Algebra II Skills Topic Outline Course Description and Philosophy Algebra II Skills is an intermediate algebra course that reviews algebra skills introduced in the first year algebra course, a well as, explore and apply the concepts of number systems, functions and graphs. The graphing calculator will be used to enhance visual understanding of these topics. This course is intended for students who have successfully completed Geometry Skills. It provides students with the essential knowledge and strategies to be successful problem solvers and further their study of mathematics. Text Reference: Hall and Fabricant, Algebra 2 with Trigonometry, 1991, Prentice Hall, Englewood Cliffs, NJ. Written 2007 1
Unit I: Essential Questions: Why is the order of operations so important? Objectives: Students will be able to: Identify real numbers Represent them on the number line Add, subtract, multiply and divide real numbers Identify the properties of real numbers Use the properties to solve linear equations Translate word phrases into algebraic symbols Set up and solve simple word problems 2
Unit I: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit I Topic 1: The Set of Real Numbers Classification Symbols Graphing on the number line Test/Quizzes Homework Class Participation Class Investigations Text Worksheets Lectures Hands-On Activities Graphing Calculator 4.1.12A1,2,3 4.1.12.B1 4.5A1,2,3 4.5B1,2,3,4 Topic 2: Operations with Real Numbers Commutative property Associative property Distributive property Identity and inverse properties Absolute value Topic 3: Variable and Expressions Order of operations Simplifying numerical and algebraic equations Topic 4: Properties of Real Numbers Commutative property Associative property Distributive property Properties of equality Topic 5: Solving Equations in One Variable Solving multi-step equations Solving equations with fractions Solving equations with absolute value Topic 6: Translating word sentences into equations Operation words Equality and inequality phrases 3
Topic 7:Problem Solving Strategies Reading the problem Plan the approach Identify the variable Choose a model Complete the work Check answer for reasonableness Differentiated Learning Activities Strategic learner: Break problems into discrete steps. Verbalize reasoning process. Provide concrete examples. Utilize alternative assessment in the form of group activities. Advanced learner: Enrichment activities. 4
Unit II: Equations and Inequalities Essential Question: How can I solve problems using equations and inequalities? Objectives: Students will be able to: Solve literal equations or formulas for one of the variables Solve linear inequalities in one variable and graph their solution sets Determine the truth or falsity of conjunctions and disjunctions Graph the solutions sets of conjunctions and disjunctions Solve compound sentences involving inequalities and graph their solution sets Solve equations involving absolute value Solve inequalities involving absolute value and graph the solution sets Solve word problems using equations and inequalities 5
Unit II: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit II Topic 1: Literal Equations and Formulas Define literal equation List several important formulas used in mathematics and other real-world situations Solve literal equations for a particular variable Restrictions in the denominator Topic 2: Solving Inequalities Trichotomy property Properties of inequality Solving multi-step inequalities Applications using inequalities Topic 3: Conjunctions and Disjunctions Define compound inequalities Define conjunctions and disjunctions Illustrate their graphs Use Venn Diagrams to illustrate union and intersection Topic 4: Solving Compound Sentences and Inequalities Determining the type of compound sentence Solving the compound inequality Graphing the solution set Topic 5: Problem Solving Strategies Making a table Making a drawing Defining the variable Looking at a simpler problem Topic 6: Absolute Value Equations Test/Quizzes Homework Class Participation Class Activities Text Worksheets Lectures Hands-On Activities Graphing Calculator 4.1.12A3 4.3.12A2,3 4.3.12C1 4.3.12D2 4.5A1,2,3,4,5 4.5B1,2,3,4, 4.5C1,2,3,4,5,6 4.5E1,2,3 Review the definition of absolute value Solve multi-step equations Verify all solutions 6
