Unit 2 Quadratic Equations and Polynomial Functions Algebra 2


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1 Number of Days: 29 10/10/16 11/18/16 Unit Goals Stage 1 Unit Description: Students will build on their prior knowledge of solving quadratic equations. In Unit 2, solutions are no longer limited to real numbers, but now include the set of complex numbers. Students will now have five strategies to use to solve quadratic equations: graphing, factoring, taking the square root, completing the square and the Quadratic Formula. Solving nonlinear systems and quadratic inequalities broadens the students knowledge of linear functions and quadratic equations. Viewing s through the lens of quadratics, students already have a familiarity with factoring, graph behavior and transformations. Extending from second degree functions to third and fourth degree functions, students work with the end behaviors of graphs, the number of zeros of an n th degree and the Fundamental Theorem of Algebra. Important for student understanding of s is that s form a system analogous to integers, namely, they are closed under the operations of addition, subtraction and multiplication. Materials: Graphing calculators, Desmos Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content Clusters Addressed [s] NCN.A Perform arithmetic operations with complex numbers. [a] NCN.C Use complex numbers in identities and equations. [m] ASSE.A Interpret the structure of expressions. Transfer Goals Students will be able to independently use their learning to Make sense of neverbeforeseen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will understand that Solutions to quadratic equations are not restricted to real numbers. There are many strategies that one can use to solve quadratic equations. One strategy may be more appropriate than another given the initial form of the equation. The skills students learn for solving quadratic and linear equations can be transferred to graphing and solving nonlinear systems. Earlier work with graphing function transformations applies to functions. Polynomials form a system analogous to the integers. They are closed under addition, subtraction and multiplication. Polynomial functions can be divided using steps that are similar to the steps used with long division. The degree of a equation designates how many roots the equation has (corollary to the Fundamental Theorem of Algebra). The xcoordinates of the points where the graphs of a system of equations or inequalities intersect are the solutions of the equation f(x) = g(x). ESSENTIAL QUESTIONS Students will keep considering Why are complex numbers necessary? How do you know how many solutions a quadratic equation has? In a contextual situation, what information can you derive from the characteristics of the graph of a function? What do functions have in common with constant, linear and quadratic functions? How does the degree of a affect the related function? How does the graph of a function reflect the solutions to the equation? Posted 9/20/16
2 [m] AAPR.A Perform arithmetic operations on s. [m] AAPR.B Understand the relationship between zeros and factors of s. [a] AAPR.C Use identities to solve problems. [s] AAPR.D Rewrite rational expressions. [m] ACED.A Create equations that describe numbers or relationships. [m] AREI.B Solve equations and inequalities in one variable. [m] AREI.D Represent and solve equations and inequalities graphically. [m] FIF.B Interpret functions that arise in applications in terms of the context. [s] FIF.C Analyze functions using different representations. [m] FBF.A Build a function that models a relationship between two quantities. [a] FBF.B Build new functions from existing functions. KNOWLEDGE Students will know The basic shapes of parent functions: y = x 2 and y = x 3. How the parameters a, b, and c affect the graph of f(x) = ax 2 + bx + c and f(x) =a f(x b) + c. The Quadratic Formula. Names for common functions. When f(k) = 0, x k is a factor. When a has a root of a + bx, there is also a root of a bx. The Fundamental Theorem of Algebra. The definition of complex numbers (a + bi) and i 2 = i. When a has a root of a + bi, there is also a root of a bi. Recognize even and odd functions from their graphs and their algebraic forms. Acquisition SKILLS Students will be skilled at and/or be able to Solve quadratic equations graphically and algebraically including those with complex solutions. Define and use the imaginary unit i. Add, subtract and multiply complex numbers. Write quadratic functions in standard and vertex form. Analyze the discriminant of the Quadratic Formula to determine a quadratic equation s number and type of solutions. Solving systems of nonlinear equations. Graph and solve quadratic inequalities and a system of quadratic inequalities. Graph functions using tables, end behavior and technology. [ACC] Use Pascal s Triangle to expand binomials. Add, subtract, multiply, and divide s. Use synthetic division to divide s by the binomial x k. Factor s. Find solutions of equations and zeros of functions using a variety of techniques including the Rational Root Theorem. Find conjugate pairs of complex zeros of functions. Describe and write an equation to represent the graph of a function which has been transformed from an original function. Use xintercepts, local minima and maxima, and whether the function is odd or even to construct a rough graph. Write functions for given sets of points. Apply functions to solve realworld problems. Use technology to find a function to model a set of data points Posted 9/20/16
