High School Math Arkansas Standards

Interpret key features of graphs and tables in terms of two quantities, which extend to function families beyond linear and quadratic, that model a relationship between the quantities in a contextual application and/or student-generated data.

Interpret the parameters of functions beyond the level of linear and quadratic in terms of a given context.

Graph functions expressed symbolically and show key features of the graph using technology.

Functions include exponential, logarithmic, and trigonometric functions.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, with or without the appropriate technology. Contextual situations may include:

Cube root (e.g., minimizing packaging on cubic boxes, geostationary satellites)

Piecewise (e.g., postage stamp function, teacher salary, GPS for distance)

Square root (distance via Pythagorean Theorem)

Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior, with or without the appropriate technology.

Graph rational functions, identifying zeros and asymptotes (vertical, horizontal, and/or oblique) when suitable factorizations are available and showing end behavior, with or without the appropriate technology.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Graph trigonometric functions showing period, midline, and amplitude.

Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Calculate the expected value for a random variable and describe the expected value as the mean of the probability distribution in context.

Create a probability distribution using theoretical probabilities; calculate the expected value.

Create a probability distribution using experimental or observational data; calculate the expected value.

Analyze the costs and benefits of possible outcomes of making a decision by assigning probabilities to particular payoff values; calculate expected values.

Analyze decisions and strategies using probability concepts.

Recognize that vector quantities have both magnitude and direction and can be represented by directed line segments.

Add and subtract matrices.

Multiply matrices, understanding that matrix multiplication for square matrices is not commutative.

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

Find and use the inverse of a matrix to solve systems of linear equations; solve 3 × 3 or greater systems of equations with technology.

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving velocity and other quantities that can be represented by vectors.

Add and subtract vectors graphically and algebraically.

Given two vectors in magnitude and direction form, determine the magnitude and direction of the sum.

Multiply a vector by a scalar graphically and analytically; reverse their direction when possible.

Compute the magnitude and direction of a vector by multiplying a vector by a scalar.

Use matrices to represent, list, describe and manipulate data with technology.

Multiply a matrix by a scalar.

Functions

Interpreting Functions: Students extend previous knowledge of functions beyond linear and quadratic.

Expected Value: Students calculate and use expected values to solve problems.

Vectors: Students represent and model vector quantities and perform operations on vectors.

Statistics & Probability

Graphing Functions: Students analyze functions using graphing.

Decisions Using Probability: Students use probability to evaluate outcomes of decisions.

Matrices: Students perform operations on matrices and use matrices in applications.

Vectors & Matrices

Use the properties of exponents to find equivalent expressions and to solve equations, including those involving rational exponents.

Sketch the graph of an exponential function given a verbal description and show key features.

Calculate and interpret the average rate of change of an exponential function represented in a table, graph, or as an equation in the context of mathematical and real-world problems.

Graph exponential and logarithmic functions with and without context, identifying key features, and determining constraints in a given context.

Graph and generalize the effect of transformations on exponential and logarithmic functions.

Transformations include: stretches, compressions, vertical shifts, and horizontal shifts

Given the graphs of exponential and logarithmic functions, explain the effects of the transformation from the parent function.

Exponentials: 𝑦 = 𝑎𝑏 𝑥 , 𝑎 ≠ 0, 𝑏 > 0, and 𝑏 ≠ 1

Logarithms: 𝑦 = 𝑙𝑜𝑔𝑏(𝑥), 𝑏 > 0, 𝑥 > 0 and 𝑏 ≠ 1

Write and solve equations from real-world problems that can be represented as a logarithmic or exponential function in one variable.

Write exponential equations that model the relationship between two quantities when given a graph, a written description, or a table of values within a mathematical or real-world context.

Write and use geometric sequences recursively and explicitly to model situations; translate between the two forms when given a graph, a description of the relationship, or two input-output pairs.

Translate between logarithmic and exponential forms of an equation.

Use properties of logarithms to simplify and evaluate logarithmic expressions, with or without technology.

Use the inverse relationship between exponents and logarithms to solve problems.

Determine the domain and range of logarithmic functions in mathematical problems.

Determine reasonable domain and range values of logarithmic functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Combine functions by addition, subtraction, multiplication, division, and composition to model the relationship between two quantities in mathematical and real-world contexts.

Decide if a function is even or odd from a graph or an algebraic expression.

Explain how restricting the domain of a function allows the creation of its inverse.

Write and graph the inverse of a given function; understand that the graph of an inverse function is a reflection of the function over the line 𝑦 = 𝑥.

Write and use arithmetic sequences recursively and explicitly to model situations; translate between the two forms when given a graph, a description of the relationship, or two input-output pairs.

Multiply a matrix by a scalar.

Add and subtract matrices.

Solve systems of linear equations in three variables using matrices; use Gaussian elimination or technology.

Write and solve equations from real-world problems that can be represented as a rational or square root function in one variable.

Sketch the graph of a polynomial function given a verbal description and show key features.

Calculate and interpret the average rate of change of polynomial functions represented in a table, graph, or as an equation in context of mathematical and real-world problems.

Graph functions with and without context, identifying key features and determining constraints in a given context.

Functions include: polynomial, rational, square root, and piecewise-defined

Graph and generalize the effect of transformations on square root, cubic, and rational functions.

Transformations include: stretches, compressions, vertical shifts, and horizontal shifts

Given a graph, explain the effects of the transformation from the parent function.

Square Roots: 𝑦 = √�

Cubics: 𝑦 = 𝑥 3

Rationals: 𝑦 = 1 �

Apply the Remainder Theorem to factor and create equivalent forms of polynomial functions.

Verify polynomial identities and use them to describe numerical relationships.

Apply understanding of rational number operations to add, subtract, multiply, and divide by nonzero rational expressions.

Rewrite simple rational expressions in different forms.

Divide polynomial expressions using inspection, long division, and synthetic division, with and without a remainder.

Solve non-linear formulas for a specified variable.

Explain why a solution to the equation 𝑓(𝑥) = 𝑔(𝑥) is the x-coordinate where the y-coordinate of 𝑓(𝑥)and 𝑔(𝑥) are the same using graphs, tables, or approximations.

Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, exponential, logarithmic, or rational and where at least one of the functions is not linear.

Find zeros of polynomial functions with a degree of 3 or higher when suitable factorizations are available in a real-world and mathematical context.

Use zeros and an understanding of multiplicity to sketch a graph of a polynomial function with a degree of 3 or higher.

Apply the Fundamental Theorem of Algebra to determine the number and potential types of roots of polynomial functions based on the degree of the polynomial.

