﻿ FCPS Program of Studies
Mathematics Curriculum
Algebra 2

The standards below outline the content for a one-year course in Algebra II. This course is designed for students who have successfully completed the standards for Algebra I and Geometry. All students preparing for postsecondary and advanced technical studies are expected to achieve the Algebra II standards. A thorough treatment of advanced algebraic concepts will be provided through the study of functions, “families of functions,” equations, inequalities, systems of equations and inequalities, polynomials, rational and radical equations, complex numbers, and sequences and series. Emphasis will be placed on practical applications and modeling throughout the course of study. Oral and written communication concerning the language of algebra, logic of procedures, and interpretation of results should also permeate the course.

These standards include a transformational approach to graphing functions. Transformational graphing uses translation, reflection, dilation, and rotation to generate a “family of graphs” from a given graph and builds a strong connection between algebraic and graphic representations of functions. Students will vary the coefficients and constants of an equation, observe the changes in the graph of the equation, and make generalizations that can be applied to many graphs.

Graphing utilities (graphing calculators or computer graphing simulators), computers, spreadsheets, and other appropriate technology tools will be used to assist in teaching and learning. Graphing utilities enhance the understanding of realistic applications through mathematical modeling and aid in the investigation and study of functions. They also provide an effective tool for solving and verifying solutions to equations and inequalities.

Last Updated: 08/17/12 10:27 AM

MTH.ALG2
Standard 1
DEVELOP SKILLS INVOLVING RATIONAL, RADICAL, AND POLYNOMIAL EXPRESSIONS

The student, given rational, radical, or polynomial expressions, will
a)   add, subtract, multiply, divide, and simplify rational algebraic expressions;
b)   add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents;
c)   write radical expressions as expressions containing rational exponents and vice versa; and
d)   factor polynomials completely.

 Essential Understandings: Computational skills applicable to numerical fractions also apply to rational expressions involving variables. Radical expressions can be written and simplified using rational exponents. Only radicals with a common radicand and index can be added or subtracted. A relationship exists among arithmetic complex fractions, algebraic complex fractions, and rational numbers. The complete factorization of polynomials has occurred when each factor is a prime polynomial. Pattern recognition can be used to determine complete factorization of a polynomial.

Benchmark 1.a
Add, Subtract, Multiply, Divide, & Simplify Rational Expressions

The student, given rational, radical, or polynomial expressions, will add, subtract, multiply, divide, and simplify rational algebraic expressions.

 Indicator 1.a.1 Add, subtract, multiply, and divide rational algebraic expressions Add, subtract, multiply, and divide rational algebraic expressions.

 Indicator 1.a.2 Simplify rational algebraic expressions Simplify a rational algebraic expression with common monomial or binomial factors.

 Indicator 1.a.3 Recognize and simplify a complex algebraic fraction Recognize a complex algebraic fraction, and simplify it as a quotient or product of simple algebraic fractions.

 Indicator 1.a.4 Perform operations on rational expressions Add, subtract, multiply, and divide rational expressions whose denominators are monomials or polynomial expressions in non-factored form.

Benchmark 1.b

The student, given rational, radical, or polynomial expressions, will add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents.

 Indicator 1.b.1 Simplify radical expressions Simplify radical expressions containing positive rational numbers and variables.

 Indicator 1.b.3 Multiply and divide radical expressions Multiply and divide radical expressions not requiring rationalizing the denominators.

 Indicator 1.b.4 Rationalize radical expressions in the denominator and then simplify Rationalize radical expressions in the denominator, with and without conjugates, and simplify.

 Indicator 1.b.5 Simplify expressions with positive or negative rational exponents Simplify expressions containing positive or negative rational exponents.

Benchmark 1.c
Write Radical Expressions with Rational Exponents & Vice Versa

The student, given rational, radical, or polynomial expressions, will write radical expressions as expressions containing rational exponents and vice versa.

