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WELCOME TQ SUMMER 2011 WORKSHOP: PROPORTIONALITY JUNE 6-17, 2011

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GROUP NORMS Be an active learner Be an active learner Be an attentive listener Be an attentive listener Be a reflective participant Be a reflective participant Be conscious of your needs Be conscious of your needs and needs of others and needs of others

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Success in Algebra What mathematics do students need to know to be successful in Algebra I?

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Pre Test Work on test items individually. Work on test items individually. Write your participant number on the test. Write your participant number on the test.

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Our Focal Point for Proportional Reasoning Students will be able to identify, analyze, and represent proportional and non-proportional linear relationships, to transition from proportional models a/b = c/d to direct variation models y = kx.

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Understanding Proportionality What do students need to know and be able to do to understand proportionality? What do students need to know and be able to do to understand proportionality? Chart ideas and be prepared to share your ideas with the whole group Chart ideas and be prepared to share your ideas with the whole group

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Texas Response to Curriculum Focal Points for Grade 8 Mathematics Representing, applying, and analyzing proportionality Students extend their understanding of proportionality to include representations on a coordinate plane and applications, including proportional changes. TEA 2009 TEA 2009

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Fish is Fish Lets read a story by Leo Lionni. Read the story individually.

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Fish is Fish What 2 or 3 messages does this story convey to you? What 2 or 3 messages does this story convey to you? Discuss these at your table. Discuss these at your table. Select someone who will share at least one message from your table with the whole group. Select someone who will share at least one message from your table with the whole group.

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PRINCIPLES OF LEARNING Principle One: Building on Prior Knowledge Building on Prior Knowledge Principle Two: Understanding requires factual knowledge and conceptual framework Understanding requires factual knowledge and conceptual framework Principle Three: Metacognition

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Principle One Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.

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Principle Two Achieving goals in mathematical understanding requires teachers to develop students knowledge networks, address students learning paths, and use multiple methods. This suggests the importance of conceptual understanding, procedural fluency, and an effective organization of knowledge.

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Principle Three Metacognition, getting students to think about their thinking, can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.

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Atlantis 2008 Unit on Proportional Reasoning What is a unit? Structure of the unit Planning and Development of the Unit

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Unit Framework Pre- Assessment Post- Assessment Unit Map/ Learning Pathway Unit Project/ Introductory Activity Unit Objectives Language and Communication Prior Knowledge/ Pre-Lesson Generalization Debugging Application Extension Connection Unit Core Introductory Elements Applied ElementsCore Elements

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Outcomes Identify and incorporate the three Principles of Learning Understand proportional reasoning and connect to linear relationships Transition from a single lesson lens to a unit learning lens Experience the unit through teacher and student lenses

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Unit Pre-Lesson Student Hat: Complete the activity individually Student Hat: Complete the activity individually Write your answers on the extra copy Write your answers on the extra copy Consider the following as you work: Consider the following as you work: What content is new and what is familiar to you as a student? What content is new and what is familiar to you as a student?

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Unit Pre-Lesson Teacher Hat: Table group discussion What preconceptions might your students have? (misconceptions, good knowledge, and missing knowledge) What preconceptions might your students have? (misconceptions, good knowledge, and missing knowledge) How does the pre-lesson connect with the three principles? How does the pre-lesson connect with the three principles? How will the pre-lesson inform classroom instruction? How will the pre-lesson inform classroom instruction?

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Pre-Lesson: Teacher Notes Review the teacher notes. Highlight areas you want to remember. What else would you add to the teacher notes?

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Looking at Student Work Analyze the student work on the Analyze the student work on the Pre-Lesson Pre-Lesson What misconceptions did you find? What misconceptions did you find? How would you address these misconceptions? How would you address these misconceptions?

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Another Problem I went to the store and bought the same number of books as records. Books cost two dollars each and records cost six dollars each. I spent $40 altogether. Assuming that the equation 2B+6R=40 is correct, what is wrong if anything, with the following student reasoning? 2B+6R=40, since B=R, I can write 2B+6B=40, then 8B=40. This last equation says 8 books is equal to $40. So, one book costs $5.2B+6R=40, since B=R, I can write 2B+6B=40, then 8B=40. This last equation says 8 books is equal to $40. So, one book costs $5.

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REFLECTIONS In your memo book, list two or three advantages to having a pre-lesson. In your memo book, list two or three advantages to having a pre-lesson. How does this connect to principle one? (refer to summary pages in first tab of notebook) (refer to summary pages in first tab of notebook)

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Time for Reading From Words to Algebra: Mending Misconceptions From Words to Algebra: Mending Misconceptions by Jack Lochhead and Jose P. Mestre (Readings are in your binder under the Resources tab.)

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Mark your Reading ! Insight ? Questions Affirmation Affirmation

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From Words to Algebra: Mending Misconceptions Table Group Discussion: How does reading the article assist you in addressing misconceptions you identified in the student work?

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From Words to Algebra: Mending Misconceptions Table Group Discussion: How will this impact your teaching?

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BIG MATHEMATICAL IDEA Goal is to transition students from proportional models to direct variation models. Goal is to transition students from proportional models to direct variation models. rational numbers proportional relationships direct variation models linear functions rational numbers proportional relationships direct variation models linear functions

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BIG IDEAS - NCSM A ratio is a multiplicative comparison of quantities A ratio is a multiplicative comparison of quantities Ratios give relative sizes of quantities being compared Ratios give relative sizes of quantities being compared Ratios can be expressed as units by finding an equivalent ratio where the second term is one. Ratios can be expressed as units by finding an equivalent ratio where the second term is one.