Solve rational and radical equations containing one variable specifying extraneous solutions.

Determine the domain and range of polynomial and rational functions in mathematical problems.

Determine reasonable domain and range values of polynomial and rational functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of polynomial functions that model a relationship between two quantities in a given context; translate between different representations of the function, especially graphs, tables, and equations.

Calculate and interpret the average rate of change of a quadratic function represented in a table, graph, or as an equation in the context of mathematical and real-world problems.

Solve quadratic equations with complex number solutions.

Represent and solve real-world problems using quadratic inequalities.

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

Sketch the graph of a quadratic function given a verbal description and show key features.

Solve a system of equations consisting of a linear equation and a nonlinear equation in two variables by choosing substitution or graphically (with or without technology) as appropriate for the system of equations.

Explain how extending the properties of integer exponents to rational exponents provides an alternative notation for radicals.

Apply the properties of exponents to translate between radical and exponential forms of expressions.

Simplify and perform operations with radical expressions with and without variables; rationalizing denominators should include conjugates.

Know there is a complex number 𝑖 and describe contexts from which complex numbers appear.

Perform the operations of addition, subtraction, multiplication, and conjugation of complex numbers.

Use polynomial identities with complex numbers.

Use data from a random sample to make inferences about a population.

Compare theoretical and empirical probabilities using simulations.

Distinguish between sample surveys, experiments, and observational studies and explain the purpose of randomization in statistical studies.

Read and explain, in context, the validity of data from outside reports by:

Identifying the variables as quantitative or categorical.

Describing how the data was collected.

Indicating any potential biases or flaws.

Identifying inferences the author of the report made from sample data.

Functions

Linear Functions and Equations

Compositions: Students compose and compare functions.

Exponential & Logarithmic Functions & Equations

Matrices: Students perform operations and matrices.

Factor Polynomials: Students factor polynomials.

Systems of Equations: Students solve systems of equations.

Rational Expressions: Students perform operations on rational expressions.

Complex Numbers: Students apply properties to complex numbers.

Statistical Experiments & Studies: Students evaluate processes for statistical experiments, make inferences, and justify conclusions from statistical studies.

Transformations: Students perform transformations in the coordinate plane.

Create & Solve: Students create and solve problems that model quadratic relationships.

Create & Solve: Students create and solve problems that model exponential and logarithmic relationships.

Arithmetic Sequences: Students use arithmetic sequences to model problems.

Graph & Key Features: Students graph and interpret key features of equations that model quadratic relationships.

Graph & Key Features: Students graph and interpret key features of exponential and logarithmic models.

Radical Expressions and Rational Exponents: Students apply properties to radical expressions and rational exponents.

Create & Solve: Students create and solve rational and polynomial equations.

Systems of Equations: Students solve systems of equations.

Logarithms: Students define and use logarithms.

Systems of Equations: Students solve systems of equations.

Graph & Key Features: Students graph and identify key features of functions.

Polynomial, Rational, & Other Functions & Equations

Quadratic Functions, Equations and Inequalities

Radicals and Complex Numbers

Statistics & Probability

Find the conjugate of a complex number; use conjugates to find quotients of complex numbers.

Generate an equivalent form of an equation for a conic section by completing the square to identify key characteristics.

Conic sections include: circles, ellipses, parabolas, and hyperbolas

Identify, graph, write, and analyze equations of each type of conic section using properties and technology when appropriate.

Conic sections include: circles, ellipses, parabolas, and hyperbolas

Properties include: symmetry, intercepts, foci, asymptotes, and eccentricity

Analyze and interpret exponential and logarithmic functions, identifying key characteristics.

Functions should be represented numerically, graphically, and algebraically.

Key features include asymptotes, end behavior, intercepts, domain, and range.

Understand and apply the inverse relationship between exponents and logarithms to solve problems.

Combine functions by addition, subtraction, multiplication, division, and composition to model the relationship between two quantities in real-world and mathematical contexts.

Verify if two functions are inverses of each other using composition of functions.

For a function with an inverse, explain how to read the ordered pairs of the inverse function when given a graph or table of values.

Construct an invertible function from a non-invertible function by restricting the domain.

Graph and generalize the effect of transformations on quadratic, absolute value, square root, cube root, cubic, and step functions.

Transformations include: stretches, compressions, vertical shifts, and horizontal shifts

Determine if a function is even, odd, or neither from a graph or an algebraic expression.

Write and use arithmetic and geometric sequences recursively and explicitly to model situations, translating between the two forms.

Forms include: when given a graph, a description of the relationship, or two input-output pairs

Use matrices to describe, list, and manipulate data in different situations.

Multiply matrices by scalars to solve real-world and mathematical problems.

Add and subtract matrices.

Multiply matrices, understanding that matrix multiplication for square matrices is not commutative.

Calculate the determinant of a square matrix to determine if it has an inverse.

Solve systems of linear equations using augmented matrices.

Analyze and interpret polynomial functions, identifying key characteristics.

Functions should be represented numerically, graphically, and algebraically.

Key features include end behavior, intercepts, domain, range, relative and absolute maximum and minimum, and intervals over which the function is increasing or decreasing.

Analyze and interpret rational functions, identifying key characteristics.

Functions should be represented numerically, graphically, and algebraically.

Key features include asymptotes (vertical, horizontal, and slant), end behavior, point discontinuities, intercepts, domain, and range.

Graph rational functions showing zeros, asymptotes, and end behavior.

Complex Numbers and Conic Sections

Exponential and Logarithmic Functions

Functions

Matrices

Solve: Students solve problems with exponential and logarithmic functions.

Sequences: Students use sequences to model and analyze mathematical situations.

Inverses: Students find inverse functions.

Complex Numbers: Students apply properties to complex numbers.

Analyze & Interpret: Students analyze and interpret exponential and logarithmic functions.

Analyze, Interpret, & Graph: Students analyze, interpret, and graph polynomial and rational functions.

Transformations: Students graph function transformations.

Operations: Students represent and perform operations with matrices.

Compositions: Students compose and compare functions.

Systems: Students use matrices to solve systems of equations.

Conic Sections: Students relate the equations and graphs of conic sections.

Polynomial & Rational Functions

Approximate the derivative:

Graphically by finding the slope of a tangent line drawn to a curve at a given point.

Numerically by using the difference quotient.

Solve application problems involving:

Optimization

Related rates

Interpret the derivative as a rate of change and varied applied contexts.

