MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
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1 Topic IV: Polynomial Functions, Expressions, and Equations Pacing Date(s) Traditional 19 12/12/16 01/20/17 Block 9 12/12/16 01/20/17 Topic IV Assessment Window 01/17/17 01/20/17 MATHEMATICS FLORIDA STATE STANDARDS (MAFS) & MATHEMATICAL PRACTICES (MP) ESSENTIAL CONTENT OBJECTIVES (from Item Specifications) MAFS.912.A-APR.1.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (MP.2, MP.7) MAFS.912.A-APR.2.3: 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. (MP.1, MP.2, MP.4, MP.5, MP.8) MAFS.912.A-APR.3.4: Prove polynomial identities and use them to describe numerical relationships. (MP.7, MP.8) MAFS.912.A-APR.3.5: 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.(+) MAFS.912.A-APR.2.2: 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). (MP.2, MP.3, MP.8) MAFS.912.A-SSE.1.1: Interpret expressions that represent a quantity in terms of its context. (MP.1, MP.2, MP.4, MP.7) MAFS.912.A-SSE.1.2: Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). (MP.2, 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.4, MP.5, MP.6, MP.7, MP.8) MAFS.912.N-CN.3.8: Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x 2i). (+) MAFS.912.N-CN.3.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (+) A. Polynomials 1. Adding and Subtracting Polynomials. 2. Multiplying Polynomials 3. The Binomial Theorem 4. Factoring Polynomials 5. Dividing Polynomials B. Polynomial Equations 1. Finding Rational Solutions of Polynomial Equations 2. Finding Complex Solutions of Polynomials Equations. I can: apply their understanding of closure to adding, subtracting, and multiplying polynomials. add, subtract, and multiply polynomials with rational coefficients. use polynomial identities to describe numerical relationships. use the structure of algebra to complete an algebraic proof of a polynomial identity. find the zeros of a polynomial function when the polynomial is in factored form. identify a rough graph of a polynomial function in factored form by examining the zeros of the function. use the x-intercepts of a polynomial function and end behavior to graph the function. graph a polynomial function using key features. use polynomial long division to divide a polynomial by a polynomial. use the Remainder Theorem to determine if (x a) is a factor of a polynomial. use the Remainder Theorem to determine the remainder of p(x)/(x a). rewrite algebraic expressions in different equivalent forms using factoring techniques (e.g., common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely) or simplifying expressions (i.e., combining like terms, using the distributive property, and using other operations with polynomials). Division of Academics - Department of Mathematics Page 1 of 10
2 INSTRUCTIONAL TOOLS Core Text Book: Houghton Mifflin Harcourt Algebra 2 Algebra 2 Honors Course Description Pacing Date(s) Traditional 19 12/12/16 01/20/17 Block 9 12/12/16 01/20/17 Topic IV Assessment Window 01/17/17 01/20/17 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 Auto-assign Test and Quizzes for intervention and enrichment: NO Homework PMT Preferences: Auto-assign for intervention and enrichment: YES Auto-assign Standard-Based for intervention Intervention and enrichment: NO Course Intervention Daily Intervention Division of Academics - Department of Mathematics Page 2 of 10
3 INSTRUCTIONAL TOOLS STANDARDS MODULES TEACHER NOTES MAFS.912.A-APR.1.1 MAFS.912.A-APR.2.2 MAFS.912.A-APR.2.3 MAFS.912.A-APR.3.4 MAFS.912.A-APR.3.5 (+) MAFS.912.A-APR.4.6 MAFS.912.A-SSE.1.1a MAFS.912.A-SSE.1.2 MAFS.912.N-CN.3.8 (+) MAFS.912.N-CN.3.9 (+) Module 6 Module 7 Algebra 2 Honors Block Schedule Suggested Pace Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day Topic Test Topic IV Assessment: Polynomial Functions, Expressions, and Equations (Also assesses Topic III) (+) Additional mathematics that students should learn in fourth credit courses or advanced courses such as calculus, advanced statistics, or discrete mathematics Topic Resources PowerPoint Available in Learning Village MODULE LESSON STANDARDS Module MAFS.912.A-APR MAFS.912.A-APR.1.1 MAFS.912.A-APR MAFS.912.A-APR.3.5 SUGGESTED PROBLEMS BY TEACHERS FOR TEACHERS* 2, 3, 8, 9, 14, 22, 24 2, 4, 6, 12, 13, , 15 17, 19, 20 NOTES / RESOURCES Illustrative Mathematics Task(s): Non-Negative Polynomials, Trina s Triangle MAFS.912.A-SSE.1.1a MAFS.912.A-SSE.1.2 MAFS.912.A-APR.2.3 MAFS.912.A-APR.2.2 MAFS.912.A-APR.4.6 1, 2, 3, 5, 8, 10, 15, 16, 19, 21, 26 1, 3, 5, 7, 9, 11 13, 16, 21, 23 GeoGebra: Factors and Zeros Illustrative Mathematics Task(s): Zeroes and factorization of a general polynomial, Zeroes and factorization of a quadratic polynomial I, The Missing Coefficient, Graphing from Factors III Module MAFS.912.A-APR.2.2 MAFS.912.A-APR MAFS.912.N-CN.3.9 (+) 1, 2, 6, 7, 14, 16, 18, 19 1, 2, 5, 6, 12 *Problems were suggested by M-DCPS teachers during May Algebra 2 PD. Division of Academics - Department of Mathematics Page 3 of 10
