Algebra+2+Curriculum+and+Standards

This is the document from the Indiana Department of Education. Our Algebra 2 courses at CHS cover the same material, but in a different order. **ALGEBRA II** Grade 11, 2 Semester, 2 Credits

** Course Overview ** //Algebra II// is a course that extends the content of Algebra I and provides further development of the concept of a function. Topics include: (1) relations, functions, equations and inequalities; (2) conic sections; (3) polynomials; (4) algebraic fractions; (5) logarithmic and exponential functions; (6) sequences and series; and (7) counting principles and probability.

Recommended Prerequisite: Algebra I  · Credits: A two credit course · Fulfills the Algebra II/Integrated Mathematics III requirement for the Core 40, Core 40 with Academic Honors and Core 40 with Technical Honors diplomas and counts as a Mathematics Course for the General Diploma · A Career Academic Sequence or Flex Credit course

**__UNIT 1: Linear Equations/Inequalities, Systems of Equations__** 9 Weeks ** Description ** The purpose of this unit is to serve as an introduction to Algebra II, including a review of necessary Algebra 1 skills, such as solving and graphing linear equations and inequalities, and systems of equations. Matrices are covered only to the extent that they are used as a method for solving systems of equations. Absolute value equations are also discussed. **Indicators mastered**: 1.2 (partial), 1.4, 1.6, 2.2, 2.3 (partial), 2.4

** Big Ideas ** 1. Multiple equations can be used to simultaneously model and solve a situation with several variables. 2. Equations, tables, graphs, and words can all be utilized interchangeably to represent the same event. 3. Equations can provide a model that can be used to analyze, draw conclusions and make predictions. 4. Equations and inequalities allow us to graphically or algebraically represent and solve problems.

** Essential Questions ** 1. How can simultaneous equations be used to model real world situations? 2. How would you determine which representation would be appropriate for a given event? 3. When would it not be a good idea to make predictions from a model? 4. When are inequalities more appropriate than equations in describing a situation? 5. What makes an answer reasonable?

** Vocabulary ** absolute value, function notation, dependent variable, independent variable, system of equations, substitution, elimination, parent function, median fit line, identity, correlation, regression, correlation coefficient, dimensions of a matrix, scalar, row reduction

**__TOPIC 1.1: Interpreting Functions__** 3 Weeks

** Description ** This topic focuses on an introduction to functions and will be expanded on later. Students should spend time mastering graphing linear functions.

**__ Learning Targets __** A2.1.1e Graph an algebraic function A2.1.2e Write an equation in function notation A2.1.4a Graph an equation by hand A2.1.4b Graph an equation using graphing technology A2.1.8a Interpret a graph of a given function in words A2.1.8b Develop a formula from a given situation A2.1.8c Create a graph from a given situation

**__TOPIC: 1.2 Reasoning with Equations and Inequalities__** 5 Weeks

** Description ** Each of these learning targets includes review of Algebra 1 concepts with increased skill needed for mastery. Topics include absolute value, linear equations and inequalities, and solving systems of equations.

**__ Learning Targets __** A2.1.6a Compare the graph of an inequality to its related equation A2.1.6b Identify the solution of an inequality from the graph A2.1.6c Draw graph of an inequality A2.2.1a Define absolute value A2.2.1b Recognize a graph of an absolute value A2.2.1c Graph absolute value inequalities A2.2.1d Write the absolute value equation or inequality given the graph A2.2.2a Solve a system of equations using substitution A2.2.2b Solve a system of equations using elimination

**__TOPIC 1.3: Vector and Matrix Quantities__** 1 Week

** Description ** The learning targets in this unit are the application of matrices to solve systems of equations.

**__ Learning Targets __** A2.2.2c Perform matrix row reduction A2.2.2d Solve a system of equations using matrices

**__Unit 1: Ongoing Mathematical Practices__**

**__Linear Models__** ** Description ** Learning targets in this unit are only focused on linear models

**__ Learning Targets __** A2.2.3a Write a system of equations based on the information in a word problem A2.2.3b Write a system of inequalities based on the information in a word problem A2.2.3c Solve a system of linear equations in the context of a word problem A2.2.4a Explain the meaning of the slope of a line in the context of the problem A2.2.4b Explain the meaning of the y-intercept of a line in the context of the problem A2.2.4c Model data using a median-fit line, more common to use best-fit line. A2.2.4d Use a linear model to make predictions

** Description ** Students should use these strategies nearly every day. These learning targets can only be assessed in conjunction with other learning targets, and should not be viewed as stand-alone targets.