Contexts include: velocity, speed, and acceleration

Find the equation of the tangent line using the definition of derivative.

Establish and apply that differentiability implies continuity, but continuity does not necessarily imply differentiability.

Compare the characteristic of graphs of 𝑓 and 𝑓’:

Generate the graph of 𝑓 given the graph of 𝑓’ and vice versa.

Establish the relationship between the increasing and decreasing behavior of 𝑓 and the sign of 𝑓’.

Identify maxima and minima as points where increasing and decreasing behavior change.

Apply the Mean Value Theorem on a given interval.

Compare the characteristic of graphs of 𝑓, 𝑓’, and 𝑓”:

Generate the graphs of 𝑓 and 𝑓′ given the graph of 𝑓” and vice versa.

Establish the relationship between the concavity of 𝑓 and the sign of 𝑓”.

Identify points of inflection as points where concavity changes.

Find derivatives of functions using:

Power rule

Product rule

Quotient rule

Find derivatives of:

An implicitly defined equation

Composite functions using chain rule

Exponential and logarithmic functions

Functions requiring the use of more than one differentiation rule

Find the equation of:

A line tangent to the graph of a function at a point

A normal line to the graph of a function at a point

Define the definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval.

If 𝑓 is a real, continuous function defined on [𝑎, 𝑏] and 𝐹 is an antiderivative of 𝑓 in [𝑎, 𝑏], then ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑏 𝑎 .

Determine the area between two curves and identify the definite integral as the area of the region bounded by two curves.

Apply the Fundamental Theorem of Calculus to solve contextual models that represent real-world phenomena.

Find the general solution to indefinite integrals.

Determine the antiderivative of a function using rules of basic differentiation, and solve problems using the techniques of antidifferentiation including but not limited to power rule and u-substitution.

Estimate definite integrals by using Riemann sums (left, right, midpoint, and trapezoidal) and identify the definite integral as a limit of Riemann sums.

Explore applications of integration.

Identify vertical asymptotes in rational and logarithmic functions by identifying locations where the function value approaches infinity; estimate limits numerically and graphically; calculate limits analytically:

Algebraic simplification

Direct substitution

One-sided limits

Rationalization

Calculate infinite limits and use the result to identify vertical asymptotes in rational and logarithmic functions.

Calculate limits at infinity and use the result to identify horizontal asymptotes in rational and exponential functions.

Calculate limits at infinity and use the result to identify unbounded behavior in rational, exponential, and logarithmic functions.

Identify and classify graphically, algebraically, and numerically if a discontinuity is removable or nonremovable; identify the three conditions that must exist in order for a function to be continuous at 𝑥 = 𝑎:

𝑓(𝑎) is defined

The limit as 𝑥 approaches 𝑎 of 𝑓(𝑥) equals 𝑓(𝑎)

The limit as 𝑥 approaches 𝑎 of 𝑓(𝑥) exists

Apply the Intermediate Value Theorem for continuous functions.

Derivatives

Integrals

Limits & Continuity

Define the Definite Integral: Students apply techniques of integration to solve problems, both theoretically and in contextual models that represent realworld phenomena.

Equation of a Tangent Line: Students use derivatives to solve problems both theoretically and in real-world context.

Identify and Calculate Limits: Students determine the limit of a function at a value numerically, graphically, and analytically.

Represent and solve real-world problems, using exponential equations in one variable.

Write exponential functions that provide a reasonable fit to data and use them to make predictions with technology.

Represent real-world problems (growth, decay, and compound interest), using exponential equations.

Construct exponential equations from geometric sequences with and without context.

Determine the domain and range of exponential functions in mathematical problems.

Determine reasonable domain and range values of exponential functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of an exponential function that models a relationship between two quantities in a given context.

Flexibly use different representations of an exponential function, including graphs, tables, and equations.

Interpret the quantities in an exponential equation in the context of a real-world problem, including growth, decay, and compound interest.

Graph exponential functions that model real-world problems (growth, decay, and compound interest), showing key attributes.

Add, subtract, and multiply polynomials; compare the system of polynomials to the system of integers when performing operations.

Simplify and perform operations with radical expressions without variables; rationalizing denominators should not include conjugates.

Simplify algebraic expressions using the laws of exponents.

Interpret the parts of expressions such as terms, factors, and coefficients in terms of a real-world context.

Explain that a function assigns each element in the domain to exactly one element in the range.

Use function notation to represent functions, understanding that if 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) represents the output of 𝑓 corresponding to the input 𝑥.

Graph functions given in function notation, understanding that the graph contains the points (𝑥, 𝑓(𝑥)).

Evaluate functions expressed in function notation for one or more elements in their domains (inputs); use function notation to describe a contextual situation.

Differentiate between real-world scenarios that can be modeled by exponential or linear functions by determining whether the relationship has a common difference or a common ratio.

Compare the growth pattern of exponential to linear or quadratic functions using graphs and tables and recognize how exponential growth exceeds other functions.

Represent and solve real-world problems, using linear expressions, equations, and inequalities in one variable.

Translate among equivalent forms of equations for linear functions, including standard, point-slope, and slopeintercept forms; recognize that each form reveals key features in a given context.

Solve systems of linear equations by substitution, elimination, and graphing with and without a real-world context; understand that the solutions will be the same regardless of the method for solving.

Solve a system of equations consisting of a linear equation and a quadratic equation in two variables graphically with the assistance of technology.

Explain why a solution to the equation 𝑓(𝑥) = 𝑔(𝑥) is the x-coordinate where the y-coordinate of 𝑓(𝑥) and 𝑔(𝑥) are the same using graphs, tables, or approximations. Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, quadratic, absolute value, and exponential.

Solve linear inequalities and systems of linear inequalities in two variables by graphing.

Write linear equations that model the relationship between two quantities and produce a graph of the equation.

Graph linear functions expressed as an equation and show intercepts of the graph without technology.

Graph absolute value functions expressed as an equation with and without technology, showing intercepts and end behavior.

Graph and generalize the effect of transformations on linear and absolute value functions.

Given the graph of a linear function, explain the effects of the transformation from the parent function, 𝑦 = 𝑥.

Construct linear functions from arithmetic sequences with and without context.

Write linear functions that provide a reasonable fit to data and use them to make predictions, with and without technology; interpret the slope and y-intercept in context.

Calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.

Compare and contrast correlation and causation in real-world problems.

Solve linear formulas for a specified variable.

Solve linear equations, linear inequalities, and absolute value equations in one variable, including those with rational number coefficients, and variables on both sides of the equal or inequality sign; solve them fluently, explaining the process used.