3 Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Assessed Grade Level Standards Standards for Mathematical Content [s] NCN.A Perform arithmetic operations with complex numbers. NCN.1 Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. NCN.2 Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. [a] NCN.C Use complex numbers in identities and equations. NCN.7 Solve quadratic equations with real coefficients that have complex solutions. [ACC] NCN.8 (+) Extend identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i). [ACC] NCN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic s. [m] ASSE.A Interpret the structure of expressions. ASSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). [m] AAPR.A Perform arithmetic operations on s. AAPR.1 Understand that s form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply s. [m] AAPR.B Understand the relationship between zeros and factors of s. AAPR.2 Know and apply the Remainder Theorem: For a p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 AAPR.3 if and only if (x a) is a factor of p(x). Identify zeros of s when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the. [a] AAPR.C Use identities to solve problems. AAPR.4 Prove identities and use them to describe numerical relationships. For example, the identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. [ACC] AAPR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. [s] AAPR.D Rewrite rational expressions. AAPR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are s with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. [m] ACED.A Create equations that describe numbers or relationships. ACED.1 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 Posted 9/20/16
4 Assessed Grade Level Standards ACED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ACED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. [m] AREI.B Solve equations and inequalities in one variable. AREI.4 Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. [m] AREI.D Represent and solve equations and inequalities graphically. AREI.11 Explain why the xcoordinates 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,, rational, absolute value, exponential, and logarithmic functions. [m] FIF.B Interpret functions that arise in applications in terms of the context. FIF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal 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. [s] FIF.C Analyze functions using different representations. FIF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph functions, identifying zeros when suitable factorizations are available, and showing end behavior. FIF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [m] FBF.A Build a function that models a relationship between two quantities. FBF.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. [a] FBF.B Build new functions from existing functions. FBF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters; [ACC] = Algebra 2 ACC only Posted 9/20/16
5 Assessment Evidence Evidence of Learning Stage 2 Unit Assessment Students will complete selected response and constructed response items to indicate level of mastery/understanding of the unit standards as outlined in this guide. [s] NCN.A The student will perform arithmetic operations with complex numbers, a + bi. The student will explain, and use in calculation, the complex number i. [a] NCN.C The student will solve quadratic equations with real coefficients that have complex solutions. [m] ASSE.A The student will use the structure of an expression to identify ways to rewrite it. For example, students will factor a equation to solve for its zeros. [m] AAPR.A The student will perform arithmetic operations on s. [m] AAPR.B The will know and apply the Remainder Theorem. The student will identify zeros of s when suitable factorizations are available. The student will use the zeros of a to construct a rough graph of the function as defined by its zeros. [a] AAPR.C [ACC] The student will use Pascal s Triangle to expand a given binomial function. [s] AAPR.D The student will use long division to rewrite a expression. [m] ACED.A The student will create equations to represent relationships between quantities and will graph equations on coordinate axes with labels and scales. The student will represent constraints in context using equations or inequalities. The student will write systems of equations and/or inequalities in a modeling context. The student will interpret solutions from a system of equations and/or inequalities as viable or nonviable in a modeling context. [m] AREI.B The student will solve equations in one variable by taking square roots, completing the square, using the Quadratic Formula, and factoring as appropriate to the initial form of the equation Posted 9/20/16
