Unit Title: Polynomial and Rational Functions Time Frame: 11 blocks Six Weeks: 2 nd & 3 rd Unit Number: 4 Curriculum Enduring Understandings (Big Ideas): Graphs provide a visual representation of relationships among data. Representing a situation in multiple ways leads to better understanding and more effective communication. Patterns are mathematical in nature and allow us to make predictions. There are many ways to solve problems, but some are more efficient than others. The student will know: General shapes and end-behavior of polynomial functions Limitations of technology as it relates to the behavior of functions around discontinuities The effects of multiple zeros on the graph of a polynomial The real zeros are the x-axis intercepts i is a symbol that represents the solution to a problem All operations on complex numbers are closed Complex and irrational zeros must appear in conjugate pairs Hierarchy of number sets All polynomials are smooth and continuous everywhere The student will be able to: Determine the key features of polynomial functions using zeros, leading coefficients, and degree Determine the key features of rational functions using zeros, asymptotes, and undefined values on the domain Sketch/graph polynomial and rational functions including key features such as zeros, maxima, minima, domain/range intervals, end-behavior, asymptotes, and holes Apply long division and synthetic division to solve polynomials Apply/use the Factor Theorem, Remainder Theorem, and Rational Roots Theorem to solve polynomials Perform operations on complex numbers including addition, subtraction, multiplication, and division Find conjugates including complex conjugates and irrational conjugates Unit Title: Polynomial and Rational Functions Unit Number 4 Page 1 of 5
Essential Questions: What predictions can be made from characteristics of the relationship? What benefits do visual representations provide? Why do we explore different solution methods? What are the benefits and shortcomings of the different solution methods? Student Understanding (student friendly TEKS): I can graph polynomial and rational functions. (taken from PC.2.F) I can determine key features of polynomial and rational functions. (taken from PC.2.I) I can determine the end-behavior of polynomial and rational functions. (taken from PC.2.J) I can determine the behavior around asymptotes of rational functions. (taken from PC.2.K) I can determine the various types of discontinuities. (taken from PC.2.L) I know the limitations of technology in producing graphs. (taken from PC.2.L) I can describe the behavior of the graph around discontinuities. (taken from PC.2.M) I can analyze situations modeled by functions to solve problems. (taken from PC.2.N) I can solve polynomial equations using a variety of techniques. (taken from PC.5.J) I can solve polynomial inequalities using a variety of techniques. (taken from PC.5.K) I can write the solution set of polynomial inequalities in interval notation. (taken from PC.5.K) I can solve rational inequalities using a variety of techniques. (taken from PC.5.L) I can write the solution set of rational inequalities using interval notation. (taken from PC.5.L) Solve polynomial inequalities by sketching and/or developing a sign chart and write the solution set in interval notation Solve rational inequalities by sketching and/or developing a sign chart and write the solution set in interval notation Identify all zeros of a polynomial function including complex zeros Determine various types of discontinuities as they relate to functions Analyze situations modeled by rational and polynomial functions to solve problems Rewrite rational equations as partial fractions (EXTENSION) Unit Title: Polynomial and Rational Functions Unit Number 4 Page 2 of 5
I can apply math in everyday life. (taken from PC.1.A) I can develop a problem solving model and justify my solutions. (taken from PC.1.B) I can select appropriate tools and techniques to solve problems. (taken from PC.1.C) I can communicate using appropriate mathematical language. (taken from PC.1.D) I can use various representations to communicate mathematical concepts. (taken from PC.1.E) I can analyze relationships to connect and communicate mathematical ideas. (taken from PC.1.F) I can use precise mathematical language in both written and oral communication. (taken from PC.1.G) TEKS: (PC.2) Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to: (F) graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions; (I) determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing; (J) analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems; (K) analyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes; (L) determine various types of discontinuities in the interval (-, ) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities; (M) describe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities; (N) analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems. (PC.5) Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to: (J) solve polynomial equations with real coefficients by applying a variety of techniques in mathematical and real-world problems; (K) solve polynomial inequalities with real coefficients by applying a variety of techniques and write the solution set of the polynomial inequality in interval notation in mathematical and real-world problems; (L) solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems. Unit Title: Polynomial and Rational Functions Unit Number 4 Page 3 of 5
(PC.1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Targeted College Readiness Standards: IA1-2, IB1, IIA1, IIB1, IIC1-2, IID1, VIIA2, VIIB1, VIIC1, VIIIA1-5, VIIIB1-2, VIIIC1, IXA1-3, IXB1-2, XA1, XB1-2 Targeted ELPs: 1A, 1C, 1D, 1E, 1F, 1H, 2C, 2D, 2E, 2G, 2I, 3D, 3E, 3F, 3G, 3H, 3I, 3J, 4C, 4D, 4F, 4G Academic Vocabulary: Language of Instruction: Quadratic Cubic Higher degree End behavior Leading coefficient Multiplicity with respect to zero Long division Synthetic division Rational roots Complex numbers Complex conjugates Irrational conjugates Conjugates Partial fractions Unit Title: Polynomial and Rational Functions Unit Number 4 Page 4 of 5
Instruction Instructional Resources: Textbook: Chapter 2.1-2.7, & 7.4(extension) Mathematics Inequality Rational function Polynomial function Zeros Factor Roots Remainder Evaluate Value of a function Non-linear inequality Undefined Conjugate pairs Asymptote Linear factorization Non-factorable quadratic Technology: Exemplar Lessons: Assessment Student self-assessment & reflection: Career Connections/Real Life Application: Research Based Instructional Strategies: Acceptable evidence or artifacts: Performance Task Unit 4a Performance Task Unit 4b Unit Title: Polynomial and Rational Functions Unit Number 4 Page 5 of 5