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Essential, Immediate Actions to Implement the Common Core State Standards for Mathematics Grades Diane J. Briars NCSM Immediate Past President Mathematics Education Consultant October 20, NCTM Regional Conference and Exposition Atlantic City, NJ

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Briars, June 2011 FYI Presentation slides will be posted on the NCSM website: mathedleadership.org or me at

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Briars, June Today’s Goals Examine critical differences between CCSS and current practice. Consider how the CCSS-M are likely to effect your mathematics program. Identify productive starting points for beginning implementation of the CCSS-M. Learn about tools and resources to support the transition to CCSS.

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Briars, June What is NCSM? International organization of and for mathematics education leaders: Coaches and mentors Curriculum leaders Department chairs District supervisors/leaders Mathematics consultants Mathematics supervisors Principals Professional developers Publishers and authors Specialists and coordinators State and provincial directors Superintendents Teachers Teacher educators Teacher leaders

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Briars, June 2011 NCSM Position Papers 1.Effective and Collaborative Teams 2.Sustained Professional Learning 3.Equity 4.Students with Special Needs 5.Assessment 6.English Language Learners 7.Positive Self-Beliefs 8.Technology mathedleadership.org

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Briars, June “ The Common Core State Standards represent an opportunity – once in a lifetime – to form effective coalitions for change.” Jere Confrey, August 2010

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Briars, June CCSS: A Major Challenge/Opportunity College and career readiness expectations Rigorous content and applications Stress conceptual understanding as well as procedural skills Organized around mathematical principles Focus and coherence Designed around research-based learning progressions whenever possible.

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Briars, June Common Core State Standards for Mathematics Introduction –Standards-Setting Criteria –Standards-Setting Considerations Application of CCSS for ELLs Application to Students with Disabilities Mathematics Standards –Standards for Mathematical Practice –Contents Standards: K-8; HS Domains Appendix A: Model Pathways for High School Courses

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Expanded CCSS and Model Pathways available at

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Briars, June What’s different about CCSS? Accountability

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Briars, June What’s different about CCSS? These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep. — CCSS (2010, p.5)

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Briars, June Assessment Consortia Partnership for the Assessment of Readiness for College and Careers (PARCC) SMARTER Balanced Assessment Consortium

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Briars, June Implementing CCSS Challenge: –CCSS assessments not available for several years ( deadline). Where NOT to start-- –Aligning CCSS standards grade-by-grade with existing mathematics standards.

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Briars, June Implementing CCSS: Where to Start? Mathematical practices Progressions within and among content clusters and domains “Key advances” Conceptual understanding Research-Informed C-T-L-A Actions Assessment tasks –Balanced Assessment Tasks (BAM) –State released tasks

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Briars, June Collaborate! Engage teachers in working in collaborative teams Grade level/course/department meetings –Common assessments –Common unit planning –Differentiating instruction Cross grade/course meetings –End-of-year/Beginning-of-year expectations

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

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

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Briars, June Conceptual Understanding – comprehension of mathematical concepts, operations, and relations Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic Competence – ability to formulate, represent, and solve mathematical problems Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. Strands of Mathematical Proficiency

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

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Briars, June The Standards for Mathematical Practice Take a moment to examine the first three words of each of the 8 mathematical practices… what do you notice? Mathematically Proficient Students…

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Briars, June The Standards for [Student] Mathematical Practice What are the verbs that illustrate the student actions for your assigned mathematical practice? Circle, highlight or underline them for your assigned practice… Discuss with a partner: What jumps out at you?

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Briars, June 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: Notice and pay attention to…

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Briars, June The Standards for [Student] Mathematical Practice On a scale of 1 (low) to 6 (high), to what extent is your school promoting students’ proficiency in the practice you discussed? Evidence for your rating? Individual rating Team rating

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

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Briars, June Standards for Mathematical Practice Describe the thinking processes, habits of mind and dispositions that students need to develop a deep, flexible, and enduring understanding of mathematics; in this sense they are also a means to an end. SP1. Make sense of problems “….they [students] analyze givens, constraints, relationships and goals. ….they monitor and evaluate their progress and change course if necessary. …. and they continually ask themselves “Does this make sense?”

