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Professional Development: Grades 9 – 12 Phase I Regional Inservice Center Summer 2011 PART A

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Components of the Course of Study High School Course Progressions/Pathways Standards for Mathematical Practice Literacy Standards for Grades 6-12 ◦ History/Social Studies, Science, and Technical Subjects The Big Picture ◦ Domains of Study and Conceptual Categories Learning Progressions/Trajectories ◦ Vertical Alignment of Content Addressing Content Shifts Early Entry Algebra I ◦ Considerations/Consequences

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Goal Domains of Study Position Statements Standards for Mathematical Practice Conceptual Categories

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Preface Acknowledgments General Introduction Conceptual Framework Position Statements ◦ Equity ◦ Curriculum ◦ Teaching ◦ Learning ◦ Assessment ◦ Technology Standards for Mathematical Practice

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Directions for Interpreting the Minimum Required Content GRADE 4 Students will: Number and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 6. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. [4-NBT1] 7. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meaning of the digits in each place using >, =, and < symbols to record the results of comparisons. [4-NBT2] 8. Use place value understanding to round multi-digit whole numbers to any place. [4-NBT3] Cluster Content Standards Content Standard Identifiers Domain

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ALGEBRA II WITH TRIGONOMETRY Students will: FUNCTIONS Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 32. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. [F-TF1] 33. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. [F-TF2] 34.Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. Content Standards Cluster Domain Content Standard Identifiers Conceptual Category

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Standards for High School Mathematics ◦ Conceptual Categories for High School Mathematics Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability ◦ Additional Coding (+) STEM Standards (*) Modeling Standards ( ) Alabama Added Content

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(+) STEM Standards Geometry 22. Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. [G-SRT9] (+)

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(*) Modeling Standards Algebra I 28.Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. [F-IF5] *

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Added Content Specific to Alabama Geometry 35. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.

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Description of Standards Relation to K-8 Content Content Progression in 9-12

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Narrative Domains and Clusters Standards for Mathematical Practice

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Appendices A-E ◦ Appendix A Table 1: Common Addition and Subtraction Situations Table 2: Common Multiplication and Division Situations Table 3: Properties of Operations Table 4: Properties of Equality Table 5: Properties of Inequality ◦ Appendix B Possible Course Progressions in Grades 9-12 Possible Course Pathways ◦ Appendix C Literacy Standards For Grades 6-12 History/Social Studies, Science, and Technical Subjects ◦ Appendix D Alabama High School Graduation Requirements ◦ Appendix E Guidelines and Suggestions for Local Time Requirements and Homework Bibliography Glossary

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Required for All Students Algebra I Geometry Algebra II with Trigonometry or Algebra II Courses Must Increase in Rigor New Courses Discrete Mathematics Mathematical Investigations Analytical Mathematics

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The Standards for Mathematical Practice

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Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.” (CCSS, 2010)

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Underlying Frameworks National Council of Teachers of Mathematics NCTM (2000M). Principles and Standards for School Mathematics. Reston, VA: Author. 5 PROCESS Standards Problem Solving Reasoning and Proof Communication Connections Representations

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Underlying Frameworks Strands of Mathematical Proficiency NRC (2001). Adding It Up. Washington, D.C.: National Academies Press. Conceptual Understanding Procedural Fluency Strategic Competence Adaptive Reasoning Productive Disposition National Research Council

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Standard 1:Make sense of problems and persevere in solving them. Standard 2:Reason abstractly and quantitatively. Standard 3:Construct viable arguments and critique the reasoning of others. Standard 4:Model with mathematics. Standard 5:Use appropriate tools strategically. Standard 6:Attend to precision. Standard 7:Look for and make use of structure. Standard 8:Look for and express regularity in repeated reasoning. The Standards for Mathematical Practice Mathematically proficient students:

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1. What does this standard look like in the classroom? 2. What will students need in order to do this? 3. What will teachers need in order to do this? Adapted from Kathy Berry, Monroe County ISD, Michigan

