Students will:

• Ask investigative questions that require collecting or acquiring data on two variables and use a data cycle to explore these questions.
• Gather or acquire data on two variables through research, surveys, observations, experiments, polls, or questionnaires.
• Use technology to represent the data on a scatterplot.
• Decide whether the relationship between two variables is best described by a linear, quadratic, exponential, or a combination of these functions.
• Use technology to find the equation(s) that best models the relationship between the two variables. These may include linear, quadratic, or exponential functions, or a combination of these.
• Use the correlation coefficient to determine how well a linear function fits the data using technology.
• Use the data, scatterplots, or equations to make predictions, decisions, and judgments.
• Assess whether the mathematical model makes sense for the situation you're exploring.
• Describe the limitations in the conclusions of a statistical investigation based on the data collection technique. Describe the potential sources of bias in the data.
• Describe why a set of data could or could not be meaningfully used to answer a statistical question, using terms like categorical, quantitative, univariate, bivariate, relationship between data sets, correlation, or causation.
• Investigate the formula for calculating the correlation coefficient r and explain how the formula leads to describing the goodness of fit of a linear relationship
• Calculate residuals and explain how the residuals can help to evaluate the goodness of fit of a linear relationship.

Students will:

• Identify the differences between the graphs of basic functions like square root, cube root, rational, exponential, and logarithmic functions.
• Write the equation for square root, cube root, rational, exponential, and logarithmic functions when given a graph, using transformations like shifting or stretching the basic function (where adjustments are simple and involve rational values).
• Graph square root, cube root, rational, exponential, and logarithmic functions when given the equation, applying simple transformations like shifting or stretching. Use technology to confirm the changes you’ve made to the functions.
• Compare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions to understand how they are similar or different.
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities. 
• Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions. 
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant. 
• Determine the location and value of absolute (global) maxima and absolute (global) minima of a function. 
• Determine the location and value of relative (local) maxima or relative (local) minima of a function. 
• For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable. 
• Describe the end behavior of a function.
• Make connections and generalizations regarding transformations and changes in function characteristics.
• Describe how the rate of change of a function changes over a given increasing or decreasing interval.
• Given a set of characteristics, create and justify a graph of a function that exhibits those characteristics.

Students will:

• Create an absolute value equation in one variable to model a contextual situation. 
• Solve an absolute value equation in one variable algebraically and verify the solution graphically. 
• Create an absolute value inequality in one variable to model a contextual situation. 
• Solve an absolute value inequality in one variable and represent the solution set using set notation, interval notation, and using a number line. 
• Verify possible solution(s) to absolute value equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
• The student will extend their understanding of absolute value equations.
• Develop strategies to solve an equation algebraically that contains more than one absolute value term
• Describe characteristics of absolute value functions, including domain, range, intercepts, extrema, and end behavior.
• Write absolute value functions as piecewise functions.
• Solve absolute value inequalities that contain additional variable terms.
• Create a linear-quadratic or quadratic-quadratic system of equations to model a contextual situation. 
• Determine the number of solutions to a linear-quadratic and quadratic-quadratic system of equations in two variables. 
• Solve a linear-quadratic and quadratic-quadratic system of equations algebraically and graphically, including situations in context. 
• Verify possible solution(s) to linear-quadratic or quadratic-quadratic system of equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Students will:

• Factor polynomials completely in one and two variables with no more than four terms over the set of integers.
• Represent and demonstrate equality of polynomial expressions written in different forms and verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.
•  Explain the meaning of i. 
• Identify equivalent radical expressions containing negative rational numbers and expressions in a + bi form. 
• Apply properties to add, subtract, and multiply complex numbers. 
• Create a quadratic equation or inequality in one variable to model a contextual situation. 
• Solve a quadratic equation in one variable over the set of complex numbers algebraically. 
• Determine the solution to a quadratic inequality in one variable over the set of real numbers algebraically. 
• Verify possible solution(s) to quadratic equations or inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
• Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation. 
• Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions.
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities. 
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant. 
• Determine the location and value of absolute (global) maxima and absolute (global) minima of a function. 
• Determine the location and value of relative (local) maxima or relative (local) minima of a function. 
•  For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable. 
• Describe the end behavior of a function. 
• Formulate investigative questions that require the collection or acquisition of bivariate data and investigate questions using a data cycle. 
• Collect or acquire bivariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires.
• Represent bivariate data with a scatterplot using technology. 
• Determine whether the relationship between two quantitative variables is best approximated by a linear, quadratic, exponential, or a combination of these functions. 
• Determine the equation(s) of the function(s) that best models the relationship between two variables using technology. Curves of best fit may include a combination of linear, quadratic, or exponential (piecewise-defined) functions.
• Use the correlation coefficient to designate the goodness of fit of a linear function using technology. 
• Make predictions, decisions, and critical judgments using data, scatterplots, or the equation(s) of the mathematical model.
• Evaluate the reasonableness of a mathematical model of a contextual situation.
• Factor completely 4th and higher degree polynomials and expressions containing other functions that can be factored by applying other methods and patterns addressed in Algebra 1 and 2.
• Divide complex numbers using complex conjugates.
• Apply properties of complex numbers to determine if a given statement is always true, sometimes true, or neve true.
• SOlve quadratic equations modeling contextual situations where the variable can be defined in different ways, including with compositions of linear and quadratic functions.
• Consider how changes to the contextual situation impact the model and make connections to transformation of functions.
• Predict the number of solutions to a system of equations, with and without the use of technology. 
• Solve linear-quadratic and quadratic-quadratic systems of equations including equations of circles and ellipses.