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Briars, June Standards for Mathematical Practice AND…. Describe mathematical content students need to learn. SP1. Make sense of problems “……. students can 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.”

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Briars, June Buttons Task Gita plays with her grandmother’s collection of black & white buttons. She arranges them in patterns. Her first 3 patterns are shown below. Pattern #1 Pattern #2 Pattern #3 Pattern #4 1.Draw pattern 4 next to pattern 3. 2.How many white buttons does Gita need for Pattern 5 and Pattern 6? Explain how you figured this out. 3.How many buttons in all does Gita need to make Pattern 11? Explain how you figured this out. 4.Gita thinks she needs 69 buttons in all to make Pattern 24. How do you know that she is not correct? How many buttons does she need to make Pattern 24?

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Briars, June Button Task 1.Individually complete parts Then work with a partner to compare your work. Look for as many ways to solve part 3 as possible. 3.Which mathematical practices are needed to complete the task?

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Briars, June Mathematics.org visits/public-lessons-numerical-patterning/218-numerical-patterning- lesson-planning?phpMyAdmin=NqJS1x3gaJqDM-1-8LXtX3WJ4e8http://www.insidemathematics.org/index.php/classroom-video- visits/public-lessons-numerical-patterning/218-numerical-patterning- lesson-planning?phpMyAdmin=NqJS1x3gaJqDM-1-8LXtX3WJ4e8 A reengagement lesson using the Button Task Francis Dickinson San Carlos Elementary Grade 5

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Briars, June 2011 Learner A Pictorial Representation What does Learner A see staying the same? What does Learner A see changing? Draw a picture to show how Learner A sees this pattern growing through the first 3 stages. Color coding and modeling with square tiles may come in handy. Verbal Representation Describe in your own words how Learner A sees this pattern growing. Be sure to mention what is staying the same and what is changing.

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Briars, June 2011 Learner B Pictorial Representation What does Learner B see staying the same? What does Learner B see changing? Draw a picture to show how Learner B sees this pattern growing through the first 3 stages. Color coding and modeling with square tiles may come in handy. Verbal Representation Describe in your own words how Learner B sees this pattern growing. Be sure to mention what is staying the same and what is changing.

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Briars, June Button Task Revisited Which of the Standards of Mathematical Practice did you see the students working with? What did Mr. Dickinson get out of using the same math task two days in a row, rather than switching to a different task(s)? How did the way the lesson was facilitated support the development of the Standards of Practice for students? What implications for implementing CCSS does this activity suggest to you?

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Briars, June 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.” Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

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Briars, June The Standards for [Student] Mathematical Practice The 8 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, the student learning environment creates great inequity in our schools…

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Briars, June Learners should Acquire conceptual knowledge as well as skills to enable them to organize their knowledge, transfer knowledge to new situations, and acquire new knowledge. Engage with challenging tasks that involve active meaning-making

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Briars, June What Are Mathematical Tasks? Mathematical tasks are a set of problems or a single complex problem the purpose of which is to focus students’ attention on a particular mathematical idea.

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Briars, June Why Focus on Mathematical Tasks? Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it; Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information; The level and kind of thinking required by mathematical instructional tasks influences what students learn; and Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students’ opportunities to learn mathematics.

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Briars, June Why Focus on Mathematical Tasks? Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it; Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information; The level and kind of thinking required by mathematical instructional tasks influences what students learn; and Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students’ opportunities to learn mathematics.

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Briars, June 2011 Increasing Task Cognitive Demand: The Handshake Problem Version 1 How many handshakes will there be if each person in your group shakes the hand of every person once? Tell who was in your group How did you get your answer? Try to show your solution on paper Version 2 How many handshakes will there be if one more person joins your group? How many handshakes will there be if two more people join your group? How many handshakes will there be if three more people join your group? Organize your data into a table. Do you see a pattern? How many handshakes will there be if 100 more people join your group? Generalization: What can be said about the relationship between the number of people in a group and the total number of handshakes? Blanton & Kaput, 2003

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Briars, June 2011 Compare the Two Versions of the Handshake Problem How are the two versions of the handshake task the same? How are they different? Blanton & Kaput, 2003

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Briars, June Cognitive Level of Tasks Lower-Level Tasks (e.g., Handshakes Version 1) Higher-Level Tasks (e.g., Handshakes Version 2) The Quasar Project

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Briars, June Lower-Level Tasks Memorization –What are the decimal equivalents for the fractions ½ and ¼? Procedures without connections –Convert the fraction 3/8 to a decimal.