**__ Learning Targets __** A2.10.1a Use a variety of problem-solving strategies, such as drawing a diagram and writing an equation A2.10.2a Decide whether a solution is reasonable in the context of the original problem A2.10.4a Justify the steps of simplifying functions and solving equations. Use properties of number systems and order of operations A2.10.5a Compare the solutions of simplified equation(s) to the solutions of the original equation(s)

**__UNIT 2: Quadratic Functions, Complex Numbers, Polynomials__** 9 Weeks

** Description ** Unit 2 advances and builds on the foundation of functions that was set in the first quarter. Quadratic functions are introduced now that linear functions have been mastered. Complex numbers expand the scope of number system beyond real solutions to equations. Polynomials are then explored, generalizing the concepts of quadratic functions to higher degrees, with a special focus on zeros and factors. **Indicators mastered**: 1.1 (partial) 1.5, 3.1, 3.2, 3.3, 3.4, 3.5, 3.7, 5.2, 5.3, 5.6, 5.7 // Indicators from previous units should be incorporated into Unit 2, to ensure that students continue to build mastery of previous topics. Students with basic understanding of Unit 1 indicators should be provided opportunities to demonstrate mastery of these indicators. //

** Big Ideas ** 1. Complex numbers provide a means of evaluating expressions that have no real solution. 2. Quadratic expressions allow us to model real world situations that do not follow a linear pattern. 3. An understanding of the factors of a polynomial can be used to illustrate characteristics of graphs and solutions of equations. 4. The shape and features of a graph provide valuable information about its corresponding equation.

** Essential Questions ** 1. Why do we need complex numbers? 2. How are the multiple representations of quadratic functions related? 3. How do quadratic equations increase our ability to understand real-world situations? 4. How can higher-order polynomials be used as tools to best describe and help explain real-world situations?

** Vocabulary ** vertex, focus, axis of symmetry, intercepts, directrix, zeros, roots, solutions, factors, complex number, imaginary numbers, complex conjugate, standard form, vertex form, maximum, minimum, binomial, trinomial, complete the square, degree, leading coefficient, long division, synthetic division, multiplicity, fundamental theorem of algebra, end behavior, local maxima/minima **__TOPIC 2.1: Quadratic Functions__** 4 Weeks

** Description ** These learning targets focus on quadratic functions and equation, including systems of equations that maybe linear or quadratic.

**__ Learning Targets __** A2.3.3a Find the real solutions to a quadratic equation A2.3.3b Find the complex solutions to a quadratic equation A2.3.4a Identify the parts of a graph of a quadratic equation Parts: vertex, focus, axis of symmetry, intercepts, directrix. Note: ACT Course Standards expect students to find domain and range as well. A2.3.4b Find the maximum or minimum value of a quadratic equation A2.3.4c Graph a quadratic function given its equation A2.3.4d Shift the graph of a quadratic function horizontally and/or vertically A2.3.4e Stretch and shrink graphs of quadratic equations A2.3.4f Interpret the zeros and maximum or minimum values of quadratic functions A2.3.5a Write an appropriate quadratic equation to model a situation A2.3.5b Interpret the solutions to a quadratic equation in relation to a given problem A2.3.7a Solve a system containing one linear and one quadratic equation A2.3.7b Solve a system containing two quadratic equations

**__TOPIC 2.2: The Complex Number System__** 1 Week

** Description ** Complex numbers will be introduced, manipulated, and applied as solutions to quadratic equations.

**__ Learning Targets __** A2.3.1a Define complex number A2.3.1b Evaluate powers of i A2.3.1c Add and subtract complex numbers A2.3.1d Multiply complex numbers A2.3.1e Divide two complex numbers A2.3.2a Explain how real numbers and complex numbers are related A2.3.2b Plot complex numbers on the coordinate plane A2.3.3b Find the complex solutions to a quadratic equation

**__TOPIC 2.3: Arithmetic with Polynomials Expressions__** 4 Weeks

** Description ** This topic explores characteristics of polynomial equations and graphs. Polynomials will be broken down by division and factoring. The relationship between zeros, roots, solutions and factors is given special attention.

