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Practicing the Standards for Mathematical Practice (SMP) Susie W. Håkansson California Mathematics Project

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Overview of Session The CDE Professional Learning Module Series K-12 SMP Module Outline of K-12 SMP Module Inside the K-12 SMP Module Conclusion

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CALIFORNIA DEPARTMENT OF EDUCATION Tom Torlakson, State Superintendent of Public Instruction Professional Learning Module Series Overview of the Common Core State Standards for California Educators

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The Common Core State Standards Professional Learning Series Overview of the Common Core State Standards for California Educators Mathematics: Kindergarten through Grade Eight Learning Progressions Mathematics: Kindergarten through Grade Twelve Standards for Mathematical Practice English Language Arts: Informational Text—Reading 4 | California Department of Education

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CALIFORNIA DEPARTMENT OF EDUCATION Tom Torlakson, State Superintendent of Public Instruction Mathematics: Kindergarten through Grade 12 Standards for Mathematical Practice (K-12 SMP) Professional Learning Module Series

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K-12 SMP Module: CMP Development Team Susie W. Hakansson Patrick Callahan Kathlan Latimer Sheri Willebrand Diane Kinch Joan Easterday Kyndall Brown Carol Cronk Joanne Rossi Becker Tsai-Tsai O-Lee Ginny Wu Patricia Dickenson

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Welcome and Overview Module Overview: Goals and Organization Thinking about Thinking Background of the CCSS for Mathematics Pre-Assessment

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Goal of Module Throughout the module, the overarching goals of the K-12 SMP module is two-fold: Deepen the teachers’ understanding of the SMP Support the learning of all students

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CCSS Mathematical Practices REASONING AND EXPLAINING 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others MODELING AND USING TOOLS 4. Model with mathematics 5. Use appropriate tools strategically SEEING STRUCTURE AND GENERALIZING 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning OVERARCHING HABITS OF MIND 1. Make sense of problems and persevere in solving them 6. Attend to precision

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Overview of Units Unit 1: Teaching and Learning the Standards for Mathematical Practice Unit 2: Overarching Habits of Mind (MP1 and MP6) Unit 3: Reasoning and Explaining (MP2 and MP3) Unit 4: Modeling and Using Tools (MP4 and MP5) Unit 5: Seeing Structure and Generalizing (MP7 and MP8) Unit 6: Summary and Next Steps

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Unit 1: Teaching and Learning the SMP Observing Students Two Types of Mathematics Standards Interaction of the Practice Standards and Content Standards Meeting the Needs of All Students Looking Ahead

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Students at Work You will see a video of a 2 nd Grade EL class engaged in mathematics. Here are some questions to think about as you watch the video: What do you see the students doing? In what ways are the students engaged? Describe your observations of students’ thinking, problem solving and interactions.

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Pre Self-Assessment Survey Take a few minutes to complete the Pre Self- Assessment Survey. This will give you an indication of your current comfort level and confidence in supporting all students to be successful in the eight SMP. We will ask users to complete this Self-Assessment Survey after they have completed the module. We expect huge increases.

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Points of Intersection Expectations that begin with the word ‘understand’ are often especially good opportunities to connect the practices to the content. - CCSS for Mathematics, p.8

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Grain Size It is not expected that a single problem would engage students with all eight practice standards, even a lesson may be too small to accomplish that. - Phil Daro, CCSS for Mathematics lead writer

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Unit 2: Overarching Habits of Mind (MP1 and MP6) Unpacking MP1 and MP6 Sense Making and Mindsets Student Self-Efficacy and Perseverance Attending to Precision Summary and Reflection

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Students Attending to Precision A student is asked if 5 = 5 is true. The student says no.

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A student is asked if the statement = is true. The student says it is false. Students Attending to Precision

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A student is asked if 8 = and = 8 are the same number sentences. Her answer is based on whether it has to make sense to kids or if it has to be what the student perceives the teacher wants. Students Attending to Precision

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Reflection There is something off in the first two of these students’ understanding of the equal sign. Why do they say the statements are not true? The third student bases her answer on whether or not it has to make sense or be what the teacher wants to hear. Why do you think she says this?