Topic 7: Absolute Value Inequalities Solve multi-step absolute value inequalities Graph on the number line Verify all solutions Topic 8: Problem Solving Using Equations and Inequalities Define arithmetic mean Use strategies discussed in topic 5 Solve various types of word problems Check all answers for reasonableness Differentiated Learning Activities Strategic learner: Use visual models. Have students work in pairs to define sets of items and use Venn Diagrams to illustrate union and intersection. Verbalize thinking process when solving problems. Advanced learner: Develop truth tables and show how union and intersection are used. Ethical Decision Making/Character Education: Discuss tolerance limits in manufacturing and how absolute values are used in their calculations. 7
Unit III: Functions and Their Graphs Essential Question: How can functions and their graphs be used to generalize solutions to problems? Objectives: Students will be able to: Graph ordered pairs on a coordinate plane Identify relations and functions Define slope of a line Identify parallel and perpendicular lines and their relation to slope Define domain and range of a function Graph the equation of a line Write the equation of a line given the slope and one point, or two points Evaluate a linear function Find the composition of two functions Find the inverse of a function 8
Unit III: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit III Topic 1: Coordinate Plane Define parts of the plane Name coordinates of given points Graph given coordinates Test/Quizzes Homework Class Participation Text Worksheets Lectures Hands-On Activities Graphing Calculator Excel Computer Program 4.3.12B1,2.3 4.3.12C1,2 4.5E1 4.5F1 Topic 2: Relations and Functions Define relations and functions Finding the domain of a function Using mapping to illustrate domain and range Vertical line test for functions Topic 3: Graphing Equations Determine if ordered pairs are solutions to linear equations Graphing lines by plotting points from a table Topic 4: Composition of Functions Evaluating functions using functional notation Defining the absolute value function Sketching the graph of the absolute value function Defining composite functions Evaluating composite functions Topic 5: Inverse Functions Definition of inverse functions Finding inverse functions by reversing domain and range and solving for Y Horizontal line test to determine if an inverse exists Graphing functions and their inverses 9
Topic 6: Slope of a Line Find the slope given two points Use the slope and y-intercept to write the equation of a line Graph the equation of a line Topic 7: Parallel and Perpendicular Lines Determine if the lines are parallel or perpendicular Write the equation of a line parallel or perpendicular to a given line Topic 8: Equations of a Line Point-slope form of a line Slope-intercept form of a line Standard form of a line Graphing lines from all forms Topic 9: Problem Solving Using Coordinate Geometry Writing a linear model to solve word problems Solving the problem Checking for reasonability Differentiated Learning Activities Strategic learner: More dependent on graphing calculator to Identify functions and their inverses. Advanced learner: Find the inverses of formulas in Geometry and Chemistry. 10
Unit IV: Systems of Equations and Inequalities Essential Question: How can a large industry, such as the auto manufacturers, organize information and solve their problems? Objectives: Students will be able to: Determine when a function is direct variation Solve direct variation problems Find the solution to a linear system of equations in two variables algebraically Determine if a linear system is consistent and independent, consistent and dependent or inconsistent Solve a linear system of equations graphically Solve word problems using systems of linear equations Graph linear inequalities in two variables Solve linear systems of inequalities by graphing Solve problems using linear programming 11
Unit IV: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit IV Topic 1: Direct Variation Determine if a function is a direct variation Find the constant of variation Solve direct variation problems Topic 2: Graphing to Solve Problems Parallel lines Intersecting lines Coinciding lines Topic 3: Solving Systems of Linear Equations by Graphing Graph the system using slopeintercept form of a line Find points of intersection Classify the system as consistent and independent, consistent and dependent or inconsistent Topic 4: Solving Systems of Linear Equations by Elimination Eliminating one variable by addition or subtraction Using multiplication to eliminate one variable Check answers Topic 5: Solving Linear Systems Using the Substitution Method Solving for one variable in terms of the other Substitute for that variable Solve the linear equation in one variable Substitute to find the other variable Check answers Test/Quizzes Homework Class Participation Text Worksheets Lectures Class Activities Graphing Calculator 4.3.12B1,2 4.3.12D2 4.5E1 12