 Indicator 1.c.1 Convert from radical notation to exponential notation Convert from radical notation to exponential notation.

 Indicator 1.c.2 Convert from exponential notation to radical notation Convert from exponential notation to radical notation.

Benchmark 1.d
Factor Polynomials Completely

The student, given rational, radical, or polynomial expressions, will factor polynomials completely.

 Indicator 1.d.1 Factor polynomials by applying general patterns Factor polynomials by applying general patterns including difference of squares, sum and difference of cubes, and perfect square trinomials.

 Indicator 1.d.2 Factor polynomials completely over the integers Factor polynomials completely over the integers.

 Indicator 1.d.3 Verify polynomial identities Verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.

Benchmark 1.e
Expand and condense logarithmic expressions.

Expand and condense logarithmic expressions.

 Indicator 1.e.1 Condense & expand logarithmic expressions using log properties Use properties of logarithms to condense and expand logarithmic expressions.

MTH.ALG2
Standard 2
INVESTIGATE/APPLY PROPERTIES OF ARITHMETIC/GEOMETRIC SEQUENCES/SERIES

The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ  and an.

 FCPS Notes: An arithmetic sequence is a linear function. A geometric sequence is an exponential function. The graphs of sequences are discrete.

 Essential Understandings: Sequences and series arise from real-world situations. The study of sequences and series is an application of  the investigation of patterns. A sequence is a function whose domain is the set of natural numbers. Sequences can be defined explicitly and recursively.

Benchmark 2.a
Investigate/Apply Properties of Arithmetic/Geometric Sequences/Series

The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ  and an.

 Indicator 2.a.1 Distinguish between a sequence and a series Distinguish between a sequence and a series.

 Indicator 2.a.2 Generalize patterns in a sequence using explicit and recursive formula Generalize patterns in a sequence using explicit and recursive formulas.

 Indicator 2.a.3 Use and interpret notations Use and interpret the notations Σ , n, nth term, and an.

 Indicator 2.a.4 Find the nth term for an arithmetic or a geometric sequence Given the formula, find an (the nth term) for an arithmetic or a geometric sequence.

 Indicator 2.a.5 Find the sum of the first n terms of an arithmetic or geometric series Given formulas, write the first n terms and find the sum, Sn, of the first n terms of an arithmetic or geometric series.

 Indicator 2.a.6 Given the formula, find the sum of a convergent infinite series Given the formula, find the sum of a convergent infinite series.

 Indicator 2.a.7 Model real-world situations using sequences and series Model real-world situations using sequences and series.

 Indicator 2.a.8 Write an explicit formula for an arithmetic or geometric sequence Write an explicit formula for an arithmetic or geometric sequence.

 Indicator 2.a.9 Use sigma notation to express an arithmetic or geometric series Express an arithmetic or geometric series in abbreviated form using sigma notation.

 Indicator 2.a.10 Find the position of a term in an arithmetic sequence Find the position of a term in an arithmetic sequence, given its value.

 Indicator 2.a.11 Represent sequences and series using a table or graph Represent sequences and series using a table or graph.

 Indicator 2.a.12 Use the arithmetic or the geometric mean to find a missing term Use the arithmetic mean or the geometric mean to find a missing term in an arithmetic or geometric sequence.

MTH.ALG2
Standard 3
PERFORM OPERATIONS ON COMPLEX NUMBERS

The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers.

 Essential Understandings: Complex numbers are organized into a hierarchy of subsets. A complex number multiplied by its conjugate is a real number. Equations having no real number solutions may have solutions in the set of complex numbers. Field properties apply to complex numbers as well as real numbers. All complex numbers can be written in the form a+bi where a and b are real numbers and i is − 1 .

Benchmark 3.a
Perform Operations on Complex Numbers

The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers.

 Indicator 3.a.1 Recognize that the square root of -1 is represented as i Recognize that the square root of –1 is represented as i.

 Indicator 3.a.2 Determine which field properties apply to the complex number system Determine which field properties apply to the complex number system.