Students will:

• Determine sums, differences, and products of polynomials in one and two variables. 
• Factor polynomials completely in one and two variables with no more than four terms over the set of integers. 
• Determine the quotient of polynomials in one and two variables, using monomial, binomial, and factorable trinomial divisors. 
• Represent and demonstrate equality of polynomial expressions written in different forms and verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.
• Explain the meaning of i. 
• Identify equivalent radical expressions containing negative rational numbers and expressions in a + bi form. 
• Apply properties to add, subtract, and multiply complex numbers. 
• Determine a factored form of a polynomial equation, of degree three or higher, given its zeros or the x-intercepts of the graph of its related function. 
• Determine the number and type of solutions (real or imaginary) of a polynomial equation of degree three or higher. 
• Solve a polynomial equation over the set of complex numbers. 
• Verify possible solution(s) to polynomial equations of degree three or higher algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions in context.
• Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation. 
• Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions.
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant. 
• Determine the location and value of absolute (global) maxima and absolute (global) minima of a function. 
• Determine the location and value of relative (local) maxima or relative (local) minima of a function. 
• For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable.
• Describe the end behavior of a function. 
• Factor completely 4th and higher degree polynomials and expressions containing other functions that can be factored by applying other methods and patterns addressed in Algebra 1 and 2.
• Divide polynomials using long division.
• Divide a polynomial by a linear binomial using synthetic division.
• Use numbers, figures, graphs, or words to explain why the difference of squares and sum/difference of cubes identities only contain a leading term and constant term.
• Determine the solution to polynomial inequalities algebraically and graphically.
• Explore theorems and properties associated with polynomials including the Remainder Theorem, Rational Root Theorem, and Descartes Rule of Signs.

Students will:

• Simplify and determine equivalent radical expressions that include numeric and algebraic radicands. 
• Add, subtract, multiply, and divide radical expressions that include numeric and algebraic radicands, simplifying the result. Simplification may include rationalizing the denominator. 
• Convert between radical expressions and expressions containing rational exponents. 
• Solve an equation containing no more than one radical expression algebraically and graphically. 
• Verify possible solution(s) to radical equations algebraically, graphically, and with technology, to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context. 
• Justify why a possible solution to an equation with a square root might be extraneous.
• Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation. 
• Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions.
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities. 
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant. 
• Determine the location and value of absolute (global) maxima and absolute (global) minima of a function. 
• Determine the location and value of relative (local) maxima or relative (local) minima of a function. 
• For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable.
• Describe the end behavior of a function. 
•  Determine the inverse of a function algebraically and graphically, given the equation of a linear or quadratic function (linear, quadratic, and square root). Justify and explain why two functions are inverses of each other. 
• Graph the inverse of a function as a reflection over the line y = x. 
• Determine the composition of two functions algebraically and graphically.
• Derive sqrt(x^2)=|x| and explain the importance of specifying the restriction on variable values when simplifying certain algebraic expressions.
• Rationalize expressions with sums and differences of radicals in the denominator using conjugates.
• Compare radical form and rational exponent form and give examples of expressions where one form might be more useful than the other.
• Solve equations containing more than one radical expression,
• Use numbers, figures, graphs, or words to describe situations where rational equations generate extraneous solutions.

Students will:

• Add, subtract, multiply, or divide rational algebraic expressions, simplifying the result. 
• Justify and determine equivalent rational algebraic expressions with monomial and binomial factors. Algebraic expressions should be limited to linear and quadratic expressions. 
• Recognize a complex algebraic fraction and simplify it as a product or quotient of simple algebraic fractions.
• Represent and demonstrate equivalence of rational expressions written in different forms.
• Create an equation containing a rational expression to model a contextual situation. 
• Solve rational equations with real solutions containing factorable algebraic expressions algebraically and graphically. Algebraic expressions should be limited to linear and quadratic expressions. 
• Verify possible solution(s) to rational equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context. 
• Justify why a possible solution to an equation containing a rational expression might be extraneous.
• Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families. 
• Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation. 
• Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions. 
• Determine when two variables are directly proportional, inversely proportional, or neither, given a table of values. Write an equation and create a graph to represent a direct or inverse variation, including situations in context.  
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities. 
• Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions. 
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant.
• For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable. 
• Describe the end behavior of a function. 
• Determine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic).
• Determine the coordinates of removable discontinuities (holes in rational functions.
• Rewrite rational expressions using polynomial long division with remainders and make connections between the graph of the original function and terms of the quotient.
• Determine and justify equivalent rational algebraic expressions, including restrictions on variables; expressions may be rewritten with factioning or long division.
• Use numbers, figures, graphs, or words to describe situations where rational equations generate extraneous solutions.
• Solve contextual problems that require more complex analytical techniques, including additional rational terms and/or higher order factorable polynomials.
• Given a set of characteristics, create and justify a graph of a function that exhibits those characteristics.

