High School Math Maryland Standards

Algebra

Arithmetic with Polynomials and Rational Expressions

PERFORM ARITHMETIC OPERATIONS ON POLYNOMIALS

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

UNDERSTAND THE RELATIONSHIP BETWEEN ZEROS AND FACTORS OF POLYNOMIALS.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Creating Equations

CREATE EQUATIONS THAT DESCRIBE NUMBERS OR RELATIONSHIPS.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Reasoning with Equations and Inequalities

UNDERSTAND SOLVING EQUATIONS AS A PROCESS OF REASONING AND EXPLAIN THE REASONING.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

SOLVE EQUATIONS AND INEQUALITIES IN ONE VARIABLE.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable.

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (𝑥𝑥 − 𝑝𝑝)2 = 𝑞𝑞 that has the same solutions. Derive the quadratic formula from this form.

Solve quadratic equations by inspection (e.g., for 𝑥𝑥2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝑎𝑎 ± 𝑏𝑏𝑏𝑏 for real numbers a and b.

SOLVE SYSTEMS OF EQUATIONS.

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

REPRESENT AND SOLVE EQUATIONS AND INEQUALITIES GRAPHICALLY.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Seeing Structure in Expressions

INTERPRET THE STRUCTURE OF EXPRESSIONS.

Interpret expressions that represent a quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret 𝑃𝑃(1 + 𝑟𝑟)𝑛𝑛 as the product of P and a factor not depending on P.

Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥𝑥4 − 𝑦𝑦4 as (𝑥𝑥2)2 − (𝑦𝑦2)2, thus recognizing it as a difference of squares that can be factored as (𝑥𝑥2 − 𝑦𝑦2) (𝑥𝑥2 − 𝑦𝑦2).

WRITE EXPRESSIONS IN EQUIVALENT FORMS TO SOLVE PROBLEMS.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15𝑡𝑡 can be rewritten as to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Functions

Building Functions

BUILD A FUNCTION THAT MODELS A RELATIONSHIP BETWEEN TWO QUANTITIES.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

BUILD NEW FUNCTIONS FROM EXISTING FUNCTIONS.

Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) by 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘𝑘𝑘(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), and 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Interpreting Functions

UNDERSTAND THE CONCEPT OF A FUNCTION AND USE FUNCTION NOTATION.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then 𝑓𝑓(𝑥𝑥) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥).

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

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by 𝑓𝑓(0) = 𝑓𝑓(1) = 1, 𝑓𝑓(𝑛𝑛 + 1) = 𝑓𝑓(𝑛𝑛) + 𝑓𝑓(𝑛𝑛 − 1) for 𝑛𝑛 ≥ 1.

INTERPRET FUNCTIONS THAT ARISE IN APPLICATIONS IN TERMS OF THE CONTEXT.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛𝑛) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

ANALYZE FUNCTIONS USING DIFFERENT REPRESENTATIONS.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

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

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Linear, Quadratic, and Exponential Models

CONSTRUCT AND COMPARE LINEAR, QUADRATIC, AND EXPONENTIAL MODELS AND SOLVE PROBLEMS.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

INTERPRET EXPRESSIONS FOR FUNCTIONS IN TERMS OF THE SITUATION THEY MODEL.

Interpret the parameters in a linear or exponential function in terms of a context.

Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials.

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Understand the relationship between zeros and factors of polynomials.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Creating Equations

Create equations that describe numbers or relationships.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve equations and inequalities in one variable.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable.

Use the method of completing the square to transform any quadratic equation in <em>x</em> into an equation of the form <em>(x - p)² = q</em> that has the same solutions. Derive the quadratic formula from this form.

Solve quadratic equations by inspection (e.g., for <em>x² = 49)</em>, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as <em>a ± bi</em> for real numbers <em>a</em> and <em>b</em>.

Solve systems of equations.

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Represent and solve equations and inequalities graphically.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the <em>x</em>-coordinates of the points where the graphs of the equations <em>y = f(x)</em> and <em>y = g(x)</em> intersect are the solutions of the equation <em>f(x) = g(x)</em>; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where <em>f(x)</em> and/or <em>g(x)</em> are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Seeing Structure in Expressions

Interpret the structure of expressions.

Interpret expressions that represent a quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions.

Building Functions

Build a function that models a relationship between two quantities.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Build new functions from existing functions.

Identify the effect on the graph of replacing <em>f(x)</em> by <em>f(x) + k, kf(x), f(kx),</em> and <em>f(x + k)</em> 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Interpreting Functions

Understand the concept of a function and use function notation.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If <em>f</em> is a function and <em>x</em> is an element of its domain, then <em>f(x)</em> denotes the output of <em>f</em> corresponding to the input <em>x</em>. The graph of <em>f</em> is the graph of the equation <em>y = f(x)</em>.

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

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by <em>f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1)</em> for <em>n ≥ 1</em>.

Interpret functions that arise in applications in terms of the context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

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

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Analyze functions using different representations.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

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

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Linear, Quadratic, and Exponential Functions

Construct and compare linear, quadratic, and exponential models and solve problems.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret expressions for functions in terms of the situation they model.

Interpret the parameters in a linear or exponential function in terms of a context.

Quantities

Reason quantitatively and use units to solve problems.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Define appropriate quantities for the purpose of descriptive modeling. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

The Real Number System

Use properties of rational and irrational numbers.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on a single count or measurement variable.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

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

Summarize, represent, and interpret data on two categorical and quantitative variables.

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

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

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Informally assess the fit of a function by plotting and analyzing residuals.

Fit a linear function for a scatter plot that suggests a linear association.

Interpret linear models.

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 of a linear fit.

Distinguish between correlation and causation.

Number and Quantity

Quantities

REASON QUANTITATIVELY AND USE UNITS TO SOLVE PROBLEMS.

Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Define appropriate quantities for the purpose of descriptive modeling. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

The Real Number System

USE PROPERTIES OF RATIONAL AND IRRATIONAL NUMBERS.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Statistics

Interpreting categorical and quantitative data

SUMMARIZE, REPRESENT, AND INTERPRET DATA ON TWO CATEGORICAL AND QUANTITATIVE VARIABLES.

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

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Informally assess the fit of a function by plotting and analyzing residuals.

Fit a linear function for a scatter plot that suggests a linear association.

INTERPRET LINEAR MODELS

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 of a linear fit.

Distinguish between correlation and causation.

Algebra

Arithmetic with Polynomials and Rational Expressions

UNDERSTAND THE RELATIONSHIP BETWEEN ZEROS AND FACTORS OF POLYNOMIALS.

Know and apply the Remainder Theorem: For a polynomial 𝒑𝒑(𝒙𝒙) and a number a, the remainder on division by 𝑥𝑥 − 𝑎𝑎 is 𝑝𝑝(𝑎𝑎) so 𝑝𝑝(𝑎𝑎) = 0 if and only if (𝑥𝑥 − 𝑎𝑎) is a factor of 𝑝𝑝(𝑥𝑥).

dentify zeros of polynomials when suitable factorizations are available; use the zeros to construct a rough graph of the function defined by the polynomial.

USE POLYNOMIAL IDENTITIES TO SOLVE PROBLEMS.

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (𝑥𝑥2 + 𝑦𝑦2)2 = (𝑥𝑥2 − 𝑦𝑦2)2 + (2𝑥𝑥𝑥𝑥)2 can be used to generate Pythagorean triples.

REWRITE RATIONAL EXPRESSIONS.

Rewrite simple rational expressions in different forms; write 𝑎𝑎(𝑥𝑥) 𝑏𝑏(𝑥𝑥) in the form 𝑞𝑞(𝑥𝑥) + 𝑟𝑟(𝑥𝑥) 𝑏𝑏(𝑥𝑥) , where 𝑎𝑎(𝑥𝑥), 𝑏𝑏(𝑥𝑥), 𝑞𝑞(𝑥𝑥), and 𝑟𝑟(𝑥𝑥) are polynomials with the degree of 𝑟𝑟(𝑥𝑥) less than the degree of 𝑏𝑏(𝑥𝑥), using inspection, long divi

Creating Equations

CREATE EQUATIONS THAT DESCRIBE NUMBERS OR RELATIONSHIPS.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Reasoning with Equations and Inequalities

UNDERSTAND SOLVING EQUATIONS AS A PROCESS OF REASONING AND EXPLAIN THE REASONING.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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

SOLVE EQUATIONS AND INEQUALITIES IN ONE VARIABLE.

Solve quadratic equations in one variable.

