheduling of projects using Program Evaluation Review Technique (PERT).

Consider problems that can be resolved by coloring graphs.

Develop methods or algorithms to analyze discrete situations.

Create and verify identification numbers.

Analyze and evaluate the mathematics behind various methods of voting and selection.

Evaluate various voting and selection processes to determine an appropriate method for a given situation.

Apply various ranking algorithms to determine an appropriate method for a given situation.

Create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money.

Use exponential functions to model change in a variety of financial situations.

Determine, represent, and analyze mathematical models for income, expenditures, and various types of loans and investments.

Analyze the chances for success or failure in order to make decisions.

Determine conditional probabilities and probabilities of compound events to make decisions in problem situations.

Use probabilities to make and justify decisions about risks in everyday life.

Model strategic interaction among rational decision-makers.

Calculate expected value to analyze mathematical fairness, payoff, and risk.

Analyze real-life situations involving strategic interactions using the mathematics of zero-sum games.

Construct a mathematical model of probabilistic situations to make mathematical assumptions.

Make decisions and solve problems using ratios, rates, and percents in a variety of real-world applications.

Apply proportions, ratios, rates, and percentages to various settings, including business, media, and consumerism.

Solve problems involving ratios in mechanical and agricultural contexts.

Use proportions to solve problems involving large quantities that are not easily measured.

Make predictions by analyzing averages and indices of large data sets through investigations of real-world contexts.

Use averages and weighted averages to make decisions.

Calculate and interpret indices.

Modeling and Analyzing Quadratic Functions

Investigating Rational and Irrational Numbers

Analyzing Linear Inequalities

Modeling Linear Functions

Use mathematically applicable situations algebraically and graphically to build and interpret arithmetic sequences as functions whose domain is a subset of the integers.

Construct and interpret the graph of a linear function that models real-life phenomena and represents key characteristics of the graph using formal notation.

Relate the domain and range of a linear function to its graph and, where applicable, to the quantitative relationship it describes. Use formal notation and set notation to describe the domain and range of linear functions.

Use function notation to build and evaluate linear functions for inputs in their domains and interpret statements that use function notation in terms of a mathematical framework.

Analyze the difference between linear functions and nonlinear functions by informally analyzing the graphs of various parent functions (linear, quadratic, exponential, absolute value, square root, and cube root parent curves).

Construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a contextual situation for which the graph serves as a model.

Use function notation to build and evaluate quadratic functions for inputs in their domains and interpret statements that use function notation in terms of a given framework. (See the Mathematical Modeling Framework and Statistical Reasoning Framework for contextual connections.)

Identify the effect on the graph generated by a quadratic function when replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

Graph and analyze the key characteristics of quadratic functions including contextual situations.

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

Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it defines. Explain what the value describes in context.

Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales.

Estimate, calculate, and interpret the average rate of change of a quadratic function and make comparisons to the average rate of change of linear functions.

Write a function defined by a quadratic expression in different but equivalent forms to reveal and explain different properties of the function.

Compare characteristics of two functions each represented in a different way.

Construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena. Use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and nonlinear functions using parent graphs.

Investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots.

Rewrite algebraic and numeric expressions involving radicals.

Using numerical reasoning, show and explain that the sum or product of rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

Create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena.

Create and solve linear inequalities in two variables to represent relationships between quantities including mathematically applicable situations; graph inequalities on coordinate axes with labels and scales.

Represent constraints of linear inequalities and interpret data points as possible or not possible.

Solve systems of linear inequalities by graphing, including systems representing a mathematically applicable situation.

Create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations.

Interpret exponential expressions and parts of an exponential expression that represent a quantity in terms of its framework. (See the Mathematical Modeling Framework and Statistical Reasoning Framework for contextual connections.)

Create exponential equations in one variable and use them to solve problems, including mathematically applicable situations.

Create exponential equations in two variables to represent relationships between quantities, including in mathematically applicable situations; graph equations on coordinate axes with labels and scales.

Represent constraints by exponential equations and interpret data points as possible or not possible in a modeling environment.

Data & Statistical Reasoning

Numerical Reasoning

Patterning & Algebraic Reasoning

Geometric & Spatial Reasoning

Functional & Graphical Reasoning

Mathematical Modeling

Mathematical Practices

Collect, analyze, and interpret univariate quantitative data to answer statistical investigative questions that compare groups to solve real-life problems; Represent bivariate data on a scatter plot and fit a function to the data to answer statistical questions and solve real-life problems.

Use statistics appropriate to the shape of the data distribution to compare and represent center (median and mean) and variability (interquartile range, standard deviation) of two or more distributions by hand and using technology.

Interpret differences in shape, center, and variability of the distributions based on the investigation, accounting for possible effects of extreme data points (outliers).

Represent data on two quantitative variables on a scatter plot and describe how the variables are related.

Interpret the slope (predicted rate of change) and the intercept (constant term) of a linear model based on the investigation of the data.

Calculate the line of best fit and interpret the correlation coefficient, � r, of a linear fit using technology. Use � r to describe the strength of the goodness of fit of the regression. Use the linear function to make predictions and assess how reasonable the prediction is in context.

Decide which type of function is most appropriate by observing graphed data.

Distinguish between correlation and causation.

Construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena. Use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and non-linear functions using parent graphs.

Use mathematically applicable situations algebraically and graphically to build and interpret arithmetic sequences as functions whose domain is a subset of the integers.

Construct and interpret the graph of a linear function that models real-life phenomena and represent key characteristics of the graph using formal notation.

Relate the domain and range of a linear function to its graph and, where applicable, to the quantitative relationship it describes. Use formal interval and set notation to describe the domain and range of linear functions.

Use function notation to build and evaluate linear functions for inputs in their domains and interpret statements that use function notation in terms of a mathematical framework.

Analyze the difference between linear functions and nonlinear functions by informally analyzing the graphs of various parent functions (linear, quadratic, exponential, absolute value, square root, and cube root parent curves).

Construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a mathematically applicable situation for which the graph serves as a model.

Use function notation to build and evaluate quadratic functions for inputs in their domains and interpret statements that use function notation in terms of a given framework.

Identify the effect on the graph generated by a quadratic function when replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

Graph and analyze the key characteristics of quadratic functions.

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

Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it defines. Explain what the value describes in context.

Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales.

Estimate, calculate, and interpret the average rate of change of a quadratic function and make comparisons to the average rate of change of linear functions.

Write a function defined by a quadratic expression in different but equivalent forms to reveal and explain different properties of the function.

Compare characteristics of two functions each represented in a different way.

Construct and analyze the graph of an exponential function to explain a mathematically applicable situation for which the graph serves as a model; compare exponential with linear and quadratic functions.

Use function notation to build and evaluate exponential functions for inputs in their domains and interpret statements that use function notation in terms of a context.

Graph and analyze the key characteristics of simple exponential functions based on mathematically applicable situations.

Identify the effect on the graph generated by an exponential function when replacing f(x) with f(x) + k, and kf(x), for specific values of k (both positive and negative); find the value of k given the graphs.

Use mathematically applicable situations algebraically and graphically to build and interpret geometric sequences as functions whose domain is a subset of the integers.

