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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Shifts that Support Teaching, Learning, and Monitoring Student Progress in Mathematics

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Session Objective Understand the role of the eight Standards for Mathematical Practice in Common Core aligned instruction. Recognize evidence of the Standards for Mathematical Practice in classroom instruction. Refine tools for grading and communicating progress with content and practice standards to familes.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Group Tasks Read the MP and A Story of Ratios examples As a group, discuss: 1. Restating the MP in student friendly terms 2. What student behaviors does the MP elicit? 3. Teaching practices that support the MP’s 4. Teaching practices that suppress the MP’s Chart the results of your discussion

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Sample Chart MP.1: Make sense of problems and persevere in solving them Student Friendly : Don’t give up on tricky problems Content: Multiple approaches to problem solving, toolbox of strategies from which to draw Engagement: Persistence, risk-taking Support Suppress xxxxx

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 1: Make sense of problems and persevere in solving them. When presented with a problem I can make a plan, carry out my plan, and evaluate its success. Before During After Explain the problem to myself Persevere Check Organize information Monitor my work Change my plan if it isn’t working out Ask myself “Does this make sense?” Have I solved a problem like this before? Is my answer correct? How do my representations connect to my algorithm? What is the question I need to answer? What is given? What is not given? What are the relationships between known and unknown quantities? What prior knowledge do I have to help me? Evaluate What worked? What didn’t work? What other strategies were used? How was my solution similar to or different from my peers?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP1 in action – Grade 7 Jake Disston works with his middle school students to deepen their understanding of the connections between different representations of functions—graphs, t-charts, and equations. In this clip, he begins the period by challenging his students to work at their tables to group the cards they have in front of them, creating as many different groupings as they can, and devising language to describe each group. Disston's colleague Jesse Ragent then asks the students to do a "matching game." He passes out sets of tables and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set."

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 2: Reason abstractly and quantitatively. I can use reasoning habits to help me contextualize and decontextualize problems. Contextualize Decontextualize I can take numbers and put them in a real-world context. I can take numbers out of context and work mathematically with them. Reasoning habits include: 1) creating an understandable representation of the problem solved, 2) considering the units involved, 3) attending to the meaning of the quantities, and 4) using properties to help solve problems.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 2 in action – Grades 5/6 Fran Dickinson leads a lesson on numeric patterning, helping students to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson describes the importance of individual think time before he asks his students to discuss the relative strength of two different approaches to a patterning task. One pair discusses the numbers within the sample strategy, and Dickinson repeats back their conversation to the whole group, telling his students, “I’ve heard two really good questions about Learner B’s strategy. One was, what are all these 3’s? and Kelcey’s question was, what about this 4? Where’s the 4 coming from?” This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).standard 1 (make sense of problems and persevere in solving them)

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 3: Construct viable arguments and critique the reasoning of others. I can make conjectures and evaluate the appropriateness of peer’s thinking. I can construct, justify, and communicate argument by I can critique the reasoning of others by considering context using examples and non- examples using objects, drawings, diagrams and actions listening comparing arguments identifying flawed logic asking questions to clarify or improve arguments

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP3 in action – 6 th grade Jacob Disston leads a lesson on connections between ideas about equations, inequalities, and expressions, helping students to use mathematical vocabulary for a purpose to describe, discuss, and work with these symbol strings. In this clip, his students have grouped themselves by the common features of the symbol strings. Disston then asks the students to explain their groupings, saying “Oliver, you had another whole category. Who are you standing with? … Why are you guys saying you guys are similar? You want to tell us?” He involves the whole class in evaluating the groups’ justifications. This clip is also indicative of standard 1 (make sense of problems and persevere in solving them) and standard 6 (attend to precision).standard 1 (make sense of problems and persevere in solving them)standard 6 (attend to precision)

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 4: Model with mathematics I can recognize math in everyday life and use math I know to real world problems. I can make assumptions and estimate to make complex problems easier identify important quantities and use tools to show their relationships evaluate my answer and make changes if needed R epresent Math Using Concrete models Pictures Real-world situations Oral language Symbols

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP4 in action – Grade 7/8 Cecilio Dimas leads a lesson on making comparisons between three different financial plans, helping students use multiple representations of mathematical problems: verbal, tabular, graphical, and algebraic generalization. In this clip, Dimas connects to the prior day’s lesson, in which the class “started a conversation about the economic status of our world [and] about making responsible decisions when we’re spending our money.” His students share that they were to represent various DVD rental plans using verbal and tabular representations.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 5: Use appropriate tools strategically I know when to use certain tools to help me explore math and deepen my understanding. I have a math toolbox and I know HOW to use math tools know WHEN to use math tools can reason: “Was the tool I used appropriate for the situation?” can reason: “Did the tool allow me to find the correct answer?”

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP5 in action – Grade 9/10 Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, Humphreys circulates around groups of students as they use rulers, protractors, and sheets of paper to make different kinds of quadrilaterals and examine and “guarantee” their properties. A group of students debates whether or not one student’s assertion about the properties of a trapezoid hold in all circumstances, using the kite “sticks” to illustrate their points. 9th/10th grade first video This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).standard 1 (make sense of problems and persevere in solving them)

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 6: Attend to precision. I can use precision when solving problems and communicating my ideas. Problem Solving I can calculate accurately. I can calculate efficiently. My answer matches what the problem asked me to do – estimate or find the exact answer. Communicating I can SPEAK, READ, WRITE, and LISTEN mathematically. I can correctly use math symbols math vocabulary units of measure

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP6 in action – Grades 9/10 Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, she orients students to the task and explains how they are to communicate their ideas to one another in their group work. This clip is also indicative ofstandard 1 (make sense of problems and persevere in solving them).standard 1 (make sense of problems and persevere in solving them)

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 7: Look for and make use of structure. I can step back and see the connections between concepts and look closely to see the patterns. AlgebraGeometry For Example: Students recognize this expression as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real number x and y.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP7 in action – Grades Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In this clip, his students work together in a group, clarifying each other's process and thinking. The female students clarify accurate steps for the male student. They discuss how to use the x- and y-intercepts to support their process. They step back and ask themselves what they are seeking in their work.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP 8: Look for and express regularity in repeated reasoning. I notice when calculations are repeated. Then, I can find more efficient methods. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP8 in action – Grade 8 Jesse Ragent asks the students to do a "matching game." He passes out sets of tables and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set." He asks students to consider distinctions, characteristics, attributes as they make their grouping decisions. Students jump right in and start the matching quite eagerly. They are justifying their matches in different ways. Some are using the math vocabulary from the prior whole class discussion, while others are just matching points from the graph with the t-chart, or checking to see if an equation will generate the points on the chart or graph. Either way, students are making connections among these different representations in ways that are meaningful to them. In the clips excerpted it seems that the students are listening to one another and letting the mathematics sway the argument rather than the force of a personality.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios MP Sorting Activity Use the posters to sort the behaviors/actions into Mathematics Practice groups.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Chalk Talk What does this mean for Common Core aligned teaching and learning at my school/district?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Student Progress What does/doesn’t the report card tell you about the student as a mathematician? What questions does this raise for you? Discuss with your table group:

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Revisions What are 3-5 characteristics of a high quality tool for reporting student progress with Common Core aligned mathematics?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Moving Forward What do we need to do? What do they need to do? What do I need to do to help us/them get there?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Biggest Takeaway Turn and Talk: What questions were answered for you? What new questions have surfaced?

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