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Algebra 2 A 1st Semester Unit 1: Sequences and Series Activities: Timeline: 1st Semester 2 Weeks Vocabulary: Arithmetic Mean Arithmetic Sequence Arithmetic Series Common Difference Common Ratio Explicit Formula Geometric Mean Recursive Formula Sequence Series Term Summation Notation Infinite Series New State Standards: A-SSE Seeing Structure in Expressions 1. Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★ F-IF Interpreting Functions 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ F-BF Building Functions 1. Write a function that describes a relationship between two quantities.★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★ F-LE Linear, Quadratic and Exponential Models 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). College Readiness: Range (13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimals Range (13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbers Range (16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step percent Range (16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems Range (20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor Range (33-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, Resources: Concepts and Skills: Students will: Generate a sequence Use a recursive formula Identify an arithmetic sequence Use an arithmetic mean Identify a geometric sequence Use a geometric mean Write and evaluate arithmetic and geometric series Write a series in summation notation Find the sum of a finite arithmetic series Find the sum of a finite geometric series Find the sum of an infinite arithmetic series Find the sum of an infinite geometric series Unit Learning Targets I can use the fundamental counting principle to count the number of ways an event Algebra II textbook: Larson, Boswell, Kanold, Stiff CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/algebra2/ Strategies: Graphing calculator applications Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders algebraic properties, and/or relationships between expressions and numbers Range (13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g) Range (13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals Range (16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers Range (20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities Range (20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations Range (28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations Range (33-36) Expressions, Equations, & Inequalities: Write expressions that require planning and/or manipulating to accurately model a situation NCTM: Numbers & Operations use number-theory arguments to justify relationships involving whole numbers. judge the reasonableness of numerical computations and their results Algebra generalize patterns using explicitly defined and recursively defined functions understand relations and functions and select, convert flexibly among, and use various representations for them use a variety of symbolic representations, including recursive and parametric equations, for functions and relations use symbolic algebra to represent and explain mathematical relationships use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts Quality Core A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressions A1D: Solve single-step and multistep equations and inequalities in one variable A1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problems H2A: Find the nth term of an arithmetic or geometric sequence H2B: Find the position of a given term of an arithmetic or geometric sequence H2C: Find sums of a finite arithmetic or geometric series H2D: Use sequences and series to solve real-world problems H2E: Use sigma notation to express sums can happen. I can use permutations to count the number of ways an event can happen. I can combination to count the number of ways can happen. I can use the binomial theorem to expand a binomial that is raised to power. I can find the theoretical and experimental probabilities. I can find geometric probabilities. I can find probabilities of unions and intersections of two events. I can use the compliments to find the probability of an event. I can find the probability of independent events. I can find the probability of dependent events. I can find the binomial probabilities and analyze binomial distributions. I can test the hypothesis. I can calculate probabilities using normal distributions. I can use normal distributions to approximate binomial distributions. Timeline: 1st Semester Unit 2: Inequalities Unit 3: Systems of Inequalities and Linear Programming New State Standards: 2 weeks Linear Inequalities A-SSE Seeing Structure in Expressions Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. a) Interpret parts of an expression, such as terms, factors, and coefficients. 