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+ CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices.

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+ Day 2 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices

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+ Welcome Back Activity Success, challenge, breakthrough Reflect on your experience using one of the assigned tasks in your classroom. Use separate post-it notes to capture your successes, challenges and/or breakthroughs Post on appropriate poster Read post-it notes on all posters and select one that resonates with you. Quick share of selected post-its.

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+ Reflection What is your current reality around classroom culture? What can you do to enhance your current reality?

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+ Outcomes: Day 2 Understand how to analyze student work with the Standards for Mathematical Practice and content standards. Understand the SBAC Assessment Claims 3 & 4 for Mathematics. Analyze and adapt a task with the integrity of the Common Core State Standards.

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Looking at Student Work Form small groups of 3 or 4 Each person selects 1 or 2 works samples to share with the group Follow the Collaboration Protocol to review student work samples and record information observations Inferences implications

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Collaboration Protocol-Looking at Student Work (55 minutes) 1. Individual review of student work samples (10 min) All participants observe or read student work samples in silence, making brief notes on the form Looking at Student Work 2. Sharing observations (15 min) The facilitator asks the group What do students appear to understand based on evidence? Which mathematical practices are evident in their work? Each person takes a turn sharing their observations about student work without making interpretations, evaluations of the quality of the work, or statements of personal reference. 3. Discuss inferences -student understanding (15 min) Participants, drawing on their observation of the student work, make suggestions about the problems or issues of students content misunderstandings or use of the mathematical practices. Adapted from: Steps in the Collaborative Assessment Conference developed by Steve Seidel and Project Zero Colleagues 4. Discussing implications-teaching & learning (10 min) The facilitator invites all participants to share any thoughts they have about their own teaching, students learning, or ways to support the students in the future. How might this task be adapted to further elicit students use of Standards for Mathematical Practice or mathematical content. 5. Debrief collaborative process (5 min) The group reflects together on their experiences using this protocol. Select one group member to be todays facilitator to help move the group through the steps of the protocol. Teachers bring student work samples with student names removed.

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Looking at student work

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What instructional strategies did you use with this lesson? What Standards for Mathematical Practices did you notice your students engaging in during this task? Using the SMP Matrix how would you describe the students in your classroom Homework Debrief….

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+ Applying your learning From previous homework: you found/created or adapted to make a rich task. Have you turned in your item for the Intel CCSS-M Item bank – digital is preferred, hard copy is better than not at all!!!!!

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+ BREAK TIME!!! 15 minutes….Go

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Assessment Claims for Mathematics Students can demonstrate progress toward college and career readiness in mathematics. Students can demonstrate college and career readiness in mathematics. Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency. Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Overall Claim (Gr. 3-8) Overall Claim (High School) Claim 1 Concepts and Procedures Claim 1 Concepts and Procedures Claim 2 Problem Solving Claim 2 Problem Solving Claim 3 Communicating Reasoning Claim 3 Communicating Reasoning Claim 4 Modeling and Data Analysis Claim 4 Modeling and Data Analysis

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Claim 2 – Problem Solving A.Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace B.Select and use tools strategically C.Interpret results in the context of the situation D.Identify important quantities in a practical situation and map their relationships. Claim 2: Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.

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Looking at SBAC Tasks: Depth of Knowledge and Mathematical Practices, two lenses What is the depth of knowledge of these tasks? Which mathematical practices do they promote?

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4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b

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Claim 3 – Communicating Reason A.Test propositions or conjectures with specific examples. B.Construct, autonomously, chains of reasoning that justify or refute propositions or conjectures. C.State logical assumptions being used. D.Use the technique of breaking an argument into cases. E.Distinguish correct logic or reasoning from that which is flawed, andif there is a flaw in the argumentexplain what it is. F.Base arguments on concrete referents such as objects, drawings, diagrams, and actions. G.Determine conditions under which an argument does and does not apply. Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.

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Looking at SBAC Tasks: Depth of Knowledge and Mathematical Practices, two lenses What is the depth of knowledge of these tasks? Which mathematical practices do they promote?