6 Evidence of Learning Stage 2 Assessment Evidence The students will recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b. [m] AREI.D The student will explain why the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) are the solutions of the equation f(x) = g(x). The student will find the solution(s) to a system of equations. [m] FIF.B The student will interpret the key features of a graph in terms of the context of a problem. The student will sketch a graph, showing the key features, given a verbal description of a realworld application. [s] FIF.C The student will graph a function expressed symbolically by hand in simple cases or using technology for more complicated cases. The student will graph functions showing zeros when suitable factorizations are available and showing end behavior. The student will factor, or complete the square, in a quadratic function to show zeros, extreme values, and symmetry of the graph. The student will interpret the key features of the graph of a function in the context of the problem. [m] FBF.A The student will determine a function that describes a relationship between two quantities. [a] FBF.B The student will identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative). The student will find the value of k given the graphs. The student will recognize even and odd functions from their graphs and algebraic expressions. For selected content, students will need to Solve complex problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Other Evidence Formative Assessment Opportunities Opening s Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Modeling Lessons (SMP 4) s Formative Assessment Lessons (FAL) Quizzes / Chapter Tests Big Ideas Math Performance s SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents Posted 9/20/16
7 Days Learning Target I will review my knowledge of function transformations in the Opening. I will find and identify the solutions of onevariable quadratic equations by Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Expectations OPENING TASK Transformers This Opening is a review of transforming the graphs of quadratic functions. Students are asked to experiment with and to identify the effect of the transformation parameters within the context of a problem. The task ends with the students identifying a, b, and c in the standard form of the quadratic equation y = ax 2 + bx + c given that a 0 and distributing binomials to derive the quadratic equation s standard form. A primary focus of Unit 2 is identifying key features of graphs. During the Opening, precise vocabulary should be used when discussing the graphs, pointing out the minimum or maximum, x and yintercepts, domain, range, and where the graph is increasing or decreasing. The use of graphing technology is encouraged. Graphing the equations related functions. (SMP 5) By taking square roots, factoring, completing the square, and/or using the Quadratic Formula. Interpreting the solutions to a quadratic equation in the context of a reallife problem. (SMP 3) Defining, using and performing operations with complex numbers. Recognizing complex solutions. (SMP 6) o What does it mean to solve a quadratic equation in one variable? o How can you use the graph of a quadratic equation to determine its real solutions? o How would knowing the zeros of a quadratic function help find the vertex of a parabola? o What are the subsets of the set of complex numbers? Do the subsets overlap? o How is the imaginary unit i defined and how is it used? o What must you add to the expression x 2 + bx to complete the square? o What must you add to the expression ax 2 + bx to complete the square given that a 1? o Can you derive a general formula for solving a quadratic equation? o What are the advantages of each of the strategies for solving a quadratic equation? o When using the quadratic formula, can you predict the number and type of solution to the given quadratic equation? Big Ideas Math Algebra 2 (Activities and Lessons) Section 3.1 Section 3.2 STEM Video: Complex Numbers Made Real Section 3.3 Section 3.4 Curriculum Intranet Transformers Two Squares Are Equal Open Middle: Quadratic Formula Illuminations: Proof without Words: Completing the Square Dynamic Activity Application: STEM Performance : Complex Numbers Made Real Posted 9/20/16
8 Days Learning Target I will understand nonlinear systems and quadratic inequalities in one and two variables by I will understand functions by Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Expectations Graphing nonlinear systems of equations. (SMP 5) Using elimination or substitution to solve a nonlinear system. Graphing quadratic inequalities in one and two variables. (SMP 5) Explaining why the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) are the solutions to the equation f(x) = g(x). (SMP 3) Solving quadratic inequalities in one and two variables. o What does it mean to solve a system of equations? o What methods have you used to solve a system of equations? o How many solutions could there be for a system with two linear equations? A linear equation and a quadratic equation? Two quadratic equations? o When might you prefer to use substitution rather than elimination when solving a nonlinear system of equations? o How can you solve a quadratic inequality? o When solving an inequality graphically, what would be a good test point to use and why? o What is the difference between the graph of a quadratic inequality in one variable and the graph of a quadratic inequality in two variables? Using graphing technology to show specific values and the end behavior of complex functions. (SMP 5) Identifying a by a name that reflects the degree of the. Evaluating a for a given value of x. Describing the end behavior of a graph based on the s degree and leading coefficient. (SMP 3) Describing intervals over the domain as increasing or decreasing. o How do you know when a function is a? o What information do you need to graph a function? o Will plotting points produce an accurate graph? o What is meant by the end behavior of a function? o How do you know the end behavior of a function? Big Ideas Math Algebra 2 (Activities and Lessons) Section 3.5 Section 3.6 Section 4.1 Curriculum Intranet A Linear and Quadratic System Application: Population and Food Supply Graphs of Power Functions Procedural Skills and Fluency: Illuminations: Function Matching Interactive Tool Posted 9/20/16