Solve quadratic equations by inspection (e.g., for 𝑥𝑥2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝑎𝑎 ± 𝑏𝑏𝑏𝑏 for real numbers a and

SOLVE SYSTEMS OF EQUATIONS.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line 𝑦𝑦 = −3𝑥𝑥 and the circle 𝑥𝑥2 + 𝑦𝑦2 = 3.

REPRESENT AND SOLVE EQUATIONS AND INEQUALITIES GRAPHICALLY.

Explain why the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

Seeing Structure in Expressions

INTERPRET THE STRUCTURE OF EXPRESSIONS.

Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥𝑥4 − 𝑦𝑦4 as (𝑥𝑥2)2 − (𝑦𝑦2)2, thus recognizing it as a difference of squares that can be factored as (𝑥𝑥2 − 𝑦𝑦2) (𝑥𝑥2 − 𝑦𝑦2).

WRITE EXPRESSIONS IN EQUIVALENT FORMS TO SOLVE PROBLEMS.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15𝑡𝑡 can be rewritten as to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

Functions

Building Functions

BUILD A FUNCTION THAT MODELS A RELATIONSHIP BETWEEN TWO QUANTITIES.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

BUILD NEW FUNCTIONS FROM EXISTING FUNCTIONS.

Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) by 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘𝑘𝑘(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), and 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Find inverse functions.

Solve an equation of the form 𝑓𝑓(𝑥𝑥) = 𝑐𝑐 for a simple function f that has an inverse and write an expression for the inverse. For example, 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 or 𝑓𝑓(𝑥𝑥) = 𝑥𝑥+1 𝑥𝑥−1 for 𝑥𝑥 ≠ 1.

Interpreting Functions

UNDERSTAND THE CONCEPT OF A FUNCTION AND USE FUNCTION NOTATION.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by 𝑓𝑓(0) = 𝑓𝑓(1) = 1, 𝑓𝑓(𝑛𝑛 + 1) = 𝑓𝑓(𝑛𝑛) + 𝑓𝑓(𝑛𝑛 − 1) for 𝑛𝑛 ≥ 1.

INTERPRET FUNCTIONS THAT ARISE IN APPLICATIONS IN TERMS OF THE CONTEXT.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

ANALYZE FUNCTIONS USING DIFFERENT REPRESENTATIONS.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as 𝑦𝑦 = (1.02)𝑡𝑡, 𝑦𝑦 = (0.97)𝑡𝑡, 𝑦𝑦 = (1.01)12𝑡𝑡, 𝑦𝑦 = (1.2)𝑡𝑡 10

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Linear, Quadratic, and Exponential Models

CONSTRUCT AND COMPARE LINEAR, QUADRATIC, AND EXPONENTIAL MODELS AND SOLVE PROBLEMS.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

For exponential models, express as a logarithm the solution to 𝒂𝒂𝒂𝒂𝒄𝒄𝒄𝒄 = 𝒅𝒅 where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

INTERPRET EXPRESSIONS FOR FUNCTIONS IN TERMS OF THE SITUATION THEY MODEL.

Interpret the parameters in a linear or exponential function in terms of a context.

Trigonometric Functions

EXTEND THE DOMAIN OF TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE.

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

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

MODEL PERIODIC PHENOMENA WITH TRIGONOMETRIC FUNCTIONS.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

PROVE AND APPLY TRIGONOMETRIC IDENTITIES.

Prove the Pythagorean identity sin2(𝜃𝜃) + cos2(𝜃𝜃) = 1 and use it to find sin(𝜃𝜃), cos(𝜃𝜃), or tan(𝜃𝜃) given sin(𝜃𝜃), cos(𝜃𝜃), or tan(𝜃𝜃) and the quadrant of the angle.

Model periodic phenomena with trigonometric functions.

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.

Know and apply the Remainder Theorem: For a polynomial <em>p(x)</em> and a number <em>a</em>, the remainder on division by <em>x - a</em> is <em>p(a)</em>, so <em>p(a) = 0</em> if and only if <em>(x - a)</em> is a factor of <em>p(x)</em>.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Use polynomial identities to solve problems.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite rational expressions.

Rewrite simple rational expressions in different forms; write <em>a(x)/b(x)</em> in the form <em>q(x) + r(x)/b(x)</em>, where <em>a(x), b(x), q(x)</em>, and <em>r(x)</em> are polynomials with the degree of <em>r(x)</em> less than the degree of <em>b(x)</em>, using inspection, long division, or, for the more complicated examples, a computer algebra system.

Creating Equations

Create equations that describe numbers or relationships

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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

Solve equations and inequalities in one variable.

Solve quadratic equations in one variable.

Solve quadratic equations by inspection (e.g., for <em>x² = 49</em>, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as <em>a ± bi</em> for real numbers <em>a</em> and <em>b</em>.

Solve systems of equations

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

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

Represent and solve equations and inequalities graphically.

Explain why the x-coordinates of the points where the graphs of the equations <em>y = f(x)</em> and <em>y = g(x)</em> intersect are the solutions of the equation <em>f(x) = g(x)</em>; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where <em>f(x)</em> and/or <em>g(x)</em> are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Seeing Structure in Expressions

Interpret the structure of expressions

Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Building Functions

Build a function that models a relationship between two quantities.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Combine standard function types using arithmetic operations.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Build new functions from existing functions.

Identify the effect on the graph of replacing <em>f(x) by f(x) + k, kf(x), f(kx)</em>, and <em>f(x + k)</em> for specific values of <em>k</em> (both positive and negative); find the value of <em>k</em> given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Find inverse functions.

Solve an equation of the form <em>f(x) = c</em> for a simple function <em>f</em> that has an inverse and write an expression for the inverse.

Interpreting Functions

Understand the concept of a function and use function notation.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Interpret functions that arise in applications in terms of the context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Analyze functions using different representations.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Use the properties of exponents to interpret expressions for exponential functions.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Linear, Quadratic, and Exponential Models

Construct and compare linear, quadratic, and exponential models and solve problems.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

For exponential models, express as a logarithm the solution to <em>ab^ct = d</em> where <em>a, c,</em> and <em>d</em> are numbers and the base <em>b</em> is 2, 10, or <em>e</em>; evaluate the logarithm using technology.

Interpret expressions for functions in terms of the situation they model.

Interpret the parameters in a linear or exponential function in terms of a context.

Trigonometric Functions

Extend the domain of trigonometric functions using the unit circle.

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

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Prove and apply trigonometric identities.

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section.

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

The Complex Number System

Perform arithmetic operations with complex numbers.

Know there is a complex number <em>i</em> such that <em>i² =-1</em>, and every complex number has the form <em>a + bi</em> with <em>a</em> and <em>b</em> real.

Use the relation <em>i² = -1</em> and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use complex numbers in polynomial identities and equations.

Solve quadratic equations with real coefficients that have complex solutions.

Quantities

Reason quantitatively and use units to solve problems.

Define appropriate quantities for the purpose of descriptive modeling.

The Real Number System

Extend the properties of exponents to rational exponents.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Conditional Probability and the Rules of Probability

Understand independence and conditional probability and use them to interpret data

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Understand the conditional probability of <em>A</em> given <em>B</em> as <em>P(A</em> and <em>B)/P(B)</em>, and interpret independence of <em>A</em> and <em>B</em> as saying that the conditional probability of <em>A</em> given <em>B</em> is the same as the probability of <em>A</em>, and the conditional probability of <em>B</em> given <em>A</em> is the same as the probability of <em>B</em>.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Find the conditional probability of <em>A</em> given <em>B</em> as the fraction of <em>B</em>'s outcomes that also belong to <em>A</em>, and interpret the answer in terms of the model.

Apply the Addition Rule, <em>P(A</em> or <em>B) = P(A) + P(B) - P(A</em> and <em>B</em>), and interpret the answer in terms of the model.

Making Inferences and Justifying Conclusions

Understand and evaluate random processes underlying statistical experiments.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Make inferences and justify conclusions from sample surveys, experiments and observational studies.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

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

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Evaluate reports based on data.

Interpreting categorical and quantitative data

Summarize, represent, and interpret data on a single count or measurement variable.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Summarize, represent, and interpret data on two categorical and quantitative variables.

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

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Number and Quantity

The Complex Number System

PERFORM ARITHMETIC OPERATIONS WITH COMPLEX NUMBERS.

Know there is a complex number i such that 𝑖𝑖2 = −1, and every complex number has the form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 with a and b real.

Use the relation 𝑖𝑖2 = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

USE COMPLEX NUMBERS IN POLYNOMIAL IDENTITIES AND EQUATIONS.

Solve quadratic equations with real coefficients that have complex solutions.

Quantities

REASON QUANTITATIVELY AND USE UNITS TO SOLVE PROBLEMS.

