Polynomials and Polynomial Functions


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1 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions Overview Number of instruction days: (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Prove polynomial identities and use them in describing relationships between numbers. Use the characteristics and structure of an expression to recognize ways to rewrite it. Know and apply the Remainder Theorem. Identify zeros of polynomials when suitable factorizations are available. Calculate and interpret the average rate of change of a polynomial function over a specified interval. 4 Model with mathematics. Model realworld situations using technology to represent regressions 5 Use appropriate tools strategically. Use graph paper and technology (TINspire) to graph and analyze polynomial functions. 7 Look for and make use of structure. Rewrite polynomial expressions in equivalent forms. Interpret key features of graphs and tables. Create and graph polynomial equations. Graph polynomial functions, identify zeros when factorizations are available, and show and describe end behavior. Recognize structural similarities between integers and polynomials. Discern patterns in factoring polynomials. Explain why the xcoordinates of the points of intersection are the solutions of the equation f(x) = g(x). Providence Public Schools D28
2 Polynomials and Polynomial Functions (1315 days) Algebra II, Quarter 1, Unit 1.4 Essential Questions How do you simplify polynomial expressions? Why is it important to rewrite polynomial expressions in different forms? What can you determine from different features of the graph of a polynomial function? What is the relationship between zeros and factors of polynomials? How is division of polynomials connected to the Remainder Theorem? How can key features of a polynomial function be identified given different representations? What realworld situations can be modeled with polynomial functions? Standards Common Core State Standards for Mathematical Content Algebra Arithmetic with Polynomials and Rational Expressions AAPR Understand the relationship between zeros and factors of polynomials AAPR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). AAPR.3 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. Use polynomial identities to solve problems AAPR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. 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 polynomials 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. Providence Public Schools D29
3 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (1315 days) Seeing Structure in Expressions ASSE Interpret the structure of expressions [Polynomial and rational] 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 ). [ASSE.2 is embedded throughout most Algebra II units.] Reasoning with Equations and Inequalities AREI Understand solving equations as a process of reasoning and explain the reasoning AREI.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. 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, polynomial, rational, absolute value, exponential, and logarithmic functions. Functions Interpreting Functions FIF Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models] FIF.4 FIF.6 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. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function] 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. D30 Providence Public Schools
4 Polynomials and Polynomial Functions (1315 days) Algebra II, Quarter 1, Unit 1.4 c. Graph polynomial 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. Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Providence Public Schools D31
5 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (1315 days) 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the wellremembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In eighth grade, students learned and applied properties of integer exponents. They also expressed numbers in scientific notation and performed operations on numbers in scientific notation. Students compared properties of two functions, each represented in a different way (algebraically, graphically, in tables, or by verbal description). They also constructed functions to model linear relationships between two quantities and qualitatively described the functional relationship between two quantities by analyzing the graph. In Algebra I, students extended the properties of exponents to rational exponents. They also performed arithmetic operations (addition, subtraction, multiplication) on linear and quadratic polynomials. They rewrote linear, quadratic, and exponential expressions in different but equivalent forms. Students have studied the distributive property since the third grade, when they used area models to multiply numbers. Students studied division with remainders in the fourth grade. Students deepened their understanding of the distributive property in the seventh grade, when they solved equations of the form p(x + q) = r. In Algebra I, students performed arithmetic operations on linear and quadratic polynomials, and they factored quadratic expressions. Students interpreted linear, quadratic, and exponential functions that arise in applications in terms of a context. They also analyzed functions and created equations to describe numbers and relationships. Additionally, they used factoring, completing the square, and the quadratic formula to solve equations involving seconddegree polynomials. Current Learning Work with polynomials is classified as major content in Algebra II according to the PARCC Model Content Frameworks. Analyzing functions using different representations is defined as supporting content. In this unit, students develop an understanding of structural similarities between the systems of polynomials and systems of integers. Students prove and use polynomial identities to describe numerical relationships and write polynomial expressions in different forms. Students connect division of polynomials with long division of integers. Students identify zeros of polynomials and make D32 Providence Public Schools