 Indicator 3.a.3 Simplify radical expressions containing negative rational numbers Simplify radical expressions containing negative rational numbers and express in a+bi form.

 Indicator 3.a.4 Simplify powers of i Simplify powers of i.

 Indicator 3.a.5 Add, subtract, and multiply complex numbers Add, subtract, and multiply complex numbers.

 Indicator 3.a.6 Place identified sets of numbers in a hierarchy of subsets Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole, and natural.

 Indicator 3.a.7 Write a real number in a+bi form Write a real number in a+bi form.

 Indicator 3.a.8 Write a pure imaginary number in a+bi form Write a pure imaginary number in a+bi form.

 Indicator 3.a.9 Divide complex numbers using complex conjugates Divide complex numbers using complex conjugates.

MTH.ALG2
Standard 4
SOLVE A VARIETY OF EQUATIONS, ALGEBRAICALLY AND GRAPHICALLY

The student will solve, algebraically and graphically,
a)   absolute value equations and inequalities;
b)   quadratic equations over the set of complex numbers;
c)   equations containing rational algebraic expressions; and
Graphing calculators will be used for solving and for confirming the algebraic solutions.

 Essential Understandings: A quadratic function whose graph does not intersect the x-axis has roots with imaginary components. The quadratic formula can be used to solve any quadratic equation. The value of the discriminant of a quadratic equation can be used to describe the number of real and complex solutions. The definition of absolute value (for any real numbers a and b, where b ≥ 0 , if | a | = b , then a = b or a = - b) is used in solving absolute value equations and inequalities. Absolute value inequalities can be solved graphically or by using a compound statement. Real-world problems can be interpreted, represented, and solved using equations and inequalities. The process of solving radical or rational equations can lead to extraneous solutions. Equations can be solved in a variety of ways. Set builder notation may be used to represent solution sets of equations and inequalities.

Benchmark 4.a
Solve Absolute Value Equations and Inequalities

The student will solve, algebraically and graphically, absolute value equations and inequalities.  Graphing calculators will be used for solving and for confirming the algebraic solutions.

 Indicator 4.a.1 Solve absolute value equations algebraically and graphically Solve absolute value equations algebraically and graphically.

 Indicator 4.a.2 Solve absolute value inequalities algebraically and graphically Solve absolute value inequalities algebraically and graphically.

 Indicator 4.a.3 Apply an appropriate equation to solve a real-world problem Apply an appropriate equation to solve a real-world problem.

 Indicator 4.a.4 Verify possible solutions to absolute value equations Verify possible solutions to absolute value equations.

 Indicator 4.a.5 Verify possible solutions to absolute value inequalities Verify possible solutions to absolute value inequalities.

 Indicator 4.a.6 Recognize absolute value equations/inequalities with no solution Recognize absolute value equations and inequalities that have no solution.

 Indicator 4.a.7 Express solutions to abs. val. ineq. in set and interval notation Express solutions to absolute value inequalities in both set and interval notation.

 Indicator 4.a.8 Given the equation, graph a piecewise or step function Given the equation, graph a piecewise or step function.

 Indicator 4.a.9 Given a graph, write an equation for a piecewise or step function Given a graph, write an equation for a piecewise or step function.

Benchmark 4.b
Solve a Quadratic Equation Over the Set of Complex Numbers

The student will solve, algebraically and graphically, quadratic equations over the set of complex numbers.  Graphing calculators will be used for solving and for confirming the algebraic solutions.

 Indicator 4.b.1 Solve a quadratic equation over the set of complex numbers Solve a quadratic equation over the set of complex numbers using an appropriate strategy.

 Indicator 4.b.2 Use the discriminant to determine the number and type of solutions Calculate the discriminant of a quadratic equation to determine the number of real and complex solutions.

 Indicator 4.b.3 Apply an appropriate equation to solve a real-world problem Apply an appropriate equation to solve a real-world problem.

