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North Country Inservice HS Mathematics Common Core State Standards Mathematics Practice and Content Standards Day 1 Friday, October 19, 2012 Presenter: Elaine Watson, Ed.D.

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Introductions Share What feeds your soul personally? What is your professional role? What feeds your soul professionally?

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Volunteers for Breaks I need volunteers to remind me when we need breaks! Every 20 minutes, we need a 2-minute movement break to help our blood circulate to our brains. Every hour we need a 5-minute bathroom break.

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Formative Assessment How familiar are you with the CCSSM?

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Setting the Stage Dan Meyers TED Talk Math Class Needs a Makeover Go to link: watsonmath.com North Country High School Math Inservice October 19, 2012

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CCSSM Equally Focuses on Standards for Mathematical Practice Standards for Mathematical Content Same for All Grade Levels Specific to Grade Level

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8 Practice Standards Look at the handout SMP Lesson Alignment Template For an electronic copy to use later, go to watsonmath.com North Country High School Math Inservice October 19, 2012

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Standards for Mathematical Practice Describe ways in which student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity

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Standards for Mathematical Practice Provide a balanced combination of Procedure and Understanding Shift the focus to ensure mathematical understanding over computation skills

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Standards for Mathematical Practice Students will be able to: 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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Video of NYC High School Piloting the CCSS Watch first 5 minutes on Math See link in watsonmath.com

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Standards for Mathematical Practice Some of the following slides on the Practice Standards have been adapted from slides presented in several online EdWeb Webinars in February through May 2012 discussing that focused on the Practice Standards by Sara Delano Moore, Ph.D.

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The 8 Standards for Mathematical Practice can be divided into 4 Categories Overarching Habits of Mind of a Mathematical Thinker (# 1 and # 6) Reasoning and Explaining (# 2 and # 3) Modeling and Using Tools (# 4 and # 5) Seeing Structure and Generalizing (# 7 and # 8)

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The 8 Standards for Mathematical Practice are fluidly connected to each other. One action that a student performs, either internally or externally, when solving a problem can take on characteristics from several of the 8 Practice Standards.

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Overarching Habits of Mind of a Mathematical Thinker 1.Make sense of problems & persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments & critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. 6. Attend to precision. Look for & make use of structure. Look for & express regularity in repeated reasoning.

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Start with Good Problems Characteristics Example from Illustrative Mathematics (F-BF.A.1.a, F-IF.B.4, F-IF.B.5 ) Context relevant to students Incorporates rich mathematics Entry points/solution pathways not readily apparent Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

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Make Sense of Problems (part I) Mathematically proficient students Explain the meaning of the problem to themselves Look for entry points to the solution Analyze givens, constraints, relationships, goals

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Mikes Canoe Trip Explain the meaning of the problem Entry points Givens, constraints, relationships, goals Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

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Persevere in Solving Them Mathematically proficient students. Plan a solution pathway Consider analogous cases and alternate forms Monitor progress and change course if necessary

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Persevere in Solving Them It's not that I'm so smart, it's just that I stay with problems longer. - Albert Einstein

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Mikes Canoe Trip Possible solution pathways/strategies Consider analogous cases & alternate forms Monitor progress and change course if needed Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

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Make Sense of Problems (part II) Mathematically proficient students Explain correspondence and search for trends Check their answers using alternate methods Continually ask themselves, Does this make sense? Understand the approaches of others

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What can teachers do? Select rich mathematical tasks Connected to rigorous mathematics content Resources for rigorous mathematical tasks can be found on www.watsonmath,.com North Country High School Math Inservice October 19, 2012 Illustrative Mathematics MARS Tasks Inside Mathematics 3 Act Math Tasks Dan Meyer Andrew Stadel Others

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What can teachers do? Ask good questions Is that true every time? Explain how you know. Have you found all the possibilities? How can you be sure? Does anyone have the same answer but a different way to explain it? Can you explain what youve done so far? What else is there to do?

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What can teachers do? Communicate to students the final solution to a problem is less important than the skills they develop during the process of finding the solution. The skills developed in working through the process are long-lasting skills that will serve them in other areas of life.

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Attend to Precision In Vocabulary In Mathematical Symbols In Computation In Measurement In Communication

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How is the teacher ensuring that students are making sense of problem and attending to precision? See Video: Discovering Properties of Quadrilaterals on Watsonmath.com

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Challenges to Precision Vocabulary Similar, adjacent Mathematical Symbols == Computation and Measurement Accurate computation Estimating when appropriate Appropriate units of measure Communication Formulate explanations carefully Make explicit use of definitions

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Mikes Canoe Trip Vocabulary Mathematical Symbols Computation & Measurement Communication Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

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Reasoning and Explaining Make sense of problems & persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments & critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for & make use of structure. Look for & express regularity in repeated reasoning.