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3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size

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Focus and Coherence through the Major and Supporting Clusters of the CCSS-M Discuss with a partner which: Major, Supporting, and/or Additional Clusters are involved in the task

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+ Which Fraction Pair is Greater? Think like a fourth grader Give one or more reasons for the comparison Try not to use models or drawings DO NOT USE cross multiplication Elementary and Middle School Mathematics VandeWalle, 2013 p. 311

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+ Ways In Which To Compare 1. Same size whole (same denominator) - B&G 2. Same number of parts (same numerator) but different sized wholes – A,D,& H 3. More than/less than one-half or one – A,D,F,G, and H 4. Closeness to one-half or one – C,E,I,J,K, and L

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+ Exploring Equivalent Fractions atics.org/pages/fractions_pro gression atics.org/pages/fractions_pro gression

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Task Analysis Protocol Sheet

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Fraction Tracks Play Fraction Tracks (access online) Video

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+ How did this activity support the big idea of equivalent fractions Reasoning about the size of a unit fraction based on the meaning of a unit fraction Turn and talk with your partners and then we will share out with the group

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+ Exploring Adding Fractions atics.org/pages/fractions_pro gression atics.org/pages/fractions_pro gression

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Justify the truth of this statement in at least two different ways.

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+ Exploring Multiplying Fractions atics.org/pages/fractions_pro gression Semi-concrete explanation Multiplying part 2 Abstract explanation Multiplying part 1

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+ Read a story and explore the math Read Beasts of Burden from The Man who Counted by Tahan 1993 Rusty Bresser in Math and Literature describes three days of activities with fifth graders, based on this story

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Paper folding I planted half my garden with vegetables this summer. One third of the half that is vegetables is planted with green beans. What fraction of the whole garden is planted with green beans?

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Math Matters When two fractions are multiplied it is based on a fraction as an operator. A fraction is operating on another number and changes the other number. The use of the word of to multiply is based on the operator interpretation of fraction: when we multiply ½ and 8 we are taking one half of eight. p. 125 Math Matters Chapin and Johnson

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+ Finding a fraction of a fraction Conceptually – with no subdivisions Pictures – YES Algorithms - NO

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+ Finding a fraction of a fraction

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+ Exploring Dividing Fractions atics.org/pages/fractions_pro gression atics.org/pages/fractions_pro gression

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+ Dividing by one –half Shauna buys a three-foot-long sandwich for a party. She then cuts the sandwich into pieces, with each piece being 1/ 2 foot long. How many pieces does she get? Phil makes 3 quarts of soup for dinner. His family eats half of the soup for dinner. How many quarts of soup does Phil's family eat for dinner? A pirate finds three pounds of gold. In order to protect his riches, he hides the gold in two treasure chests, with an equal amount of gold in each chest. How many pounds of gold are in each chest? Leo used half of a bag of flour to make bread. If he used 3 cups of flour, how many cups were in the bag to start ? Illustrative Math Example

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VideoCathy Humphreys Defending Reasonableness: Division of Fractions 1 ÷ 2/3 Use the Standards for Mathematical Practice Matrix to reflect on where this classroom falls on the continuum and be ready to discuss any activities you could use to move this classroom forward on the scale

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Standards for Mathematical Practice Matrix

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Claim 4 – Modeling and Data Analysis A.Apply mathematics to solve problems arising in everyday life, society, and the workplace. B.Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. C.State logical assumptions being used. D.Interpret results in the context of a situation. E.Analyze the adequacy of and make improvement to an existing model or develop a mathematical model of a real phenomenon. F.Identify important quantities in a practical situation and map their relationships. G.Identify, analyze, and synthesize relevant external resources to pose or solve problems. Claim 4: Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.

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Looking at SBAC Performance Task: Depth of Knowledge and Mathematical Practices, three lenses What are the specific content standards in this performance task? What is the depth of knowledge of these tasks? Which mathematical practices do they promote?

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Planting Tulips DOMAINS: Operations and Algebraic Thinking, Number and OperationsFractions, Measurement and Data

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+ Using your materials modify or create a rich task or one that is more conceptually focused for your students

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+ Applying your learning From previous homework: you found/created or adapted to make a rich task. Have you turned in your item for the Intel CCSS-M Item bank – digital is preferred, hard copy is better than not at all!!!!!

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Where to Look for Resources for Mathematics RMC website: OPSI website: Sandys Math wiki:

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+ Reflection What is your current reality around classroom culture? What can you do to enhance your current reality?

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+ Wrap up Activity Evaluations Thank You!

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