 Indicator 4.b.4 Recognize that the quad form can be derived from completing the square Recognize that the quadratic formula can be derived by applying the completion of squares to any quadratic equation in standard form.

Benchmark 4.c
Solve Equations Containing Rational Algebraic Expressions

The student will solve, algebraically and graphically, equations containing rational algebraic expressions.  Graphing calculators will be used for solving and for confirming the algebraic solutions.

 Indicator 4.c.1 Solve equations containing rational algebraic expressions Solve equations containing rational algebraic expressions with monomial or binomial denominators algebraically and graphically.

 Indicator 4.c.2 Apply an appropriate equation to solve a real-world problem Apply an appropriate equation to solve a real-world problem.

 Indicator 4.c.3 Identify vertical and horizontal asymptotes Identify vertical asymptotes and horizontal asymptotes of a rational function from the equation and the graph.

Benchmark 4.d

The student will solve, algebraically and graphically, equations containing radical expressions.  Graphing calculators will be used for solving and for confirming the algebraic solutions.

 Indicator 4.d.1 Solve an equation containing radical expressions Solve an equation containing a radical expression algebraically and graphically.

 Indicator 4.d.2 Verify possible solutions to an equation Verify possible solutions to an equation containing rational or radical expressions.

 Indicator 4.d.3 Apply an appropriate equation to solve a real-world problem Apply an appropriate equation to solve a real-world problem.

 Indicator 4.d.4 Identify possible extraneous solutions when solving equations Identify possible extraneous solutions when solving equations.

Benchmark 4.e
Solve exponential and logarithmic equations

Solve exponential and logarithmic equations.

 Indicator 4.e.1 Solve exponential equations Solve exponential equations.

 Indicator 4.e.2 Solve logarithmic equations Solve logarithmic equations.

 Indicator 4.e.3 Solve exponential & logarithmic equations using properties of logs Solve exponential and logarithmic equations using properties of logarithms.

MTH.ALG2
Standard 5
SOLVE NONLINEAR SYSTEMS OF EQUATIONS

The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically.  Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions.

 Essential Understandings: Solutions of a nonlinear system of equations are numerical values that satisfy every equation in the system. The coordinates of points of intersection in any system of equations are solutions to the system. Real-world problems can be interpreted, represented, and solved using systems of equations.

Benchmark 5.a
Solve Nonlinear Systems of Equations

The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically.  Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions.

 Indicator 5.a.1 Predict number of solutions to a nonlinear system of two equations Predict the number of solutions to a nonlinear system of two equations.

 Indicator 5.a.2 Solve a linear-quadratic system of two equations Solve a linear-quadratic system of two equations algebraically and graphically.

 Indicator 5.a.4 Solve a circle-linear system of two equations graphically Solve a circle-linear system of two equations graphically.

 Indicator 5.a.5 Solve a circle-quadratic system of two equations graphically Solve a circle-quadratic system of two equations graphically.

 Indicator 5.a.6 Solve a circle-linear system of two equations algebraically Solve a circle-linear system of two equations algebraically.

 Indicator 5.a.7 Solve a circle-quadratic system of two equations algebraically Solve a circle-quadratic system of two equations algebraically.

 Indicator 5.a.8 Sketch graph of a conic section (parabola or hyperbola) in (h, k) form Sketch the graph of a conic section (parabola or hyperbola) in (h, k) form.

 Indicator 5.a.9 Write the equation of a conic section in (h, k) form from a graph Write the equation of a conic section (parabola or hyperbola) in (h, k) form from a graph.

 Indicator 5.a.10 Use conic sections to model practical problems Use conic sections to model practical problems.

MTH.ALG2
Standard 6
RECOGNIZE MULTIPLE REPRESENTATIONS OF FUNCTIONS

The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions.

 FCPS Notes: (Honors)  Parametric equations are used to express two dependent variables, x and y, in terms of an independent variable (parameter), t.