Define appropriate quantities for the purpose of descriptive modeling.

The Real Number System

EXTEND THE PROPERTIES OF EXPONENTS TO RATIONAL EXPONENTS.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1 3 to be the cube root of 5 because we want to hold, so must equal 5.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Statistics

Interpreting categorical and quantitative data

SUMMARIZE, REPRESENT, AND INTERPRET DATA ON TWO CATEGORICAL AND QUANTITATIVE VARIABLES.

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

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Circles

UNDERSTAND AND APPLY THEOREMS ABOUT CIRCLES.

Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

FIND ARC LENGTHS AND AREAS OF SECTORS OF CIRCLES.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Congruence

EXPERIMENT WITH TRANSFORMATIONS IN THE PLANE.

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

UNDERSTAND CONGRUENCE IN TERMS OF RIGID MOTIONS.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

PROVE GEOMETRIC THEOREMS.

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

MAKE GEOMETRIC CONSTRUCTIONS.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Geometric measurement and dimension

EXPLAIN VOLUME FORMULAS AND USE THEM TO SOLVE PROBLEMS.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

VISUALIZE RELATIONSHIPS BETWEEN TWO-DIMENSIONAL AND THREEDIMENSIONAL OBJECTS.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Expressing Geometric Properties with Equations

TRANSLATE BETWEEN THE GEOMETRIC DESCRIPTION AND THE EQUATION FOR A CONIC SECTION.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

USE COORDINATES TO PROVE SIMPLE GEOMETRIC THEOREMS ALGEBRAICALLY

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point lies on the circle centered at the origin and containing the point (0, 2).

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Modeling with Geometry

APPLY GEOMETRIC CONCEPTS IN MODELING SITUATIONS

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Similarity, Right Triangles, and Trigonometry

UNDERSTAND SIMILARITY IN TERMS OF SIMILARITY TRANSFORMATIONS.

Verify experimentally the properties of dilations given by a center and a scale factor.

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

PROVE THEOREMS INVOLVING SIMILARITY.

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

DEFINE TRIGONOMETRIC RATIOS AND SOLVE PROBLEMS INVOLVING RIGHT TRIANGLES.

Understand 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 right triangles in applied problems.

Understand and apply theorems about circles.

Circles

Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Find arc lengths and areas of sectors of circles.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Congruence

Experiment with transformations in the plane.

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems.

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Make geometric constructions

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Geometric Measurement and Dimension

Explain volume formulas and use them to solve problems.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Modeling with Geometry

Apply geometric concepts in modeling situations.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations.

Verify experimentally the properties of dilations given by a center and a scale factor.

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity.

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Define trigonometric ratios and solve problems involving right triangles.

Understand 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 right triangles in applied problems.

Algebraic Thinking

Reasoning with Data, Statistics, and Probability

Geometric Reasoning

LOOK FOR AND MAKE USE OF STRUCTURE TO REWRITE EXPRESSIONS IN EQUIVALENT FORMS AND REASON ABOUT THEIR PROPERTIES.

Rewrite and interpret linear and exponential expressions to explain key properties of a relationship.

Interpret components of linear and exponential expressions (e.g., coefficients, constants, bases, and exponents) as single entities and explain their meaning in context.

Rewrite linear and exponential expressions in different equivalent forms to highlight and explain properties of the relationship (e.g., growth or decay rate, initial value, rate of change).

Compare equivalent forms of linear and exponential expressions to determine which is most useful for understanding or solving a problem in context.

MAKE SENSE OF AND SOLVE EQUATIONS, INEQUALITIES, AND SYSTEMS OF EQUATIONS OR INEQUALITIES.

Create and solve one-variable equations, justify the solution process, and interpret the solution(s) in context.

Create linear and exponential equations in one variable to represent relationships between quantities.

Solve linear equations in one variable using flexible and efficient strategies. Justify the solution process using logical steps and explanations. Interpret the meaning of the solution in context.

Solve absolute value equations in one variable where the absolute value expression is isolated on one side of the equation. Justify the solution process and interpret the solution(s) in context.

Solve exponential equations where both sides have comparable exponential expressions, requiring at most two steps to rewrite expressions to the same base, and explain the reasoning using properties of exponents.

Create and solve one-variable inequalities, justify the solution process, and interpret the solutions in context.

Create inequalities in one variable to represent constraints or conditions.

Solve linear inequalities in one variable, including those with variables on both sides, using flexible and efficient strategies. Represent the solution set using a number line and inequality notation.

Solve absolute value inequalities in one variable where the absolute value expression is isolated on one side of the inequality. Represent the solution set using compound inequalities and a number line.

Interpret the meaning of the solution set in the context of the problem, including what values are reasonable within the given situation.

Create linear and exponential equations in two variables, graph them, and use those graphs to represent and solve problems in context.

Create linear and exponential equations in two variables to model relationships, and graph them on the coordinate plane.

Use the understanding that the graph of an equation represents the set of all its solutions to identify solutions, describe patterns, and interpret the relationship between variables.

Create, solve, and analyze systems of two linear equations in two variables in context.

Create systems of linear equations to represent relationships.

Solve systems of linear equations using algebraic methods (i.e., substitution and linear combination).

Approximate or verify solutions to systems of equations by graphing and identifying the point of intersection.

Interpret the meaning of the solution in context.

Create linear inequalities in two variables to represent constraints in context and graph the solution set as a half-plane within the coordinate plane.

Represent and analyze systems of two linear inequalities in two variables in context.

Create systems of linear inequalities in two variables to represent constraints and relationships.

Graph the solution set to a system of linear inequalities as the intersection of the corresponding half-planes.

Interpret the meaning of the solution set in context and determine whether given ordered pairs are solutions to the system.

Model and solve optimization problems using systems of linear inequalities.

Represent contexts involving constraints and an objective to optimize (i.e., maximize or minimize) using a system of linear inequalities.

Graph the feasible region and identify solutions that satisfy all constraints.

Identify and justify the optimal solution(s) based on the objective, and interpret the meaning in context.

Use technology to solve systems of equations consisting of a linear equation and an exponential in two variables and interpret the solution within a given context.

REASON ABOUT FUNCTIONS.

Identify, represent, and analyze functions in terms of their domain and range.

Determine whether a relationship is a function by verifying that each element of the domain is assigned to exactly one element of the range using tables, graphs, and equations.

Determine whether a function is one-to-one by analyzing tables, graphs, or equations, and justify the reasoning based on the relationship between inputs and outputs.

Use function notation to evaluate functions for specific input values, x, and describe how the output changes with respect to the input. Interpret the meaning of specific values and patterns in context.

Represent the graph of a function as the set of ordered pairs (x, ) and explain how it illustrates the relationship between domain and range.

Calculate or estimate and interpret the average rate of change over a specified interval of linear and exponential functions presented graphically, symbolically, or in a table.

Compare key properties (i.e., rates of change, intercepts, patterns of growth) of two functions, one linear and one exponential, when each is represented in a different way (i.e., algebraically, graphically, numerically in tables, or by narrative descriptions).

MODEL WITH FUNCTIONS.

Analyze and model arithmetic and geometric sequences using recursive and explicit representations.

Identify whether a sequence is arithmetic or geometric and create an explicit rule to describe the pattern.

Rewrite a recursive representation of an arithmetic or geometric sequence to an explicit rule.

Model contexts involving arithmetic or geometric patterns using an explicit or recursive rule, and use the rule to solve problems.

Represent functions using tables and graphs, and interpret key features in context. Key features include: domain, intercepts, intervals of increase and decrease, positive and negative values, end behavior, asymptote, and points of transition between pieces.

Analyze linear and exponential functions by identifying patterns in tables and graphs. Distinguish between constant and proportional growth.

Extend the understanding of functions to include piecewise-defined and absolute value functions. Represent them in tables and graphs and explain how absolute value functions can be represented as a specific case of piecewise functions.

Relate key features of a function’s graph and table to the characteristics of a context, and justify the appropriateness of a function model based on that context.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Given a relationship in context where the rate of change over equal intervals is constant, identify it as linear and create a linear function to model the relationship.

Given a relationship in context where a quantity changes by a constant percent per unit interval relative to another, identify it as exponential and create an exponential function to model the relationship.

Analyze how changes to a function’s input or output affect the graph and its meaning in context.

Identify how replacing with and changes the graph of a linear, exponential, or absolute value function. Describe the direction of the change and determine the value of k when given a graph.

Explain how changing the input or output of a function affects the quantities represented in context. Use the structure and features of the parent function to justify how these changes impact the initial context.