6 Polynomials and Polynomial Functions (1315 days) Algebra II, Quarter 1, Unit 1.4 connections between zeros of polynomials and solutions of polynomial equations. Students also understand and apply the Remainder Theorem. Students identify appropriate types of polynomial functions to model a situation, adjust parameters to improve the model, and compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. They find and interpret features of graphs of polynomial functions, including zeros, complex roots, minima, maxima, rate of change, and end behavior. This unit is a critical area for Algebra II students. Future Learning In Precalculus, students will perform arithmetic operations with complex numbers. They will also represent complex numbers and their operations on the complex plane. Students will also graph rational functions, identify zeros and asymptotes when suitable factorizations are available, and show end behavior. Mastery of these concepts will be required in Precalculus and AP Calculus. Additional Findings There are no additional findings for this unit. Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Rewrite polynomial expressions in different equivalent forms by factoring or combining like terms. Use long division to divide and simplify polynomial expressions. Use synthetic division to divide and simplify polynomial expressions. Determine whether a binomial is a factor of polynomial by using synthetic substitution. Apply the Remainder Theorem when evaluating functions. Providence Public Schools D33
7 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (1315 days) Analyze the characteristics of graphs and tables of polynomial functions. Solve polynomial equations with and without technology. Determine the number and types of roots of polynomials. Calculate the average rate of change of a function. Use multiple representations to determine the characteristics of polynomial graphs. Build polynomial functions to relate problem situations. Graph polynomial function derived from given roots or solutions. Identify the roots, extreme values and symmetry of a quadratic equation and interpret these in a realworld situation using factoring. Find the number and types of zeros of polynomial functions. Instruction Learning Objectives Students will be able to Apply the rules of polynomial identities when simplifying expressions. Generate equivalent polynomials expressions. Use long division to divide and simplify polynomial expressions. Use synthetic division to divide, simplify polynomial expressions. Apply the Remainder Theorem using synthetic division to evaluate a function. Determine whether a binomial is a factor of polynomial by using synthetic substitution. Determine the number and or types of roots for a polynomial functional. Read, understand, and interpret characteristics of graphs and tables of polynomial functions. Sketch a graph of a polynomial function given the zeros. Use graphing technology to find the solution(s) of polynomial equations. D34 Providence Public Schools
8 Polynomials and Polynomial Functions (1315 days) Algebra II, Quarter 1, Unit 1.4 Calculate and determine the average rate of change of a function over a specified interval. Use the degree of the polynomial to identify the key characteristics. Use a variety of techniques (including factoring) to solve polynomials. Build polynomial equations to relate and solve problem situations. Demonstrate understanding of concepts and skills learned in this unit. Resources Algebra 2, Glencoe McGrawHill, 2010: Student/Teacher Editions Sections (pp ) Extend 63 Algebra Lab, Polynomial Functions and Rate of Change (pp. 356) Chapter 6 Resources Masters Glencoe McGrawHill Online Interactive Classroom CD (PowerPoint Presentations) Exam View Assessment Suite Representing Polynomials: See the Supplementary Materials section of this binder for notes for this activity. Graphic Organizer: Synthetic Division. See the Supplementary Materials section of this binder for notes for this activity. Graphic Organizer: Finding the zeros for a Polynomial Function. See the Supplementary Materials section of this binder for notes for this activity. TINspire Teacher Software Education.TI.com: End Behavior of Polynomial Functions. See the Supplementary Materials Section of this Binder for the Student and Teacher materials. Education.TI.com: Polynomials Factors, Roots, and Zeros. See the Supplementary Materials Section of this Binder for the Student and Teacher materials. Providence Public Schools D35
9 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (1315 days) Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials TINspire Graphing Calculator, gridded chart paper, markers, glue, construction paper and sentence strips Instructional Considerations Key Vocabulary depressed polynomial synthetic division synthetic substitution end behavior Remainder Theorem Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: binomial, coefficient, degree, monomial, terms, degree of a polynomial, trinomial, long division. roots, zeros, polynomial function, binomial, coefficient, intercepts, intervals of increasing, decreasing, positive, negative, relative maximums and minimums; symmetries, zero product property, degree of a polynomial, completing the square, and solutions. The Mathematics Assessment Project, also known as Math Shell, provides a variety of projects for formative and summative assessment that may help make knowledge and reasoning visible to students. These CCSSM aligned resources include a variety of lessons focused on concept development and problem solving. It also includes numerous tasks designed for expert, apprentice and novice learners. The Representing Polynomials lesson is intended to help teachers assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in: Recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials. Recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, f(x), f(x)... D36 Providence Public Schools
10 Polynomials and Polynomial Functions (1315 days) Algebra II, Quarter 1, Unit 1.4 Use the link below to access detailed information for the Representing Polynomials lesson: The lesson resources are also provided in the supplementary materials section of this curriculum frameworks binder. Post essential questions for this unit around the classroom at the beginning of the unit. Use posted Essential Questions upon completion of an objective to review and close the lesson. Essential Questions can also be as an Exit Ticket to formatively assess student understanding. Living word walls will assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, both new and reinforced, and have pictures, examples and/or diagrams to accompany the definitions. Additionally, on page 388 at the beginning of Chapter 6, the foldable organizer has students use nonlinguistic representations. Using physical models while summarizing and taking notes helps students make sense of the attributes of polynomials and polynomial functions. The Glencoe resources listed above should provide ample choices to engage students in operations with polynomials. The Math in Motion and Personal Tutors animations on Glencoe.com also offer additional visual support for students struggling with these concepts. Have the class discuss the following: Given 4x 2 x, identify the leading coefficient, degree, and base. Explain what the degree tells you about possible solutions. Then break students into groups of two or three. Give each group of students two different polynomials (i.e., 3x 4 + 4x 2 + x 3 and 3x 2 3) written on large strips of