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Unit 3: Reasoning and Explaining (MP2 and MP3) Unpacking MP2 and MP3 Beginning to Reason: Definitions and Conjectures Explaining and Justifying Identifying Flaws in Reasoning Making Arguments More Viable Summary

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Beginning to Reason It is important in mathematics to give clear and logical explanations. Please take a few minutes to write down your ideas What is the definition of an even number? Explain why the sum of two even numbers is always even.

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Beginning to Reason Over 300 students from grade five through high school geometry responded to these prompts. Consider the clarity and logic of the following student generated definitions. Does the student’s grade level impact the logic and/or clarity?

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5 th Grade Students’ Responses What is the definition of an even number? A number that ends in 2, 4, 6, 8, 0. If you count by 2s, you will eventually make it to that number. It can be split in half equally. To find out which ones are even and which ones are odd, I think this one … 2, 4, 6, 8. Every other one is going to be even. To find out which ones are even and which ones are odd, I think this one … 2, 4, 6, 8. Every other one is going to be even.

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5 th Grade Student’s Response What is the definition of an even number? 4 is an even number because if you divide it by 2, it will have an equal set of numbers (see diagram below)

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Middle School Students’ Responses What is the definition of an even number? The definition of an even number is like, they are multiplying by 2, because it goes to 2, 4, 6, 8 and etc. That’s what I like about math. You skip the odd number and that’s the even number You skip the odd number and that’s the even number A number that can be evenly divided by half

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High School Students’ Responses What is the definition of an even number Usually not a prime number except two, and is not odd. An even number, such as two, can always be divisible and stays a composite number. The pattern of all numbers is odd, even, odd, even, etc. So every other number is even. If a number ends in an even number (2, 4, 6, 8 and technically 0) that number is automatically even. Because when you go on a date, you go with another person and there’s two of you. And if you go on a double date, there’s four of you. But if there is a third wheel, because their date didn’t show up that’s just awkward and odd … no pun intended.

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Reflection Discuss what argument strategies (e.g., example, counter-examples, non-examples) are evident in the student definitions. What do you think should be included in a definition to be considered complete and clear?

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Unit 4: Modeling and Using Tools (MP4 and MP5) Unpacking MP4 and MP5 Introduction to Modeling with Mathematics (MP4) Modeling with Mathematics at the Different Grade Spans Introduction to Using Appropriate Tools Strategically (MP5) Use of Tools at Different Grade Spans Summary and Reflection

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PISA Rubric Three Levels Level one problems are those that are directly translatable from a context. An example is a simple word problem from which students can formulate an equation.

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PISA Rubric Three Levels Level two problems are those where a model can be modified to satisfy changed conditions. Such problems allow students to study patterns and relationship between quantities, and represent these patterns and relationships using words, numbers, symbols and pictures. These can be problems that have multiple solution strategies, but usually have only one correct solution.

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PISA Rubric Three Levels Level three problems have no predetermined solution. Such problems require students to make assumptions about the context, develop a strategy to solve the problem, check their answers, present results, and possibly revise their solution strategies and begin the process over again.

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Modeling Different Grade Spans Grades K-2: Connie has 13 marbles. Juan has 5 marbles. How many more marbles doe Connie have than Juan? Ellen walks 3 miles an hour. How many hours will it take her to walk 15 miles? The giraffe in the zoo is 3 times as tall as the kangaroo. The kangaroo is 6 feet tall. How tall is the giraffe? These are examples of Level 1 modeling tasks.

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Modeling Different Grade Spans Grades 3-5: Lamar wants a new toy truck that sells for $25. Lamar has $3 dollars now. Create a plan that would help Lamar buy his truck three weeks from today. This is an example of a Level 3 modeling task.

