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Advanced Topics Unit B: Trig Functions Unit Overview Math Florida Standards Content Standards Students will understand the construction of the unit circle, work with trigonometric functions graphically and algebraically and apply trigonometric identities and properties. Textbook Resources Prentice Hall Geometry Pearson Success Net Section: 8-2 Glencoe McGraw-Hill Pre-calculus textbook McGraw-Hill Sections: 4-1, 4-2, 4-3, 4-4, 4-6, 5-1, 5-2, 5-3, 5-4, 5-5 Additional Lessons: Deriving the Unit Circle from special right triangles (F-TF 1.3) Mathematics Formative Assessment System Tasks The system includes tasks or problems that teachers can implement with their students, and rubrics that help the teacher interpret students' responses. Teachers using MFAS ask students to perform mathematical tasks, explain their reasoning, and justify their solutions. Rubrics for interpreting and evaluating student responses are included so that teachers can differentiate instruction based on students' strategies instead of relying solely on correct or incorrect answers. The objective is to understand student thinking so that teaching can be adapted to improve student achievement of mathematical goals related to the standards. Like all formative assessment, MFAS is a process rather than a test. Research suggests that well-designed and implemented formative assessment is an effective strategy for enhancing student learning. MAFS.912.F-TF.1.3 MAFS.912.F-TF.1.4 MAFS.912.F-TF.2.5 MAFS.912.F-TF.2.6 MAFS.912.F-TF-2.7 MAFS.912.F-TF.3.8 MAFS.912.F-TF.3.9 MAFS.912.F-IF.3.7 Standards for Mathematical Practice MAFS.K12.MP.2.1 MAFS.K12.MP.7.1 MAFS.K12.MP.8.1 Other Resources As the Wheel Turns Task Applications of Trigonometric Functions Regents Prep Mathematics Formative Assessment System Tasks This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Advanced Topics Unit B: Trig Functions Unit Scale (Multidimensional) (MDS) The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1, “What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards. 4.0 In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these standards. Examples: 3.0 2.0 Explore composition of trigonometric functions with polynomial, rational, exponential, and/or logarithmic functions. The Student will: Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. (MAFS.912.F-TF.1.3) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. MAFS.912.F-TF.1.4) Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (MAFS.912.F-TF.2.6) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. (MAFS.912.F-TF.2.7) Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios. (MAFS.912.F-TF.3.8) Prove the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent and use these formulas to solve problems. (MAFS.912.F-TF.3.9) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (MAFS.912.F-IF.3.7) e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. The student will recognize or recall specific vocabulary, such as: Unit circle, trigonometric function, sine, cosine, tangent, symmetry, midline, period, frequency, phase shift, end behavior, x- and y-intercepts, amplitude The student will perform basic processes, such as: Identify key information on the unit circle Identify some features of trigonometric functions Use trigonometric properties and identities 1.0 With help, partial success at 2.0 content but not at score 3.0 content This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Advanced Topics Unit B: Trig Functions Unpacking the Standard: What do we want students to Know, Understand and Do (KUD): The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question #1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are essential for mastery. Domain: Functions: Trigonometric Functions Cluster: Model periodic phenomena with trigonometric functions (Major Cluster) Standard: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5) Understand “Essential understandings,” or generalizations, represent ideas that are transferable to other contexts. Real world situations can be modeled using trigonometric functions. Know Declarative knowledge: Facts, vocab., information Sine, cosine, tangent, amplitude, frequency, midline, period, phase shift Given multiple graphs and/or practical application problems, determine what trigonometric function would best fit the information Do Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts. Knowledge Utilization Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard? Recognize parent trigonometric function visually, Graph parent trigonometric functions This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Advanced Topics Unit B: Trig Functions Uni-Dimensional, Lesson Scale: The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1, “Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level, not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard. Standard: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5) Score Learning Progression I can… Write the equation of an appropriate trigonometric function from a graph and/or data given Sample Tasks A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally with respect to time, in days. He carefully collected data and graphed his resulting equation. From the graph, determine the amplitude, period, horizontal shift and vertical shift. Write the equation of the graph. 4.0 3.5 I can do everything at a 3.0, and I can demonstrate partial success at score 4.0. This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Advanced Topics Unit B: Trig Functions I can… Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally with respect to time, in days. He carefully collected data and graphed his resulting equation. From the graph, determine the amplitude, period, horizontal shift and vertical shift. Which trigonometric function would best model the graph? Defend your answer. 3.0 2.5 I can do everything at a 2.0, and I can demonstrate partial success at score 3.0. I can… State key components of a trigonometric graph o Amplitude, frequency, midline, phase shift A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally with respect to time, in days. He carefully collected data and graphed his resulting equation. From the graph, determine the amplitude, period, horizontal shift and vertical shift. 2.0 1.0 I need prompting and/or support to complete 2.0 tasks. This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Advanced Topics Unit B: Trig Functions Sample High Cognitive Demand Tasks: These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?” and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors). MAFS Mathematical Content Standard(s) Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5) Design Question 1; Element 1 (MAFS.K12.MP.2.1) Reason abstractly and quantitatively. MAFS Mathematical Practice(s) Design Question 1; Element 1 Marzano’s Taxonomy (MAFS.K12.MP.3.1) Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.4.1) Model with mathematics. Knowledge Utilization Teacher Notes Questions to develop mathematical thinking, possible misconceptions/misunderstandings , how to differentiate/scaffold instruction, anticipate student problem solving strategies This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Advanced Topics Unit B: Trig Functions Task *These tasks can either be teacher created or modified from a resource to promote higher order thinking skills. Please cite the source for any tasks. This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015