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Briars, June Higher-Level Tasks Procedures with connections –Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5. Doing mathematics –Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine: a)The decimal part of area that is shaded; b)The fractional part of area that is shaded.

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Briars, June Implementation Issue Do all students have the opportunity to engage in mathematical tasks that promote students’ attainment of the mathematical practices on a regular basis?

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Briars, June Opportunities for all students to engage in challenging tasks? Examine tasks in your instructional materials: –Higher cognitive demand? –Lower cognitive demand? Where are the challenging tasks? Do all students have the opportunity to grapple with challenging tasks? Examine the tasks in your assessments: –Higher cognitive demand? –Lower cognitive demand?

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Briars, June 2011 Insidemathematics.org

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Briars, June 2011 Insidemathematics.org 50

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Briars, June 2011 “Algebrafying” Instructional Materials Transforming problems with a single numerical answer to opportunities for pattern building, conjecturing, generalizing, and justifying mathematical facts and relationships. Blanton & Kaput, 2003, p.71

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Briars, June 2011 Increasing Task Cognitive Demand: The Handshake Problem Version 1 How many handshakes will there be if each person in your group shakes the hand of every person once? Tell who was in your group How did you get your answer? Try to show your solution on paper Version 2 How many handshakes will there be if one more person joins your group? How many handshakes will there be if two more people join your group? How many handshakes will there be if three more people join your group? Organize your data into a table. Do you see a pattern? How many handshakes will there be if 100 more people join your group? Generalization: What can be said about the relationship between the number of people in a group and the total number of handshakes? Blanton & Kaput, 2003

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Briars, June 2011 Algebrafy This Task Jack wants to save to buy a CD player that costs $35. He makes $5 per hour babysitting. How many hours will he need to work in order to buy the CD player? Blanton & Kaput, 2003

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Briars, June 2011 Creating A Classroom Culture and Practices that Promote Algebraic Reasoning Incorporate conjecture, argumentation, and generalization in purposeful ways so that students consider arguments as ways to build reliable knowledge Respect and encourage these activities as standard daily practice, not as occasional “enrichment” separate from the regular work of learning and practicing arithmetic Blanton & Kaput, 2003, p.74

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Briars, June Leading with the Mathematics Practices Build upon/extend work on NCTM Processes and NRC Proficiencies Phase in implementation Consider relationships among the practices Analyze instructional tasks in terms of opportunities for students to regularly engage in practices.

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Briars, June Standards for Mathematical Content Counting and Cardinality (K) Operations and Algebraic Thinking (K-5) Number and Operations in Base Ten (K-5) Measurement and Data (K-5) Geometry (K-HS) Number and Operations — Fractions (3-5) Ratios and Proportional Relationships (6-7) The Number System (6-8) Expressions and Equations (6-8) Statistics and Probability (6-HS) Functions (8-HS) Number and Quantity (HS) Algebra (HS) Modeling (HS)

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Briars, June Grade Level Standards

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Briars, June Progressions within and across Domains Daro, 2010 Number and Operations ―Fractions Algebra Expressions and Equations The Number System Operations and Algebraic Thinking Number and Operations―Base Ten K High School

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Briars, June Key Advances 1.Operations and the problems they solve 2.Properties of operations: Their role in arithmetic and algebra 3.Mental math and “algebra” vs. algorithms 4.Units and unitizing a.Unit fractions b.Unit rates 5.Defining similarity and congruence in terms of transformations 6.Quantities-variables-functions-modeling 7.Number-expression-equation-function 8.Modeling Daro, 2010

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Briars, June 2011 Write a word problem that could be modeled by a + b = ca × b = p

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Briars, June 2011 Word problems for a + b = c or a x b = p Result unknown; e.g = ?, 3 x 6 = ? –Mike has 8 pennies. Sam gives him 3 more. How many does Mike have now? –There are 3 bags with 6 plums in each bag? How many plums are there in all?