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Modeling Different Grade Spans Grades 6-8: The physicist Enrico Fermi enjoyed posing questions like the ones below to his students and colleagues. What is the weight of all the trash produced in your house in a year? What do you think is the volume of gasoline your car uses in a year? How does this compare to the volume of liquid (water, soda, coffee, etc.) you drink in a year? These are examples of Level 3 modeling tasks.

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Modeling Different Grade Spans Grades 9-12: Suppose a building has 5 floors (1-5), occupied by offices. The ground floor (0) is not used for business purposes. Each floor has 80 people working on it, and there are 4 elevators available. Each elevator can hold 10 people at one time. The elevators take 3 seconds to travel between floors and average 22 seconds on each floor when someone enters and exits. If all of the people arrive at work at about the same time and enter the elevator on the ground floor, how should the elevators be used to get the people to their offices as quickly as possible? This is an example of a Level 3 modeling task.

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Modeling Different Grade Spans Grades 9-12: Note: To make this problem more open-ended, allow students to decide how many people work on each floor, the times when people arrive in the morning, and how long the elevator takes to travel between floors, as well as how long it remains on each floor.

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Unit 5: Seeing Structure and Generalizing (MP7 and MP8) Unpacking MP7 and MP8 Structure, Repeated Reasoning, and Generalization Making Sense of a Growing Pattern Geometry Examples of Structure and Generalization Performance Tasks and Student Work Summary and Reflection

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Consecutive Sums Some numbers can be written as a sum of consecutive positive integers. For example: 6= =4+5+6 = Which numbers have this property? Explain. Let’s look at what might be expected of students at each grade span when working on this problem.

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Grades K-2 Write the first 10 numbers as a sum of other numbers. Which of these sums contain only consecutive numbers?

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Grades 3-5 Some numbers can be written as a sum of consecutive positive integers. For example: 6= =4+5+6 = In small groups, find all numbers from that can be written as a consecutive sum. Look for patterns as you work. Conjecture which numbers can and which cannot be expressed as a consecutive sum. How can some of the sums be used to find others?

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Grades 6-8 Some numbers can be written as a sum of consecutive positive integers. For example: 6= =4+5+6 = Which numbers have this property? Explain. Sally made this conjecture. “Powers of two cannot be expressed as a consecutive sum.” Agree or disagree and explain your reasoning.

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Grades 9-12 Some numbers can be written as a sum of consecutive positive integers. For example: 6= =4+5+6 = Exactly which numbers have this property? When investigating this problem, Joe made the following conjecture: “A number with an odd factor can be written as a consecutive sum and the odd factor will be the same as the number of terms.” Agree or disagree with this statement and explain your reasoning.

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Reflection For each of the grade spans, how will understanding the structure of the number system support students as they work on this problem? How will your students use repeated reasoning?

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Unit 6: Summary and Next Steps Summary Take Action: Planning the Next Steps

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Post Self-Assessment Survey This is the same survey that you completed at the beginning of this session. You have only seen a glimpse of this module. Even with this limited introduction, we hope that you have gained some understanding of the SMP.

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End of Module Post-AssessmentGlossaryResourcesAcknowledgements Module Evaluation

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Going Online Go to Select California’s Common Core State Standards (right side of page) Then select CCSS Professional Learning Modules On the top right is the module list.

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GSDMC Student Awards You are invited to the TODOS San Diego Area Student Awards beginning at 5:00 PM this afternoon in the Starboard Room. Students, their families, and nominating teachers will be recognized. Refreshments will be served. Support for this annual heartwarming event is being provided by TODOS, GSDMC, and Texas Instruments. You can recommend your students (K-12) for next year’s awards!

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Speaker Evaluation Strongly Disagree Text your message to this Phone Number: Strongly Agree Disagree NeutralAgree ___ ___ ___ ___________ ” Speaker was well- prepared and knowledgeable Speaker was engaging and an effective presenter Session matched title and description in program book Other comments, suggestions, or feedback Example: 545 Great session! ” “ 31165

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Thanks! For more information, contact : Susie W. Håkansson

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