Compare characteristics of two functions each represented in a different way.

Solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena

Solve real-life problems involving slope, parallel lines, perpendicular lines, area, and perimeter.

Apply the distance formula, midpoint formula, and slope of line segments to solve real-world problems.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain applicable, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities domains.

Use units of measure (linear, area, capacity, rates, and time) as a way to make sense of conceptual problems; identify, use, and record appropriate units of measure within the given framework, within data displays, and on graphs; convert units and rates using proportional reasoning given a conversion factor; use units within multi-step problems and formulas; interpret units of input and resulting units of output.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Define appropriate quantities for the purpose of descriptive modeling.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots.

Rewrite algebraic and numeric expressions involving radicals.

Using numerical reasoning, show and explain that the sum or product of rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

Create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena.

Create and solve linear inequalities in two variables to represent relationships between quantities including mathematically applicable situations; graph inequalities on coordinate axes with labels and scales.

Represent constraints of linear inequalities and interpret data points as possible or not possible.

Solve systems of linear inequalities by graphing, including systems representing a mathematically applicable situation.

Build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations.

Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context.

Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression.

Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena.

Represent constraints by quadratic equations and interpret data points as possible or not possible in a modeling framework.

Create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations.

Interpret exponential expressions and parts of an exponential expression that represent a quantity in terms of its framework.

Create exponential equations in one variable and use them to solve problems, including mathematically applicable situations.

Create exponential equations in two variables to represent relationships between quantities, including in mathematically applicable situations; graph equations on coordinate axes with labels and scales.

Represent constraints by exponential equations and interpret data points as possible or not possible in a modeling environment.

Patterning & Algebraic Reasoning

Geometric & Spatial Reasoning

Functional & Graphical Reasoning

Mathematical Modeling

Mathematical Practices

Apply limit notation and characteristics of continuity to analyze behaviors of functions.

Estimate limits from graphs and tables of values.

Find limits of sums, differences, products, and quotients using substitution.

Represent asymptotic behavior using limits.

Find limits of rational functions using algebraic techniques.

Demonstrate continuity at a point using the definition and limit notation.

Apply the Intermediate Value Theorem to a function over a closed interval.

Relate limits and continuity to the derivative as a rate of change and apply it to a variety of situations including modeling contexts.

Interpret the derivative as an instantaneous rate of change that is a two-sided limit of an average rate of change.

Demonstrate and apply the relationship between differentiability and continuity.

Apply the concept of derivative geometrically, numerically, and analytically.

Find the derivatives of sums, products, quotients, and composite functions.

Find the derivatives of a variety of relations.

Calculate higher order derivatives.

Apply derivatives to situations in order to draw conclusions including curve analysis and modeling rates of change in applications.

Calculate the slope of a curve at a point.

Write the equation of the tangent line to a curve at a point and use it to obtain a local linear approximation of a value near the point of tangency.

Identify intervals where functions are increasing, decreasing, and constant by using the relationship between the function and the sign of its first derivative.

Identify points of inflection and intervals of concavity of a function by using the second derivative of a function.

Compare characteristics of f, f’, and f” graphically, numerically, analytically, and with technology.

Apply Mean Value Theorem.

Apply Extreme Value Theorem.

Apply the derivative to real-world problems to find both local and absolute extrema, with and without technology.

Model rates of change in applied situations.

Analyze the relationship between the derivative and the integral using the Fundamental Theorem of Calculus.

Use Riemann sums to approximate values of definite integrals.

Find the exact value of a definite integral using geometric formulas on a coordinate plane.

Demonstrate the use of properties of definite integrals.

Apply the Fundamental Theorem of Calculus as an interpretation of the accumulation in the rate of change of a function as equivalent to the change in the antiderivative over the interval.

Apply Fundamental Theorem of Calculus to indefinite integrals to represent the family of antiderivatives.

Apply integration by substitution to definite and indefinite integrals.

Interpret a definite integral as a limit of Riemann sums.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Apply the definite integral and indefinite integral to contextual situations.

Find a particular curve in a family of antiderivatives using an initial condition.

Solve separable differential equations and use them to model real-world problems.

Apply definite integrals to find the area between two curves.

Apply definite integrals to find the average value of a function over a closed interval.

Data & Statistical Reasoning

Geometric & Spatial Reasoning

Functional & Graphical Reasoning

Patterning & Algebraic Reasoning

Numerical Reasoning

Mathematical Modeling

Mathematical Practices

Make sense of and reason about variation in data using graphs, tables and probability models to solve problems and draw appropriate conclusions from solutions.

Represent univariate data on the real number line.

Calculate the conditional probability of A given B.

Calculate, compare, and interpret shape, center, and spread of two or more univariate data sets, accounting for possible effects of extreme data points.

Summarize categorical data for two categories in two-way frequency tables using relative frequencies in the context of the data.

Represent bivariate data on a scatter plot and describe how the variables are related in terms of strength and direction.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

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

Distinguish between correlation and causation, and interpolation and extrapolation.

Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events.

Use the two-way frequency table to calculate conditional probabilities.

Define, build and interpret functions that arise in various contexts by applying knowledge of the characteristics of the different families of functions, and analyze the effects of parameters.

Define a function through maps, sets, equations and graphs using function notation.

Identify and sketch by hand the parent graph of functions expressed algebraically and show key characteristics of the graph using technology.

Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function.

Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

Compare characteristics of two functions each represented in a different way.

Construct linear and exponential functions, given a graph, a description of a relationship, or two input-output pairs.

Construct arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect linear functions to arithmetic sequences and exponential functions to geometric sequences.

Identify the effect on the parent graph of replacing f(x) by f(x) + k, kf(x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Reason deductively and inductively about figures and their properties and make sense of geometric situations using measurements in real-world contexts.

Use the distance formula, midpoint formula or slope to verify simple geometric properties.

Use coordinates to compute perimeters of polygons, circumference of circles and areas of triangles, rectangles and circles.

Informally derive the formulas for the volume and surface area of a cylinder, sphere, prism, pyramid, and cone.

Use formulas for finding the volume and surface area of spheres, right and oblique prisms, cylinders, pyramids, and cones.

Apply the Pythagorean Theorem and trigonometric ratios to solve problems involving right triangles.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration, and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Utilize exact and approximate calculations to quantify real-world phenomena and solve problems.

Through multi-step/multi-operational problems, perform mathematical operations on real numbers demonstrating fluency using the order of operations.

Represent and solve problems using proportional reasoning with ratios, rates, proportions, and scaling.

Apply the rules of exponents to simplify numerical expressions, extending the properties of exponents to rational exponents.

Perform mathematical operations on real numbers to include numerical radical expressions and complex fractions.

Estimate solutions to problems with real numbers and use the estimates to assess the reasonableness of results in the context of the problem.

Construct expressions, equations, and inequalities, and use them to represent and solve problems by choosing appropriate procedures and interpreting solutions in context.

Create equations in one variable and use them to solve problems.

Create inequalities in one variable and use them to solve problems.

Using multiple representations, solve equations and inequalities and use the solutions to draw reasonable conclusions about a situation being modeled, including possible constraints.

Solve quadratic equations using a variety of methods.

Rearrange literal equations to highlight a specified variable using the same reasoning as in solving equations.