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Analyze givens, constraints, relationships Make conjectures Plan solution pathways Make meaning of the solution Monitor and evaluate their progress Change course if necessary Ask themselves if what they are doing makes sense

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Make sense of quantities and relationships Able to decontextualize ◦ Abstract a given situation ◦ Represent it symbolically ◦ Manipulate the representing symbols Able to contextualize ◦ Pause during manipulation process ◦ Probe the referents for symbols involved

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Construct arguments Analyze situations Justify conclusions Communicate conclusions Reason inductively Distinguish correct logic from flawed logic Listen to/Read/Respond to other’s arguments and ask useful questions to clarify/improve arguments

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Apply mathematics to solve problems from everyday life situations Apply what they know Simplify a complicated situation Identify important quantities Map math relationships using tools Analyze mathematical relationships to draw conclusions Reflect on improving the model

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Consider and use available tools Make sound decisions about when different tools might be helpful Identify relevant external mathematical resources Use technological tools to explore and deepen conceptual understandings

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Communicate precisely to others Use clear definitions in discussions State meaning of symbols consistently and appropriately Specify units of measurements Calculate accurately & efficiently

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Discern patterns and structures Use strategies to solve problems Step back for an overview and can shift perspective

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Notice if calculations are repeated Look for general methods and shortcuts Maintain oversight of the processes Attend to details Continually evaluates the reasonableness of their results

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The Standards for [Student] Mathematical Practice SMP1: Explain and make conjectures… SMP2: Make sense of… SMP3: Understand and use… SMP4: Apply and interpret… SMP5: Consider and detect… SMP6: Communicate precisely to others… SMP7: Discern and recognize… SMP8: Note and pay attention to…

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This task gives students the chance to: Find relationships between graphs, equations, tables, and rules. Explain reasoning for answers. Algebra Task 3 Sorting Functions

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Algebra Task 3 Sorting Functions Sorting Functions

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Algebra Task 3 Sorting Functions Sorting Functions

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Algebra – 2008 Copyright © 2008 by Noyce Foundation. All rights reserved. The information provided in the following slides is for professional development only.

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Making connections between different algebraic representations: graphs, equations, verbal rules, and tables Understanding how the equation determines the shape of the graph Developing a convincing argument using a variety of algebraic concepts Being able to move from specific solutions to thinking about generalizations Algebra – 2008 Copyright © 2008 by Noyce Foundation. All rights reserved.

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Standard 1:Make sense of problems and persevere in solving them. Standard 2:Reason abstractly and quantitatively. Standard 3:Construct viable arguments and critique the reasoning of others. Standard 4:Model with mathematics. Standard 5:Use appropriate tools strategically. Standard 6:Attend to precision. Standard 7:Look for and make use of structure. Standard 8:Look for and express regularity in repeated reasoning. The Standards for Mathematical Practice Mathematically proficient students:

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Student B Student A

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The Standards for [Student] Mathematical Practice “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” Stein, Smith, Henningsen, & Silver, 2000 “The level and kind of thinking in which students engage determines what they will learn.” Herbert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

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But, WHAT TEACHERS DO with the tasks matters too! The Mathematical Tasks Framework Tasks as they appear in curricular materials Tasks are set up by teachers Tasks are enacted by teachers and students Student Learning Stein, Grover, & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen, & Silver (2000)

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Standards for [Student] Mathematical Practice The Standards for Mathematical Practice place an emphasis on student demonstrations of learning… Equity begins with an understanding of how the selection of tasks, the assessment of tasks, and the student learning environment create inequity in our schools…

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Leading with the Mathematical Practice Standards You can begin by implementing the 8 Standards for Mathematical Practice now Think about the relationships among the practices and how you can move forward to implement BEST PRACTICES Analyze instructional tasks so students engage in these practices repeatedly

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