Topic 6: Mathematical Models Using Systems of Equations Use a table to organize information Define variables Set up a linear system Solve the system Check answers Topic 7: Solving Linear Systems of Inequalities Define open and closed boundaries Graph linear inequalities using shading Solve the system by graphing both inequalities Use the graphing calculator to graph and shade the linear inequalities Topic 8:Linear Programming Define constraints, feasible region, and objective function Use graphing systems of linear equations to solve linear programming problems Test vertex points in the objective function to determine maximums or minimums Differentiated Learning Activities Strategic learner: Practice with graphing calculator. Work in pairs on linear programming activity. Advanced learner: Define the constraints and objective function for more difficult problems. Ethical Decision Making/Character Education: Discuss constraints, such as employee benefits and salaries when making decisions to optimize profit and minimize costs. 13
Unit V: Matrices Essential Question: How can matrices be used in communications? Objectives: Students will be able to: Find the sum and difference of two matrices and perform scalar multiplication Find the product of two matrices if it exists Solve problems using matrices Solve systems of equations using matrices 14
Unit V: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit V Topic 1: Finding the Sum and Difference of Two Matrices Naming matrices by dimensions Properties of matrix addition Adding and subtracting matrices Properties of scalar multiplication Topic 2: Matrix Multiplication Defining matrix multiplication Properties of matrix multiplication The dimensions of the product matrix Test/Quizzes Homework Class Participation Text Worksheets Lectures Class Activities Graphing Calculator 4.1.12B3 4.4.12D1 4.5.B.1,2,3,4 4.5E1,2,3 Topic 3: Using Matrices to Solve Problems Directed graphs Communications between company employees Two step communications Topic 4: Solving Systems of Linear Equations Using Matrices Setting up a matrix equation Entering a matrix into the graphing calculator Using the inverse of a matrix to solve the system Differentiated Learning Activities Strategic learner: Use matrices to illustrate the inventory of two sporting goods stores. Advanced learner: Prove that multiplication of matrices is not commutative. 15
Unit VI: Polynomials Essential Question: What type of function does the path of a projected arrow take? Objectives: Students will be able to: Review properties of exponents Classify, add, and subtract polynomials Multiply polynomials including special products Factor polynomials Solve polynomial equations by factoring Solve problems using polynomial equations 16
Unit VI: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit VI Topic 1: Exponents and Monomials Review properties of exponents Define monomials Degree of a monomial Operations with monomials Test/Quizzes Homework Class Participation Text Worksheets Lectures Class Activities Graphing Calculator 4.3.12D1,2 4.5B1,2,3,4 4.5C6 Topic 2: Polynomials: Classifications Classify polynomials Degree of a polynomial Adding and subtracting polynomials Topic 3: Multiplying Polynomials Monomial times a polynomial Binomial times a binomial Perfect square trinomials Difference of two squares Topic 4: Factoring Polynomials Greatest Common Factors Perfect square trinomials Difference of squares Sum and difference of cubes Topic 5: Factoring Quadratic Trinomials Define a quadratic trinomial Factor by reverse FOIL Topic 6: Factoring Polynomials Completely Factor by grouping Combined forms of factoring Topic 7: Solving Polynomial Equations Solve by factoring Zero product property Double roots 17
Topic 8: Problem Solving Using Polynomial Equations Geometry related problems Physics related problems Set up a polynomial equation Solve by factoring Check answers Differentiated Learning Activities Strategic learner: Use puzzle worksheets to enhance factoring skills. Advanced learner: Factor higher power polynomials by quadratic methods. Ethical Decision Making/Character Education: Review academic integrity policy before the midterm exam.. 18