Solve inequalities in one variable graphically and algebraically.

Using multiple methods, create and solve systems of linear equations and inequalities.

Solve a simple system of equations consisting of a linear and a quadratic equation in two variables. algebraically and graphically.

Abstract Reasoning

Mathematical Modeling

Mathematical Practices

Solve contextual, mathematical problems involving first-order differential equations to explain real-life phenomena.

Classify differential equations by order and linearity.

Solve separable differential equations for general solutions and initial value problems.

Solve first-order linear differential equations and initial value problems using integrating factors.

Use modeling or numerical methods to approximate solutions of first-order differential equations in context.

Draw direction fields containing solutions curves for first-order differential equations by hand and using modeling.

Solve first-order linear differential equations that apply to various real-world models including falling bodies, mixtures, population and the logistic equation, continuously compounded interest, and other physics applications.

Solve contextual, mathematical problems involving second and higher order differential equations to explain real-life phenomena.

Determine whether a first- or second-order differential equation has a unique solution over a given interval by applying the Existence and Uniqueness Theorem.

Determine ordinary points, recurrence relations, and change the index as they relate to series solutions to ordinary differential equations.

Find series solutions to first and second-order non-linear initial value problems.

Solve second-order linear homogeneous and non-homogeneous differential equations by finding characteristic equations, using the method of undetermined coefficients and variation of parameters.

Solve second-order differential equations that apply to various real-world models.

Use vector function notation when discussing the structure of solution sets for homogeneous systems as it pertains to the Wronskian.

Determine the existence and uniqueness of solutions for second-order linear differential equations, determine a fundamental set of solutions, and verify that two solutions form a fundamental set by taking the Wronskian.

Determine the structure of solution set to higher-order differential equations, apply the basic Existence and Uniqueness Theorem to higher-order differential equations, and use the generalizations of the Wronksian for higher order differential equations.

Solve higher-order constant coefficient homogeneous differential equations.

Solve special case non-homogeneous second order ordinary differential equations including Cauchy-Euler Equations.

Find a second linearly dependent solution using reduction of order when given a solution to a non-homogeneous second-order differential equation.

Solve contextual, mathematical problems involving systems of differential equations to explain real-life phenomena.

Determine whether a contextual situation results in a system of differential equations and apply the basic existence and uniqueness results for the corresponding initial value problems.

Solve constant coefficient homogeneous systems using eigenvalues and eigenvectors. Solve systems with real, distinct eigenvalues, as well as those with repeated and imaginary eigenvalues.

Draw phase portraits for solutions of homogeneous systems with constant coefficients.

Solve non-homogeneous systems of ordinary differential equations using the method of undetermined coefficients and variation of parameters.

Determine which non-linear systems are locally linear and identify the behavior of the system about each critical point.

Plot locally linear systems.

Use population models derived from locally linear systems.

Solve contextual, mathematical problems using Laplace transforms to explain real-life phenomena.

Use the integral definition to perform Laplace transforms for functions.

Use a Laplace table to accurately and efficiently identify Laplace transforms.

Perform inverse Laplace transforms using a variety of techniques.

Solve first- and second-order differential equations using Laplace transforms that apply to fields such as electrical and mechanical engineering.

Write piecewise functions as compositions of step (Heaviside) functions.

Find the general uniqueness and existence of solutions for step functions, and use Laplace transforms to find solutions to step functions.

Find the Laplace transform of the Dirac delta function.

Solve linear systems of differential equations using Laplace transforms.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Abstract Reasoning – Integration in Engineering

Abstract Reasoning – Partial Differentiation in Engineering

Abstract Reasoning – Algebra in Engineering

Abstract Reasoning – Impact of Engineering in Mathematics

Mathematical Modeling

Mathematical Practices

Using the engineering design process, apply mathematical concepts and procedures to solve problems in engineering contexts and research the impact of engineering and technological advancement on mathematics and society.

Build new mathematical knowledge through problem solving that involves the engineering design process.

Present a technical design, using computer-generated model, for an assigned design project utilizing the appropriate scientific units (US standards and SI units).

Use connections among mathematics, technology, and engineering in contextual situations.

Describe the issues of necessity that have influenced innovation and technological development.

Explain the impact of key persons and historical events and their impact on engineering and society.

Investigate the educational requirements and professional expectations associated with engineering career paths.

Solve problems that arise in mathematics and in engineering contexts.

Apply and adapt a variety of appropriate strategies to solve problems.

Use visual and written communication to organize, record, and articulate coherent, mathematical thinking and to express basic design elements.

Monitor and reflect on the process of mathematical problem solving and interpret solutions that arise in engineering contexts.

Produce multiple representations for mathematics presented in engineering contexts.

Select, apply, and translate among mathematical representations to solve problems that arise in engineering contexts.

Use mathematical representations to model and interpret physical and engineering phenomena.

Demonstrate fundamentals of technical sketching using computer-generated visuals in appropriate mathematical scaling.

Using the engineering design process, express spatial and functional relationships with vectors, functions, and analytic geometry in three dimensions, and use these relationships to solve real-life, mathematical problems and to explain engineering-based phenomena.

Determine the equations of lines and surfaces using vectors and 3D graphing.

Apply dot and cross products of vectors to express equations of planes, parallelism, perpendicularity, angles.

Describe the role of vectors in engineering applications, such as modeling the velocity of moving objects or static forces on structures and objects.

Evaluate matrices and apply their properties to solve problems expressed as matrix equations.

Compute limits of scalar and vector-valued functions.

Identify and graph level curves of multivariate functions.

Find the regions of continuity of multivariate functions.

Define, describe, and represent the differentiation of functions of two independent variables and differential vectors to solve contextual, mathematical problems and to explain engineering-based phenomena.

Compute the first and second partial derivatives of a function.

Use the general chain rule to determine the partial derivatives of composite functions.

Compute and apply the gradient of multivariable functions.

Solve engineering optimization problems by applying partial differentiation or Lagrange multipliers.

Utilize partial derivatives in developing the appropriate system balances in engineering problems.

Interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain engineering-based phenomena.

Manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates.

Evaluate and apply line integrals that are independent of path.

Apply properties of integrals to calculate and represent area, volume, or mass.

Use integrals of vectors to define and apply the gradient, divergence, and the curl.

Interpret the theorems of Green, Stokes, and Gauss and apply them to the study of real-world phenomena.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals. Code Expectation

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

PC Algebraic & Geometric Reasoning

PC Geometric & Spatial Reasoning

AA Geometric & Spatial Reasoning

PC Functional & Graphical Reasoning

PC Patterning & Algebraic Reasoning

AA Patterning & Algebraic Reasoning

AA Functional & Graphical Reasoning

AA Data & Statistical Reasoning

PC Mathematical Modeling

PC Mathematical Practices

Communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn.

When collecting and considering data, critically evaluate ethics, privacy, potential bias, and confounding variables along with their implications for interpretation in answering a statistical investigative question. Implement strategies for organizing and preparing big data sets.

Distinguish between population distributions, sample data distributions, and sampling distributions. Use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language.

Calculate and interpret z-scores as a measure of relative standing and as a method of standardizing units.

Given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology.