Find and interpret the inverse of a linear function in context.

Show that a linear function is one-to-one and therefore has an inverse that is also a function.

Find the inverse of a linear function represented with an equation or a table.

Interpret the meaning of the inverse in context as a relationship that maps outputs of the original function back to their corresponding inputs.

MAKE SENSE OF STATISTICAL INQUIRY.

Distinguish between correlation and causation.

Understand and apply statistics as a process for making inferences about population parameters using random samples.

Explain the importance of randomization in selecting samples to ensure unbiased and representative data.

Analyze the representativeness of a sample and its implications for making inferences about the population.

Analyze and interpret how data are represented and used in arguments or reports.

Identify key components of data-based arguments, including claims, visualizations (e.g., graphs, tables), and the statistical language used to support conclusions.

Evaluate the accuracy and effectiveness of visualizations in representing data and supporting a claim or conclusion.

Describe how data were collected and whether the method (e.g., sampling, use of randomization) supports valid conclusions.

Plan statistical investigations.

Formulate statistical questions that can be answered with data.

Determine effective data collection methods that enhance accuracy, validity, and representativeness of the data.

DESCRIBE, ANALYZE, AND COMPARE DATA USING VISUAL AND NUMERICAL REPRESENTATIONS TO MODEL SITUATIONS AND DRAW INFERENCES.

Analyze and compare one-variable data visualizations on the real number line to draw conclusions.

Compare multiple data visualizations to identify the unique statistical information each provides and their usefulness in different contexts.

Analyze and compare the shape, center, and spread of data sets using appropriate visualizations.

Construct and interpret two-way frequency tables to analyze associations between two categorical variables.

Use the tables to decide if events are independent

Use the tables to find joint and marginal probabilities.

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

Fit a linear or exponential function to data; use functions fitted to data to solve problems in the context of the data.

Informally assess the fit of a linear or exponential function visually by analyzing the graph and analytically by evaluating the appropriateness of the chosen model in representing the data and context.

Analyze and interpret the key components of linear and exponential models in the context of the data with and without technology.

Interpret slope (rate of change) and the intercept (constant term) of a linear model to describe patterns and relationships in the data.

Analyze the growth factor or average rate of change in exponential models to describe patterns, interpret trends, and make predictions in context.

REASON ABOUT TRANSFORMATIONS AND CONGRUENCE.

Apply definitions of rotations, reflections, and translations to transform figures and use these transformations to verify congruence.

Describe rotations, reflections, and translations as functions that take points in the plane as inputs and give corresponding points as outputs.

Verify experimentally that rigid transformations preserve distance, angle measure, and parallelism, and use this understanding to justify congruence.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

CONJECTURE ABOUT AND VERIFY GEOMETRIC RELATIONSHIPS.

Apply triangle congruence theorems (SSS, SAS, ASA, AAS, HL) to reason logically about geometric diagrams, including verifying congruence and solving for unknown measures.

Understand and apply theorems and criteria for parallel and perpendicular lines.

Use theorems about parallel lines cut by a transversal to identify and apply angle pair relationships, including corresponding, alternate interior, alternate exterior, and consecutive interior angles.

Verify the slope criteria for parallel and perpendicular lines and apply them to solve geometric problems (e.g., determine when lines are parallel or perpendicular and find equations of lines that satisfy given geometric conditions).

Use angle relationships and slope criteria to verify properties of parallelograms.

Use angle relationships from parallel lines and transversals, along with triangle congruence theorems, to verify properties of parallelograms, including congruent opposite sides, congruent opposite angles, and diagonals that bisect each other.

Apply the slope criteria for parallel and perpendicular lines to verify when a quadrilateral is a parallelogram in a coordinate plane.

Algebraic Thinking

Reasoning with Data, Statistics, and Probability

Geometric Reasoning

LOOK FOR AND MAKE USE OF STRUCTURE TO REWRITE EXPRESSIONS IN EQUIVALENT FORMS AND REASON ABOUT THEIR PROPERTIES.

Rewrite and interpret quadratic expressions to explain key properties of a relationship.

Factor quadratic expressions to reveal the zeros of the function, and explain their significance in context.

Complete the square to identify the maximum or minimum value of a relationship and interpret that value in context.

Compare different forms of a quadratic expression (standard, factored, vertex) to determine which form is most useful for interpreting a given situation.

Given a polynomial of degree 3 or higher in factored form, use the Zero Product Property to identify its zeros and sketch a graph of the function.

Add, subtract, and multiply polynomials to rewrite expressions in equivalent forms for clarity, efficiency, or problem-solving applications. Use these operations to represent and analyze polynomial relationships in context. Multiplication of polynomials includes multiplying a binomial and a trinomial.

MAKE SENSE OF AND SOLVE EQUATIONS AND SYSTEMS OF EQUATIONS.

Create and solve quadratic equations in one variable.

Create quadratic equations in one variable to represent relationships between quantities.

Flexibly choose and apply methods to solve quadratic equations based on their initial form (i.e., solve equations by recognizing perfect squares (e.g., ), taking square roots, factoring when possible, completing the square, or applying the quadratic formula when necessary). Justify the selected method by reasoning about the structure of the equation and interpret the solution in context when applicable.

Represent complex solutions as for real numbers a and b.

Solve radical equations in one variable that contain a single radical term. Explain why some solutions may be extraneous and justify whether a given solution is valid based on a given context.

Create, solve, and analyze systems consisting of a linear and a quadratic equation in two variables in context.

Create systems consisting of a linear and a quadratic equation to represent relationships.

Solve these systems exactly and approximately using algebraic methods (i.e., substitution), graphical methods, or tables.

Interpret the meaning of the solution(s) in context.

Use technology to solve systems of equations that include a combination of linear, exponential, quadratic, and/or radical equations. Interpret the solution(s) within a given context.

REASON ABOUT FUNCTIONS.

Analyze and compare the behavior of functions across different families.

Compare linear, quadratic, exponential, polynomial (degree of 3 or higher), and radical functions by analyzing differences in their patterns of change and key characteristics (e.g., symmetry, end behavior). Explain how these differences affect the modeling of various contexts.

Analyze how the number and type of solutions to equations vary across function families and explain how those solutions are represented and interpreted in graphs and equations.

Calculate or estimate and interpret the average rate of change over a specified interval for exponential, quadratic, and radical functions presented graphically, symbolically or as a table.

Compare properties of two quadratic functions each represented in a different way (algebraically, graphically, numerically in tables, or by narrative descriptions), to analyze differences and similarities between the relationships the functions model.

MODEL WITH FUNCTIONS.

Write a quadratic function to model relationships between quantities given a narrative description, table of values, or graph.

Represent and interpret quadratic functions using tables and graphs and apply them in context.

Create tables and graphs to represent quadratic relationships and identify key features. Key features include: intercepts/zeros, symmetry, intervals of increase and decrease, relative extrema, end behavior, domain (if restricted in context), and range.

Relate key features in the table and graph to characteristics of a context.

Justify the appropriateness of a quadratic model by analyzing trends in the table or graph and explaining how the model’s features support predictions or contextual interpretations.

Find and interpret the inverse of quadratic and exponential functions using algebraic and graphical representations.

Restrict the domain of a quadratic to make it one-to-one and express its inverse as a square root function.

Express the inverse of an exponential function as a logarithmic function and evaluate expressions involving logarithms.

Graph a function and its inverse and analyze features (e.g., domain and range, intercepts, points of intersection) and symmetry across the line y = x.

Interpret the meaning of inverse functions in context.

Represent and interpret higher degree polynomial (degree 3 or higher), and radical functions in context using tables and graphs.

Create tables and graphs to represent higher degree polynomial functions and identify key features. Key features include: intercepts/zeros, symmetry, intervals of increase and decrease, relative extrema, and end behavior.

Create tables and graphs to represent radical functions and identify key features. Key features include: domain restrictions, intercepts, intervals of increase and decrease, and end behavior.

Relate key features of higher-degree polynomial and radical functions to contexts.

Analyze how transformations affect the graph of a polynomial function given in vertex form and interpret their meaning in context.

Identify how the graph of a quadratic function changes when is replaced by , , and and determine the value of k from the graph. Describe the type of transformation.

Explain the meaning of each transformation by connecting the structure of the transformed function to the quantities it represents (e.g., how changes affect the vertex, direction of opening, and rate of change).

MAKE SENSE OF STATISTICAL INQUIRY.

Evaluate and refine statistical study designs and critically analyze data-based claims to improve the quality and validity of inferences.

Assess the limitations of a sample or study design in terms of bias, randomness, and representativeness.