Unit VII: Rational Expressions Essential Question: How does a computer convert very large numbers and decimals into simpler forms? Objectives: Students will be able to: Define zero and negative exponents Write numbers in scientific notation Simplify rational expressions Multiply and divide rational expressions Add and subtract rational expressions Simplify complex rational expressions Solve rational equations Solve problems using rational equations 19
Unit VII: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit VII Topic 1: Negative Exponents and Scientific Notation Zero as an exponent Define negative exponents Using scientific notation to compute with very large or small numbers Test/Quizzes Homework Class Participation Text Worksheets Lectures Class Activities Graphing Calculator 4.3.12A2 4.1.12B2 4.3.12C1 4.5F3 Topic 2: Simplifying Rational Expressions Restrictions of the denominator Using rules of exponents Using factoring Topic 3: Multiplying and Dividing Rational Expressions Review rules for multiplying and dividing numerical fractions Using rules of exponents Using factoring Topic 4: Adding and Subtracting Rational Expressions Review rules for adding and subtracting numerical fractions Like denominators Finding the least common denominator Topic 5: Complex Rational Expressions Using the least common denominator and the distributive property Reducing with factoring Topic 6: Solving Rational Equations Factor denominators Finding the least common denominator Multiplying both sides by the LCD 20
Combining like terms Solve the resultant equation Check for extraneous roots Topic7: Using Rational Equations to Solve Problems Work problems Rate problems Percent mixture problems Differentiated Learning Activities Strategic learner: Use graphing calculator to check for restrictions. Advanced learner: Write word problems that involve rational equations. Share solutions with the class. 21
Unit VIII: Irrational and Complex Numbers Essential Question: How can I evaluate the square root of -1? Objectives: Students will be able to: Simplify an nth root Find the nth root Add, subtract, multiply and divide radical expressions Write radical expressions in simplest form Solve radical equations Use the Distance and Midpoint Formulas Simplify square roots using powers of i Add, subtract, multiply and divide complex numbers 22
Unit VIII: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit VIII Topic 1: Roots and Radicals Principal roots Define index, root and radicand Odd and even indexes Solving polynomial equations by extraction Test/Quizzes Homework Class Participation Text Worksheets Lectures Class Activities Graphing Calculator 4.1.12A1,2,3 4.1.12.B1 4.5D1 Topic 2: Multiplying and Dividing Radicals Like indexes Multiplying or dividing coefficients Adding or subtracting exponents Rules for simplest radical form Topic 3: Adding and Subtracting Radicals Like radicands Define conjugate Multiplying and dividing binomials that contain radicals Special products Topic 4: Solving Radical Equations Isolate the radical Separate two radicals Squaring both sides Check answers Topic 5: Distance and Midpoint Formulas Finding the distance between two points in the plane Finding the coordinates of the midpoint of a segment Topic 6: Imaginary Numbers Definition Powers of i Simplify square roots of negative 23
numbers Topic 7: Adding and subtracting complex numbers Standard form of a complex number Pure imaginary numbers Adding and subtracting like parts Absolute value of a complex number Topic 8: Multiplying and Dividing Complex Numbers FOIL Special products Using the conjugate to divide Differentiated Learning Activities Strategic learner: Review rules of exponents. Advanced learner: Prove that (4,3), (-6,6) and (-3,-4) form and isosceles triangle. 24
Unit IX: Quadratic Functions Essential Question: How can I solve a quadratic equation that doesn t factor? Objectives: Students will be able to: Graph quadratic functions in standard form Find the vertex, direction, intercepts and axis of symmetry Solve quadratic equations by the Quadratic Formula Use the discriminant to find the nature of the roots Make a connection between the graph and the nature of the roots Find a quadratic equation given the roots 25