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2. Reason abstractly & quantitatively Mathematics in and out of context Working with symbols as abstractions Quantitative reasoning requires number sense Using properties of operations and objects Considering the units involved Attending to the meaning of quantities, not just computation

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Construct viable arguments Understand and use assumptions, definitions, and prior results Think about precision (MP6) Make conjectures and build logical progressions to support those conjectures Not just two column proofs in high school Analyze situations by cases Positive values of X and negative values of X Two-digit numbers vs three-digit numbers Recognize & use counter-examples Maximum area problem

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How do we help children learn how to reason and explain? Provide rich problems where multiple pathways and solutions are possible Celebrate multiple pathways to the same answer Monitor students as they work to choose approaches to share with the whole class Provide plenty of opportunities for students to talk to each other Recognize the difference between a viable argument and opinion Provide scaffolds for thembut not too many!

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How do we help children learn how to reason and explain? Provide plenty of opportunities for students to talk to each other. Create a classroom culture in which all students feel safe to express their thinking Make sure students recognize the difference between a viable argument and an opinion Create a classroom culture where its safe to critique each other in a respectful way Provide scripts (sentence frames)for them to use such as those from Accountable Talk (see resources on watsonmath.com)

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Teacher Moves in Group Discussion By scaffolding students' responses and contributions, teachers can quickly make a difference in the level of rigor and productivity in classroom talk. Teachers can bring everyone's attention to a key point By "marking" a student's contribution "that's an important point By asking the student to repeat the remarkor restating it in their own wordsand indicating why the point is important. From ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine OConnor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

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Teacher Moves in Group Discussion If someone asks a thought-provoking question, the teacher might turn the question back to the group Good question, what do you think? as a way to encourage students to push their own thinking. By citing facts and posing counterexamples, teachers can challenge students to elaborate or clarify their arguments "but what about...? From ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine OConnor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

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Using Properties to See Structure Properties of Operations Commutative Property of multiplication and addition Distributive Property of multiplication over addition Identity Property of multiplication and addition Properties of Equality Transitive Property (if a=b and b=c, then a=c) Properties of Inequality Exactly one of the following is true: a > b, a = b, a < b

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Van Hiele Levels of Geometric Thinking shows how students increasing see more structure in shapes as they mature mathematically Level 0 (Pre-recognition) Students do not yet see shapes clearly enough to compare with prototypes Level 1 (Visualization) Students understand shapes by comparing to prototypes Students do not see properties Students make decisions based on perception, not reasoning Level 2 (Analysis) Students see shapes as collections of properties Students do not identify necessary and sufficient properties

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Van Hiele Levels (cont) Level 3 (Abstraction) Students see relationships among figures and properties Students can create meaningful definitions and reason informally Level 4 (Deduction) Students can construct proofs Students understand necessary & sufficient conditions Level 5 (Rigor) Students can understand non-Euclidean systems Students can use indirect proof and formal deduction

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Look for and express regularity in repeated reasoning Focus on computation here 1 3 = Examining points on a line and slope (1,2), m=3 (y-2)/(y-1) = 3 Attending to intermediate results

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Look for and express regularity in repeated reasoning These practices are about seeing the underlying mathematical principles and generalizations. These practices have more subtlety, and are often hard to distinguish between each other.

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Pyramid of Pennies Work through Dan Meyers Pyramid of Pennies Problem See link on watsonmath.comwatsonmath.com Use your SMP Lesson Planning Template and fill out what Math Practices you used when solving the problem.

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Content Standard Activity Work with a partner and on your own laptop, go to Illustrative Mathematics on watsonmath.com Go to the HS Standards Check out the illustrations. Which SMPs would they reinforce? Which illustrations would you use in your classroom? Check out the modeling standards (those with a star) Why do you think these standards were chosen as modeling standards?

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Last But Not Least Formative Assessment Watch the two videos on watsonmath.com:watsonmath.com My Favorite No Daily Assessment with Tiered Exit Cards

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Resources For resources used and/or discussed in this presentation, go to: www.watsonmath.com North Country High School Math Inservice October 19, 2012 Check out other resources available on watsonmath.com: archives of past posts resource links in the right hand column Contact Elaine at elaine.watson0729@gmail.comelaine.watson0729@gmail.com