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Briars, June 2011 Word problems for a + b = c Change or part unknown; e.g., 5 + ? = 8 –Mike has 5 pennies. Sam gives him some more. Now he has 8. How many did he get from Sam? Start unknown; e.g., ? + 3 = 8 –Mike has some pennies. He gets 3 more. Now he has 11. How many did he have at the beginning?

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Briars, June 2011 Word problems for a x b = p Number of groups (a) or number in each group (b) unknown? –If 18 plums are to be packed 6 to a bag, then how many bags are needed? –If 18 plums are to packed into 6 bags, then how many plums will be in each bag? –Contexts: Equal groups? Arrays? Area? Size comparison? Scale factor or smaller quantity unknown –Crista earned $18 babysitting. Ann only earned $6. How many times as much money did Crista earn than Ann? –Crista earned $18 babysitting. That was 3 times as much as Ann earned. How much money did Ann earn?

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Common Addition and Subtraction Situations

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Common Multiplication and Division Situations

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Briars, June 2011 Does It Always Work? Looking for “Key Words” or “Clue Words” is sometimes taught to children to help them choose the operation to solve a word problem. Does it always work? Try to solve each type of problem in the table using “key words.” Did you always get the correct answer using this method? For which problems does it work? For which problems, if any, doesn’t it work?

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Briars, June 2011 Operations and the Problems They Solve To what extent do your instructional materials provide opportunities for students to develop proficiency with the full range of situations that can be modeled by multiplication/division or ratios/rates? To what extent are students expected to write equations to describe these situations? 68

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Briars, June Key Advances 1.Operations and the problems they solve 2.Properties of operations: Their role in arithmetic and algebra 3.Mental math and “algebra” vs. algorithms 4.Units and unitizing a.Unit fractions b.Unit rates 5.Defining similarity and congruence in terms of transformations 6.Quantities-variables-functions-modeling 7.Number-expression-equation-function 8.Modeling Daro, 2010

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Briars, June 2011 Mental Math and “Algebra” vs. Algorithms = = 29 x 12 = 70

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Operations and Algebraic Thinking Numbers and Operations in Base Ten Fractions 1 Understand and apply properties of operations and the relationship between addition and subtraction. Use place value understanding and properties of operations to add and subtract. 2 3 Understand properties of multiplication and the relationship between multiplication and division. Use place value understanding and properties of operations to perform multi-digit arithmetic. A range of algorithms may be used. 4 Use place value understanding and properties of operations to perform multi-digit arithmetic. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 5 Perform operations with multi-digit whole numbers and with decimals to hundredths. Fluently multiply multi-digit whole numbers using the standard algorithm. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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Operations and Algebraic Thinking Numbers and Operations in Base Ten Fractions 1 Understand and apply properties of operations and the relationship between addition and subtraction. Use place value understanding and properties of operations to add and subtract. 2 3 Understand properties of multiplication and the relationship between multiplication and division. Use place value understanding and properties of operations to perform multi-digit arithmetic. A range of algorithms may be used. 4 Use place value understanding and properties of operations to perform multi-digit arithmetic. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 5 Perform operations with multi-digit whole numbers and with decimals to hundredths. Fluently multiply multi-digit whole numbers using the standard algorithm. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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Operations and Algebraic Thinking Numbers and Operations in Base Ten Fractions 1 Understand and apply properties of operations and the relationship between addition and subtraction. Use place value understanding and properties of operations to add and subtract. 2 3 Understand properties of multiplication and the relationship between multiplication and division. Use place value understanding and properties of operations to perform multi-digit arithmetic. A range of algorithms may be used. 4 Use place value understanding and properties of operations to perform multi-digit arithmetic. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 5 Perform operations with multi-digit whole numbers and with decimals to hundredths. Fluently multiply multi-digit whole numbers using the standard algorithm. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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Briars, June 2011 Strategies vs. Algorithms Computational strategy: –Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. Computational algorithm: –A set of predetermined steps applicable to a class of problems that gives the correct results in every case when the steps are carried out correctly. Which methods are strategies? Algorithms? 74