Model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population.

Given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability.

Summarize and evaluate reports based on data for appropriateness of study design, analysis methods, and statistical measures used.

Explore and analyze structures and patterns for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena.

Find the inverse of exponential and logarithmic functions using equations, tables, and graphs, limiting the domain of inverses where necessary to maintain functionality, and prove by composition or verify by inspection that one function is the inverse of another.

Analyze, graph, and compare exponential and logarithmic functions.

Use the definition of a logarithm, logarithmic properties, and the inverse relationship between exponential and logarithmic functions to solve problems in context.

Create exponential equations and use logarithms to solve mathematical, applicable problems for which only one variable is unknown.

Create and interpret logarithmic equations in one variable and use them to solve problems.

Create, interpret, and solve exponential equations to represent relationships between quantities and analyze the relationships numerically with tables, algebraically, and graphically.

Create, interpret, and solve logarithmic equations in two or more variables to represent relationships between quantities.

Explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena.

Rewrite radical expressions as expressions with rational exponents. Extend the properties of integer exponents to rational exponents.

Solve radical equations in one variable, and give examples showing how extraneous solutions may arise.

Analyze and graph radical functions.

Create, interpret and solve radical equations with one unknown value and use them to solve problems that model real-world situations.

Create, interpret, and solve radical equations in two or more variables to represent relationships between quantities.

Extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-world phenomena.

Graph and analyze quadratic functions in contextual situations and include analysis of data sets with regressions.

Use the structure of an expression to factor polynomials, including the sum of cubes, the difference of cubes, and higher-order polynomials that may be expressed as a quadratic within a quadratic.

Using all the zeros of a polynomial function, list all the factors and multiply to write a multiple of the polynomial function in standard form.

Define complex numbers i such that i^2 = –1 and show that every complex number has the form a + bi where a and b are real numbers and that the complex conjugate is a − bi.

Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use the structure of an expression to factor quadratics.

Write and solve quadratic equations and inequalities with real coefficients and use the solution to explain a mathematical, applicable situation.

Solve systems of quadratic and linear functions to determine points of intersection.

Create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations.

Identify the number of zeros that exist for any polynomial based upon the greatest degree of the polynomial and the end behavior of the polynomial by observing the sign of the leading coefficient.

Identify zeros of polynomial functions using technology or pre-factored polynomials and use the zeros to construct a graph of the function defined by the polynomial function. Analyze identify key features of these polynomial functions.

Analyze the behaviors of rational functions to model applicable, mathematical problems.

Rewrite simple rational expressions in equivalent forms.

Add, subtract, multiply and divide rational expressions, including problems in context and express rational expressions in irreducible form.

Graph rational functions, identifying key characteristics.

Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

Develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle.

Define the three basic trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane.

Apply understanding of the angle measures and coordinates of the unit circle to solve practical, real-life problems involving trigonometric equations.

Represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications.

Use matrices to represent data, and perform mathematical operations with matrices and scalars, demonstrating that some properties of real numbers hold for matrices, but that others do not.

Rewrite a system of linear equations using a matrix representation.

Use the inverse of an invertible matrix to solve systems of linear equations.

Utilize linear programming to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as solutions or non-solutions under the established constraints in real-world problems.

Manipulate, prove, and apply trigonometric identities and equations to solve contextual mathematical problems.

Apply the fundamental trigonometric identities to simplify expressions and verify other identities.

Use sum, difference, double-angle, and half-angle formulas for sine, cosine, and tangent to establish other identities and apply them to solve problems.

Solve trigonometric equations arising in modeling contexts.

Prove and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Determine the area of an oblique triangle.

Represent and model vector quantities to solve problems in contextual situations.

Represent vector quantities as directed line segments; represent magnitude and direction of vectors in component form using appropriate mathematical notation.

Add and subtract vectors and multiply vectors by a scalar to find the resultant vector.

Add and subtract vectors on a coordinate plane using different methods.

Solve contextual vector problems, such as those involving velocity, force, and other quantities.

Sketch the graph of a curve represented parametrically, indicating the direction of motion.

Apply parametric equations to contextual problems.

Analyze the behaviors of rational and piecewise functions to model contextual mathematical problems.

Graph piecewise-defined functions, including step functions and absolute value functions.

Describe characteristics by interpreting the algebraic form and graph of a piecewise-defined function.

Represent the limit of a function using both the informal definition and the graphical interpretation in the context of piecewise-defined functions; interpret limits expressed in analytic notation.

Divide polynomials using various methods.

Graph rational functions and identify key characteristics.

Represent the behavior of a rational function using limit notation for vertical and horizontal asymptotes and end behavior.

Represent the limit of a function using both the informal definition and the graphical interpretation in the context of rational functions; interpret limits expressed in analytic notation.

Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

Perform partial fraction decomposition of rational functions using non-repeated linear factors.

Utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions.

Use the concept of a radian as the ratio of the arc length to the radius of a circle to establish the existence of 2π radians in one revolution.

Utilize right triangles on the unit circle to determine the values of the six trigonometric ratios for π/6​ , π/4​ , and π/3​ . Use reflections of the triangles as reference angles to establish known values in all four quadrants of the coordinate plane.

Define the six trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane. Interpret radian measures of angles as a rotation both counterclockwise and clockwise around the unit circle.

Derive the fundamental trigonometric identities.

Determine the value(s) of trigonometric functions for a set of given conditions.

Graph and write equations of trigonometric functions using period, phase shift, and amplitude in modeling contexts.

Classify the six trigonometric functions as even or odd and describe the symmetry.

Restrict the domain of a trigonometric function to create an invertible function and graph the inverse function. Evaluate inverse trigonometric expressions.

Analyze the behaviors of conic sections and polar equations to model contextual mathematical problems.

Identify and graph different conic sections given the equations in standard form.

Identify different conic sections in general form and complete the square to convert the equation of a conic section into standard form.

Define polar coordinates and relate polar coordinates to Cartesian coordinates.

Classify special polar equations and apply to contextual situations.

Graph equations in the polar coordinate plane with and without the use of technology.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Demonstrate how sequences and series apply to mathematical models in real-life situations.

Demonstrate that sequences are functions whose domain is the set of natural numbers.

Represent sequences graphically, numerically, and symbolically.

Determine the limit of a sequence if it exists.

Demonstrate that a series is the sum of the sequence and represent series graphically, numerically, and symbolically.

Describe the behavior of a series in terms of the limit of its partial sums.

Derive and use the sum formula of a finite geometric series to solve contextual problems to model real-life situations.

Derive and use the sum formula of an infinite geometric series to solve contextual problems to model real-life situations.

Data & Statistical Reasoning

Probabilistic Reasoning

Geometric & Spatial Reasoning

Patterning & Algebraic Reasoning

Mathematical Modeling

Mathematical Practices

Examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena.

Construct and summarize categorical data for two categories in two-way frequency tables.

Use categorical data in two-way frequency tables to calculate and interpret probabilities based on the investigation.

Experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena.

Use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations.

Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines.

Use geometric descriptions of rigid motions to draw the transformed figures and to predict the effect on a given figure. Describe a sequence of transformations from one figure to another and use transformation properties to determine congruence.

Explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. Use congruency criteria for triangles to solve problems and to prove relationships in geometric figures.

Establish facts between angle relations and generate valid arguments to defend facts established. Prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena.

Use the undefined notions of point, line, line segment, plane, distance along a line segment, and distance around a circular arc to develop and use precise definitions and symbolic notations to prove theorems and solve geometric problems.

Classify quadrilaterals in the coordinate plane by proving simple geometric theorems algebraically.

Make formal geometric constructions with a variety of tools and methods.

Prove and apply theorems about lines and angles to solve problems.

Use geometric reasoning to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Describe dilations in terms of center and scale factor and use these terms to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena.

Verify experimentally the properties of dilations.

Given two figures, use and apply the definition of similarity in terms of similarity transformations.

Use the properties of similarity transformations to establish criterion for two triangles to be similar. Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Construct formal proofs to justify and apply theorems about triangles.

Examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine and cosine to solve geometric problems and to model and explain real-life phenomena.

Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Use trigonometric ratios and the Pythagorean Theorem to solve for sides and angles of right triangles in applied problems.

Explore the concept of a radian measure and special right triangles.

Explore and interpret a radian as the ratio of the arc length to the radius of a circle.

Explore and explain the relationship between radian measures and degree measures and convert fluently between degree and radian measures.

Use special right triangles on the unit circle to determine the values of sine, cosine, and tangent for 30° (π/6​ ), 45° (π/4​ ) and 60° (π/3​ ) angle measures. Use reflections of triangles to determine reference angles and identify coordinate values in all four quadrants of the coordinate plane.

Examine and apply theorems involving circles; describe and derive arc length and area of a sector; and model and explain real-life situations involving circles.

Identify and apply angle relationships formed by chords, tangents, secants and radii with circles.

Using similarity, derive the fact that the length of the arc (arc length) intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. Solve mathematically applicable problems involving applications of arc length and area of sector.

Write and graph the equation of circles in standard form.

Develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri’s principle; solve realistic problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena.

Use volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems including right and oblique solids.

Use geometric shapes, their measures, and their properties to describe objects and approximate volumes.

Apply concepts of density based on area and volume in modeling situations.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain mathematically applicable problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a mathematically applicable situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Interpret the structure of polynomial expressions and perform operations with polynomials within a geometric framework.

Interpret polynomial expressions of varying degrees that represent a quantity in terms of its given geometric framework.

Perform operations with polynomials and prove that polynomials form a system analogous to the integers in that they are closed under these operations.

Using algebraic reasoning, add, subtract, and multiply single variable polynomials.

Solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions.

Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events. Apply the Addition Rule conceptually, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answers in context.

Apply and interpret the general Multiplication Rule conceptually to independent events of a sample space, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)] using contingency tables or tree diagrams.

Use conditional probability to interpret risk in terms of decision-making and investigate questions such as those involving false positives or false negatives from screening tests.

Define permutations and combinations and apply this understanding to compute probabilities of compound events and solve meaningful problems.

Interpret the probability distribution for a given random variable and interpret the expected value.

Develop a probability distribution for variables of interest using theoretical and empirical (observed) probabilities and calculate and interpret the expected value.

Calculate the expected value of a random variable and interpret it as the mean of a given probability distribution.

Compare the payoff values associated with the probability distribution for a random variable and make informed decisions based on expected value and measures of variability.

Probability

Probability Reasoning

Geometric Measurement

Shapes & Properties

Geometric & Spatial Reasoning

Function Families

Functional & Graphical Reasoning

Graphing

Ratios & Rates

Variable Equations & Inequalities

Expressions

Patterns

Patterning & Algebraic Reasoning

Comparisons

Computational Fluency

Numbers (rational numbers and irrational numbers)

Numerical Reasoning

5-12 Learning Progressions

All rational numbers

Operations with radicals

All numbers in The Real Number System

Operations with real numbers (rational and irrational)

Multiplication of irrational numbers

Rate of change (slope)

Intercept

Distributions of two or more data sets

Arithmetic sequences

Geometric sequences

Exponential expressions

Quadratic expressions

Expressions of varying degrees

Add, subtract, multiply single variable polynomials

Adding, Subtracting and Multiplying Polynomials

Factoring and expanding polynomials

Exponential equations

Quadratic equations

Equations of parallel and perpendicular lines

Analyze and solve linear inequalities

Equations involving geometric measurement

Convert units and rates given a conversion factor

Side ratios of similar triangles

Trigonometric ratios

Linear functions with function notation

Exponential functions

Quadratic functions

Systems of linear inequalities

Equations of circles in standard form

Linear functions with function notation

Parent graphs of function families

Exponential functions

Quadratic functions

Function notation to represent transformations

Develop and use precise definitions to prove theorems and solve geometric problems

Formal proofs & theorems about triangles

Trigonometric ratios (Sin, Cos, & Tan)

Prove slope criteria for parallel and perpendicular lines

Transform polygons using rotations, reflections, dilations, and translations.

Congruence and trans-formations

Triangle congruence

Use congruence to prove relationships in geometric figures

Similarity and dilations

Similar triangles

Use similarity to prove relationships in geometric figures

Use distance formula, midpoint formula, and slope to calculate perimeter and area of triangles and quadrilaterals

Volumes of prisms, cones, cylinders, pyramids, and spheres

Approximate volumes of irregular objects

Approximate density of irregular objects

Categorical data & two-way frequency tables

Interpret probabilities in context

Logical, Mathematical & Investigative Reasoning – 20TH Century Mathematics

Logical, Mathematical & Investigative Reasoning – Modern Mathematics

Logical, Mathematical & Investigative Reasoning – Mathematics of the Classical Era

Logical, Mathematical & Investigative Reasoning – Mathematics in the Middles Ages

Logical, Mathematical & Investigative Reasoning – Ancient Greek Mathematics

Numerical Reasoning – Origins of Mathematics

Mathematical Modeling

Mathematical Practices

Engage in the mathematical and cultural accomplishments of the ancient Greeks in order to grasp the foundational aspects of modern mathematics.

Prove statements in a deductive system by using its definitions, postulates, and axioms

Describe the cultural aspects of Greek society that influenced the way mathematics developed in ancient Greece.

Describe the theories for the rise of intellectual thought in ancient Greece and the factors involved in its collapse.

Analyze factors involved in the rise and fall of ancient Greek society.

Prove the first five propositions in Book I of Euclid's <em>Elements</em>.

Construct a regular pentagon with a straight-edge and compass.

Compute the areas of regular polygons by Heron's formulas.

Translate Greek geometric algebra into modern algebraic notation.

Find the first four perfect numbers using Euclid's formula.

Justify statements concerning figurate numbers using both graphical (as in the manner of the Greeks) and algebraic methods.

Solve systems of linear and nonlinear equations using Diophantus' method.

Explain the distinction made between number and magnitude, commensurable and incommensurable, and arithmetic and logistic, the cultural factors inherent in this distinction, and the logical crisis that occurred concerning incommensurable (irrational) magnitudes.

Engage in the mathematical and cultural accomplishments of the world's societies in the 5th century through the 15th century in order to grasp the foundational aspects of modern mathematics.