 Essential Understandings: The graphs/equations for a family of functions can be determined using a transformational approach. Transformations of graphs include translations, reflections, and dilations. A parent graph is an anchor graph from which other graphs are derived with transformations.

Benchmark 6.a
Recognize Multiple Representations of Functions

The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions.

 Indicator 6.a.1 Recognize graphs of parent functions Recognize graphs of parent functions.

 Indicator 6.a.2 Identify the graph of the transformed function Given a transformation of a parent function, identify the graph of the transformed function.

 Indicator 6.a.3 Given equation & using a transformational approach, graph a function Given the equation and using a transformational approach, graph a function.

 Indicator 6.a.4 Given the graph of a function, identify the parent function Given the graph of a function, identify the parent function.

 Indicator 6.a.5 Identify the transformations of a function to determine the equation Given the graph of a function, identify the transformations that map the preimage to the image in order to determine the equation of the image.

 Indicator 6.a.6 Given the graph, write the equation of the function Using a transformational approach, write the equation of a function given its graph.

 Indicator 6.a.7 Write the equation of a linear function in point-slope form Write the equation of a linear function in point-slope form.

 Indicator 6.a.8 Use parametric equations to represent a linear or quadratic function Use parametric equations to represent a linear or quadratic function.

MTH.ALG2
Standard 7
INVESTIGATE AND ANALYZE FUNCTIONS, ALGEBRAICALLY & GRAPHICALLY

The student will investigate and analyze functions algebraically and graphically. Key concepts include
a)   domain and range, including limited and discontinuous domains and ranges;
b)   zeros;
c)   x- and y-intercepts;
d)   intervals in which a function is increasing or decreasing;
e)   asymptotes;
f)   end behavior;
g)   inverse of a function; and
h)   composition of multiple functions.
Graphing calculators will be used as a tool to assist in investigation of functions.

 Essential Understandings: Functions may be used to model real-world situations. The domain and range of a function may be restricted algebraically or by the real-world situation modeled by the function. A function can be described on an interval as increasing, decreasing, or constant. Asymptotes may describe both local and global behavior of functions. End behavior describes a function as x approaches positive and negative infinity. A zero of a function is a value of x that makes f ( x )   equal zero. If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function. Exponential ( y = a x ) and logarithmic ( y = log ⁡ a x ) functions are inverses of each other. Functions can be combined using composition of functions.

Benchmark 7.a
Determine the Domain and Range of Functions

The student will investigate and analyze functions algebraically and graphically. Key concepts include domain and range, including limited and discontinuous domains and ranges.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.a.1 Identify the domain and range of a function Identify the domain and range of a function presented algebraically or graphically.

 Indicator 7.a.2 Describe restricted/discontinuous domains and ranges Describe restricted/discontinuous domains and ranges.

 Indicator 7.a.3 Write domain and range of a function using interval notation Write domain and range of a function using interval notation.

Benchmark 7.b
Determine the Zeros of Functions

The student will investigate and analyze functions algebraically and graphically. Key concepts include zeros.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.b.1 Identify the zeros of a function Identify the zeros of a function presented algebraically or graphically.

 Indicator 7.b.2 Use long division to find all zeroes of a polynomial Use long division to find all zeroes of a polynomial.

 Indicator 7.b.3 Use synthetic division/substitution to find all zeros Use synthetic division/substitution to find all zeros.

Benchmark 7.c
Determine the X- and Y-Intercepts of Functions

The student will investigate and analyze functions algebraically and graphically. Key concepts include x- and y-intercepts.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.c.1 Identify the intercepts of a function Identify the intercepts of a function presented algebraically or graphically.

Benchmark 7.d
Determine Intervals in Which a Function is Increasing or Decreasing

The student will investigate and analyze functions algebraically and graphically. Key concepts include intervals in which a function is increasing or decreasing.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.d.1 Identify intervals on which the function is increasing and decreasing Given the graph of a function, identify intervals on which the function is increasing and decreasing.