**__ Learning Targets __** A2.1.1a Recognize the graph of a polynomial function A2.1.1d Graph a polynomial function. For higher-order polynomials (eg: degree 4), emphasis should be placed on //sketching// a graph based on zeros. A2.1.5a Define a zero of a function A2.1.5b Locate the zeros of a function on a graph A2.1.5c Identify the zeros of a function that is written in factored form A2.1.5d Find the zeros of a function written in standard form A2.1.5e Write the equation of a function given the zeros A2.1.5f Sketch a function given the zeros A2.5.2a Divide polynomials by others of lower degree: Synthetic Division, Long Division A2.5.3a Factor a polynomial completely: GCF, factor by grouping, difference of two squares, sum/difference of two cubes, trinomials A2.5.3b Solve polynomial equations by factoring A2.5.4a Find the approximate solutions to a polynomial equation using graphing technology A2.5.5a Write a polynomial equation to model a word problem A2.5.5b Solve a polynomial equation in the context of a word problem A2.5.6a Write a polynomial equation given its solutions A2.5.7a Describe the relationship between solutions, zeros, x-intercepts and factors A2.5.7b Find the solutions to an equation given the factors A2.5.7c Solve a polynomial given some of the factors

**__Unit 2: Ongoing Mathematical Practices__**

** Description ** Students should use these strategies nearly every day. These learning targets can only be assessed in conjunction with other learning targets, and should not be viewed as stand-alone targets.

**__ Learning Targets __** A2.10.1b Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards A2.10.2a Decide whether a solution is reasonable in the context of the original problem A2.10.4a Justify the steps of simplifying functions and solving equations. Use properties of number systems and order of operations A2.10.5a Compare the solutions of simplified equation(s) to the solutions of the original equation(s) A2.10.6a Show that statements are false by finding a counterexample

**__UNIT 3: Rational, Exponential and Logarithmic Functions__** 9 Weeks

** Description ** Unit 3 is filled with two primary topics: algebraic fractions (rational functions) and exponential/logarithmic functions. Each topic has a number of required skills and concepts that were not introduced in any previous math course, requiring an entire quarter to thoroughly cover everything. **Indicators mastered**: 1.1 (partial) 3.6, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8 // Indicators from previous units should be incorporated into Unit 3, to ensure that students continue to build mastery of previous topics. Students with basic understanding of Unit 1 and 2 indicators should be provided opportunities to demonstrate mastery of these indicators. //

** Big Ideas ** 1. Algebraic fractions can be manipulated in a similar manner to simple numerical fractions. 2. Exponentials and logarithms are inverse operations that can be manipulated and rewritten. 3. The characteristics of exponential and logarithmic functions and their representations are useful in solving real-world problems. 4. Fractional equations are useful in solving problems involving direct and inverse variation.

** Essential Questions ** 1. Why do valid algebraic steps sometimes produce extraneous solutions? 2. Why do we need both exponential and logarithmic equations? 3. How can exponential and logarithmic functions be used as tools to best describe and help explain real-world situations? 4. How can rational functions be used as tools to best describe and help explain real-world situations?

** Vocabulary ** direct variation, joint variation, inverse variation, rational expression, undefined, extraneous solution, asymptote, exponential growth, exponential decay, inverse relation, exponential form, logarithmic form, natural logarithms, rational exponent, radical function

**__TOPIC 3.1: Arithmetic with Polynomials and Rational Expressions__** 4 Weeks

** Description ** The focus of this topic is rational expressions and equations. Negative exponents will be reviewed, and rational exponents will be introduced. Application will largely take the form of direct, inverse and joint variation problems. **__ Learning Targets __** A2.1.1b Recognize the graph of an algebraic function A2.1.1c Recognize the graph of a rational function A2.1.1e Graph an algebraic function A2.1.1f Graph a rational function A2.1.4a Graph an equation by hand A2.1.4b Graph an equation using graphing technology A2.6.1a Simplify an expression with negative exponents A2.6.1b Simplify an expression with fractional exponents A2.6.2a Find a common denominator between algebraic fractions A2.6.2b Perform basic operations on algebraic fractions A2.6.2c Simplify algebraic fractions A2.6.3a Simplify complex fractions A2.6.4a Solve equations involving algebraic fractions A2.6.5a Write a fractional equation to model a given situation A2.6.5b Solve a word problem involving fractional equations A2.6.6a Compare direct, inverse and joint variation A2.6.6b Solve a problem involving direct variation A2.6.6c Solve a problem involving inverse variation A2.6.6d Solve a problem involving joint variation A2.6.6e Identify the type of variation present in a problem

