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

2010 Alabama Course of Study: Mathematics

College- and Career-Ready Standards

Professional Development: Grades 9 – 12 Phase I

Regional Inservice CenterSummer 2011

PART A

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

Topics for Today

Components of the Course of Study

Goal

Domains of Study

Position Statements

Standards for Mathematical Practice

Conceptual Categories

Preface Acknowledgments General Introduction Conceptual Framework Position Statements

◦ Equity◦ Curriculum◦ Teaching◦ Learning◦ Assessment◦ Technology



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

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

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


(+) 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]


(+)

(*) Modeling Standards

Algebra I

28. Relate the domain of a function to itsgraph and, where applicable, to the quantitative relationship it describes. [F-IF5]


*

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.



•Description of Standards

•Relation to K-8 Content

•Content Progression in 9-12


•Narrative

•Domains and Clusters

•Standards for Mathematical Practice

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


High School Course Progressions

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

High School Course Pathways

?? Questions ??

The Standards for Mathematical Practice

2010 Alabama Course of Study: Mathematics

College- and Career-Ready Standards


“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)

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

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

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.


Mathematically proficient students:

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

QUESTIONS TO CONSIDER…

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

Standard 1: Make sense of problems and persevere in solving them.What do mathematically proficient students do?

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

Standard 2: Reason abstractly and quantitatively. What do mathematically proficient students do?

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

Standard 3: Construct viable arguments and critique the reasoning of others. What do mathematically proficient students do?

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

Standard 4: Model with mathematics. What do mathematically proficient students do?

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

Standard 5: Use appropriate tools strategically. What do mathematically proficient students do?

Communicate precisely to others Use clear definitions in discussions State meaning of symbols consistently and

appropriately Specify units of measurements Calculate accurately & efficiently

Standard 6: Attend to precision. What do mathematically proficient students do?

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

perspective

Standard 7: Look for and make use of structure. What do mathematically proficient students do?

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

Standard 8: Look for and express regularity in repeated reasoning. What do mathematically proficient students do?

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…

Mathematical Practic

e

Mathematical Conten

t

CONNECTION and BALANCE

www.insidemathematics.org

This task gives students the chance to:

• Find relationships between graphs, equations, tables, and rules.

• Explain reasoning for answers.

Algebra Task 3 Sorting Functions

Algebra Task 3

Sorting Functions

www.insidemathematics.org

The mathematics of this task: 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.

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.


Mathematically proficient students:

Student B

Student A

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

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 Learnin

g

Stein, Grover, & Henningsen (1996)Smith & Stein (1998)

Stein, Smith, Henningsen, & Silver (2000)

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…

Leading with theMathematical Practice

StandardsYou 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

?? Questions ??

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

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