Topic V: Rational Functions, Expressions, and Equations MATHEMATICS FLORIDA STATE STANDARDS (MAFS) & MATHEMATICAL PRACTICES (MP) ESSENTIAL CONTENT Pacing OBJECTIVES Date(s) Traditional 12 11/29/17 12/14/17 Block 6 11/29/17 12/14/17 Topic V Assessment Window 12/07/17 12/14/17 MAFS.912.F-IF.3.7d: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (MP.5, MP.6, MP 7) MAFS.912.F-BF.2.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. (MP.2, MP.4, MP.5, MP.6) MAFS.912.A-REI.1.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (MP.1, MP.2. MP.3, MP.7) MAFS.912.A-APR.4.6: 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. (MP.2, MP.5, MP 7, MP.8) MAFS.912.A-APR.4.7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. (+) MAFS.912.A-REI.1.1: 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. (MP.1, MP.2, MP.3, MP 7) A. Rational Functions a. Graphing Simple Rational Functions b. Graphing More Complicated Rational Functions. B. Rational Expressions and Equations a. Adding and Subtracting Rational Expressions b. Multiplying and Dividing Rational Expressions c. Solving Rational Equations. I can: identify intercepts, asymptotes, and end behavior of a rational function. graph rational functions using key features. determine the value of k when given a graph of the function and its transformation. identify differences and similarities between a function and its transformation. identify a graph of a function given a graph or a table of a transformation and the type of transformation that is represented. graph by applying a given transformation to a function. identify ordered pairs of a transformed graph. complete a table for a transformed function. recognize even and odd functions from their graphs and equations. solve a rational equation in one variable complete an algebraic proof to explain steps for solving a rational equation. construct a viable argument to justify a solution method. rewrite a rational expression as the quotient in the form of a polynomial added to the remainder divided by the divisor. Division of Academics - Department of Mathematics Page 1 of 10
INSTRUCTIONAL TOOLS Core Text Book: Houghton Mifflin Harcourt Algebra 2 Algebra 2 Honors Course Description Pacing Date(s) Traditional 12 12/12/16 01/20/17 Block 6 12/12/16 01/20/17 Topic V Assessment Window 12/07/17 12/14/17 RECOMMENDED INSTRUCTIONAL DESIGN AND PLANNING CONTINUUM Before During After During the lesson: Activate (or supply) prior knowledge and/or spiral back o Warm ups, Bell Ringers, Openers, etc. Tailor lesson experiences to the different needs and ability of the learners. Clarify vocabulary and mathematical notation. Incorporate a variety of higher order questions to encourage and increase critical thinking skills. Continuously check for student understanding and provide feedback. Provide opportunities for students to develop selfassessment and to reflect about their understanding and work. Bring closure to the lesson so that the students can articulate what they have learned. Prior to the lesson: Outline content standard(s). Determine learning targets. Anticipate student understanding and misconceptions. Determine prerequisite skills. Plan for learning experiences that target Rigor o Conceptual Understanding o Procedural Fluency o Application Determine the task students will demonstrate to reach the desired learning targets. Plan instructional delivery methods that will maximize initial engagement and sustain it throughout the lesson. Decide how students will reflect upon, self-assess, and set goals for their future learning. After the lesson: Analyze evidence of student learning to develop intervention, enrichment, and future instruction. Discuss results of assessments with students. Engage students in reflective processes and goal setting. Engage in self-reflection to adapt/modify teaching strategies to improve instruction. Unit Resources Unit Tests A, B, and C Performance Assessment Module Resources Module Test B Common Core Assessment Readiness Advanced Learners Challenge Worksheets Algebra 2 Honors H.M.H. Resources Unit Resources Math in Careers Video Assessment Readiness (Mixed Review) Lesson Resources Lessons Work text/interactive Student Edition Practice and Problem Solving: A/B Advanced Learners - Practice and Problem Solving: C PMT Preferences: Auto-assign for intervention and enrichment: NO Test Auto-assign and Quizzes for intervention and enrichment: NO Homework PMT Preferences: Auto-assign for intervention and enrichment: YES Standard-Based Intervention Course Intervention Daily Intervention Division of Academics - Department of Mathematics Page 2 of 10