Unit IX: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit IX Topic 1: Graphing a Quadratic Function Axis of symmetry Vertex point Direction of the parabola Zeros Y-intercept and mirror point Topic 2: The Quadratic Formula Definition Connection to standard form Using the formula to solve quadratic equations Real and complex roots Storing the quadratic formula on the calculator Topic 3: Discriminant Definition Nature of the roots Relationship between the nature of the roots and the graph Topic 4: Problem Solving Using Quadratic Equations Geometry problems Number problems Check for extraneous roots Test/Quizzes Homework Class Participation Class Activities Text Worksheets Lectures Hands-On Activities Graphing Calculator 4.3.12B2,3,4 4.3.12C1 4.3.12D5 Differentiated Learning Activities Strategic learner: Use note cards for formulas. Verbalize thinking process when solving problems. Advanced learner: Have students develop a connection between the vertex point and the midpoint of the segment connecting the x-intercepts of a quadratic function. 26
Unit X: Exploring Other Functions Essential Question: How does interest get compounded? Objectives: Students will be able to: Graph exponential functions using the graphing calculator Solve exponential equations Graph logarithmic equations using the graphing calculator Evaluate logarithms using the graphing calculator Graph and evaluate sine and cosine functions using the graphing calculator 27
Unit X: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit X Topic 1: Exponential and Logarithmic Functions Define Domain and range Increasing and degreasing Review inverse functions Graphs using the calculator Evaluating these functions Change of base formula Problem solving using exponential functions Topic 2: Sine and Cosine Functions Define Domain and range Graphing sine and cosine curves Amplitude and period Evaluating these functions using the calculator Inverse of sine and cosine Finding the inverse using the calculator Test/Quizzes Homework Class Participation Class Activities Teacher s notes Worksheets Lectures Hands-On Activities Graphing Calculator 4.3.12.B1,2,4 4.3.12C1,2 4.5C3 4.5E1 Differentiated Learning Activities Strategic learner: Use visual models. Use note cards for formulas. Verbalize thinking process when solving problems. Advanced learner: Have students translate sine and cosine curves. 28
Unit XI: Probability and Statistics Essential Question: Can I use the mean, median and mode interchangeably? Objectives: Students will be able to: Find the mean, median and mode of a set of data Construct a histogram from a frequency distribution Use the Fundamental Counting Principle Specify a sample space Calculate simple probabilities 29
Unit XI: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit XI Topic 1: Mean, Median and Mode Define and discuss differences Using these measures of central tendency accurately Frequency distribution Constructing a histogram Topic 2: Counting Principle Define factorials Tree diagrams Using the Counting Principle Define permutations and combinations Using permutations and combinations with the graphing calculator to solve problems Topic 3: Probability Define probability Specify a sample space for an experiment (tree diagram) Calculate the probability that a specific event will occur Calculate probability of dependent and independent events Test/Quizzes Homework Class Participation Class Activities Text Teacher s notes Worksheets Lectures Hands-On Activities Graphing Calculator 4.4.12A1,2,3 4.4.12B1,2,3,4 4.4.12C1,2 Differentiated Learning Activities Strategic learner: Use visual models. Verbalize thinking process when solving problems. Advanced learner: Calculate permutations without the use of the calculator. Ethical Decision Making/Character Education: Discuss the probability of winning a card game. 30
Unit XII: Patterns Essential Question: Is there a shortcut to finding the nth term in a sequence? Objectives: Students will be able to: Write the rule of a sequence as a recursive formula Find specific terms of an arithmetic sequence Find specific terms of a geometric sequence 31
Unit XII: Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit XII Topic 1: Looking for Patterns Define finite and infinite sequences Define recursive formula Find a recursive formula for a given sequence Topic 2: Arithmetic and Geometric Sequences Define arithmetic and geometric sequences Calculate the arithmetic and geometric means of given sequences Find the nth term of an arithmetic or geometric sequence Test/Quizzes Homework Class Participation Class Activities Text Teacher s notes Worksheets Lectures Hands-On Activities Graphing Calculator 4.3.12A1,2,3 4.5B1,2,3,4 Differentiated Learning Activities Strategic learner: Use visual models. Verbalize thinking process when solving problems. Advanced learner: Research the Fibonacci sequence and Tower of Hanoi. Ethical Decision Making/Character Education: Review academic integrity policy before final exam. 32