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Accentuate the negative without eliminating the positives Beth Schefelker Bridget Schock Connie Laughlin Hank Kepner Kevin McLeod October 26, 2012

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Sharing of Professional Practice Turn and Talk Select one of the problems you did with students: and 135 – 97 Share your successes and challenges of capturing student thinking. What did you learn about student use of representations? Number line Associating decomposed and recomposed numbers

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CCSSM Connection Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.3. Apply properties of operations as strategies to add and subtract. 2 Examples: If = 11 is known, then = 11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition.) 1.OA.4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.

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Content Goals Explore how the number system expands across the grades and how operations are extended in such a way that the properties of operations remain unchanged. Analyze and solve mathematical modeling situations and/or problems using single and multi-step word problem structures. Extend understanding of whole number operations to fractions, ratios and proportions. Focus on Math Practice Standards 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.

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Learning Intention & Success Criteria Learning Intention We are learning to extend the operations of addition and subtraction as the number system expands. Success Criteria We will be successful when we can use addition and subtraction of integers to solve real world problems.

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Importance of Context With integers, students often get confused as to which number is bigger or which direction they are moving when they are doing operations, so having a context is particularly important. VdW, 479

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Avoidance of Context In what ways do we skirt around the use of everyday language to avoid using negative numbers? Simpson shot 2-under on the day, finishing the tournament at 1-over, one stroke ahead of Michael Thompson and Graeme McDowell U.S. Open

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Common Contexts for Integers Linear Representations Altitude Timelines Quantity Representations Golf scores (above/below par) Debit/Credit

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The Bus Problem I got on the Center Street bus at Sherman going east. I counted 15 people on the bus. At 35 th Street some people got on and some people got off. I couldnt see exactly how many people got off and on, but when the bus left 35 th Street, there were 10 people on the bus. I wondered what could have happened? How many people could have gotten off? How many could have gotten on?

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The Next Day on the Bus I got on the Center Street bus at Sherman going east. I counted 26 people on the bus. At 35 th Street some people got on and some people got off. The bus continued to 27 th Street. Again, I couldnt see exactly how many people got off and on, but when the bus left 27 th Street there were 14 people on the bus. What could the net changes have been at 35 th Street and 27 th Street?

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Important Mathematical Ideas Consider the work displayed on the charts. What were some of the important mathematical ideas that surfaced as you made sense of the bus problem and represented your thinking?

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CCSSM Connection Apply and extend previous understanding of numbers to the system of rational numbers. 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real- world contexts, explaining the meaning of 0 in each situation. Apply and extend previous understanding of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

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Kevins Joke of the Day A biologist, a physicist, and a mathematician are sitting on a bench looking at the house across the street. They see two people walk into it, and, some time later, see three people walk out. The biologist says, "They must have reproduced while they were inside." The physicist says, "No, our initial observation must have been in error." Finally, the mathematician says, "If one more person goes inside, then the house will be empty."

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Learning Intention & Success Criteria Learning Intention We are learning to extend the operations of addition and subtraction as the number system expands. Success Criteria We will be successful when we can use addition and subtraction of integers to solve real world problems.

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Making Connections How did the earlier work we did with addition and subtraction support our work with integers?

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Professional Practice Use the textbooks at your school Examine the lessons for addition and subtraction of integers. What contexts are used? What suggestions would you offer to enrich the lesson? Create a situation that could be used as a lesson or a CABS.

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