Determine the domain and range of linear functions in mathematical problems.

Determine reasonable domain and range values of linear functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of a linear and absolute value functions that models a relationship between two quantities in a given context.

Flexibly use different representations of a linear function, including graphs, tables, and equations.

Calculate and interpret the rate of change of a linear function represented in a table, graph, or as an equation in context of real-world and mathematical problems.

Represent and solve real-world problems using quadratic expressions and equations in one variable.

Graph quadratic functions given as an equation or in function notation, labeling key attributes, without technology.

Graph and describe the effect of transformations on quadratic functions.

Given the graph of a quadratic function, explain the effects of the transformation from the parent function, y = x 2 .

Write quadratic functions that provide a reasonable fit to data and use them to make predictions with technology.

Write quadratic equations with real number solutions that model the relationship between two quantities and produce a graph of the equation.

Solve quadratic equations with real number solutions, containing one variable, including those with variables on both sides of the equal sign. Equations should be solved by:

Graphing,

Factoring (including perfect square trinomials and difference of squares binomials),

Using the quadratic formula,

Completing the square, or

Taking the square root.

Determine the domain and range of quadratic functions in mathematical problems.

Determine reasonable domain and range values of quadratic functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of a quadratic function that models a relationship between two quantities in a given context.

Flexibly use different representations of a quadratic function, including graphs, tables, and equations.

Explain how each form of a quadratic expression (standard, factored, and vertex form) identifies different key attributes, using the different forms to interpret quantities in context.

Use factoring and completing the square to create equivalent forms of quadratic functions to reveal key attributes.

Use box plots and histograms to determine the statistics appropriate to the shape of the data distribution; compare the center and spread of two or more data sets.

Summarize data from two categorical variables in a frequency table; interpret relative frequencies in the context of the data, recognizing data trends and associations.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.

Exponential Functions & Equations

Expressions

Functions

Linear Functions, Equations, & Inequalities

Statistical Relationships: Students explore linear statistical relationships.

Statistical Relationships: Students explore quadratic statistical relationships.

Polynomials, Roots, & Exponent Laws: Students simplify algebraic and numerical expressions.

Construct & Compare: Students construct and compare linear, quadratic, and exponential models and solve problems.

Interpret Key Features: Students interpret key features of equations that model linear relationships.

Graphing: Students graph exponential functions.

Bivariate Data: Students will investigate patterns of association in bivariate data.

Interpret Key Features: Students interpret key features of equations that model exponential relationships.

Create & Solve: Students create and solve problems that model exponential relationships.

Statistical Relationships: Students explore exponential statistical relationships.

Systems of Equations & Inequalities: Students solve systems of equations and inequalities.

Create & Solve: Students create and solve equations that model quadratic relationships.

Graphing & Transformations: Students graph linear functions, equations, and inequalities.

Interpret Key Features: Students interpret key features of equations that model quadratic relationships.

Create & Solve: Students create and solve equations that model linear relationships.

Domain & Range, Function Notation: Students understand the concept of a function, domain and range, and use function notation; students use function notation to solve problems.

Numerical Data: Students summarize and describe distributions.

Graphing & Transformations: Students graph quadratic functions and explore different transformations of 𝑓(𝑥) = 𝑥 2 .

Quadratic Functions & Equations

Statistics & Probability

Represent and solve real-world problems, using exponential equations in one variable.

Write exponential functions that provide a reasonable fit to data and use them to make predictions with technology.

Represent real-world problems (growth, decay, and compound interest), using exponential equations.

Construct exponential equations from geometric sequences with and without context.

Determine the domain and range of exponential functions in mathematical problems.

Determine reasonable domain and range values of exponential functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of an exponential function that models a relationship between two quantities in a given context.

Flexibly use different representations of an exponential function, including graphs, tables, and equations.

Interpret the quantities in an exponential equation in the context of a real-world problem, including growth, decay, and compound interest.

Graph exponential functions that model real-world problems (growth, decay, and compound interest), showing key attributes.

Students create and solve problems that model exponential relationships.

Students interpret key features of equations that model exponential relationships.

Students graph exponential functions.

Students explore exponential statistical relationships.

Students simplify algebraic and numerical expressions.

Add, subtract, and multiply polynomials; compare the system of polynomials to the system of integers when performing operations.

Simplify and perform operations with radical expressions without variables; rationalizing denominators should not include conjugates.

Simplify algebraic expressions using the laws of exponents.

Interpret the parts of expressions such as terms, factors, and coefficients in terms of a real-world context.

Explain that a function assigns each element in the domain to exactly one element in the range.

Use function notation to represent functions, understanding that if f is a function and x is an element of its domain, then f(x) represents the output of f corresponding to the input x.

Graph functions given in function notation, understanding that the graph contains the points (x,f(x)).

Evaluate functions expressed in function notation for one or more elements in their domains (inputs); use function notation to describe a contextual situation.

Differentiate between real-world scenarios that can be modeled by exponential or linear functions by determining whether the relationship has a common difference or a common ratio.

Compare the growth pattern of exponential to linear or quadratic functions using graphs and tables and recognize how exponential growth exceeds other functions.

Students understand the concept of a function, domain and range, and use function notation; students use function notation to solve problems.

Students construct and compare linear, quadratic, and exponential models and solve problems.

Represent and solve real-world problems, using linear expressions, equations, and inequalities in one variable.

Translate among equivalent forms of equations for linear functions, including standard, point-slope, and slope-intercept forms; recognize that each form reveals key features in a given context.

Solve systems of linear equations by substitution, elimination, and graphing with and without a real-world context; understand that the solutions will be the same regardless of the method for solving.

Solve a system of equations consisting of a linear equation and a quadratic equation in two variables graphically with the assistance of technology.

Explain why a solution to the equation f(x) = g(x) is the x-coordinate where the y-coordinate of f(x) and g(x) are the same using graphs, tables, or approximations. Include cases where f(x) and/or g(x) are linear, quadratic, absolute value, and exponential.

Solve linear inequalities and systems of linear inequalities in two variables by graphing.

Write linear equations that model the relationship between two quantities and produce a graph of the equation.

Graph linear functions expressed as an equation and show intercepts of the graph without technology.

Graph absolute value functions expressed as an equation with and without technology, showing intercepts and end behavior.

Graph and generalize the effect of transformations on linear and absolute value functions.<ul><li>Transformations include: stretches, compressions, vertical, and horizontal</li></ul>

Given the graph of a linear function, explain the effects of the transformation from the parent function, y=x.

