# Algebra 2 Curriculum

## Family-facing version of the Algebra 2 curriculum

## Quarterly Overview of Algebra 2

The objectives and outcomes for each unit are common across FCPS and based on the Virginia Standards of Learning. The pacing by quarter and by week provides an example of how the curriculum can be organized throughout the year. Teacher teams may adjust the pacing or order of units to best meet the needs of students.

The unit on permutations and combinations is flexible. Teachers can choose to teach it at any time during the school year when it best fits with other math topics and as the calendar allows. It is likely to be taught in Quarter 2.

## Units and Details

**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.

**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.

**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.

**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.

**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.

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

**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.
- 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 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.
- 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).

**Students will:**

- Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families.
- Compare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions.
- 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.
- 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.

**Students will:**

- Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families.
- 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 the intervals on which the graph of a function is increasing, decreasing, or constant.
- 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.

**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.

The unit on permutations and combinations is flexible. Teachers can choose to teach it at any time during the school year when it best fits with other math topics and as the calendar allows. It is likely to be taught in Quarter 2.

**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.

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

## Virginia Department of Education Resources

## Assessments

Student assessments are part of the teaching and learning process.

- Teachers give assessments to students on an ongoing basis to
- Check for understanding
- Gather information about students' knowledge or skills.

- Assessments provide information about a child's development of knowledge and skills that can help families and teachers better plan for the next steps in instruction.

For testing questions or additional information about how schools and teachers use test results to support student success, families can contact their children's schools.

In Fairfax County Public Schools (FCPS), tests focus on measuring content knowledge and skill development.