**__TOPIC 3.2: Linear, Quadratic, and Exponential Models__** 5 Weeks

** Description ** This topic focuses on radicals, exponentials, and logarithms (both equations and inequalities). Since exponential and logarithmic functions are being introduced for the first time, extensive time and effort should be devoted toward those topics. Logarithmic properties will be explored.

**__ Learning Targets __** A2.1.4a Graph an equation by hand A2.1.4b Graph an equation using graphing technology A2.3.6a Solve an equation that contains radical expressions A2.3.6b Identify extraneous solutions A2.7.1a Recognize an exponential function given its graph A2.7.1b Recognize when an exponential function will be increasing or decreasing A2.7.1c Graph an exponential function given its equation A2.7.2a Recognize the simple laws of logarithms Ex: True or false: log x 7 = 7 log x. True or false: log (x + 7) = log x + log 7 A2.7.2b Prove simple laws of logarithms A2.7.2c Apply the properties of logarithms to expand or contract a logarithmic expression A2.7.3a Describe the relationship between exponents and logarithms A2.7.3b Simplify expressions involving both exponents and logarithms Ex: log 10 7 (10) A2.7.4a Solve exponential equations A2.7.4b Solve logarithmic equations A2.7.4c Solve exponential inequalities A2.7.4d Solve logarithmic inequalities A2.7.5a Convert a logarithm to a different base A2.7.6a Simplify logarithmic expressions A2.7.6b Evaluate a logarithm using the properties of logarithms A2.7.7a Use a calculator to find the decimal approximation of natural and common logarithmic expressions A2.7.8a Write an exponential equation to model a given situation A2.7.8b Solve word problems involving exponential growth or decay

**__ Unit 3: Ongoing Mathematical Practice __**

** Description ** Students should use these strategies nearly every day. These learning targets can only be assessed in conjunction with other learning targets, and should not be viewed as stand-alone targets.

**__ Learning Targets __** A2.10.1a Use a variety of problem-solving strategies, such as drawing a diagram and writing an equation A2.10.2a Decide whether a solution is reasonable in the context of the original problem A2.10.3a Decide if a given algebraic statement is true always, sometimes or never A2.10.4a Justify the steps of simplifying functions and solving equations. Use properties of number systems and order of operations A2.10.5a Compare the solutions of simplified equation(s) to the solutions of the original equation(s)

**__UNIT 4: Conic Sections, Probability and Sequences & Series__** 9 Weeks

** Description ** Unit 4 contains assorted advanced algebra topics. The focus is largely on: 1) building new functions from existing functions; 2) conic sections; 3) counting principles and probability; and 4) sequences and series. The steady progression of more complicated functions over the course of the year hits its peak with building functions and exploring conics. The remainder of the year is devoted to topics in discrete math. **Indicators mastered**: 1.1, 1.2, 1.3, 1.7, 4.1, 4.2, 5.1, 8.1, 8.2, 8.3, 8.4, 9.1, 9.2 // Indicators from previous units should be incorporated into Unit 4, to ensure that students continue to build mastery of previous topics. Students with basic understanding of Unit 1, 2, and 3 indicators should be provided opportunities to demonstrate mastery of these indicators. //

** Big Ideas ** 1. Basic mathematical operations can be performed on given functions resulting in an infinite number of possible new functions. 2. Counting methods can be used to determine possible outcomes as well as the likelihood of an event occurring. 3. Circles, ellipses, parabolas, and hyperbolas can be found from cross sections of a double-napped cone, but have distinctly different characteristics and equations. 4. Numerical patterns can be quantified and used to make predictions.

** Essential Questions ** 1. When might it be useful to combine functions instead of working with two separate functions? 2. How can probability be used to make predictions or draw conclusions? 3. How can spatial relationships be described by careful use of geometric language? 4. How do geometric relationships help to solve problems and/or make sense of phenomena? 5. How can patterns, relations, and functions be used as tools to best describe and help explain real-world patterns?