2 weeks System of Inequalities/ Linear Programming Vocabulary: Inequalities Expressions Equations Substitution One step equations Two step equations Like terms Terms Factors Coefficients Degree Distributive Closure Commutative Associative Identities Inverse Properties Leading Coefficient Literal Equation Constraints Linear Programming Objective Function Feasible Region A-CED Creating Equations Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. A-REI Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Represent and solve equations and inequalities graphically 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★ I can find expected values of collections of outcomes. Activities: Concepts and Skills: Students will: Find distance and midpoint Solve equations and inequalities for word problems Write and solve equations and inequalities for word problems Simplify expressions and interpret their parts Identify and use algebraic properties of equality Solve literal equations (for a specified variable) Solve system of equation and inequalities using various methods (including graphing) Solve linear programming problems Solve absolute value inequalities Multiple forms for linear equations including point slope Learning Targets College Readiness: (Range 13-15) Basic Operations and Applications: Perform one-operation computation with whole numbers and decimals (Range 13-15) Basic Operations and Applications: Solve problems in one or two steps Resources: I can evaluate algebraic expressions. Algebra II textbook: Larson, Boswell, Kanold, Stiff CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/algebra2/ Strategies: Graphing calculator applications Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders Vertices Optimization Maximum Minimum Compound Inequality Absolute Value Intersection Interval Notation Set Notation using whole numbers (Range 16-19) Basic Operations and Applications: Solve some routine two-step arithmetic problems (Range 30-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers (Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor (Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g) (Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals (Range 16-19) Expressions, Equations, & Inequalities: Substitute whole numbers for unknown quantities to evaluate expressions (Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers (Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x) (Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities (Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions (Range 20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations (Range 20-23) Expressions, Equations, & Inequalities: Perform straightforward word-tosymbol translations (Range 24-27) Expressions, Equations, & Inequalities: Solve real-world problems using first degree Equations (Range 24-27) Expressions, Equations, & Inequalities: Write expressions, equations, or inequalities with a single variable for common pre-algebra settings (e.g., rate and distance problems and problems that can be solved by using proportions) (Range 24-27) Expressions, Equations, & Inequalities: Solve first-degree inequalities that do not require reversing the inequality sign (Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations (Range 28-32) Expressions, Equations, & Inequalities: Write expressions, equations, and inequalities for common algebra settings (Range 28-32) Expressions, Equations, & Inequalities: Solve linear inequalities that require reversing the inequality sign (Range 33-36) Expressions, Equations, & Inequalities: Write expressions that require planning and/or manipulating to accurately model a situation (Range 33-36) Expressions, Equations, & Inequalities: Write equations and inequalities that require planning, manipulating, and/or solving NCTM: Number and Operations: Develop a deeper understanding of very large and very small numbers and of various representations of them; Use number-theory arguments to justify relationships involving whole numbers. Algebra: understand the meaning of equivalent forms of expressions, equations, inequalities, I can simplify algebraic expressions by combining like terms. I can solve linear equations I can use linear equations to solve real world problems. I can rewrite equations with more than one variable I can rewrite common formulas I can solve simple inequalities I can solve compound inequalities I can solve absolute value equations and inequalities. and relations; write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases; Use symbolic algebra to represent and explain mathematical relationships judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. Draw reasonable conclusions about a situation being modeled Quality Core: A1D: Solve single-step and multistep equations and inequalities in one variable A1F: Write linear equations in standard form and slope-intercept form when given two points, a point and the slope, or the graph of the equation A1G: Graph a linear equation using a table of values, x- and y-intercepts, or slope-intercept form A1H: Find the distance and midpoint between two points in the coordinate plane A1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problems D1A: Solve linear inequalities containing absolute value D1B: Solve compound inequalities containing “and” and “or” and graph the solution set D2A: Graph a system of linear inequalities in two variables with and without technology to find the solution set to the system D2B: Solve linear programming problems by finding maximum and minimum values of a function over a region defined by linear inequalities Timeline: Unit 4: Matrices I can use absolute value equations and inequalities to solve real world problems. Activities: 1st Semester 2 Weeks Vocabulary: New State Standards: N-VM Vector & Matrix Quantities 6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network 7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 8. (+) Add, subtract, and multiply matrices of appropriate dimensions. Resources: Concepts and Skills: Students will: Write the dimensions of a matrix Identify a matrix element Use identity and inverse Algebra II textbook: Larson, Boswell, Kanold, Stiff CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Augmented Matrix Determinant Equal Matrices Matrix Matrix Addition Matrix Element Matrix Equation Matrix Multiplication Row Operations Scalar Multiplication Variable Matrix Zero Matrix Square Matrix Inverse Matrix 9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. A-REI Reasoning with Equations & Inequalities 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. 