Construct linear functions from arithmetic sequences with and without context.

Write linear functions that provide a reasonable fit to data and use them to make predictions, with and without technology; interpret the slope and y-intercept in context.

Calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.

Compare and contrast correlation and causation in real-world problems.

Solve linear formulas for a specified variable.

Solve linear equations, linear inequalities, and absolute value equations in one variable, including those with rational number coefficients, and variables on both sides of the equal or inequality sign; solve them fluently, explaining the process used.

Determine the domain and range of linear functions in mathematical problems.

Determine reasonable domain and range values of linear functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of a linear and absolute value functions that models a relationship between two quantities in a given context.

Flexibly use different representations of a linear function, including graphs, tables, and equations.

Calculate and interpret the rate of change of a linear function represented in a table, graph, or as an equation in context of real-world and mathematical problems.

Students create and solve equations that model linear relationships.

Students interpret key features of equations that model linear relationships.

Students solve systems of equations and inequalities.

Students graph linear functions, equations, and inequalities.

Students explore linear statistical relationships.

Represent and solve real-world problems using quadratic expressions and equations in one variable.

Graph quadratic functions given as an equation or in function notation, labeling key attributes, without technology.

Graph and describe the effect of transformations on quadratic functions.<ul><li>Transformations include: stretches, compressions, vertical, and horizontal</li></ul>

Given the graph of a quadratic function, explain the effects of the transformation from the parent function, y = x².

Write quadratic functions that provide a reasonable fit to data and use them to make predictions with technology.

Write quadratic equations with real number solutions that model the relationship between two quantities and produce a graph of the equation.

Solve quadratic equations with real number solutions, containing one variable, including those with variables on both sides of the equal sign. Equations should be solved by:<ul><li>Graphing,</li><li>Factoring (including perfect square trinomials and difference of squares binomials),</li><li>Using the quadratic formula,</li><li>Completing the square, or</li><li>Taking the square root.</li></ul>

Determine the domain and range of quadratic functions in mathematical problems.

Determine reasonable domain and range values of quadratic functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

Interpret the key features of a quadratic function that models a relationship between two quantities in a given context.

Flexibly use different representations of a quadratic function, including graphs, tables, and equations.

Explain how each form of a quadratic expression (standard, factored, and vertex form) identifies different key attributes, using the different forms to interpret quantities in context.

Use factoring and completing the square to create equivalent forms of quadratic functions to reveal key attributes.

Students create and solve equations that model quadratic relationships.

Students interpret key features of equations that model quadratic relationships.

Students graph quadratic functions and explore different transformations of f(x) = x².

Students explore quadratic statistical relationships.

Use box plots and histograms to determine the statistics appropriate to the shape of the data distribution; compare the center and spread of two or more data sets.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.

Summarize data from two categorical variables in a frequency table; interpret relative frequencies in the context of the data, recognizing data trends and associations.

Students summarize and describe distributions.

Students will investigate patterns of association in bivariate data.

Apply the precise definition and standard geometric notation for a circle to understand geometric relationships.

Recognize and apply relationships between angles, radii, and chords, tangents, and secants including:<ul><li>The relationship between central, inscribed, and circumscribed angles,</li><li>Inscribed angles on a diameter are right angles,</li><li>The radius of a circle is perpendicular to the tangent where the radius intersects the circle, and</li><li>The relationship of angles and segments formed by chords, secants and/or tangents to a circle.</li></ul>

Use the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems.

Use the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.

Explain why the formulas for the area and circumference of a circle work using dissection and informal limit arguments.

Write the equation of a circle, given the radius and center, where the center is at the origin or another point.

Identify the center and radius of a circle, given the equation of a circle, where the center is at the origin or another point.

Apply the equation of a circle to solve real-world problems.

Students explore and use circle relationships to solve problems.

Students solve problems involving the equation of a circle.

Find the volume and surface area of complex three-dimensional figures composed of prisms, pyramids, cones, cylinders, and spheres.

Calculate the area of triangles and rectangles when given the vertices, including using the distance formula and decomposing figures.

Describe reflectional and rotational symmetry as they apply to a rectangle, parallelogram, trapezoid, or regular polygon.

Calculate probabilities as a proportion of area in a geometric context.

Use three-dimensional geometric figures and their measures to model real-world objects and solve problems.

Explain why the formulas for the volume and surface area of a cylinder, pyramid, and cone work.

Apply the Pythagorean Theorem to determine missing measurements in a three-dimensional figure.

Identify the three-dimensional figure generated by rotating a two-dimensional figure.

Apply theorems about quadrilaterals, including those involving angles, diagonals, and sides to solve problems.

Prove that a given quadrilateral is a parallelogram, rhombus, rectangle, square, kite, or trapezoid, and apply these relationships to solve problems.

Prove and apply theorems about triangles including:<ul><li>Angle-Sum Theorem,</li><li>Exterior Angle Theorem,</li><li>Isosceles Triangle Theorem and its converse,</li><li>Midsegment Theorem,</li><li>Proportionality Theorem,</li><li>Inequality Theorem and its converse, and</li><li>Geometric Mean Theorem.</li></ul>

Calculate the perimeter of polygons when given the vertices, including using the distance formula.

Students explore and solve problems involving three-dimensional figures.

Students explore and solve problems involving two-dimensional figures.

Students determine probability in geometric contexts.

Use precise definitions and standard geometric notation for angles, perpendicular lines, parallel lines, and line segments based on the undefined notions of point, line, and distance along a line.

Make formal geometric constructions with a variety of tools and methods including:<ul><li>Congruent segments and angles,</li><li>Segment and angle bisectors,</li><li>Perpendicular lines and perpendicular bisectors of a line segment,</li><li>Parallel lines, and</li><li>An equilateral triangle, a square, and a regular hexagon inscribed in a circle.</li></ul>

Determine the point that cuts a line segment into a specified ratio on a number line and a coordinate plane, including finding the midpoint.

Derive the distance and midpoint formulas and use the formulas, including the slope formula, to verify geometric relationships on a coordinate plane.

Prove and apply slope criteria of parallel and perpendicular lines to solve problems.

Write an equation of a line that is parallel or perpendicular to a given line and passing through a given point.

Prove and apply theorems about lines and angles including:<ul><li>Vertical angles,</li><li>Angles formed by parallel lines cut by a transversal, and</li><li>Points on a perpendicular bisector.</li></ul>

Students use precise definitions and various construction tools to create geometric figures.

Students reason about geometric figures using the coordinate plane.