Benchmark 7.e
Determine the Asymptotes of Functions

The student will investigate and analyze functions algebraically and graphically. Key concepts include asymptotes.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.e.1 Find the equations of vertical and horizontal asymptotes of functions Find the equations of vertical and horizontal asymptotes of functions.

 Indicator 7.e.2 Graph rational functions Graph rational functions showing zeroes, removable discontinuities (holes), and vertical and horizontal asymptotes.

Benchmark 7.f
Determine the End Behavior of Functions

The student will investigate and analyze functions algebraically and graphically. Key concepts include end behavior.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.f.1 Describe the end behavior of a function Describe the end behavior of a function.

 Indicator 7.f.2 Describe end behavior of a function as x approaches pos & neg infinity Describe the end behavior of a function as x approaches positive and negative infinity.

Benchmark 7.g
Determine the Inverse of a Function

The student will investigate and analyze functions algebraically and graphically. Key concepts include the inverse of a function.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.g.1 Find the inverse of a function Find the inverse of a function.

 Indicator 7.g.2 Graph the inverse of a function as a reflection across the line, y = x Graph the inverse of a function as a reflection across the line, y = x.

 Indicator 7.g.3 Investigate exponential and logarithmic functions Investigate exponential and logarithmic functions, using the graphing calculator.

 Indicator 7.g.4 Convert between logarithmic and exponential forms of an equation Convert between logarithmic and exponential forms of an equation with bases consisting of natural numbers.

Benchmark 7.h
Determine the Composition of Multiple Functions

The student will investigate and analyze functions algebraically and graphically. Key concepts include composition of multiple functions.  Graphing calculators will be used as a tool to assist in investigation of functions.

 Indicator 7.h.1 Find the composition of two functions Find the composition of two functions.

 Indicator 7.h.2 Use composition of functions to verify two functions are inverses Use composition of functions to verify two functions are inverses.

 Indicator 7.h.3 Find the composition of multiple functions Find the composition of multiple functions.

 Indicator 7.h.4 Find the value of a composition of multiple functions Find the value of a composition of multiple functions for a given element from the domain.

 Indicator 7.h.5 Find the domain and range of a piecewise function Find the domain and range of a piecewise function.

MTH.ALG2
Standard 8
INVESTIGATE AND DESCRIBE RELATIONSHIPS

The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

 FCPS Notes: Imaginary roots are complex conjugates and come in pairs.

 Essential Understandings: The Fundamental Theorem of Algebra states that, including complex and repeated solutions, an nth degree polynomial equation has exactly n roots (solutions.) The following statements are equivalent:- k is a zero of the polynomial function f;- (x - k) is a factor of f(x);- k is a solution of the polynomial equation f(x) = 0; and- k is an x-intercept for the graph of y = f(x).

Benchmark 8.a
Investigate and Describe Relationships

The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

 Indicator 8.a.1 Describe relationships among solutions/zeros/intercepts/factors Describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

 Indicator 8.a.2 Define a polynomial function, given its zeros Define a polynomial function, given its zeros.

 Indicator 8.a.3 Determine a factored form of a polynomial expression Determine a factored form of a polynomial expression from the x-intercepts of the graph of its corresponding function.

 Indicator 8.a.4 Identify zeros of multiplicity greater than 1 For a function, identify zeros of multiplicity greater than 1 and describe the effect of those zeros on the graph of the function.

 Indicator 8.a.5 Determine the number of real solutions and nonreal solutions Given a polynomial equation, determine the number of real solutions and nonreal solutions.

 Indicator 8.a.6 Use long division to find all zeroes of a polynomial function Use long division to find all zeroes of a polynomial function.

 Indicator 8.a.7 Use synthetic division/substitution to find all zeros Use synthetic division/substitution to find all zeros.

MTH.ALG2
Standard 9
COLLECT/ANALYZE DATA TO MAKE PREDICTIONS & SOLVE PRACTICAL PROBLEMS

The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.