9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). College Readiness: (Range 13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimals (Range 13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbers (Range 16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step percent (Range 16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems (Range 33-36) Numbers: Concepts &Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers (Range 13-15) Numbers: Concepts &Properties: Perform one-operation computation with whole numbers and decimals (Range 13-15) Expressions, Equations, &Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g) (Range 13-15) Expressions, Equations, &Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals (Range 16-19) Expressions, Equations, &Inequalities: Solve one-step equations having integer or decimal answers (Range 20-23) Expressions, Equations, &Inequalities: Add and subtract simple algebraic expressions (Range 20-23) Expressions, Equations, &Inequalities: Solve routine first-degree equations (Range 28-32) Expressions, Equations, &Inequalities: Manipulate expressions and equations (Range 28-32) Expressions, Equations, &Inequalities: Find solutions to systems of linear equations NCTM: Numbers & Operations understand vectors and matrices as systems that have some of the properties of the real-number system matrices Subtract matrices Determine equal matrices Find unknown matrix elements Use scalar products Solve matrix equations with scalars Multiply matrices Determine if matrix multiplication is defined Verify matrix inverses Evaluate determinant of 2X2 matrix Find an inverse matrix Solve a matrix equation Evaluate determinant of 3X3 matrix Use technology to solve matrix problems Write a system as a matrix equation Solve a system of two equations Solve a system of three equations Use Cramer’s Rule Write an augmented matrix Write a system from and augmented matrix Edmodo Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/algebra2/ Strategies: Graphing calculator applications Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases judge the reasonableness of numerical computations and their results Algebra: understand relations and functions and select, convert flexibly among, and use various representations for them interpret representations of functions of two variables understand the meaning of equivalent forms of expressions, equations, inequalities, and relations write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. Quality Core: A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressions A1D: Solve single-step and multistep equations and inequalities in one variable A1E: Solve systems of two linear equations using various methods, including elimination, substitution, and graphing B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problems D1C: Solve algebraically a system containing three variables F1A: Evaluate and simplify polynomial expressions and equations I1A: Add, subtract, and multiply matrices I1B: Use addition, subtraction, and multiplication of matrices to solve real-world problems I1C: Calculate the determinant of 2 × 2 and 3 × 3 matrices I1D: Find the inverse of a 2 × 2 matrix I1E: Solve systems of equations by using inverses of matrices and determinants I1F: Use technology to perform operations on matrices, find determinants, and find inverses Timeline: Unit 5: Factoring Quadratics and Complex Numbers Activities: 2nd Semester New State Standards: 4 weeks Vocabulary: A-SSE Seeing Structure in Expressions Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. ★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. Resources: Concepts and Skills: Students will: Define parts of a quadratic function Find greatest common factors Factor quadratics Algebra II textbook: Larson, Boswell, Kanold, Stiff CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quadratic Function Factor GCF Monomial Binomial Trinomial Perfect Square Trinomial Difference of Squares 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). College Readiness: (Range 16-19) Number Concepts and Properties: Recognize one-digit factors of a number (Range 24-27) Number Concepts and Properties: Work with numerical factors (Range 28-32) Number Concepts and Properties: Apply number properties involving prime factorization (Range 28-32) Number Concepts and Properties: Apply number properties involving even/odd numbers and factors/multiples (Range 20-23) Expressions, Equations, and Inequalities: Multiply two binomials* (Range 24-27) Expressions, Equations, and Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) * (Range 28-32) Expressions, Equations, and Inequalities: Manipulate expressions and equations NCTM: Algebra: Understand the meaning of equivalent forms of expressions Use symbolic algebra to represent and explain mathematical relationships Quality Core: B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problems C1A: Identify complex numbers and write their conjugates C1C: Simplify quotients of complex numbers F1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping) Include graphical representations of factoring Students will: Solve by factoring Solve by finding square roots Solving by graphing Simplify radicals using i Perform operations on complex numbers (including rationalize the denominator) Factor using imaginary numbers Finding complex solutions Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/algebra2/ Strategies: Graphing calculator applications Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders