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Essential, Immediate Actions to Implement the Common Core State Standards for Mathematics Grades 3 - 5 Diane J. Briars NCSM Immediate Past President Mathematics.

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Presentation on theme: "Essential, Immediate Actions to Implement the Common Core State Standards for Mathematics Grades 3 - 5 Diane J. Briars NCSM Immediate Past President Mathematics."— Presentation transcript:

1 Essential, Immediate Actions to Implement the Common Core State Standards for Mathematics Grades 3 - 5 Diane J. Briars NCSM Immediate Past President Mathematics Education Consultant October 20, 2011 2011 NCTM Regional Conference and Exposition Atlantic City, NJ

2 Presentation slides will be posted on the NCSM website:
FYI Presentation slides will be posted on the NCSM website: mathedleadership.org or me at

3 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. Our goals for today are to provide fundamental information about CCSS-M and also provide some suggestions for how to get started in your local implementation efforts.

4 What is NCSM? www.mathedleadership.org
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 For those of you who are not familiar with NCSM, we are an international organization of mathematics education leaders at all levels. Our mission is to provide leaders like you with professional learning opportunities that will help you support and sustain improved student achievement in your setting.

5 PRIME provides the ADULT ACTIONS necessary to address these issue
The PRIME Leadership Framework is written and design for all mathematics leadership types: principals, math specialists, math coordinators, math coaches, curriculum coordinators, lead teachers, math supervisors; mentor teachers. A critical component to improve mathematics learning is the continuous content and pedagogy support needed by teachers on a regular basis. 5

6 NCSM Position Papers Effective and Collaborative Teams
Sustained Professional Learning Equity Students with Special Needs Assessment English Language Learners Positive Self-Beliefs Technology mathedleadership.org

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

8 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. CCSS poses both major challenges, but also a unique opportunity, to focus energy on improving mathematics education. The bullets describe major features of CCSS . . Increased rigor will certainly be a challenge; but note that that is accompanied by— Increase emphasis on conceptual understanding—something that many of us have been arguing for for a long time. Organization around core mathematics principles such as place value and properties of operations; calling for students to use these principles, not just memorize them. We’ll see specific examples of this shortly. More focus at each grade level in K-8; to provide instructional time needed for students to develop conceptual understanding. Finally, progressions across grades based on research whenever possible. Large body of research for early number concepts, much less guidance from research as proceed up the grades. When there was not a cognitive research base, writers looked to curriculum documents from other countries for guidance. “Research” in quotes.

9 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

10 Expanded CCSS and Model Pathways available at www. mathedleadership
Expanded CCSS and Model Pathways available at

11 What’s different about CCSS?
Accountability

12 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)

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

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

15 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

16 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

17 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) The next six slides describe the connections between the new Standards for Mathematical Practice and their predecessors… the NCTM Process Standards and the NRC’s Stands of Mathematical Proficiency. Review each of the slides and ask for any questions from participants.

18 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.

19 Underlying Frameworks
Strands of Mathematical Proficiency Strategic Competence Adaptive Reasoning Conceptual Understanding Productive Disposition Procedural Fluency Turn and talk, what does the graphic communicate about the nature of the 5 strands NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.

20 Strands of Mathematical Proficiency
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.

21 Standards for Mathematical Practice
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Refer participants to the handout with the description of the CCSS Standards for Mathematical Practices.

22 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…

23 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?

24 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…

25 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

26 Standards for Mathematical Practice
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Refer participants to the handout with the description of the CCSS Standards for Mathematical Practices.

27

28 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?” AND also suggest the types of mathematical activities they will need to engage in to learn mathematics content. We will see examples of both today.

29 Standards for Mathematical Practice
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.” Help participants to understand that the Practices are dense statements that describe what students need to know…

30 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 # Pattern # Pattern #4 Draw pattern 4 next to pattern 3. How many white buttons does Gita need for Pattern 5 and Pattern 6? Explain how you figured this out. How many buttons in all does Gita need to make Pattern 11? Explain how you figured this out. 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? Let participants know that we will use this task to explore the CCSS Content and Practice Standards. First we will do the task and discuss it, then we will look to see how a teacher used this task in his fifth grade. Pass out copies of the Button Task to teachers. Consider modeling the pattern on a document camera with counters as you preview the four questions, then move to the next slide for directions on how to proceed with the task. Make graph paper, colored counters or square tiles available to teachers to use as they work on the task.

31 Button Task Individually complete parts 1 - 3.
Then work with a partner to compare your work. Look for as many ways to solve part 3 as possible. Which mathematical practices are needed to complete the task? Have participants follow the directions on the slide, working first individually, then in pairs on the Button Task. Whole Group Discussion: Depending on available time, consider selecting two or three papers with interesting solution strategies and or representations to share with participants before you begin the discussion of question 3. Don’t linger on the discussion of the task solution strategies, remember the focus of today’s session is the Standards for Mathematical Practice so save time for that part of the discussion. Chart the comments from participants regarding both the mathematical content and practices needed to successfully complete the task. You may also what to push the conversation by asking which elements of the task and/or the way the task was facilitated, triggered students to use specific practices.