0 INSTRUCTIONAL TOOLS STANDARDS MODULES TEACHER NOTES MAFS.912.A-REI.1.1 MAFS.912.A-REI.1.2 MAFS.912.A-APR.4.6 Module 8 Module 9 Algebra 2 Honors Block Schedule Suggested Pace Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 MAFS.912.A-APR.4.7(+) 8.1 8.2 9.1 9.2 9.3 Topic Test MAFS.912.F-BF.2.3 MAFS.912.F-IF.3.7d Topic Resources PowerPoint Available in Learning Village Topic V Assessment: Rational Functions, Expressions, and Equations (+) Additional mathematics that students should learn in fourth credit courses or advanced courses such as calculus, advanced statistics, or discrete mathematics MODULE LESSON STANDARDS Module 8 8.1 MAFS.912.F-IF.3.7d MAFS.912.F-BF.2.3 8.2 MAFS.912.F-IF.3.7d SUGGESTED PROBLEMS BY TEACHERS FOR TEACHERS* YOUR TURN 10 HOMEWORK AND PRACTICE 10, 11, 13, 15 HOMEWORK AND PRACTICE 1, 2, 4, 6, 7 NOTES / RESOURCES Illustrative Mathematics Task(s): Graphing Rational Functions GeoGebra: Rational Functions, Transformations of the parent Rational Function 9.1 MAFS.912.A-APR.4.7 HOMEWORK AND PRACTICE 9, 10, 13, 20, 21 Module 9 9.2 MAFS.912.A-APR.4.7 HOMEWORK AND PRACTICE 3, 4, 6, 8, 12 9.3 MAFS.912.A-REI.1.2 MAFS.912.A-REI.1.1 HOMEWORK AND PRACTICE 3, 4, 5, 9, 10 Illustrative Mathematics Task(s): Basketball, Canoe Trip, An Extraneous Solution *Problems were suggested by M-DCPS teachers during May Algebra 2 PD. Division of Academics - Department of Mathematics Page 3 of 10
INSTRUCTIONAL TOOLS MODELING CYCLE ( ) The basic modeling cycle involves: 1. Identifying variables in the situation and selecting those that represent essential features. 2. Formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables. 3. Analyzing and performing operations on these relationships to draw conclusions. 4. Interpreting the results of the mathematics in terms of the original situation. 5. Validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable. 6. Reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle. http://www.cpalms.org/standards/mafs_modeling_standards.aspx Vocabulary: Asymptote, constant of variation, parent function, rational function, closure, extraneous solutions, rational expression, reciprocal. Connections to Redesigned SAT Passport to Advanced Math: Solve an equation in one variable that contains radicals or contains the variable in the denominator of a fraction. The equation will have rational coefficients, and the student may be required to identify when a resulting solution is extraneous. Rewrite simple rational expressions. Students will add, subtract, multiply, or divide two rational expressions or divide two polynomial expressions and simplify the result. The expressions will have rational coefficients. SAT Practice STEM Lessons - Model Eliciting Activity N/A STEM Lessons CPALMS Perspectives Videos Professional/Enthusiasts N/A Expert N/A Division of Academics - Department of Mathematics Page 4 of 10
MATHEMATICS FLORIDA STANDARDS MATHEMATICAL PRACTICES DESCRIPTION MAFS.K12.MP.1 Make sense of problems and persevere in solving them. Explain the meaning of a problem and looking for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution and plan a solution pathway. Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Monitor and evaluate their progress and change course if necessary. Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Check answers to problems using a different method, and continually ask, Does this make sense? Identify correspondences between different approaches. MAFS.K12.MP.2 Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Decontextualize to abstract a given situation and represent it symbolically. Contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols Create a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them. Know and be flexible using different properties of operations and objects. MAFS.K12.MP.3 Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Analyze situations by breaking them into cases, and can recognize and use counterexamples. Justify their conclusions, communicate them to others, and respond to the arguments of others. Reason inductively about data, making plausible arguments that take into account the context from which the data arose. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Determine domains to which an argument applies. MAFS.K12.MP.4 Model with mathematics. Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Apply what they know and feel comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. Analyze relationships mathematically to draw conclusions. Interpret 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. Division of Academics - Department of Mathematics Page 5 of 10