Propose modifications to sampling methods or data collection strategies to enhance the validity and generalizability of conclusions.

Identify potential sources of bias or misuse in data collection, representation, or modeling.

Recognize when correlation is misused to imply causation.

DESCRIBE, ANALYZE, AND COMPARE DATA USING VISUAL AND NUMERICAL REPRESENTATIONS TO MODEL SITUATIONS AND DRAW INFERENCES.

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

Fit a quadratic, higher degree polynomial, or radical function to data. Use functions fitted to data to solve problems in the context of the data.

Informally assess the fit of a quadratic, higher degree polynomial, or radical function visually by analyzing the graph and analytically by evaluating the appropriateness of the chosen model in representing the data and context.

Construct and interpret two-way frequency tables to analyze associations between two categorical variables.

Use the tables to decide if events are independent

Approximate conditional probabilities and explain their meaning in context, including the relationship between conditional probability and independence.

REASON ABOUT TRANSFORMATIONS AND SIMILARITY.

Describe the effect of dilations on two-dimensional figures using coordinates, including how scale factor and center of dilation impact side lengths and angle measures.

Use similarity transformations to determine and justify triangle similarity.

Determine whether two triangles are similar by identifying a sequence of similarity transformations (rotations, reflections, translations, and dilations) that relates one triangle onto another and aligns corresponding parts.

Justify triangle similarity by explaining how similarity transformations establish that corresponding angles are congruent and corresponding sides are proportional.

Apply properties of similar triangles to solve geometric problems in context.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

REASON ABOUT RIGHT TRIANGLE RELATIONSHIPS.

Use similarity of right triangles to explain and define trigonometric ratios (sine, cosine, and tangent) for acute angles as properties of the angles, based on side ratios.

Justify the side length relationships in special right triangles (30°-60°-90° and 45°-45°-90°).

Use geometric reasoning (e.g., dissecting an equilateral triangle or square) to explain why the side length relationship in special right triangles hold.

Describe the consistent ratios of side lengths in these triangles (e.g., 1:1: and 1: :2), and explain how these ratios relate to the angle measures.

Use trigonometric ratios, the Pythagorean Theorem, and special right triangle relationships to solve right triangles in applied contexts.

MAKE SENSE OF CIRCLES AND THEIR PROPERTIES.

Apply similarity to develop and use formulas involving arc length, radian measure, and sector area.

Explain and demonstrate, using similarity, that arc length is proportional to the radius of a circle.

Calculate the radian measure of a central angle as the ratio of arc length to radius.

Apply the formula for the area of a sector using radian measure.

Create the equation of a circle given the center and radius; identify the center and radius of a circle given the equation.

MODEL WITH GEOMETRY.

Use geometric shapes, their measures, and their properties to represent real world objects, and solve related authentic modeling and design problems.

Rearrange geometric formulas to isolate and interpret a quantity of interest in context (e.g., solving for height in a volume formula), using reasoning aligned with solving equations and modeling with mathematics.

REASON WITH RATIONAL EXPONENTS.

Apply the properties of exponents to generate equivalent numerical expressions involving radicals and rational exponents.

REASON ABOUT THE COMPLEX NUMBER SYSTEM.

Use the knowledge that to write complex numbers in standard form, , and use them to represent solutions to quadratics that have no real solutions.

Number and Operation Sense

Algebra

Algebra

ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSION

ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS

Understand the relationship between zeroes and factors of polynomials.

Know and apply the Remainder Theorem: For a polynomial 𝒑𝒑(𝒙𝒙) and a number a, the remainder on division by 𝒙𝒙 − 𝒂𝒂 is 𝒑𝒑(𝒂𝒂), so 𝒑𝒑(𝒂𝒂) = 𝟎𝟎 if and only if (𝒙𝒙 − 𝒂𝒂) is a factor of 𝒑𝒑(𝒙𝒙).

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Use polynomial identities to solve problems.

Use polynomial identities to solve problems.

Know and apply the Binomial Theorem for the expansion of (𝒙𝒙 + 𝒚𝒚)𝒏𝒏 in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

Know and apply the Binomial Theorem for the expansion of (𝒙𝒙 + 𝒚𝒚)𝒏𝒏 in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

Rewrite rational expressions.

Rewrite simple rational expressions in different forms.

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions.

CREATING EQUATIONS

Create equations that describe numbers or relationships.

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

Create polynomial equations given roots.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

REASONING WITH EQUATIONS AND INEQUALITIES

Solve equations and inequalities in one variable.

Solve equations and inequalities on one variable.

Solve systems of equations.

Solve systems of equations comprised of various combinations of all algebraic and transcendental functions in two variables.

SEEING STRUCTURE IN EXPRESSIONS

SEEING STRUCTURE IN EXPRESSIONS

SEEING STRUCTURE IN EXPRESSIONS

Interpret the structure of expressions.

Interpret the structure of expressions.

Use the structure of an expression to identify ways to rewrite it.

Analyze the structure of the general form of a second degree equation, 𝑨𝑨𝒙𝒙𝟐𝟐 + 𝑩𝑩𝑩𝑩𝑩𝑩 + 𝑪𝑪𝒚𝒚𝟐𝟐 + 𝑫𝑫𝑫𝑫 + 𝑬𝑬𝑬𝑬 + 𝑭𝑭 = 𝟎𝟎, to identify the conic section represented by the equation.

Write expressions in equivalent forms to solve problems.

Write expressions in equivalent forms to solve problems.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Use the properties of exponents to transform expressions for exponential functions. For example, the expression 𝟏𝟏. 𝟏𝟏𝟏𝟏𝒕𝒕 can be rewritten as to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Choose and produce an equivalent form of a second degree equation, 𝐴𝐴𝑥𝑥2 + 𝐶𝐶𝑦𝑦2 + 𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 + 𝐹𝐹 = 0, to reveal and explain properties of the conic section represented by the equation.

Translate between the standard and general form, 𝐴𝐴𝑥𝑥2 + 𝐶𝐶𝑦𝑦2 + 𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 + 𝐹𝐹 = 0, of a second degree equation.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Express the sums in a series using sigma notation.

Determine the sum, if it exists, of an infinite geometric series.

Functions

Functions

Functions

Functions

BUILDING FUNCTIONS

BUILDING FUNCTIONS

Build a function that models a relationship between two quantities.

Write a function that describes a relationship between two quantities, including more complex functions.

Compose functions. For example, if 𝑇𝑇(𝑦𝑦) is the temperature in the atmosphere as a function of height, and ℎ(𝑡𝑡) is the height of a weather balloon as a function of time, then is the temperature at the location of the weather balloon as a function of time.

Build new functions from existing functions.

Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) by 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘𝑘𝑘(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), and 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Find inverse functions.

Verify by composition that one function is the inverse of another.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

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

Build inverse trigonometric functions.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Build a function that models a relationship between two quantities.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Write sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

INTERPRETING FUNCTIONS

INTERPRETING FUNCTIONS

INTERPRETING FUNCTIONS

Understand the concept of function and use function notation.

Understand the concept of function and use function notation.

Understand the concept of a function and use function notation.

Understand the concept of limit of a function.

Extend evaluating functions to include operations with composite functions, e.g. 𝑓𝑓(𝑥𝑥 + 2) − 𝑓𝑓(𝑥𝑥).

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Interpret functions that arise in application in terms of context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities; sketch graphs showing key features given a verbal description of the relationship.

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

Analyze functions using different representations.

Analyze functions using different representations.

Analyze functions using different representations.

Describe the behavior of a sequence.

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

Determine the end behavior of the graph of a polynomial function using the degree and leading coefficient.

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

Interpret the behavior of the graph of a function using the concept of limits.

Estimate limits algebraically, numerically and graphically.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables or by a verbal description).

INEAR, QUADRATIC, AND EXPONENTIAL MODELS

LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

Construct and compare linear, quadratic, and exponential models and solve problems.

Construct and compare linear, quadratic, and exponential models and solve problems.

Distinguish between situations that can be modeled with exponential functions and logistic functions.

Use properties of logarithms, including both common and natural logarithms, to rewrite and solve exponential models.

Interpret expressions for functions in terms of the situation they model.

Interpret the parameters in a logistic function in terms of a context.

Build and interpret logistic functions to model real-world problems.

Sketch and analyze the graphs of logistic functions.

Compare and contrast the exponential, logarithmic, and logistic models.

Apply understanding of logarithmic and logistic functions to solve real-world problems.

TRIGONOMETRIC FUNCTIONS

Extend the domain of trigonometric functions using the unit circle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Use special triangles to determine geometrically the values of sine, cosine, tangent for 𝜋𝜋 3 , 𝜋𝜋 4 , 𝑎𝑎𝑎𝑎𝑎𝑎 𝜋𝜋 6 and use the unit circle to express the values of sine, cosine

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions.