Translate medieval mathematical problems that involve linear, quadratic, or cubic equations into modern notation and solve them in a variety of ways.

Use historical multiplication and division algorithms.

Use Khayyam's geometric construction to find a solution to a cubic equation.

Identify cyclic quadrilaterals and find associated lengths by Ptolemy's Theorem.

Investigate the relationships among the sides and angles of a spherical triangle.

Describe the algebraic and geometric contributions of Islamic mathematicians in the Middle Ages.

Describe the algebraic and geometric contributions of Chinese mathematicians in the Middle Ages.

Describe the transition of Hindu-Arabic numerals from regional use in the 10th century to wide-spread use in the 15th century.

Describe the transmission of ideas from the Greeks, through the Islamic peoples, to medieval Europe.

Describe the influence of the Catholic Church and Charlemagne on the establishment of mathematics as one of the central pillars of education.

Engage in the mathematical accomplishments of Europe in the 15th century through the early 17th century in order to grasp the foundational aspects of modern mathematics.

Use historical multiplication and division algorithms.

Use Cardano's cubic formula to find a solution to a cubic equation.

Explain the cultural factors that encouraged the development of algebra in 15th century Italy, and how this development influenced mathematical thought throughout Europe.

Identify the works of Galileo, Copernicus, and Kepler as a landmark in scientific thought, describe the conflict between their explanation of the workings of the solar system and then-current perspectives, and contrast their works to those of Aristotle.

Describe the mathematical contributions of Fermat, Pascal, and Descartes.

Engage in the mathematical and cultural accomplishments of the world's societies in the late 17th century through the early 20th century in order to grasp the foundational aspects of modern mathematics.

Determine tangents to quadratic curves using the algebraic techniques of Fermat, Barrow, and Newton.

Explain how the ancient Greek pattern of material axiomatics evolved into abstract axiomatics.

Solve simple linear congruences of the form ax = b mod m.

Use Fermat's Little Theorem and Euler's Theorem to simplify expressions of the form a^k mod m.

Use Gauss' Law of Quadratic Reciprocity to determine quadratic residues of two odd primes; i.e., solve quadratic congruences of the form x² = p mod q.

Verify that the real primes which can be expressed as the sum of two squares are no longer prime in the field of Gaussian integers.

Describe the mathematical contributions of Newton, Euler, and Gauss.

Explore the history of African American mathematicians in the 17th, 18th, and 19th centuries and describe their contributions to mathematics.

Explore the history of female mathematicians in the 17th, 18th, and 19th centuries and describe their contributions to mathematics.

Describe the influence the French Revolution had on mathematics education.

Prove that the summit angles of an isosceles birectangle are congruent, but that it is impossible to prove they are right without referring to the parallel postulate or one of its consequences.

Compare and contrast the hypotheses of the acute angle (Hyperbolic), the right angle (Euclidean), and the obtuse angle (Spherical).

Prove that under the hypothesis of the acute angle, similarity implies congruence.

Describe the societal factors that inhibited the development of non-Euclidean geometry.

Add, subtract, and multiply two quaternions.

Investigate abstract algebra and group-theoretic concepts.

Identify whether a given set with a binary operation is a group.

Investigate and describe modern mathematicians and their contributions to mathematics.

Investigate the implications of infinite sets of real numbers.

Compare and contrast denumerable and nondenumerable sets.

Identify algebraic and transcendental numbers.

Describe the mathematical contributions of Cantor.

Describe the implications of Klein's Erlangen Programme and Gödel's Incompleteness Theorem on the nature of mathematical discovery and proof.

Explore the history of 20th century African American mathematicians and describe their contributions to mathematics.

Explore the history of 20th century female mathematicians and describe their contributions to mathematics.

Explore the history of 20th century Indian, Asian, Hispanic, Latin American mathematicians and describe their contributions to mathematics.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals. Code Expectation

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Explore and use historical number systems and computational methods.

Use historical number systems to represent quantities.

Use historical multiplication and division algorithms.

Decompose fractions of the form 2/(pq) using the Egyptian method as recorded by Ahmes (Ahmose) in the Rhind Papyrus.

Compute lengths, areas, and volumes according to historical formulas

Describe the limitations of the Babylonian, Roman, Egyptian (hieratic and hieroglyphic), Chinese, and Greek number systems as compared to Hindu-Arabic numerals

Identify the number system and notation used by a society as an influence on the types of mathematics developed by that society.

Solve linear equations using the method of false position.

Translate ancient mathematical problems that involve linear, quadratic, or cubic equations into modern notation and solve them in a variety of ways.

Probabilistic Reasoning

Patterning & Algebraic Reasoning

Geometric & Spatial Reasoning

Abstract & Digital Reasoning

Mathematical Modeling

Mathematical Practices

Investigate and describe real-life problems in linear algebra using an object-oriented programming language.

Utilize sets, lists, dictionaries, indexing, and tuples in programming languages.

Show and explain how to program and apply modules and control statements in programming languages.

Program input and output features to read from and write to files in a programming assignment.

Solve contextual, mathematical problems involving vectors to explain real-life phenomena.

Use coordinates to represent points in n dimensions and define and use arithmetic operations on n-dimensional points.

Use vector operations to program simple authentication schemes.

Use vectors to find and interpret geometrical relationships between points in two and three dimensions, such as distance, and generalize these relationships to higher dimensions using n-dimensional vectors.

Interpret adding, scaling, and linear combinations of vectors geometrically and algebraically.

Find and use the dot product of two n-dimensional vectors.

Use properties of the dot product to prove statements about vectors and to solve problems in context.

Use the triangle inequality in n-dimensions.

Find and use the cross product of two 3-dimensional vectors.

Represent and perform vector operations using programming language classes that define the use of vectors.

Apply perfect secrecy, all-or-nothing secret sharing, and solving lights out games to vectors over GF(2).

Solve contextual, mathematical problems involving matrices as geometric transformations and to explain real-life phenomena.

Given a 2-by-2 or 3-by-3 linear transformation matrix, describe the transformation a geometric figure undergoes.

Find matrices that represent scalings, reflections, and rotations of geometric figures.

Find a matrix that represents a combination of transformations.

Find the image of a point under a transformation.

Find the area of a polygon given its coordinates using matrices; find the area of the image of a polygon after a transformation.

Write code to perform transformations in two-dimensional geometry using matrix operations.

Define functions from n dimensions to m dimensions as vectors and/or matrices.

Find the image and preimage of a linear map using matrices; determine whether the linear map is one-to-one.

Find and interpret geometrically the set of preimages of a vector under a given matrix.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Solve contextual, mathematical problems involving matrices to explain real-life phenomena.

Represent a linear system of three equations in three variables as an augmented matrix and reduce the matrix to row-echelon form.

Solve a matrix equation using inverses; find all solutions to a matrix equation given one solution and the kernel.

Improve the simple authentication scheme over GF(2).

Show and explain how threshold secret sharing works in conjunction with Gaussian elimination through programming.

Write code utilizing error-correcting concepts.

Interpret the nature of the solution of a system from its row-echelon form, and if there are infinitely many solutions, express them as a vector equation.

Determine whether a vector is a linear combination of other given vectors; find the linear combination of vectors that results in a given vector.