Students will:

• Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families. 
• Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation. 
• Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions. 
• Compare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions.
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities. 
• Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions.
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant. 
• For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable. 
• Describe the end behavior of a function.
• Determine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic). 
• Determine the inverse of a function algebraically and graphically, given the equation of a linear or quadratic function (linear, quadratic, and square root). Justify and explain why two functions are inverses of each other. 
• Graph the inverse of a function as a reflection over the line y = x. 
• Determine the composition of two functions algebraically and graphically.
• Represent, solve, and interpret the solution to exponential and logarithmic equations.

Students will:

• Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families. 
• Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation. 
• Graph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Use technology to verify transformations of the functions. 
• Determine when two variables are directly proportional, inversely proportional, or neither, given a table of values. Write an equation and create a graph to represent a direct or inverse variation, including situations in context. 
• Compare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions.
• Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities. 
• Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions. 
• Determine the intervals on which the graph of a function is increasing, decreasing, or constant. 
• Determine the location and value of absolute (global) maxima and absolute (global) minima of a function.
• Determine the location and value of relative (local) maxima or relative (local) minima of a function. 
• For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable. 
• Describe the end behavior of a function. 
• Determine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic).
• Determine the inverse of a function algebraically and graphically, given the equation of a linear or quadratic function (linear, quadratic, and square root). Justify and explain why two functions are inverses of each other. V
• Graph the inverse of a function as a reflection over the line y = x. 
• Determine the composition of two functions algebraically and graphically.
• Formulate investigative questions that require the collection or acquisition of bivariate data and investigate questions using a data cycle. 
• Collect or acquire bivariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires. 
• Represent bivariate data with a scatterplot using technology. 
• Determine whether the relationship between two quantitative variables is best approximated by a linear, quadratic, exponential, or a combination of these functions. 
• Determine the equation(s) of the function(s) that best models the relationship between two variables using technology. Curves of best fit may include a combination of linear, quadratic, or exponential (piecewise-defined) functions. 
• Use the correlation coefficient to designate the goodness of fit of a linear function using technology. 
• Make predictions, decisions, and critical judgments using data, scatterplots, or the equation(s) of the mathematical model. 
• Evaluate the reasonableness of a mathematical model of a contextual situation.
• Describe how the rate of change of a function changes over a given increasing or decreasing interval.
• Given a set of characteristics, create and justify a graph of a function that exhibits those characteristics.
• Describe the domain and range of a composition of two functions where one or both of the original functions have domain restrictions.
• Describe the limitations in the conclusions of a statistical investigation based on the data collection technique. Describe potential sources of bias in the data.
• Describe why a set of data could or could not be meaningfully used to answer a statistical question, using terms like categorical, quantitative, univariate, bivariate, relationship between data sets, correlation, or causation.

Students will:

• Formulate investigative questions that require the collection or acquisition of a large set of univariate quantitative data or summary statistics of a large set of univariate quantitative data and investigate questions using a data cycle. 
• Collect or acquire univariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires. 
• Examine the shape of a data set (skewed versus symmetric) that can be represented by a histogram, and sketch a smooth curve to model the distribution. 
• Identify the properties of a normal distribution. 
• Describe and interpret a data distribution represented by a smooth curve by analyzing measures of center, measures of spread, and shape of the curve. 
• Calculate and interpret the z-score for a value in a data set. 
• Compare two data points from two different distributions using z-scores. 
• Determine the solution to problems involving the relationship of the mean, standard deviation, and z-score of a data set represented by a smooth or normal curve. 
• Apply the Empirical Rule to answer investigative questions.
• Compare multiple data distributions using measures of center, measures of spread, and shape of the distributions. 
• Solve contextual problems with increased complexity related to z-scores and normal distributions, including problems related to statistical inference.

Students will:

• Compare and contrast permutations and combinations to count the number of ways that events can occur. 
• Calculate the number of permutations of n objects taken r at a time. 
• Calculate the number of combinations of n objects taken r at a time. 
• Use permutations and combinations as counting techniques to solve contextual problems. 
• Calculate and verify permutations and combinations using technology.
• Derive the formulas for permutation and combination using counting principles and factorials.
• Solve more complex problems involving probability, combinations, and permutations.

In this unit, teachers will provide differentiated opportunities for students to review previous content and/or explore content at a deeper level.