Choose trigonometric functions to model real world phenomena.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology and interpret them in context.

Prove and apply trigonometric identities.

Use trigonometric identities to rewrite expressions and as a tool when solving trigonometric equations.

Geometry

Geometry

Geometry

GEOMETRIC MEASUREMENT AND DIMENSION

Visualize relationships between two-dimensional and three-dimensional objects.

Identify the shapes of two-dimensional cross-sections of a right double cone.

EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS

EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS

Translate between the geometric description and the equation for a conic section.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

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

Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Polar Coordinates.

Plot points on a polar coordinate grid.

Understand the relationship between polar coordinates and Cartesian coordinates.

Convert between polar and rectangular coordinates.

Polar Equations.

Convert equations between polar and rectangular forms.

Graph polar equations by hand and using technology.

Solve systems of polar equations.

SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY

Apply trigonometry to general triangles.

Prove the Laws of Sines and Cosines and use them to solve problems.

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying, resultant forces).

Number and Quantity

Number and Quantity

Number and Quantity

Number and Quantity

Clarifications/Examples:

Sequences and Series

Identify real zeros of polynomials.

Logarithmic/Exponential Functions

Clarifications/Examples:

Function families to which this standard applies:

Indicate with an arrow on the curve the direction in which the curve is traced as t increases.

Clarifications/Examples:

Logarithmic/Exponential Functions

Trigonometric Functions

Identify any function as an even, an odd function or neither given a graphic or algebraic representation.

Logarithmic/Exponential Functions

Clarifications/Examples:

Refer to the wording of the standard.

Clarifications/Examples:

Rational Functions

Refer to the wording of the standard.

Sketch polynomials of higher degree from factored form, using x- and y-intercepts, end behavior and degree.

Explore systems comprised of functions from two different function families (e.g. Solve the system 𝑦𝑦 = 𝑥𝑥2 and 𝑦𝑦 = 2 cos 𝑥𝑥).

Clarifications/Examples:

Logarithmic/Exponential Functions

Use various algebraic techniques for evaluating limits.

Function families to which this standard applies:

Trigonometric Functions

Logarithmic/Exponential Functions

Refer to the wording of the standard.

Make connections between the structure of a polar equation and the shape of the corresponding graph.

Trigonometric Functions

Function families to which this standard applies:

Write the domain of trigonometric functions under transformations (e.g. 𝑦𝑦 = tan 𝑥𝑥 versus 𝑦𝑦 = tan(2𝑥𝑥)).

Clarifications/Examples:

Polynomial Functions

Limits

Graph inverse trigonometric functions and identify the related principal values.

Discuss relationships between domain, range, and asymptotes.

Include equations and inequalities that contain combinations of various algebraic and transcendental functions.

Clarifications/Examples:

Clarifications/Examples:

Include sequences that do not have a simple defining equation.

Clarifications/Examples:

Refer to the wording of the standard.

Factor polynomials to simplify rational expressions.

Function families to which this standard applies:

Rational Functions

Understand the concept of eccentricity of conic sections, and understand the eccentricity of the ellipse and the hyperbola.

Rational Functions

Function families to which this standard applies:

Build from unit circle.

Use the Factor Theorem.

Use the algebra rules for finite sums to evaluate expressions written using sigma notation

Logarithmic/Exponential Functions

Clarifications/Examples:

Clarifications/Examples:

Understand the similarities and differences between linear functions and arithmetic sequences.

Refer to the wording of the standard.

Logarithmic/Exponential Functions

Clarifications/Examples:

Function families to which this standard applies:

Use the double angle and half angle identities for trigonometric functions to simplify, verify, and solve expressions and equations involving sine, cosine, and tangent.

Include parabolas that have a horizontal axis of symmetry and parabolas that have a vertical axis of symmetry.

Clarifications/Examples:

Function families to which this standard applies:

Rational Functions

Understand the concept of eccentricity of conic sections, and understand the eccentricity of the parabola.

Trigonometric Functions

Refer to the wording of the standard.

Clarifications/Examples:

Trigonometric Functions

Clarifications/Examples:

Determine resultant vectors and interpret them in context.

Radical Power Functions

Rewrite complex fractions as rational expressions .

Function families to which this standard applies:

Trigonometric Functions

Trigonometric Functions

Radical Power Functions

Clarifications/Examples:

Radical Power Functions

Polar Coordinate System and Polar Equations

Refer to the wording of the standard.

Trigonometric Functions

Radical Power Functions

Refer to the wording of the standard.

Clarifications/Examples:

Rational Functions

Clarifications/Examples:

Logarithmic/Exponential Functions

Radical Power Functions

Understand that a single point on the polar coordinate plane has more than one set of polar coordinates that can be used to identify the location of the point.

Function families to which this standard applies:

Refer to the wording of the standard.

Polynomial Functions

Clarifications/Examples:

Clarifications/Examples:

Clarifications/Examples:

Trigonometric Functions

Emphasize Pythagorean identities.

Clarifications/Examples:

Function families to which this standard applies:

Rational Functions

Clarifications/Examples:

Solve exponential equations using logarithms.

Define the six trigonometric functions in terms of coordinates from the unit circle.

Clarifications/Examples:

Interpret the parameters A, B, and C in expressions of the form 𝑦𝑦 = 𝐶𝐶 1+𝐴𝐴𝑒𝑒−𝐵𝐵𝐵𝐵, in terms of a context.

Explain how to recognize asymptotic behavior given various representations of a function (algebraic, numeric and graphic).

Refer to the wording of the standard.

Rational Functions

Given the numeric or graphic representation of an invertible function, produce the graphic and numeric representation of the inverse.

Clarifications/Examples:

Write position vectors from initial and terminal points.

Estimate points of inflection.

Clarifications/Examples:

Clarifications/Examples:

Radical Power Functions

Identify transformations from parent functions.

Clarifications/Examples:

Radical Power Functions

Use equations and inequalities that arise from any type of function to solve problems.

Include composite functions (e.g. 𝑓𝑓(𝑥𝑥) = sin(2𝑥𝑥)).

Clarifications/Examples:

Factor expressions to include using the sum and difference of cubes (e.g. Factor 𝑥𝑥6 − 27𝑦𝑦3).

Radical Power Functions

Understand the relationship between right triangle trigonometric ratios and trigonometric functions.

Polynomial Functions

Identify key features of a conic section.

Discuss the domain of all function types including composite and inverse functions.

Limit to 𝑛𝑛 < 6.

Rational Functions

Polynomial Functions

Radical Power Functions

Clarifications/Examples:

Function families to which this standard applies:

Interpret the key features of the graph of any function in terms of context.

Polynomial Functions

Clarifications/Examples:

Vectors and Matrices

Clarifications/Examples:

Clarifications/Examples:

Clarifications/Examples:

Refer to the wording of the standard.

Rational Functions

Refer to the wording of the standard.

Clarifications/Examples:

Logarithmic/Exponential Functions

Function families to which this standard applies:

Clarifications/Examples:

Understand and apply the locus definitions of the ellipse and the hyperbola.

Polynomial Functions

Trigonometric Functions

Clarifications/Examples:

Function families to which this standard applies:

Function families to which this standard applies:

Clarifications/Examples:

Use graphs to estimate limits.

Rational Functions

Refer to the wording of the standard.

Clarifications/Examples:

Function families to which this standard applies:

Understand the concept of eccentricity of conic sections, and understand the eccentricity of the circle.

Rational Functions

Refer to the wording of the standard.

Polynomial Functions

Analyze vectors in terms of their horizontal and vertical components.

Include equations that contain composite expressions.

Function families to which this standard applies:

Understand and use properties of limits.

Clarifications/Examples:

Use limits to reveal asymptotic or unbounded behavior.

Graph all six trigonometric functions.

Rational Functions

Recognize how the coefficients of the terms transform the conic section.

Create equations and inequalities in one variable involving all algebraic and transcendental functions and piecewise defined functions that combine different types of functions.

Function families to which this standard applies:

Function families to which this standard applies:

Function families to which this standard applies:

Clarifications/Examples:

Clarifications/Examples:

Refer to the wording of the standard.

Refer to the wording of the standard.

Write logarithmic functions as inverses of exponential functions, including both common and natural logarithms.

Use the algebra rules for finite sums to evaluate expressions written using sigma notation .

Clarifications/Examples:

Refer to the wording of the standard.