9 Days 35 Learning Target I will perform arithmetic operations on s by Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Expectations Adding, subtracting, and multiplying s. [ACC] Using Pascal s Triangle to expand binomials. Using long division to divide s by other s. Using synthetic division to divide s by binomials of the form x k. Using the Remainder Theorem to evaluate a function. (SMP 6) Factoring s. Using a given factor to factor a cubic completely. o [ACC] What is the pattern that generates Pascal s Triangle? o Why does (a + b) 2 a 2 + b 2? o What are at least three methods to expand (x + 3) 3? o [ACC] Compare the n th row of Pascal s Triangle with the expansion of (a + b) n. o How can you use the factors of a to solve a division problem involving a? o Explain is the division process taught in elementary school? o How are synthetic division and traditional long division similar? Different? o What is the advantage of using the Remainder Theorem to evaluate a function? How else could you evaluate the function? o How can synthetic division help find solutions for a of a higher degree? o What is the Zero Product Property? When is it important? o How can a graph be used to help factor a? o Are all quadratics factorable? o When is a completely factored? o When should a be factored by grouping? o If you can t remember the special factoring patterns, what can you do? Big Ideas Math Algebra 2 (Activities and Lessons) Section 4.2 Section 4.3 Section 4.4 Curriculum Intranet The Missing Coefficient Procedural Skills and Fluency: Illuminations: Polynomial Puzzler Posted 9/20/16
10 Days 34 Learning Target I will solve equations in one and two variables by Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Expectations Factoring the equation. Graphing the related function. (SMP 5) Using the Rational Root Theorem to find the actual roots of a function. [ACC] Using the Fundamental Theorem of Algebra and its corollary. Finding conjugate pairs of complex zeros of functions. Using Descartes s Rule of Signs. o What is the maximum possible number of real zeros that a function can have? o How does knowing one solution of a equation help you to find the other solutions? o What does the graph of a function tell you about the function? o Does a quartic equation always have an even number of real solutions? o How do you know if a equation has imaginary solutions? o o Why does the Irrational Conjugates Theorem make sense? Which is the better way to find the solutions of a function: graphing or solving? Big Ideas Math Algebra 2 (Activities and Lessons) Section 4.5 Section 4.6 Curriculum Intranet Graphing from Roots Procedural Skills and Fluency: Solving Quadratics Methods Matching Cards 34 I will understand functions by Describing transformations of functions. Writing the transformation of a function when given the graph of that transformed function. (SMP 3) Using xintercepts to sketch a function. Use the Location Principle to identify zeros of functions. (SMP 6) Finding turning points and local minima and maxima of graphs of functions. Identifying even and odd functions. o In the graph of f(x) = af(x b) + c, what effect do the parameters a, b, and c have on the graph of f(x)? o How do the degree and the leading coefficient of a function affect the shape of its graph? o How many turning points can the graph of a function have? o Is it possible to sketch the graph of a cubic with no turning points? o How does the Location Principle help you to hone in on the real zero(s) of a function? o How do you know if a function is even or odd? Section 4.7 Section 4.8 STEM Video: Quonset Huts Transforming the Graph of a Function Application: STEM Performance : Quonset Huts Posted 9/20/16
11 Days Learning Target I will use functions to solve reallife problems by I will check my understanding of graphing functions by participating in the FAL. I will prepare for the unit assessment on quadratic equations and functions by... Learning Plan Stage 3 Suggested Sequence of Key Learning Events and Instruction Expectations Writing a to model a given relationship between two quantities. (SMP 5) Writing a function using finite differences. o What are some ways to find a model for reallife data? o How many points are necessary to determine a line? A quadratic? A cubic? o How do you know when a set of data can be modeled by a cubic function? o What is the purpose of a function which models the given data set? FORMATIVE ASSESSMENT LESSON  Representing Polynomials Graphically Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Unit Assessment (LBUSD Math Intranet, Assessment) Big Ideas Math Algebra 2 (Activities and Lessons) Section 4.9 Curriculum Intranet Procedural Skills and Fluency: Throwing Baseballs Application: Illuminations: Determining Functions Using Regression: What s the Function Representing Polynomials Graphically At this point, all standards addressed in the High School SBAC Interim Assessment Block Algebra and Functions (Linear Functions) have been covered. This block may now be administered Posted 9/20/16
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