4 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. Vocabulary: Binomial, monomial, polynomial, synthetic division, trinomial, cubic function, polynomial function, root, multiplicity. Connections to Redesigned SAT Passport to Advanced Math: Add, subtract, and multiply polynomial expressions and simplify the result. The expressions will have rational coefficients. Understand the relationship between zeros and factors of polynomials, and use that knowledge to sketch graphs. Students will use properties of factorable polynomials to solve conceptual problems relating to zeros, such as determining whether an expression is a factor of a polynomial based on other information provided. SAT Practice N/A STEM Lessons - Model Eliciting Activity STEM Lessons CPALMS Perspectives Videos Professional/Enthusiasts Base 16 Notation in Computing Expert Jumping Robots and Quadratics Division of Academics - Department of Mathematics Page 4 of 10
5 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
6 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 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
7 Domain: Algebra: Arithmetic with Polynomials & Rational Expressions STANDARD CODE Cluster 1: Perform arithmetic operations on polynomials MAFS.912.A-APR.1.1 STANDARD DESCRIPTION Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Cluster 2: Understand the relationship between zeros and factors of polynomials MAFS.912.A-APR.2.2 MAFS.912.A-APR.2.3 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). 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. Cluster 3: Use polynomial identities to solve problems MAFS.912.A-APR.3.4 MAFS.912.A-APR.3.5 Prove polynomial identities and use them to describe numerical relationships. 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. Context Complexity: Level 2: Basic Applications of Skills and Concepts 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. Context Complexity: Level 2: Basic Applications of Skills and Concepts Division of Academics - Department of Mathematics Page 7 of 10
8 Domain: Algebra: Seeing Structure in Expressions STANDARD CODE Cluster 1: Interpret the structure of expressions STANDARD CODE MAFS.912.A-SSE.1.1a MAFS.912.A-SSE.1.2 Interpret expressions that represent a quantity in terms of its context. Context Complexity: Level 2: Basic Applications of Skills and Concept Use the structure of an expression to identify ways to rewrite it. For example, see x4- y4 as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). Context Complexity: Level 2: Basic Applications of Skills and Concepts Domain: Number & Quantity: The Complex Number STANDARD CODE Cluster 3: Use complex numbers in polynomial identities and equations. STANDARD CODE MAFS.912.N-CN.3.8 MAFS.912.N-CN.3.9 Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x 2i). Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Division of Academics - Department of Mathematics Page 8 of 10
9 TECHNOLOGY TOOLS CPALM RESOURCES LESSON PLANS Wonka's Golden Ticket! (Polynomial Operations) Building Connections Sorting Equations and Identities Manipulating Polynomials Using algebra tiles and tables to factor trinomials (less guess and check!) Hip to be (completing the) Square VIRTUAL MANIPULATIVE Curve Fitting Number Cruncher Data Flyer Function Flyer PROBLEM-SOLVING TASK Zeroes and factorization of a non-polynomial function The Physics Professor Equivalent Expressions TUTORIAL Addition and Subtraction of Polynomials Multiplying and Dividing Monomials Division of Polynomials GRAPHING CALCULATOR CORRELATION TEXAS INSTRUMENT MATH ACTIVITY TITLE McLaurin Polynomials Application of Polynomials Zeros of Polynomials Multiplying Polynomials Exploring Polynomials: Factors, Roots, and Zeros Division of Academics - Department of Mathematics Page 9 of 10
10 GIZMOS CORRELATION GIZMO TITLE Addition of Polynomials Polynomials and Linear Factors Dividing Polynomials Using Synthetic Division Modeling the Factorization of ax 2 +bx+c Distance-Time and Velocity-Time Graphs Quadratics in Polynomial Form TOPIC IV DISCOVERY EDUCATION CORRELATION VIDEO TITLE Math Factor: Factoring and Graphing Polynomial Functions Functions, Domain, and Range -- Burning Calories MATH EXPLANATION TITLE Algebra II: Division of Polynomials Example 2: Independent and Dependent Variables--Swimming MATH OVERVIEW Solving Polynomial Equations Algebra II: Graphing Square and Cube Root Functions Division of Academics - Department of Mathematics Page 10 of 10
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Topic IV: Polynomial Functions, Expressions, and Equations Pacing Date(s) Traditional 18 10/29/18 11/28/18 Block 9 10/29/18 11/28/18 Topic IV Assessment Window 11/16/18 11/28/18 MATHEMATICS FLORIDA STATE
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