Recognize and factor an expression in quadratic form (e.g. 𝑒𝑒2𝑥𝑥 − 9𝑒𝑒𝑥𝑥 + 14, 2 cos2 𝑥𝑥 − 3 cos 𝑥𝑥 + 1, 𝑥𝑥6 − 9𝑥𝑥3 + 8).

Understand the connection between the formula for the general term 𝑎𝑎𝑛𝑛 of a given sequence and the related function that describes the relationship between the term number of the sequence and the value of the term.

Limit to binomials with variable coefficient of one or constants less than four.

Clarifications/Examples:

Function families to which this standard applies:

Rational Functions

Use information about end behavior of a polynomial to sketch and identify the possible degree of a polynomial.

Rational Functions

Clarifications/Examples:

Clarifications/Examples:

Clarifications/Examples:

Clarifications/Examples:

Radical Power Functions

Refer to the wording of the standard.

Trigonometric Functions

Clarifications/Examples:

Understand and apply the locus definition for the parabola.

Generate the terms of a sequence given a formula for the nth term of the sequence.

Refer to the wording of the standard.

Clarifications/Examples:

Rational Functions

This could be addressed in a unit on vectors or polar coordinates.

Trigonometric Functions

Logarithmic/Exponential Functions

Logarithmic/Exponential Functions

Clarifications/Examples:

Polynomial Functions

Polynomial Functions

number and type of discontinuities

Clarifications/Examples:

Clarifications/Examples:

Emphasize operations on rational expressions.

Function families to which this standard applies:

Understand the connection between modulus of a complex number and the magnitude of a vector.

Extend student understanding by using more complex situations.

Find inverses of polynomials of the form 𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 − ℎ)𝑛𝑛 + 𝑘𝑘.

Polynomial Functions

Trigonometric Functions

Determine whether a system is in equilibrium.

Rewrite trigonometric expressions based on algebraic structures .

Trigonometric Functions

Use the graph of a sequence to intuitively determine if the sequence converges or diverges.

Clarifications/Examples:

Refer to the wording of the standard.

Clarifications/Examples:

Clarifications/Examples:

Use factors of polynomials to simplify/analyze rational expressions and the graphs of the related functions.

Understand why a particular form of an expression would reveal properties such as zeros, extrema, intercepts, etc.

Clarifications/Examples:

Include piecewise defined functions comprised of different types of functions.

Clarifications/Examples:

Radical Power Functions

Discuss conjugates and how their structure can help when simplifying expressions, verifying identities and solving equations.

Rational Functions

Identify different types of discontinuities.

asymptotes and holes

Determine the best function to represent data on two-quantitative variables by analyzing the context of the data; the behavior of the scatterplot and the fit of the function to the scatterplot.

Function families to which this standard applies:

Rational Functions

Clarifications/Examples:

Polynomial Functions

Factor polynomial expressions, of degree three and higher, completely, over the complex number system (e.g. Factor 𝑥𝑥4 − 3𝑥𝑥2 − 28 to (𝑥𝑥2 + 4) (𝑥𝑥2 − 7) and then to .

Clarifications/Examples:

Polynomial Functions

Clarifications/Examples:

Evaluate trigonometric functions using reference angles.

Function families to which this standard applies:

Solve equations in one variable algebraically, numerically and/or graphically.

Polynomial Functions

Clarifications/Examples:

Refer to the wording of the standard.

Radical Power Functions

Rational Functions

Clarifications/Examples:

Rational Functions

Trigonometric Functions

Function families to which this standard applies:

Clarifications/Examples:

Connect the geometric definition of the parabola to its algebraic equation.

Polynomial Functions

Refer to the wording of the standard.

Clarifications/Examples:

Function families to which this standard applies:

Rewrite rational functions to reveal discontinuities.

Explore algebraic, numeric and graphical methods for solving systems of equations.

Polynomial Functions

Recognize when a function is the composition of two simpler functions (e.g. 𝑓𝑓(𝑥𝑥) = 𝑒𝑒sin𝑥𝑥).

Understand the connection between the Binomial Theorem and an infinite series.

Refer to the wording of the standard.

Find one-sided limits.

Derive the standard form of a parabola, centered at the origin.

Clarifications/Examples:

Clarifications/Examples:

Trigonometric Functions

Rational expressions have no restrictions on degree of the numerator or denominator.

Clarifications/Examples:

Logarithmic/Exponential Functions

Polynomial Functions

Clarifications/Examples:

Clarifications/Examples:

Interpret the reflection of a function over the line 𝑦𝑦 = 𝑥𝑥 as a representation of the inverse of a function.

Model situations involving multiple vectors.

Describe end behavior using appropriate notation, i.e. given an equation, as 𝑥𝑥 → ±∞, 𝑓𝑓(𝑥𝑥) → ±∞.

Graph and analyze the functions that represent a given sequence.

Connect the geometric definition of the circle to its algebraic equation.

Clarifications/Examples:

Trigonometric Functions

Function families to which this standard applies:

Refer to the wording of the standard.

Refer to the wording of the standard.

Radical Power Functions

Refer to the wording of the standard.

Function families to which this standard applies:

Polynomial Functions

Construct a formula for the general term 𝑎𝑎𝑛𝑛 of a given sequence.

Solve real world problems that involve finite series.

Use a table of values to estimate a limit.

Recognize how the coefficients of the terms transform the conic section.

Refer to the wording of the standard.

Clarifications/Examples:

Use right triangles to derive formulas for the direction and magnitude of a vector.

Clarifications/Examples:

Trigonometric Functions

Refer to the wording of the standard.

Function families to which this standard applies:

Use the Remainder Theorem to determine zeros (roots) of a polynomial.

Clarifications/Examples:

Use eccentricity to write equations of parabolas.

Restrict solutions to [0 , 2π ) .

Use the sum of an infinite geometric series to express a repeating decimal as a rational number.

Solve real world problems that involve infinite series.

Logarithmic/Exponential Functions

Clarifications/Examples:

Derive the standard form of an ellipse and a hyperbola, centered at the origin.

Understand that the value of a function at a given point may not be the same as the limit as the function approaches the given point.

Function families to which this standard applies:

Connect prior learning from Algebra 2 to the new learning in Precalculus.

Polynomial Functions

Rational Functions

Clarifications/Examples:

Understand and apply the locus definition for the circle.

Function families to which this standard applies:

Understand vocabulary and notation associated with the study of vectors (magnitude, direction, scalar, components, unit vector, resultant force).

Use eccentricity to write equations of ellipses and hyperbolas.

Refer to the wording of the standard.

Trigonometric Functions

Logarithmic/Exponential Functions

Clarifications/Examples:

Represent constraints of equations that contain composite expressions .

Use the Conjugate Root Theorem as it applies to irrationals.

Polynomial Functions

Trigonometric Functions

Clarifications/Examples:

Understand when limits fail to exist.

Logarithmic/Exponential Functions

Understand why constraints exist for certain equations and inequalities and for systems of equations and inequalities.

Explore rate of change over various size intervals.

Trigonometric Functions

Radical Power Functions

Function families to which this standard applies:

Clarifications/Examples:

Clarifications/Examples:

Radical Power Functions

Produce the standard form of a second degree equation given the general form.

Clarifications/Examples:

Function families to which this standard applies:

Polynomial Functions

Identify the domain and range of a composite function.

Function families to which this standard applies:

Radical Power Functions

zeros

Produce an equivalent form of a rational expression to reveal information about the behavior of the graph of the related function.

Clarifications/Examples:

Logarithmic/Exponential Functions

Connect the geometric definitions of the ellipse and hyperbola to their algebraic equations.

Parametric Equations

Algebra

Understand similarities and differences between exponential functions and geometric sequences.

Clarifications/Examples:

Clarifications/Examples:

Logarithmic/Exponential Functions

Limit to 𝑛𝑛 ≤ 5, and binomials with variables coefficient of one, or constants less than four.

Logarithmic/Exponential Functions

Build prerequisite skills needed to simplify difference quotient (e.g. Given 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 3𝑥𝑥 + 1 find 𝑓𝑓(𝑥𝑥 + ℎ)).

Include a variety of sequences that are neither arithmetic nor geometric.

Function families to which this standard applies:

Understand the relationship between the degree of a polynomial and the number and nature of the zeros of the polynomial.

Use eccentricity to write equations of circles.

Function families to which this standard applies:

Determine if a sequence is increasing, decreasing, or monotonic.

Radical Power Functions

Radical Power Functions

Add vectors symbolically and graphically.

Radical Power Functions

Evaluate the discriminant, 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴, of the general form of a second degree equation, 𝐴𝐴𝑥𝑥2 + 𝐵𝐵𝐵𝐵𝐵𝐵 + 𝐶𝐶𝑦𝑦2 + 𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 + 𝐹𝐹 = 0, to determine if the graph of the equation is a circle, an ellipse, a hyperbola or a parabola.

Refer to the wording of the standard.

Describe the motion of a particle with position ( ) xy , as t varies in a given interval.

Clarifications/Examples:

Note: Read the blog, Inverse Functions: We’re Teaching it all Wrong, before teaching inverses.

Justify solution methods.

Clarifications/Examples:

Include special cases of plane/cone intersections that result in degenerate conic sections.

Logarithmic/Exponential Functions

Refer to the wording of the standard.

Prove trigonometric identities, including Pythagorean identities and even and odd identities, using a variety of strategies. Verify identities graphically.

Logistics Growth

Function families to which this standard applies:

Trigonometric Functions

Precalculus A

Refer to the wording of the standard.

Describe the concavity on intervals.

Understand the connection between rewriting rational expressions and using long division when factoring polynomials.

Logarithmic/Exponential Functions

Trigonometric Functions

Trigonometric Functions

Include rotations of 90°, 180°, and 270°, and reflections across x-axis, y-axis, and over the line 𝑦𝑦 = 𝑥𝑥.

Clarifications/Examples:

Write the domain of all six trigonometric functions.

Polynomial Functions

Function families to which this standard applies:

Function families to which this standard applies:

Logarithmic/Exponential Functions

Topics in Analytic Geometry

Clarifications/Examples:

Clarifications/Examples:

Function families to which this standard applies:

Model in context.

Refer to the wording of the standard.

Clarifications/Examples:

Polynomial Functions

Polynomial Functions

Trigonometric Functions

Precalculus B

Students should be able to identify existence of complex roots.

Clarifications/Examples:

Trigonometric Functions

Trigonometric Functions

Radical Power Functions

Clarifications/Examples:

Clarifications/Examples:

Apply the Rational Root Theorem.

Find the partial sums of a series defined using sigma notation.

Trigonometric Functions

Polynomial Functions

Polynomial Functions

Refer to the wording of the standard.

Use limits to analyze functions for intervals of continuity or points of discontinuity.

Rational Functions

Trigonometric Functions

Function families to which this standard applies:

Graphs could include circles, lines, rose curves, cardioids, lemniscates, limaçons and spirals.

Clarifications/Examples:

Clarifications/Examples:

Emphasize double angle identities are used most frequently in Calculus.

Clarifications/Examples:

Polynomial Functions

Clarifications/Examples:

THE COMPLEX NUMBER SYSTEM

THE COMPLEX NUMBER SYSTEM

Perform arithmetic operations with complex numbers.

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

Represent Complex Numbers and their operations on the complex plane.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Use complex numbers in polynomial identities and equations.

Extend polynomial identities to the complex numbers. For example, rewrite + 2 x 4 as ( ) ( ) x ix i +− 22 .

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

OPERATIONS AND ALGEBRAIC THINKING

Write and interpret numerical expressions.

Use the notation for the factorial of a non-negative integer, n!, to evaluate expressions.

VECTOR AND MATRIX QUANTITIES

Represent and model with vector quantities.

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (𝐞𝐞. 𝐠𝐠. 𝐯𝐯, |𝐯𝐯|, ‖𝐯𝐯‖, 𝒗𝒗)

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.

Perform operations on vectors.

Add and subtract vectors.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

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

Understand vector subtraction 𝐯𝐯 − 𝐰𝐰 as 𝐯𝐯 + (−𝐰𝐰), where −𝐰𝐰 is the additive inverse of 𝐰𝐰, with the same magnitude as 𝐰𝐰 and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

Multiply a vector by a scalar.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c v v cv cv ( ) ( ) xy x y ,, = .

Compute the magnitude of a scalar multiple 𝑐𝑐𝐯𝐯 using ‖𝑐𝑐𝐯𝐯‖ = |𝑐𝑐|𝐯𝐯. Compute the direction of 𝑐𝑐𝐯𝐯 knowing that when |𝑐𝑐|𝐯𝐯 ≠ 0, the direction of 𝑐𝑐𝐯𝐯 is either along 𝐯𝐯 (for 𝑐𝑐 > 0) or against 𝐯𝐯 (for 𝑐𝑐 > 0).

Determine the dot product of two vectors.

Perform operations on matrices and use matrices in applications.

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.

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

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

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Add, subtract, and multiply matrices of appropriate dimensions.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Parametric Equations

CREATING EQUATIONS

Creating equations that describe numbers or relationships.

Create a single equation, using rectangular coordinates, that is equivalent to a pair of parametric equations.

Given a data set, create a parametric equation and a single equation using rectangular coordinates to fit the data.

INTERPRETING PARAMETRIC EQUATIONS

Analyze parametric equations.

Sketch the curve defined by parametric equations.

Use parametric equations to model and solve motion problems.

Statistics

INTERPRETING CATEGORICAL AND QUANTITATIVE DATA

Summarize, represent, and interpret data on two categorical and quantitative variables.

Fit a function to data represented by a scatterplot; use functions fitted to data to solve problems in the context of the data.

Rational Functions

Polynomial Functions

Function families to which this standard applies:

Make connections between the nature of the roots of an equation and the behavior of the graph of the function.

Apply knowledge of Fundamental Theorem of Algebra to solving polynomial equations of degree 3 and higher.

Clarifications/Examples:

Rational Functions

Polynomial Functions

Function families to which this standard applies:

building a polynomial given one complex root.

analyzing the graph of a polynomial.

solving polynomial equations with degrees greater than or equal to two.

Apply the Complex Conjugate Theorem when:

Clarifications/Examples:

Conditional Probability and the Rules of Probability

UNDERSTAND INDEPENDENCE AND CONDITIONAL PROBABILITY AND USE THEM TO INTERPRET DATA.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or", "and", "not").

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities and use this characterization to determine if they are independent.

Understand the conditional probability of A given B as 𝑃𝑃(𝐴𝐴 and 𝐵𝐵)/𝑃𝑃(𝐵𝐵), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

USE THE RULES OF PROBABILITY TO COMPUTE PROBABILITIES OF COMPOUND EVENTS.

Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A and interpret the answer in terms of the model.

Apply the Addition Rule, 𝑃𝑃 (𝐴𝐴 or 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵) − 𝑃𝑃(𝐴𝐴 and 𝐵𝐵), and interpret the answer in terms of the model.

Apply the general Multiplication Rule in a uniform probability model, 𝑃𝑃(𝐴𝐴 and 𝐵𝐵) = 𝑃𝑃(𝐴𝐴)𝑃𝑃(𝐵𝐵|𝐴𝐴) = 𝑃𝑃(𝐵𝐵)𝑃𝑃(𝐴𝐴|𝐵𝐵), and interpret the answer in terms of the model.

Use permutations and combinations to compute probabilities of compound events and solve problems.

Making Inferences and Justifying Conclusions

UNDERSTAND AND EVALUATE RANDOM PROCESSES UNDERLYING STATISTICAL EXPERIMENTS.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Introduce sampling distributions as a means to explore variability in sample statistics and ultimately evaluate a claim about a population.

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

MAKE INFERENCES AND JUSTIFY CONCLUSIONS FROM SAMPLE SURVEYS, EXPERIMENTS AND OBSERVATIONAL STUDIES.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relatesto each.

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

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Evaluate reports based on data.

Conductstatistical investigations.

Conduct observational studies.

Conductstatistical experiments.

Interpreting Categorical and Quantitative Data

SUMMARIZE, REPRESENT, AND INTERPRET DATA ON A SINGLE COUNT OR MEASUREMENT VARIABLE.

Represent data with plots on the real number line (dot plots, histograms and box plots).

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

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

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

SUMMARIZE, REPRESENT, AND INTERPRET DATA ON TWO CATEGORICAL AND QUANTITATIVE VARIABLES.

Summarize categorical data for two categoriesin two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

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

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Informally assess the fit of a function by plotting and analyzing residuals.

Fit a linear function for a scatter plot that suggests a linear association.

INTERPRET LINEAR MODELS.

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 of a linear fit.

Distinguish between correlation and causation.

Using Probability to Make Decisions

CALCULATE EXPECTED VALUES AND USE THEM TO SOLVE PROBLEMS.

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 of a random variable; interpret it as the mean of the probability distribution.

Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

USE PROBABILITY TO EVALUATE OUTCOMES OF DECISIONS.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

Evaluate and compare strategies based on expected values. For example, compare a high- deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

Use probabilitiesto make fair decisions (e.g., drawing by lots, using a random number generator).

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).