** Vocabulary ** composite function, piece-wise graph, one-to-one, horizontal line test, conic sections, focus, vertices, major axis, minor axis, conjugate axis, transverse axis, probability, combination, permutation, fundamental counting principal, factorial, outcome, sample space, event, complements, trial, experiment, geometric sequence, arithmetic sequence, geometric series, arithmetic series, summation notation, infinite, recursive formulas, explicit formula, partial sum, binomial theorem

**__TOPIC 4.1: Building Functions__** 3 Weeks

** Description ** The learning targets in this topic cover basic operations and compositions of functions. Prior knowledge of graphing will be applied to exploring piece-wise functions

**__ Learning Targets __** A2.1.1b Recognize the graph of an algebraic function A2.1.2a Add and subtract pairs of functions A2.1.2b Multiply pairs of functions A2.1.2c Divide pairs of functions A2.1.2d Determine the domain of a function A2.1.3a Describe how to create a new function through the composition of two functions A2.1.3b Perform a composition of a function A2.1.7a Recognize a piece-wise function A2.1.7b Sketch a function with a limited domain A2.1.7c Graph a piece-wise function A2.1.7d Write a piece-wise function given a graph

**__TOPIC 4.2: Expressing Geometric Properties with Equations__** 2 Weeks

** Description ** An investigation in algebraic equations and the geometric representation of conic sections.

**__ Learning Targets __** A2.4.1a Identify parts of a circle A2.4.1b Identify the parts of an ellipse A2.4.1c Identify parts of a hyperbola A2.4.1d Compare circles, ellipses, parabolas and hyperbolas A2.4.1f Write the equation of a conic based on given information A2.4.2a Classify a conic section given its equation A2.4.2b Graph a conic section from its equation

**__TOPIC 4.3: Conditional Probability and the Rules of Probability__** 2 Weeks

** Description ** The fundamental counting principle, permutations and combinations will be covered, and then applied to probability. Combinations will also be applied to expanding the power of a binomial.

** __Learning Targets__ ** A2.5.1a Explain the binomial theorem A2.5.1b Expand a binomial expression raised to a positive power, using the binomial theorem A2.9.1a Find the number of ways to order a group of a certain size by applying the basic counting principle "Basic counting principle" is also called the "fundamental counting principle." A2.9.1b Define a combination Ex. "Taking a subgroup out of a larger group, in a method where order doesn't matter." A2.9.1c Define a permutation Ex. "Taking a subgroup out of a larger group, in a method where order matters." A2.9.1d Determine whether a situation describes a permutation or a combination A2.9.1e Evaluate a problem utilizing either permutations or combinations A2.9.2a Determine the probability of a given outcome, utilizing the basic counting principle Note: ACT course standards will expect students to know: mutually exclusive, nonmutual exclusive, independent, dependent, unions, intersections, complements, and conditional probability. A2.9.2b Determine the probability of a given outcome, utilizing combinations A2.9.2c Determine the probability of a given outcome, utilizing permutations

**__TOPIC 4.4: Sequences and Series__** 2 Weeks

** Description ** This topic applies algebraic concepts to create arithmetic & geometric sequences and series from patterns.

**__ Learning Targets __** A2.8.1a Define an arithmetic sequence A2.8.1b Define a geometric sequence A2.8.1c Define arithmetic series A2.8.1d Define geometric series A2.8.1e Compare arithmetic and geometric sequences A2.8.1f Determine whether a given sequence is arithmetic or geometric A2.8.2a Find the specified term of an arithmetic sequence. Formulas might be useful. A2.8.2b Find the specified term of a geometric sequence. Formulas might be useful. A2.8.3a Find the partial sum of an arithmetic series. Formulas might be useful. A2.8.3b Find the partial sum of a geometric series. Formulas might be useful A2.8.4a Solve a word problem involving the application of a sequence A2.8.4b Solve a word problem involving the application of a series

**__Unit 4: Ongoing Mathematical Processes__**

** Description ** Students should use these strategies nearly every day. These learning targets can only be assessed in conjunction with other learning targets, and should not be viewed as stand-alone targets.

**__ Learning Targets __** A2.10.1b Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards A2.10.2a Decide whether a solution is reasonable in the context of the original problem A2.10.6a Show that statements are false by finding a counterexample