 FCPS Notes: For the r-value to appear on the graphing calculator after calculating a regression, the diagnostics feature must be turned on.

 Essential Understandings: Data and scatterplots may indicate patterns that can be modeled with an algebraic equation. Graphing calculators can be used to collect, organize, picture, and create an algebraic model of the data. Data that fit polynomial ( f ( x ) = a n x n + a n − 1 x n − 1 + ... + a 1 x + a 0  where n  is a nonnegative integer, and the coefficients are real numbers), exponential ( y = b x ), and logarithmic ( y = log ⁡ b x ) models arise from real-world situations.

Benchmark 9.a
Collect & Analyze Data to Make Predictions & Solve Practical Problems

The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.

 Indicator 9.a.1 Collect and analyze data Collect and analyze data.

 Indicator 9.a.2 Investigate scatterplots to determine if patterns exist Investigate scatterplots to determine if patterns exist, and then identify the patterns.

 Indicator 9.a.3 Find an equation for the curve of best fit for data Find an equation for the curve of best fit for data, using a graphing calculator. Models will include polynomial, exponential, and logarithmic functions.

 Indicator 9.a.4 Make predictions, using data, scatterplots, or curve of best fit Make predictions, using data, scatterplots, or the equation of the curve of best fit.

 Indicator 9.a.5 Determine the model that would best describe the data Given a set of data, determine the model that would best describe the data.

 Indicator 9.a.6 Determine how well a regression curve approximates data points Use the correlation coefficient, r, from the graphing calculator to determine how well a regression curve approximates data points.

MTH.ALG2
Standard 10
SOLVE PROBLEMS INVOLVING DIRECT, JOINT, & INVERSE VARIATIONS

The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.

 FCPS Notes: A direct variation graphs as a linear function that passes through the origin. An inverse variation graphs as a rational function.

 Essential Understandings: Real-world problems can be modeled and solved by using inverse variation, joint variation, and a combination of direct and inverse variations. Joint variation is a combination of direct variations.

Benchmark 10.a
Solve Problems Involving Direct, Joint, and Inverse Variations

The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.

 Indicator 10.a.1 Translate "y varies jointly as x and z" as y=kxz Translate "y varies jointly as x and z" as y = k x z .

 Indicator 10.a.2 Translate "y is directly proportional to x" as y=kx Translate "y is directly proportional to x" as y = k x .

 Indicator 10.a.3 Translate "y is inversely proportional to x" as y=k/x Translate "y is inversely proportional to x" as y = k x .

 Indicator 10.a.4 Determine the value of the constant of proportionality Given a situation, determine the value of the constant of proportionality.

 Indicator 10.a.5 Use combinations of direct/joint/inverse variation to solve problems Set up and solve problems, including real-world problems, involving inverse variation, joint variation, and a combination of direct and inverse variations.

 Indicator 10.a.6 Identify inverse or direct variation Identify inverse or direct variation given a graph, equation, table, or real-world application.

MTH.ALG2
Standard 11
IDENTIFY AND APPLY PROPERTIES OF A NORMAL DISTRIBUTION

The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

 Essential Understandings: A normal distribution curve is a symmetrical, bell-shaped curve defined by the mean and the standard deviation of a data set.  The mean is located on the line of symmetry of the curve. Areas under the curve represent probabilities associated with continuous distributions. The normal curve is a probability distribution and the total area under the curve is 1. For a normal distribution, approximately 68 percent of the data fall within one standard deviation of the mean, approximately 95 percent of the data fall within two standard deviations of the mean, and approximately 99.7 percent of the data fall within three standard deviations of the mean. The mean of the data in a standard normal distribution is 0 and the standard deviation is 1. The standard normal curve allows for the comparison of data from different normal distributions. A z-score is a measure of position derived from the mean and standard deviation of data. A z-score expresses, in standard deviation units, how far an element falls from the mean of the data set. A z-score is a derived score from a given normal distribution. A standard normal distribution is the set of all z-scores.