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Briars, June 2011 Multiplication Algorithms 75 CCSS Numbers and Operations in Base-Ten Progression, April 2011

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Briars, June 2011 Multiplication Algorithms 76 CCSS Numbers and Operations in Base-Ten Progression, April 2011

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Multiplication Algorithms 77 CCSS Numbers and Operations in Base-Ten Progression, April 2011

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Multiplication Algorithms 78 CCSS Numbers and Operations in Base-Ten Progression, April 2011

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Briars, June Key Advances 1.Operations and the problems they solve 2.Properties of operations: Their role in arithmetic and algebra 3.Mental math and “algebra” vs. algorithms 4.Units and unitizing a.Unit fractions b.Unit rates 5.Quantities-variables-functions-modeling 6.Number-expression-equation-function 7.Modeling 8.Practices Daro, 2010

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Briars, June 2011 Units Matter What fractional part is shaded? 80 CCSS Number and Operations-Fractions Progression, 8/2011

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Briars, June Units are things you count Objects Groups of objects ¼unit fractions Numbers represented as expressions Daro, 2010

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Briars, June Units add up 3 pennies + 5 pennies = 8 pennies 3 ones + 5 ones = 8 ones 3 tens + 5 tens = 8 tens 3 inches + 5 inches = 8 inches 3 ¼ inches + 5 ¼ inches = 8 ¼ inches 3(1/4) + 5(1/4) = 8(1/4) 3(x + 1) + 5(x+1) = 8(x+1) Daro, 2010

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Briars, June 2011 Benefits of Unit Fraction Approach Increase understanding of fractions Apply whole number concepts and skills to fractions 83

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Briars, June 2011 Unitizing links fractions to whole number arithmetic Students’ expertise in whole number arithmetic is the most reliable expertise they have in mathematics It makes sense to students If we can connect difficult topics like fractions and algebraic expressions to whole number arithmetic, these difficult topics can have a solid foundation for students. Daro, 2011

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Briars, June 2011 Fraction Equivalence Grade 3: –Fractions of areas that are the same size, or fractions that are the same point (length from 0) are equivalent –Recognize simple cases: ½ = 2/4 ; 4/6 = 2/3 –Fraction equivalents of whole numbers 3 = 3/1, 4/4 =1 –Compare fractions with same numerator or denominator based on size in visual diagram

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Briars, June 2011 Fraction Equivalence Grade 4: –Explain why a fraction a/b = na/nb using visual models; generate equivalent fractions –Compare fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½.

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Briars, June 2011 Fraction Equivalence Grade 5: –Use equivalent fractions to add and subtract fractions with unlike denominators

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Briars, June 2011 Grade 4: Comparing Fractions 1.Which is closer to 1, 4/5 or 5/4? How do you know? 2.Which is greater, 3/7 or 3/5? How do you know? 3.Which is greater, 3/7 or 5/9? How do you know? 88 CCSS Number and Operations-Fractions Progression, 8/2011

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Briars, June Key Advances 1.Operations and the problems they solve 2.Properties of operations: Their role in arithmetic and algebra 3.Mental math and “algebra” vs. algorithms 4.Units and unitizing a.Unit fractions b.Unit rates 5.Defining congruence and similarity in terms of transformations 6.Quantities-variables-functions-modeling 7.Number-expression-equation-function 8.Modeling Daro, 2010

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Briars, June Implementing CCSS: Where to Start? Mathematical practices Progressions within and among content clusters and domains “Key advances” Conceptual understanding Research-Informed C-T-L-A Actions Assessment tasks –Balanced Assessment Tasks (BAM) –State released tasks

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Briars, June Reflections What next steps will you take to implement CCSS? Who will you need to work with? What support/information/resources will you need?

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Briars, June 2011 Thank You! mathedleadership.org

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