MATHEMATICS FLORIDA STANDARDS MATHEMATICAL PRACTICES DESCRIPTION MAFS.K12.MP.5 Use appropriate tools strategically. 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. Make sound decisions about when each of the tools appropriate for their grade or course might be helpful, recognizing both the insight to be gained and their limitations. Example: High school students analyze graphs of functions and solutions using a graphing calculator. Detect possible errors by strategically using estimation and other mathematical knowledge. Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. Use technological tools to explore and deepen their understanding of concepts MAFS.K12.MP.6 Attend to precision. Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning. State the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Be careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. MAFS.K12.MP.7 Look for and make use of structure. Discern a pattern or structure. Example: In the expression x 2 + 9x + 14, students can see the 14 as 2 7 and the 9 as 2 + 7. Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Step back for an overview and shift perspective. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. 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. MAFS.K12.MP.8 Look for and express regularity in repeated reasoning. Notice if calculations are repeated, and look both for general methods and for shortcuts. Example: Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and(x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. Maintain oversight of the process, while attending to the details as they work to solve a problem. Continually evaluate the reasonableness of their intermediate results. Division of Academics - Department of Mathematics Page 6 of 10
Domain: Algebra: Reasoning with Equations & Inequalities Cluster 1: Understand solving equations as a process of reasoning and explain the reasoning MAFS.912.A-REI.1.1 STANDARD DESCRIPTION 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. Content Complexity Rating: Level 3: Strategic Thinking and Complex Reasoning MAFS.912.A-REI.1.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Content Complexity Rating: Level 3: Strategic Thinking and Complex Reasoning Domain: Algebra: Arithmetic with Polynomials & Rational Expressions Cluster 4: Rewrite rational expressions MAFS.912.A-APR.4.6 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. Content Complexity Rating: Level 2: Basic Application of Skills and Concept MAFS.912.A-APR.4.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Content Complexity Rating: Level 2: Basic Application of Skills and Concept Division of Academics - Department of Mathematics Page 7 of 10
Domain: Functions: Building Functions Cluster 2: Build new functions from existing functions MAFS.912.F-BF.2.3 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. Content Complexity Rating: Level 2: Basic Application of Skills and Concept Domain: Functions: Interpreting Functions Cluster 3: Analyze functions using different representations MAFS.912.F-IF.3.7d Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Content Complexity Rating: Level 2: Basic Application of Skills and Concept Division of Academics - Department of Mathematics Page 8 of 10
TECHNOLOGY TOOLS CPALM RESOURCES LESSON PLANS Justly Justifying A Rational Representation Preserving Our Marine Ecosystems Exploring Systems with Piggies, Pizzas and Phones Predicting Your Financial Future VIRTUAL MANIPULATIVE Data Flyer PROBLEM-SOLVING TASK How does the solution change? Combined Fuel Efficiency Dimes and Quarters TUTORIAL N/A GRAPHING CALCULATOR CORRELATION TEXAS INSTRUMENT MATH ACTIVITY TITLE Asymptotes & Zeros Families of Functions Asymptotes and Zeros of Rational Functions GIZMOS CORRELATION GIZMO TITLE General Form of a Rational Function Division of Academics - Department of Mathematics Page 9 of 10
TOPIC V DISCOVERY EDUCATION CORRELATION VIDEO TITLE Part One: Rational Functions MATH EXPLANATION TITLE Algebra II: Solving Rational Equations and Inequalities MATH OVERVIEW Algebra II: Graphing Rational Expressions Division of Academics - Department of Mathematics Page 10 of 10