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2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence Unifying Concept: Polynomial, Rational, and Radical Functions Mathematics Content Focus: Students will determine the best method to simplify and New book Chapters 3, 5 solve rational equations, and will justify each step used Old book Chapters 6, 7, 9 in their solution(s). Students will also apply the meaning of the zeroes, yintercept and asymptotes of a rational function to the context a real world problem. Students will sketch a graph of the radical function using the key features of the graph to include: zeroes, end behavior, and asymptotes, and be able to justify each step used to solve an equation. Rational Functions Mathematical Practice Focus: -Operations Mathematically Proficient Students… -Solving Rational Equations 1. Make sense of problems and persevere in solving -Graphing them. 2. Reason abstractly and quantitatively. Radical Functions 3. Construct viable arguments and critique the -Properties of Exponents reasoning of others. -Solving Radicals Equations 4. Model with mathematics. -Graphing 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Target Standards A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 (polynomial – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). only) A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = (polynomial g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., and rational using technology to graph the functions, make tables of values, or find successive only) approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. ★ F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs (polynomial and tables in terms of the quantities, and sketch graphs showing key features given a verbal only) description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★ F-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder Polynomials -Operations -Factoring -Graphing/Transformations -Remainder Theorem -Solving equations 8/28/2015 9:53 AM Curriculum Instruction and Professional Development Math Department Page 1 2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence A-APR.B.3 on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-CED.A.1 (rational only) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Quarter Major Clusters Arizona considers Major Clusters as groups of related standards that require greater emphasis than some of the others due to the depth of the ideas and the time it takes to master these groups of related standards. A-SSE.A Interpret the structure of expressions Essential Concepts Essential Questions Complicated expressions can be interpreted by viewing How does using the structure of an expression parts of the expression as single entities. help to simplify the expression? Structure within expressions can be identified and used Why would you want to simplify an expression? to factor or simplify the expression. A-SSE.B Write expressions in equivalent forms to solve problems Essential Concepts Essential Questions The solutions of quadratic equations are the x-intercepts of the parabola or zeros of quadratic functions. Factoring methods and the method of completing the square reveal attributes of the graphs of quadratic functions. Factoring a quadratic reveals the zeros of the function. Completing the square in a quadratic equation reveals the maximum or minimum value of the function. Properties of exponents are used to transform expressions for exponential functions. What are the solutions to a quadratic equation and how do they relate to the graph? What attributes of the graph will factoring and completing the square reveal about a quadratic function? How are properties of exponents used to transform expressions for exponential functions? Why would you want to transform an expression for an exponential function? A-REI.A Understand solving equations as a process of reasoning and explain the reasoning Essential Concepts Essential Questions Simple rational and radical equations can have Give an example of a simple rational or radical extraneous solutions. equation that has an extraneous solution and explain why it is an extraneous solution. A-REI.D Represent and solve equations and inequalities graphically Essential Concepts Essential Questions Solving a system of equations algebraically yields an Why are the x-coordinates of the points where exact solution; solving by graphing or by comparing the graphs of the equations y = f(x) and y = g(x) tables of values yields an approximate solution. intersect equal to the solutions of the equation f(x) = g(x)? The x-coordinates of the points where the graphs of the Why does graphing or using a table give equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). approximate solutions? In what situations would you want an exact solution rather than an approximate solution or vice versa? F-IF.B Interpret functions that arise in applications in terms of the context Essential Concepts: Essential Questions: Key features of a graph or table may include intercepts; How can you describe the shape of a graph? intervals in which the function is increasing, decreasing How can you relate the shape of a graph to the or constant; intervals in which the function is positive, meaning of the relationship it represents? negative or zero; symmetry; maxima; minima; and end How would you determine the appropriate behavior. domain for a function describing a real-world Given a verbal description of a relationship that can be situation? modeled by a function, a table or graph can be constructed and used to interpret key features of that 8/28/2015 9:53 AM Curriculum Instruction and Professional Development Math Department Page 2 2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence function. The intervals over which a function is increasing, decreasing or constant, positive, negative or zero are subsets of the function’s domain. Graphs can be described in terms of their relative maxima and minima; symmetries; end behavior; and periodicity. F-IF.C Analyze functions using different representations Essential Concepts: Essential Questions: Key features of a graph or table may include intercepts; How can you compare properties of two intervals in which the function is increasing, decreasing functions if they are represented in different or constant; intervals in which the function is positive, ways? negative or zero; symmetry; maxima; minima; end How do different forms of a function help you to behavior; asymptotes; domain; range and periodicity. identify key features? The graph of a trigonometric function shows period, How do you determine which type of function amplitude, midline and asymptotes. best models a given situation? The graph of a polynomial function shows zeros and end behavior. A function can be represented algebraically, graphically, numerically in tables, or by verbal descriptions. N-RN.A Extend the properties of exponents to rational exponents Essential Concepts: Essential Questions: Rational exponents are exponents that are fractions. How do you use properties of rational exponents to simplify and create equivalent Properties of integer exponents extend to properties of forms of numerical expressions? rational exponents. Why are rational exponents and radicals Properties of rational exponents are used to simplify related to each other? and create equivalent forms of numerical expressions. Given an expression with a rational exponent, Rational exponents can be written as radicals, and how do you write the equivalent radical radicals can be written as rational exponents. expression? A-APR.B Understand the relationship between zeros and factors of polynomials Essential Concepts: Essential Questions: The Remainder theorem says that if a polynomial p(x) is How can you determine whether x – a is a divided by x a , then the remainder is the value of the factor of a polynomial p(x)? Why does this work? polynomial evaluated at a. How do you determine how many zeros a Saying that x – a is a factor of a polynomial p(x) is polynomial function will have? equivalent to saying that p(a) = 0, by the zero property of multiplication. What information do you need to sketch a rough graph of a polynomial function? Any polynomial of degree n can be factored into n binomials of the form x – c, with possibly complex How are the zeros of a polynomial related to its values for c. graph? If p(a) = 0, then a is a zero of p. Extension: Why is it true that p(x)/(x – a) has a remainder of p(a)? If a is a zero of p, then a is an x-intercept of the graph of y = p(x). The values and multiplicity of the zeros of a polynomial, along with the end behavior, can be used to sketch a graph of the function defined by the polynomial. A-CED.A Create equations that describe numbers or relationships Essential Concepts: Essential Questions: Equations and inequalities can be created to represent How do you translate from real-world situations and solve real-world and mathematical problems. into mathematical equations and inequalities? Relationships between two quantities can be How do you determine if a situation is best 8/28/2015 9:53 AM Curriculum Instruction and Professional Development Math Department Page 3 2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence represented through the creation of equations in two variables and graphed on coordinate axes with labels and scales. Solutions are viable or not in different situations depending upon the constraints of the given context. 8/28/2015 9:53 AM represented by an equation, an inequality, a system of equations or a system of inequalities? Why would you want to create an equation or inequality to represent a real-world problem? How are graphs of equations and inequalities similar and different? How do you determine if a given point is a viable solution to a system of equations or inequalities, both on a graph and using the equations? Curriculum Instruction and Professional Development Math Department Page 4