Students solve problems involving parallel and perpendicular lines.

Apply the properties of special right triangles (30°-60°-90° and 45°-45°-90°) to solve real-world and mathematical problems.

Prove and apply the Pythagorean Theorem and its converse.

Explain how the definitions for trigonometric ratios are developed by similarity and how the side ratios in right triangles are properties of the angles in the triangle.

Explain the relationship between the sine and cosine of complementary angles and use them to solve problems.

Determine the sine, cosine, and tangent ratios of acute angles given the side lengths of right triangles.

Use trigonometric ratios (sine, cosine, and tangent) to calculate missing side lengths and angle measures in a right triangle, including applications of angles of elevation and depression; include real-world and mathematical problems.

Students explore right triangles and apply the Pythagorean Theorem.

Students apply trigonometric ratios to solve problems.

Given two figures, apply the definition of similarity in terms of a dilation to identify similar figures, proportional sides, and corresponding congruent angles.

Develop and apply the criteria of similarity for triangles (AA~, SAS~, and SSS~) to solve problems and prove geometric relationships.

Use transformations to prove all circles are similar.

Explain, using rigid motion transformations, why two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Develop and apply the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) to solve problems and prove geometric relationships.

Students use similarity criteria to solve problems.

Students apply congruence criteria to solve problems.

Describe rotations, reflections, and translations as functions that take points in the coordinate plane as inputs and give other points as outputs; write in prime notation.

Compare transformations that preserve distance and angle (rotations, reflections, and translations) to those that do not (dilations) to develop definitions for congruence and similarity.

Apply understanding of angles, circles, perpendicular lines, parallel lines, and line segments to develop definitions for rotations, reflections, and translations.

Use geometric constructions to represent rotations, reflections, translations, and dilations in the plane with a variety of tools and methods.

Given two congruent figures, identify the sequence of transformations that maps one figure to another.

Students transform figures on the coordinate plane.

Students transform figures and make geometric constructions.

Right Triangles

Similarities & Congruence

Construct & Compare

Similarity

Geometry

Plane

Special Right Triangles & Pythagorean Theorem

Linear Functions, Equations, & Inequalities

Parallel & Perpendicular Lines

Define & Construct

Numerical Data

Circle Relationships

Lines & Angles

Statistical Relationships

Expressions

Trigonometry Ratios

Algebra I

Statistics & Probability

Three-Dimensional

Quadratic Functions & Equations

Bivariate Data

Interpret Key Features

Functions

Create & Solve

Systems of Equations & Inequalities

Create & Solve

Coordinate Plane

Interpret Key Features

Geometric Figures

Domain & Range, Function Notation

Create & Solve

Equation of a Circle

Polynomials, Roots, & Exponent Laws

Graphing

Coordinate Geometry

Graphing & Transformations

Statistical Relationships

Graphing & Transformations

Geometric Probability

Two-Dimensional

Exponential Functions & Equations

Transformations

Triangle Congruence

Statistical Relationships

Circles

Interpret Key Features

Conic Sections

Functions

Derive Equations: Students derive equations for conic sections.

Vector Operations: Students perform operations involving vectors.

Solve & Graph: Students explore, solve, and sketch the graphs of periodic trigonometric functions.

Explore Graphing: Students graph and interpret functions.

Solve Problems: Students derive and apply functions.

Vector Quantities: Students recognize, model, and write vector quantities.

Systems of Equations & Inequalities: Students solve systems of equations and inequalities involving conic sections.

Identities, Formulas, & Laws: Students develop and apply identities, formulas, and laws using trigonometry.

Radians: Students understand, explain, and describe radian measure.

Unit Circle: Students use the unit circle to express and find exact values for trigonometric functions.

Explore Equations: Students identify, analyze, and sketch the graphs of the conic sections and relate their equations and graphs.

Matrix Operations: Students represent and perform operations with matrices.

Derive the general form of the equation of a circle using the Distance Formula or Pythagorean Theorem.

Derive the equation of a parabola given a focus and directrix.

Derive the equations of ellipses and hyperbolas given the foci using the Distance Formula.

Find the equations for the asymptotes of a hyperbola.

Generate an equivalent form of an equation for a conic section by completing the square to identify key characteristics.

Conic sections include: circles, ellipses, parabolas, and hyperbolas

Identify, graph, write, and analyze equations of each type of conic section using properties and technology when appropriate.

Conic sections include: circles, ellipses, parabolas, and hyperbolas

Properties include: symmetry, intercepts, foci, asymptotes, and eccentricity

Solve systems of equations and inequalities involving conics and other types of equations, with and without technology.

Equations include: conic-conic and conic-linear

Understand that sequences are functions, sometimes defined recursively, whose domains are a subset of the integers.

Derive the formula for the sum of a finite geometric series; apply the formula to solve conceptual problems.

Apply the Binomial Theorem for the expansion of (𝑎 + 𝑏) 𝑛 in powers of 𝑎 and 𝑏 for a positive integer 𝑛, where 𝑎 and 𝑏 are any number.

Build functions to model real-world contexts using algebraic operations on functions and composition, with and without appropriate technology.

Graph power and polynomial functions, identify zeros (when suitable factorizations are available), and show end behavior.

Graph rational functions, identify zeros, holes and asymptotes (when suitable factorizations are available), and show end behavior.

Asymptotes include: horizontal, vertical, and oblique

Graph exponential and logarithmic functions; show intercepts and end behavior.

Compare key features of two functions each represented in a different way.

Representations include: algebraic, graphic, numeric in tables, and verbal descriptions

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

Develop and apply the Law of Sines and the Law of Cosines to solve real-world and mathematical problems including finding unknown measurements in right and non-right triangles.

Define and use reciprocal functions, cosecant, secant, and cotangent to solve problems.

Explain whether a trigonometric function is even or odd and recognize the periodicity of the graph using the unit circle.

Graph trigonometric and inverse trigonometric functions and show period, midline, and amplitude.

Select a trigonometric function that models real-world contexts.

Explain how restricting the domain of a trigonometric function allows the creation of its inverse.

Solve and evaluate the solution of trigonometric equations in real-world contexts; interpret the solution in terms of its context.

Recognize that some trigonometric equations have infinitely many solutions and be able to state a general formula to represent the infinite solutions.

Calculate and interpret the average rate of change over a specified interval of a trigonometric function represented in a table, graph, or as an equation in the context of real-world and mathematical problems.

Convert between radian and degree measure.

Explain how the unit circle can be used to model sine, cosine, tangent, secant, cosecant, and cotangent for all real numbers.

Construct special right triangles on the unit circle to find the exact values of sine, cosine, tangent for 𝜋 3 , 𝜋 4 , 𝜋 6 , and 𝜋 2 .

Use the unit circle to express the values of sine, cosine, and tangent for 𝜋– 𝑥,𝜋 + 𝑥, and 2𝜋– 𝑥 in terms of their exact values for 𝑥, where 𝑥 is one of these values: 𝜋 3 , 𝜋 4 , 𝜋 6 , and 𝜋 2 .

Develop the Pythagorean identity, 𝑠𝑖𝑛2 (𝜃) + 𝑐𝑜𝑠2 (𝜃) = 1.

Apply the Pythagorean identity to find the remaining trigonometric functions when given 𝑠𝑖𝑛(𝜃), 𝑐𝑜𝑠(𝜃), or 𝑡𝑎𝑛(𝜃) and the quadrant of the angle.

Develop addition, subtraction, double, and half-angle formulas for sine, cosine, and tangent and use them to solve problems, including verifying other identities.

Develop the formula for the area of a triangle, 𝐴 = ( 1 2 ) 𝑎𝑏 𝑠𝑖𝑛 𝐶 , using trigonometry.

Recognize that vector quantities have both magnitude and direction and can be represented by directed line segments.

Work with 2 × 2 matrices as transformations of the plane; interpret the absolute value of the determinant in terms of area.

Write vector quantities using appropriate symbols indicating magnitude and direction.

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving velocity and other quantities that can be represented by vectors.

Add and subtract vectors graphically and algebraically.

Multiply a vector by a scalar graphically and analytically; reverse their direction when possible.

Use matrices to list, describe, and manipulate data with and without technology.

Multiply matrices, understanding that matrix multiplication for square matrices is not commutative.

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Trigonometry

Vectors & Matrices

Business Financial Literacy

Modeling

Statistical: Students draw conclusions, make decisions, and communicate based on understanding using statistical information.

Business & Economics: Students understand the principles and mathematics in business as it applies to economics.

Number, Ratio, & Probability: Students use number sense and proportional reasoning in real-world settings to make and communicate decisions in order to draw conclusions based on quantitative analysis.

Credit & Debt: Students apply mathematics to make informed credit and debt decisions.

Probabilities: Students apply probabilistic reasoning to draw conclusions, to make decisions, and to evaluate outcomes of decisions.

Problem Solving: Students use appropriate mathematical models to solve problems involving everyday life, workplace, and society.

Bivariate Data Sets: Students use bivariate data sets to solve problems.

Investment: Students apply mathematics to make informed investment decisions.

Statistics & Probability

Employment & Income: Students apply mathematics to make informed employment and income decisions.

Numerical Reasoning

Personal Financial Literacy

Use real-world data to determine how a product or service can be profitable in a community.

Determine fixed and variable expenses of running a business.

Calculate indices and solve problems using common indices:

Consumer price index

Cost of living index

Determine what constitutes an index

Analyze how stock market averages and indices are calculated with technology.

Research how inflation changes the value of the U.S. Dollar over time.

Prepare for employment by analyzing job skills.

Use mathematical models to:

Demonstrate understanding of the meaning of a solution in context.

Identify when insufficient information is given to solve a problem.

Analyze mathematical models, describe limitations, and suggest improvements.

Use mathematical models created with spreadsheets or other tools to:

Estimate solutions for contextual questions.

Identify patterns.

Identify how changing parameters affect results.

Use mathematical models to make decisions about purchases.

Select models for a given set of bivariate data sets; justify the choice.

Represent and use mathematical models for bivariate data sets to answer questions, draw conclusions, and make decisions.

Solve real-world problems and interpret results involving calculations with percentages, decimals, and fractions.

Problem types include: conversions, percent change (absolute vs relative), and percent of quantities

Use estimation in real-world situations.

Numeric and contextual benchmarks:

Identify appropriate numeric benchmarks for estimating calculations.

Identify appropriate contextual benchmarks to compare to other numbers.

Compare magnitudes of numbers in context in different forms.

Use dimensional analysis to solve problems involving multiple units of measurement.

Solve real-world problems requiring interpretation and comparison of various representations of rates and ratios.

Distinguish between proportional and non-proportional real-world situations; when appropriate, apply proportional reasoning.

Represent and analyze mathematical models for various types of income.

Represent and analyze various types of income deductions and employment forms.

Analyze expenses to create a household budget utilizing food, shelter, transportation, utilities, insurance, savings, and other expenses.

Analyze various investment instruments for:

Purposes

Advantages

Disadvantages

Risks

Analyze the characteristics of various types of loans.

Apply appropriate models to determine the impact of the relationship among loan rates, the term of a loan, the principal amount of a loan, and payments.

Examine consumer protection, bankruptcy, and credit and debt management services for ways in which they affect household budgeting.

Create charts, tables, and graphs of real-world data with and without technology.

Analyze and interpret charts, tables, and graphs using real-world data.

Compare and contrast charts, tables, and graphs using real-world data.

Analyze statistical information from studies, surveys, and polls to make informed judgements as to the validity of claims or conclusions.

Make decisions about data summarized numerically using measures of center:

Compare measures of center of two or more data sets.

Interpret the differences in context.

Justify the use of a chosen measure.

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

Evaluate the validity of claims based on experimental and theoretical probabilities.

Apply rules of counting and probability to compute probabilities of compound real-world events:

Addition Rule of Probability

Multiplication Rule of Probability

Fundamental Counting Principle

Permutation and combinations

Visual representations

Collecting Data & Data Bias

Displaying & Describing Distributions of Data

Making Inferences & Justifying Conclusion

Independence & Conditional Probability: Students understand and use independence and conditional probability to interpret data.

Data Representation: Students represent raw data in tabular and graphical form to describe features of the data and summarize trends.

Surveys, Experiments, & Observational Data: Students make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Expected Values: Students calculate and use expected values of random variables to solve problems.

Compute Probability of Compound Events: Students use the rules of probability to compute probabilities of compound events.

Outcomes of Decisions: Students evaluate outcomes of decisions using probability.

Data Collection: Students explore best practices of collecting data while identifying possible sources of bias in data collection methods.

Use Probability to Make Decisions

Conditional Probability & Rules of Probability

Describe how to take a simple random sample using technology or a random number table.

Use randomization strategies to ensure random selection processes are fair.

Understand that certain types of sampling methods may lead to bias, such as convenience and voluntary samples.

Distinguish between categorical and quantitative data.

Determine if there is an association between two quantitative variables using the correlation coefficient and scatter plots.

Model real-world data using least squares regression techniques.

Estimate a population mean or proportion from a sample survey; develop a margin of error through the use of simulation models for random sampling.

Calculate the standardized test statistic and p-value for a test about a population proportion and a population mean; determine if the sample data provides convincing evidence against a parameter claim.

Compare two treatment groups in an experiment and determine if the difference in parameters is significant by calculating the standardized test statistics and p-value.

Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Calculate the expected value for a discrete random variable; describe the expected value as the mean or typical value of the probability distribution in context.

Create a probability distribution of a discrete random variable using theoretical probabilities and use the probability distribution to calculate the probability of an event.

Create a probability distribution for a discrete random variable using experimental or observational data; calculate the expected value.

Use probability density models such as the normal curve and uniform density curve to model real-world data; calculate probabilities of continuous random variables using these models.

Analyze the costs and benefits of possible outcomes of making a decision by assigning probabilities to particular payoff values of a discrete random variable and calculate expected values.

Determine unions or intersections of events in a sample space; determine complements of events.

Identify whether or not two events are mutually exclusive / disjoint.

Apply the general Multiplication Rule, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) = 𝑃(𝐵)𝑃(𝐴|𝐵) and interpret the answer.

Compute the probability of compound events and solve problems using combinations, permutations, Venn Diagrams, and Tree Diagrams.

Identify the two components that make up a legitimate probability model/distribution.

Determine if two events, A and B, are independent when given the probabilities of A and B.

Calculate and use conditional probabilities to determine if events are independent.

Create and analyze two-way frequency tables of data to calculate marginal, joint, and conditional probabilities.

Using a two-way table, determine if two events are independent.

Explain conditional probability and independence using everyday language in a variety of real-world contexts.

Find the conditional probability of 𝐴 given 𝐵, 𝑃(𝐴|𝐵), and interpret the answer in terms of the model, including two-way frequency tables and Venn diagrams.

Apply the Addition Rule, 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵) and interpret the answer.

Algebraic Relationships

Geometry

Mathematical Processes & Modeling

Measurement

Visualizing Solid Shapes: Students will extend understanding of solid figures with and without technology.

Model with Triangles: Students will apply geometric reasoning to triangles in real-world scenarios.

Model with Functions: Students use mathematical concepts of algebra to explain linear and nonlinear applications in real-world scenarios.

Mathematical Processes & Modeling: Students use mathematical processes and models to acquire, demonstrate, and communicate mathematical understanding in real-world scenarios.

Measurement & Measurement Tools: Students apply measurement and use measurement tools in real-world scenarios.

Model with Data: Students use data to make decisions and predictions.

Model with Geometric Figures: Students will extend geometric reasoning and model with geometric figures.

Comparison: Students use number sense and proportional reasoning to draw conclusions and communicate results.

Proportional Reasoning: Students understand and reason about relationships between quantities.

Model with Transformations: Students will apply transformations to real-world scenarios.

Model with Estimation: Students use estimation to solve real-world problems and assess the reasonableness of a solution.

Numerical & Proportional Reasoning

Analyze and apply rate of change in terms of real-world scenarios.

Use concepts of systems of equations and inequalities to model and solve real-world scenarios.

Use linear programming with or without the use of technology to:

Maximize or minimize (optimize) linear objective function in real-world scenarios.

Determine the reasonableness of solutions.

Collect and organize data, independently and as a collaborative team, to create appropriate graphical representations of real-world scenarios.

Interpret graphical representations.

Make predictions and decisions based on representations.

Analyze results based on representations.

Create, interpret, and analyze best-fit models of linear and exponential functions to solve real-world scenarios.

Interpret the constants, coefficients, and bases in the context of the data.

Check the model for best fit and use the model, where appropriate, to draw conclusions or make predictions.

Identify common geometric figures in order to identify what formulas are needed to solve situational problems.

Compute measurements of common geometric figures such as area, surface area, volume, perimeter, and circumference for real-world scenarios.

Analyze how changing dimensions will affect the perimeter, circumference, area, surface area, or volume in real-world scenarios.

Determine the role angles play in a situational problem.

Draw and interpret technical drawings involving house plans, engineering drawings, or fashion design with or without the use of technology.

Views include: auxiliary, orthographic, and isometric

Use trigonometric ratios to calculate angles and lengths of sides in real-world scenarios.

Apply right-triangle relationships using Pythagorean Theorem, special right triangles, and trigonometry in real world scenarios.

Demonstrate mastery of manipulating 2D and 3D figures.

Use cross-sections of 3D shapes to relate to 2D figures.

Use revolutions of 2D shapes to create a 3D object or space.

Describe the transformation of polygons in the coordinate plane as they relate to real-world scenarios:

Transformations include: translation, reflection, rotation, and dilation

Apply mathematics to problems arising in everyday life, workplace, and society.

Use mathematical processes with algebraic formulas, numerical techniques, and graphs to solve real-world scenarios.

Create mathematical models and use problem-solving skills, independently and as a collaborative team, for real-world scenarios to:

Analyze given information or data

Identify patterns or relationships

Formulate a plan or strategy

Estimate solutions

Determine a solution

Justify a solution and its reasonableness

Describe limitations

Identify how results are affected by changing parameters

Suggest improvements

Select appropriate tools and techniques to solve problems.

Demonstrate effective use of resources.

Use precise mathematical language and multiple representations to organize, record, and communicate mathematical ideas or solutions to solve real-world scenarios independently and collaboratively.

Convert between and within the metric system and the U.S. customary system in real-world scenarios.

Demonstrate mastery of utilizing measuring devices:

Apply accurate readings of both metric and the U.S. customary measuring devices to a problem situation.

Select and use appropriate measuring devices and understand the limitations of such devices for realworld scenarios.

Determine and use appropriate unit labels for real-world scenarios.

Unit labels include: length, weight, capacity, distance, temperature, time, surface area, volume, area, and perimeter

Use estimation to identify the most reasonable mathematical solution.

Use estimation and precision in real-world scenarios.

Solve real-world problems and interpret results involving calculations with percentages, decimals, and fractions.

Calculations include: conversions, percent change, and percent of quantities

Recognize, set up, and solve proportions from real-world scenarios.

Utilize real-world scenarios requiring interpretation and comparison of various representations of rates, ratios, and proportions including scale drawings.

Compare magnitudes of numbers in context in different forms.

Use dimensional analysis to solve problems involving multiple units of measurement.