paper. Individually, students should compare the two polynomials to help them decide what similarities and differences are present. Students then write the similarities and differences on individual sentence strips. Then have the group members discuss their strips together. When they agree on what they consider to be the accurate and complete list of similarities and differences, have students place their strips into a Venn diagram. Groups will share their finished task. Each group will evaluate the other groups work. Students should use their prior knowledge of previously studied functions to extend their conceptual understanding of polynomial functions. Students should be encouraged to use technology and connect the study of polynomial functions to earlier work with quadratic functions. Provide opportunities for students to display their results on gridded chart paper as they analyze functions and their graphs using multiple representations. The more representations that students see of applications of polynomial functions, the better connected their understanding of the concept. Students should be able to discern from a graph the solutions of a polynomial and write that polynomial and give all its characteristics. A group of students could also look at a graph and be given the task of writing their own polynomials given one or two roots. Have students then share out their polynomials with the whole group, giving their justification for the polynomial they have constructed. Algebra II students should learn how to find zeros from factored form and should also learn why this works (the zero product property). Technology can also be used to support this learning, including Providence Public Schools D37
11 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (1315 days) mathematical notations indicating exact or approximate solutions. The End Behavior of Polynomial Functions activity on education.ti.com provides the opportunity for students to use technology to find the similarities and differences among functions. The Polynomials Factors, Roots, and Zeros activity uses technology to analyze multiple representation s of polynomial functions. Additional TINspire resources can be found using the TINspire Teacher Software on your work computer. The content tab of the TINspire desktop software contains links to the Algebra 2, Glencoe McGrawHill, 2010 textbook. These resources are accessible by chapter and section. Students could use the song The Twelve Days of Christmas to use a realworld example of a cubic polynomial (constant third difference). The table below shows the cubic relationship. Number of Days Total Gifts Received Problems 1. Doctors examine cardiac output in potential heart attack patients by checking the concentration of dye after a known amount is injected in a vein near the heart. In a healthy heart, the amount of dye in the bloodstream after t seconds can be expressed by the function f(t) = 0.006t t t t. Explain how the roots of this equation can be used in cardiology. Include an explanation of what the roots of this equation reveal about the concentration of dye in a healthy heart. 2. The average amount P (in dollars), donated by a company annually to any charitable organization, can be modeled by the equation P = 20(t 4 2.4t 3 28t t +756), where t = 0 represents the year What is the trend for the donations of this company? 3. Jim is saving a percentage of his income each month for a vacation. His savings for the last 6 months can be modeled by the equation S = 0.333m m m m 26.66, where S represents amount saved, given as a percentage, and m represents the number of months. Explain Jim s savings plan. 4. An estimated calorie requirement (in kilocalories) of physically active males and females can be modeled by the equations C m = 0.002x x x x and C f = x x x x , where C m represents calorie requirements of males, C f represents calorie requirements of females, and x represents age. What are the limitations of this model? D38 Providence Public Schools
12 Polynomials and Polynomial Functions (1315 days) Algebra II, Quarter 1, Unit The minimum and maximum mean temperature (in degrees Fahrenheit) of a particular city for the months January to December can be modeled by the equation T min = 0.04x x x x and T max = 0.042x x x x , respectively. What are the limitations of this model? 6. A garden bed is filled with potting soil in the shape of a rectangular prism. The length is depth squared and the width is 3 times the depth. The gardener plans to change the dimensions of the bed. The new volume of potting soil can be modeled by the equation V(h) = 2h 4 + 9h h h + 16, where h is the depth of the potting soil. What are the limitations of this equation? 7. The weight w, in pounds, of a patient during a 7week illness is modeled by the cubic equation w(t) = 0.1n 3 0.6n , where n is the number of weeks since the patient became ill. What trends in the patient s weight does the graph suggest? Is it reasonable to assume the trend will continue indefinitely? The following rubric could be used to evaluate students work on the problems above. Points Graph 5 Graph contains an appropriate title, scales are uniform, and axes are labeled. 5 Roots of the function are accurate and labeled. 10 Intervals of the function are accurate and indicated. 10 x and yintercepts are accurate and labeled. Explanation 20 Explain the similarities of zeros, roots, solutions, and xintercepts. 10 Explain rates of change between the intervals. 10 Explain all local maximum and minimum points. 10 Explain all the points of inflection and end behavior of the function. Justify 10 Appropriate domain and range of the function. 10 An accurate solution to its given problem. 100 TOTAL Providence Public Schools D39
13 Algebra II, Quarter 1, Unit 1.4 Polynomials and Polynomial Functions (1315 days) In this unit students connect division of polynomials with longg division of integers. They will learn how to find the quotient of two polynomials through synthetic division and long division. Students should be able to explain how synthetic division can be used to write the factorization of a polynomial. Be sure to connect division of polynomials by monomials to operations with rational numbers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. This unit develops the structural similarities between the system of polynomials andd the system of integers. Students draw on analogies between polynomial arithmetic and baseten computation, focusing on properties of operations, particularly the distributive property. Consider using a synthetic division matching activity as a Doo Now or an exit activity. In this activity, students match the polynomials on the left with their roots on the right. Sample tables are providedd below. Notes D40 Providence Public Schools
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