Benchmark 11.a
Identify and Apply Properties of a Normal Distribution

The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

 Indicator 11.a.1 Identify the properties of a normal probability distribution Identify the properties of a normal probability distribution.

 Indicator 11.a.2 Describe impact of standard deviation & mean on normal distribution Describe how the standard deviation and the mean affect the graph of the normal distribution.

 Indicator 11.a.3 Use standard normal distribution and z-scores to compare data sets Compare two sets of normally distributed data using a standard normal distribution and z-scores.

 Indicator 11.a.4 Represent probability as area under the curve Represent probability as area under the curve of a standard normal probability distribution.

 Indicator 11.a.5 Determine probabilities or percentiles based on z-scores. Use the graphing calculator or a standard normal probability table to determine probabilities or percentiles based on z-scores.

MTH.ALG2
Standard 12
Compute and distinguish between permutations and combinations

The student will compute and distinguish between permutations and combinations and use technology for applications.

 Essential Understandings: The Fundamental Counting Principle states that if one decision can be made n ways and another can be made m ways, then the two decisions can be made nm ways.  Permutations are used to calculate the number of possible arrangements of objects. Combinations are used to calculate the number of possible selections of objects without regard to the order selected.

Benchmark 12.a
Compute and distinguish between permutations and combinations

The student will compute and distinguish between permutations and combinations and use technology for applications.

 Indicator 12.a.1 Compare and contrast permutations and combinations Compare and contrast permutations and combinations.

 Indicator 12.a.2 Calculate the number of combinations of n objects taken r at a time Calculate the number of combinations of n objects taken r at a time.

 Indicator 12.a.3 Calculate the number of permutations of n objects taken r at a time Calculate the number of permutations of n objects taken r at a time.

 Indicator 12.a.4 Use permutations and combinations to solve real-world problems Use permutations and combinations as counting techniques to solve real-world problems.

MTH.ALG2
Standard 13
USE MATRIX MULTIPLICATION TO SOLVE PRACTICAL PROBLEMS

The student will use matrix operations to solve practical problems including systems of equations.

 FCPS Notes: When an n x n matrix is multiplied by its inverse, the product is an n x n identity matrix.

 Essential Understandings: Matrices can be used to model and solve practical problems. Matrices are a convenient shorthand for solving systems of equations. Matrices can model a variety of linear systems. Solutions of a linear system are values that satisfy every equation in the system.

Benchmark 13.a
Use Matrix Multiplication to Solve Practical Problems

The student will use matrix multiplication to solve practical problems including systems of equations.

 Indicator 13.a.1 Find the product of two 2 x 2 matrices by hand Find the product of two 2 x 2 matrices by hand.

 Indicator 13.a.2 Find the product of 2 matrices greater than 2 x 2 w/a graphing calc Find the product of two matrices greater than 2 x 2 using a graphing calculator.

 Indicator 13.a.3 Use matrix multiplication to solve practical problems Use matrix multiplication to solve practical problems.

 Indicator 13.a.4 Represent and solve a system of linear equations in matrix form Represent and solve a system of no more than three linear equations in matrix form.

 Indicator 13.a.5 Find the inverse of a matrix with a graphing calculator Find the inverse of a matrix with a graphing calculator.

 Indicator 13.a.6 Find the determinant of a matrix with a graphing calculator Find the determinant of a matrix with a graphing calculator.

 Indicator 13.a.7 Identify the identity matrix and its properties Identify the identity matrix and its properties.

Essential - Standard, benchmark, or indicator from the VDOE Standards of Learning document. In the absence of VDOE standards for a given course, content subject to testing such as AP and IB can be labeled Essential.
Expected - Standard, benchmark, or indicator added by the FCPS Program of Studies to provide a context, a bridge, or an enhancement to the Essential SBIs.
Extended - Standard, benchmark, or indicator added by the FCPS Program of Studies generally used to differentiate instruction for advanced learners (Honors/GT)