32 www.Inside Mathematics.org
A reengagement lesson using the Button Task Francis Dickinson San Carlos Elementary Grade 5 Source of the video is the website Inside Mathematics. Development of the website was funded by the Noyce Foundation with a grant to the Silicon Valley Mathematics Initiative in California. The Noyce Foundation and the Silicon Valley Mathematics Initiative have graciously granted NCSM permission to utilize the website for this resource. 1. First select and show approximately 2 minutes segment from the Lesson Planning Video Clip beginning at 3:00 and ending at 4:49 This segment shows the teacher describing the lesson he has designed and his mathematical goals are for his students. Look for notes about the teacher, his classroom, and school on the website. 2. Next, distribute the samples of student work from Learner A and Learner B provided with this package. Allow teachers to look over the two samples of student work (Learner A and Learner B) and consider the nature of mathematics content and the mathematical practices students might be engaged in as they complete this task. Next, show approximately 2 minutes segment from the Lesson 2 Video Clip beginning at 0:00 minutes and ending at 1:50 minutes. In this segment participants will see the teacher launch the task. Next share a segment or two of students trying to make sense of the sample student work. The video segments from 3:04 to 4:40 showing two girls working on the task and another segment from 5:40 to 6:07 with two boys are nice samples. Again, the question for participants to consider as they watch the video is the nature of mathematics content and the mathematical practices students are engaged in as they complete this task. 4. Finally, show approximately 2 minutes segment from the Closure Video Clip beginning at 0:00 minutes and ending at 2:00 minutes. If you have a bit more time in the session you may want to play this video all the way to the end for a total of approximately 5:10 minutes. Then proceed to the focus questions on the next slide.

33 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.

34 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.

35 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? 1. Preview the focus questions with teachers, then let them consider them individually before you faciliatate a whole group discussion.

36 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

37 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…

38 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

39 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.

40 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.

41 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.

42 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

43 Compare the Two Versions of the Handshake Problem
How are the two versions of the handshake task the same? How are they different? Make the point that version 2 is an algebrafied version of the original task. PUSH on what it asks students to do – look for patterns, find a relationship, justify your approach. Blanton & Kaput, 2003

44 Cognitive Level of Tasks
Lower-Level Tasks (e.g., Handshakes Version 1) Higher-Level Tasks (e.g., Handshakes Version 2) Page 269 Stein and Smith -- example of task at each of the four levels. The Quasar Project

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

46 Higher-Level Tasks Procedures with connections Doing mathematics
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: The decimal part of area that is shaded; The fractional part of area that is shaded.

47 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?

48 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:

49 Insidemathematics.org

50 Insidemathematics.org

51 “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 Page 72 provides a simple method for doing this. Start with a typical numerical task and generalize it to a sequence of problems, each of which can be solved by an accompanying number sentence. The result is a sequence of number sentences that can be used to determine a generalized statement that captures the entire sequence.

52 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

53 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? As a group ALGEBRAFY IT. Blanton & Kaput, 2003

54 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

55 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.

56 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)

57 Grade Level Standards

58 Progressions within and across Domains
Number and Operations ―Fractions Algebra Expressions and Equations The Number System Operations and Algebraic Thinking Number and Operations―Base Ten K- 5 6-8 High School Daro, 2010

59 Key Advances Operations and the problems they solve
Properties of operations: Their role in arithmetic and algebra Mental math and “algebra” vs. algorithms Units and unitizing Unit fractions Unit rates Defining similarity and congruence in terms of transformations Quantities-variables-functions-modeling Number-expression-equation-function Modeling Daro, 2010

60 Write a word problem that could be modeled by
a + b = c a × b = p

61 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?

62 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?

63 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?

64 Common Addition and Subtraction Situations

65 Common Addition and Subtraction Situations

66 Common Multiplication and Division Situations

67 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?

68 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?

69 Key Advances Operations and the problems they solve
Properties of operations: Their role in arithmetic and algebra Mental math and “algebra” vs. algorithms Units and unitizing Unit fractions Unit rates Defining similarity and congruence in terms of transformations Quantities-variables-functions-modeling Number-expression-equation-function Modeling Daro, 2010

70 Mental Math and “Algebra” vs. Algorithms
= = 29 x 12 =

71 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 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.

72 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 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.

73 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 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.

74 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?

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

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

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

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

79 Key Advances Operations and the problems they solve
Properties of operations: Their role in arithmetic and algebra Mental math and “algebra” vs. algorithms Units and unitizing Unit fractions Unit rates Quantities-variables-functions-modeling Number-expression-equation-function Modeling Practices Daro, 2010

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

81 Units are things you count
Objects Groups of objects 1 10 100 ¼ unit fractions Numbers represented as expressions Daro, 2010

82 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 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

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

84 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

85 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

86 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.

87 Fraction Equivalence Grade 5:
Use equivalent fractions to add and subtract fractions with unlike denominators

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

89 Key Advances Operations and the problems they solve
Properties of operations: Their role in arithmetic and algebra Mental math and “algebra” vs. algorithms Units and unitizing Unit fractions Unit rates Defining congruence and similarity in terms of transformations Quantities-variables-functions-modeling Number-expression-equation-function Modeling Daro, 2010

90 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

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

92 Thank You! mathedleadership.org


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