Interpret linear dependence of vectors geometrically.

Find the kernel of a matrix and explore the relationship between the kernel, the orthogonality of the vectors in the kernel, and the linear dependence of the rows/columns.

Add two matrices, multiply a matrix by a scalar, find the transpose of a matrix.

Determine when matrix multiplication is defined, and if defined, multiply two matrices by considering the matrix product as a dot product of a group of vectors.

Determine when the inverse of a square matrix exists, and if it exists, find it by augmenting the identity matrix to the matrix and then use row operations.

Decompose a matrix into its symmetric and skew-symmetric parts; decompose a matrix into its LU factorization.

Solve contextual, mathematical problems using vector spaces to explain real-life phenomena.

Determine whether a given set of vectors generates a vector space.

Find an orthogonal basis for a given basis or subspace by applying the Gram-Schmidt orthonormalization process.

Perform QR factorization of a matrix to solve matrix equations.

Apply the method of least squares to find the line or parabola of best fit to approximate data in context.

Apply the grow-and-shrink algorithm in the minimum spanning forest problem in GF(2).

Apply the Exchange Lemma to image perspective rendering.

Use bases to represent images and sounds as wavelets; perform wavelet transformation, implementation, and decomposition through programming.

Program a Fast Fourier Transform to store a sequence of amplitude samples.

Apply the Rank Theorem to demonstrate the simple authentication scheme.

Justify whether a subset of a vector space is a subspace.

Determine whether a given vector is in the linear span of a set of vectors.

Determine whether two vector subspaces are orthogonal; find the orthogonal component of a given subspace.

Determine whether a set of vectors is a basis for a vector space.

Find the dimension of a vector space; find the dimensions of the row space, column space, and kernel for a given matrix; find the rank of a matrix.

Find a matrix representing a linear map.

Determine the change of representation for a linear transformation given two different bases on a vector space.

Determine if two matrices are similar; determine if two matrices are orthogonal.

Solve contextual, mathematical problems using eigenvalues and eigenvectors to explain real-life phenomena.

Evaluate the determinant of a matrix along any row or column and use a recursive procedure for evaluating a determinant for matrices larger than 3-by-3.

Find the dimension of the eigenspace corresponding to the eigenvalues of a symmetric matrix.

Determine an orthogonal matrix that diagonalizes a given matrix.

Apply eigenvalues and eigenvectors to problems in context.

Justify properties of the determinant.

Calculate the determinant of the product of two matrices; calculate the determinant of the transpose of a matrix.

Determine if a matrix has a nonzero determinant and extend the nonzero determinant property to problems involving linear dependency, rank, and matrix inverses.

Extend the definition and geometric interpretation of the cross product to n – 1 vectors in n dimensions.

Use Cramer’s Rule to solve a system of linear equations.

Find the characteristic polynomial of a matrix and interpret the characteristic polynomial geometrically.

Find the eigenvalues and eigenvectors of a matrix and interpret them geometrically.

Use a basis of eigenvectors to create a change of basis matrix.

Using probabilistic and quantitative reasoning, solve contextual, mathematical problems using Markov chains to explain real-life phenomena.

Model a finite random process using transition matrices in a Markov chain.

Simulate the different stages of a Markov chain using random numbers.

Use matrix algebra to calculate the probability of future states of a Markov chain.

Determine the attractor for a regular Markov chain.

Use transition matrices to identify absorbing states of a Markov chain.

Apply Markov chains in context.

Write a program to model the probabilities of real-life phenomena using a Markov chain.

Abstract Reasoning & Probabilistic Decision-Making – Fair Representation

Abstract Reasoning & Probabilistic Decision-Making – Simulations

Probabilistic Reasoning – Probabilistic Models

Abstract Reasoning & Probabilistic Decision-Making – Binomial, Geometric, and Poisson Distributions

Abstract Reasoning & Probabilistic Decision-Making – Normal Distributions

Abstract Reasoning & Deterministic Decision-Making – Optimal Paths

Abstract Reasoning & Deterministic Decision-Making – Optimal Locations

Abstract Reasoning & Deterministic Decision-Making – Linear Programming

Mathematical Modeling

Mathematical Practices

Solve contextual, mathematical problems involving linear programming and use the mathematics as a model to make decisions about real life phenomena.

Use advanced linear programming to make decisions and interpret results in real-life contexts.

Distinguish among continuous, integer, and binary contexts

Model and interpret results of a contextual problem with three or more variables using linear programming.

Solve problems with three or more variables using technology and principles of linear programming.

Examine cause and effect of contextual changes.

Solve contextual, mathematical problems involving optimal locations and use the mathematics as a model to make decisions about real life phenomena.

Find the optimal median location in a one-dimensional context.

Find the optimal median location in a rectilinear context.

Find the optimal location given three equally weighted, noncollinear points

Find the optimal location in a set covering context.

Solve contextual, mathematical problems involving optimal paths and use the mathematics as a model to make decisions about real life phenomena.

Relate context to a network representation.

Apply appropriate recursive algorithms.

Examine alternate decisions in response to contextual changes.

Solve contextual, mathematical problems with normal distributions to make appropriate decisions.

Use properties of normal distributions to make decisions about optimization and efficiency.

Calculate, analyze and interpret theoretical and empirical probabilities using standardized and non-standardized data.

Consider contextual factors and investigate issues within the decision-making process.

Apply techniques to quality control settings.

Solve contextual, mathematical problems using other distributions (e.g., binomial, geometric, and Poisson) as well as simulations to make appropriate decisions.

Calculate theoretical and empirical probabilities using standardized and non-standardized data.

Analyze and interpret the probabilities in terms of context.

Consider contextual factors and investigate issues within the decision-making process.

Use simulations to make appropriate decisions.

Use technology to simulate a real-world situation.

Analyze, evaluate, and interpret results of simulations.

Examine alternate decisions in response to contextual changes of simulations.

Using quantitative reasoning, determine fair methods to reflect the wishes of a larger population with representatives.

Develop and analyze fair methods for voting.

Develop and analyze fair methods for apportioning representatives.

Develop fair methods for setting voting district boundaries.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Use probabilistic models to make appropriate decisions.

Use program evaluation review technique (PERT) to investigate completion times of a project.

Develop and apply transition matrices to make predictions using Markov Chains.

Apply queuing theory

Consider contextual factors and investigate issues within the decision-making process.

Abstract & Quantitative Reasoning

Patterning & Algebraic Reasoning

Mathematical Modeling

Mathematical Practices

Find the partial derivatives of the composition of functions using the general chain rule.

Define, describe, and represent the differentiation of functions of two independent variables and differential vectors to solve contextual, mathematical problems and to explain real-life phenomena.

Approximate the partial derivatives at a point of a function defined by a table of data.

Find expressions for the first and second partial derivatives of a function.

Use the total differential to approximate mathematical models.

Represent the partial derivatives of a system of two functions in two variables using the Jacobian.

Apply partial differentiation to problems of optimization, including problems requiring the use of the Lagrange multiplier.

Find the family of solutions and the envelope of the family of solutions to differential equations, including Clairaut equations.

Define and apply the gradient, the divergence, and the curl in terms of differential vector operations.

Interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain real-life phenomena.

Integrate functions of the form z = f(x, y) or w = f(x, y, z) through various techniques.

Use, evaluate, and interpret double and triple integrals in terms of volume and mass.

Represent and evaluate integrals of vector functions as double and triple integrals.

Apply line and surface integral to functions representing real-world phenomena.

Solve first-order exact differential equations.

Use Green’s Theorem to evaluate line integrals in the plane; use Stokes’ Theorem to evaluate line integrals in space.

Determine whether a line integral is independent of path and use line integrals in context.

Use Gauss’ Divergence Theorem to evaluate surface integrals.

Define and apply the gradient, the divergence, and the curl in terms of integrals of vector functions.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Express spatial and functional relationships with vectors, functions, and analytic geometry in three dimensions, and use these relationships to solve contextual, mathematical problems.

Represent equations of lines in space using vectors.

Express the analytic geometry of three dimensions in terms of the dot product and cross product of vectors.

Use a linear system of equations to determine whether two planes intersect in a single point or a line, or whether they do not intersect at all.

Evaluate functions of two independent variables at a point in the plane.

Graph the level curves of functions of two independent variables.

Investigate the continuity of functions of two independent variables in terms of the limits of such functions as (x, y) approaches a given point in the plane.

Determine points or regions of discontinuity of functions of two independent variables.

Patterning & Algebraic Reasoning

Geometric & Spatial Reasoning

Algebraic & Geometric Reasoning

Functional & Graphical Reasoning

Mathematical Modeling

Mathematical Practices

Manipulate, prove, and apply trigonometric identities and equations to solve contextual mathematical problems.

Apply the fundamental trigonometric identities to simplify expressions and verify other identities.

Use sum, difference, double-angle, and half-angle formulas for sine, cosine, and tangent to establish other identities and apply them to solve problems.

Solve trigonometric equations arising in modeling contexts.

Prove and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Determine the area of an oblique triangle.

Represent and model vector quantities to solve problems in contextual situations.

Represent vector quantities as directed line segments; represent magnitude and direction of vectors in component form using appropriate mathematical notation.

Add and subtract vectors and multiply vectors by a scalar to find the resultant vector.

Add and subtract vectors on a coordinate plane using different methods.

Solve contextual vector problems, such as those involving velocity, force, and other quantities.

Sketch the graph of a curve represented parametrically, indicating the direction of motion.

Apply parametric equations to contextual problems.

Analyze the behaviors of rational and piecewise functions to model contextual mathematical problems.

Graph piecewise-defined functions, including step functions and absolute value functions.

Describe characteristics by interpreting the algebraic form and graph of a piecewise-defined function.

Represent the limit of a function using both the informal definition and the graphical interpretation in the context of piecewise-defined functions; interpret limits expressed in analytic notation.

Divide polynomials using various methods.

Graph rational functions and identify key characteristics.

Represent the behavior of a rational function using limit notation for vertical and horizontal asymptotes and end behavior.

Represent the limit of a function using both the informal definition and the graphical interpretation in the context of rational functions; interpret limits expressed in analytic notation.

Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

Perform partial fraction decomposition of rational functions using non-repeated linear factors.

Utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions.

Use the concept of a radian as the ratio of the arc length to the radius of a circle to establish the existence of 2π radians in one revolution.

Utilize right triangles on the unit circle to determine the values of the six trigonometric ratios for π/6​ , π/4​ , and π/3​ . Use reflections of the triangles as reference angles to establish known values in all four quadrants of the coordinate plane.

Define the six trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane. Interpret radian measures of angles as a rotation both counterclockwise and clockwise around the unit circle.

Derive the fundamental trigonometric identities.

Determine the value(s) of trigonometric functions for a set of given conditions.

Graph and write equations of trigonometric functions using period, phase shift, and amplitude in modeling contexts.

Classify the six trigonometric functions as even or odd and describe the symmetry.

Restrict the domain of a trigonometric function to create an invertible function and graph the inverse function. Evaluate inverse trigonometric expressions.

Analyze the behaviors of conic sections and polar equations to model contextual mathematical problems.

Identify and graph different conic sections given the equations in standard form.

Identify different conic sections in general form and complete the square to convert the equation of a conic section into standard form.

Define polar coordinates and relate polar coordinates to Cartesian coordinates.

Classify special polar equations and apply to contextual situations.

Graph equations in the polar coordinate plane with and without the use of technology.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Demonstrate how sequences and series apply to mathematical models in real-life situations.

Demonstrate that sequences are functions whose domain is the set of natural numbers.

Represent sequences graphically, numerically, and symbolically.

Determine the limit of a sequence if it exists.

Demonstrate that a series is the sum of the sequence and represent series graphically, numerically, and symbolically.

Describe the behavior of a series in terms of the limit of its partial sums.

Derive and use the sum formula of a finite geometric series to solve contextual problems to model real-life situations.

Derive and use the sum formula of an infinite geometric series to solve contextual problems to model real-life situations.

Data & Statistical Reasoning

Mathematical Modeling

Mathematical Practices

Formulate statistical investigative questions of interest to students that can be answered with data.

Formulate statistical investigative questions about a population using samples taken from the population.

Formulate comparative and associative statistical investigative questions for surveys, observational studies, and experiments to compare two or more groups or to investigate the association of two or more variables.

Formulate multivariable statistical investigative questions.

Formulate inferential statistical investigative questions regarding association and prediction.

Collect data by designing and implementing a plan to address the formulated statistical investigative question.

Apply an appropriate data-collection plan when collecting primary or secondary data for the statistical investigative question of interest.

Distinguish between surveys, observational studies, and experiments.

Design sample surveys, experiments, and observational studies using accepted practices.

Distinguish between random selection and random assignment and identify their impact on conclusions.

Describe potential sources and effects of bias and confounding variables.

Describe and adhere to the ethical use of data (e.g., sensitive information, privacy, and living subjects).

Identify when data can be generalized to a target population.

Analyze data by selecting and using appropriate graphical and numerical methods.

Summarize quantitative or categorical data using tables, graphical displays, and numerical summary statistics.

Summarize and describe relationships among multiple variables.

Use sampling distributions developed through simulation to describe the sample-to-sample variability of sample statistics.

Use sampling distributions to compute simulated p-values.

Describe the relationship between two quantitative variables by interpreting correlation (r) and a least-square regression line (using technology).

Use simulations to investigate associations between two categorical variables and to compare groups.

Interpret the results of the analysis, making connections to the formulated statistical investigative question.

Use statistical evidence from analyses to answer the formulated statistical investigative questions.

Interpret the impact of outliers, missing values, or erroneous values on the results.

Use and interpret the p-value to determine whether the estimate for a population characteristic is plausible.

Interpret a given margin of error associated with an estimate of a population characteristic.

Explain the impact of multiple variables on one another.

Apply mathematics to real-life situations; model real-life phenomena using mathematics.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or the humanities.

Using abstract and quantitative reasoning, make decisions about information and data from a real-life situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Display perseverance and patience in problem-solving. Demonstrate skills and strategies needed to succeed in mathematics, including critical thinking, reasoning, and effective collaboration and expression. Seek help and apply feedback. Set and monitor goals.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.