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BIG IDEAS – NCSM, continued A proportion is a relationship between relationships. A proportion is a relationship between relationships. If two quantities vary proportionally, the ratio of corresponding terms is constant. If two quantities vary proportionally, the ratio of corresponding terms is constant. If two quantities vary proportionally, the constant ratio can be expressed in lowest terms ( a composite unit) or as a unit amount; the constant ratio is the slope of the related linear function. If two quantities vary proportionally, the constant ratio can be expressed in lowest terms ( a composite unit) or as a unit amount; the constant ratio is the slope of the related linear function. Ratio is a reflectively abstracted constant ratio. Ratio is a reflectively abstracted constant ratio.

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LEARNING PATHWAY

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REFLECTIONS In your memo book, describe one thing you will do differently to identify and address students misconceptions. In your memo book, describe one thing you will do differently to identify and address students misconceptions. Share an insight that you have identified in the past two days, that will help you clarify misconceptions.

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Atlantis Mission One Student Hat: Complete the activity Student Hat: Complete the activity and applications in groups of two and applications in groups of two Write your answers on the extra Write your answers on the extra copy. copy. Use your resources as needed. Use your resources as needed.

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Atlantis Mission One Teacher Hat What concepts and skills are used in this mission? What concepts and skills are used in this mission? What misconceptions might your students have? What misconceptions might your students have? How did the applications support student learning? How did the applications support student learning? How does the pre-lesson connect with mission one and the three principles? How does the pre-lesson connect with mission one and the three principles?

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Atlantis Mission Two Teacher Hat What concepts and skills are used in this mission? What concepts and skills are used in this mission? What misconceptions might your students have? What misconceptions might your students have? How does the Mission 1 connect with Mission 2 and the three principles? How does the Mission 1 connect with Mission 2 and the three principles?

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Mission One & Two Teacher Notes Take five minutes to review the teacher notes. Highlight areas you want to remember. What else would you add to the teacher notes?

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Reconnecting with Missions 1 & 2 As a teacher, what connections do you make in regard to mathematical content in the Mission 1&2 activities? As a teacher, what connections do you make in regard to mathematical content in the Mission 1&2 activities? How will these impact your teaching? How will these impact your teaching?

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Time for Reading Proportional Reasoning: Student Misconceptions and Strategies for Teaching

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Discussion Questions 1. Why do you think 90% of adults do not reason proportionally, according to Lamon 2007? 2. How are multiplicative situations different from additive situations? 3. What are the key steps to proportional reasoning? 4. What is the difference between ratio & rate?

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Implications for Teaching In what ways will the PCK tool on proportional reasoning impact your teaching? As a table group, discuss, chart, and post your ideas on the wall

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Atlantis Mission Three Student Hat: Complete the activity in groups of two Think about different ways Think about different ways students might solve the problems. students might solve the problems. Write your answers on the extra copy. Write your answers on the extra copy.

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Atlantis Mission Three Teacher Hat What concepts and skills are used in this mission? What concepts and skills are used in this mission? What misconceptions might your students have? What misconceptions might your students have? How does Mission Three connect with the three principles? How does Mission Three connect with the three principles?

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Mission Three Teacher Notes Take five minutes to review the teacher notes. Highlight areas you want to remember. What else would you add to the teacher notes?

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Reflections What are the conceptual ideas in Mission Three? Mission Three? In your memo book, answer: How might transitioning from a single lesson lens to a unit (sequence of lessons) learning lens impact teaching and learning in your classroom? How might transitioning from a single lesson lens to a unit (sequence of lessons) learning lens impact teaching and learning in your classroom?

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Time for Reading Proportional Reasoning: Student Misconceptions and Strategies for Teaching Pp. 7- 13

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Discussion Questions What are the common student error strategies in solving proportional reasoning tasks? What are the common student error strategies in solving proportional reasoning tasks? Does the way a problem is presented affect how student interact with the problem? Explain how. Does the way a problem is presented affect how student interact with the problem? Explain how. Could additive methods be used in proportional situation? Could additive methods be used in proportional situation? What are the most common correct strategies in solving proportional tasks? What are the most common correct strategies in solving proportional tasks?

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Mission Four Student Hat: Complete the activity in groups of two at your table Complete the activity in groups of two at your table Write your answers on the extra copy Write your answers on the extra copy

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Atlantis Mission Four Teacher Hat How is Mission Four different than the other missions? How is Mission Four different than the other missions? What different levels of thinking, cognitive demands, are required to complete Mission Four? What different levels of thinking, cognitive demands, are required to complete Mission Four? What are the common characteristics of proportional relationships? How do they appear in each representation? What are the common characteristics of proportional relationships? How do they appear in each representation?

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Teacher Notes Take five minutes to review the teacher notes. Highlight areas you want to remember. What else would you add to the teacher notes?

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Mission Five Student Hat: Complete the activity in groups of two at your table Complete the activity in groups of two at your table Write your answers on the extra copy Write your answers on the extra copy

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Atlantis Mission Five Teacher Hat What concepts and skills are used in this mission? What concepts and skills are used in this mission? What misconceptions might your students have? What misconceptions might your students have? How does Mission Five connect with the three principles? How does Mission Five connect with the three principles?

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Mission Six Student Hat: Complete the activity in groups of two at your table Complete the activity in groups of two at your table Write your answers on the extra copy Write your answers on the extra copy

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Atlantis Mission Six Teacher Hat What concepts and skills are used in this mission? What concepts and skills are used in this mission? What misconceptions might your students have? What misconceptions might your students have? How does Mission Six connect with the three principles? How does Mission Six connect with the three principles?

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What are Key Characteristics of a Proportional Thinker?

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Key Characteristics of a Proportional Thinker Knowing and understanding the idea of relationship Knowing and understanding the idea of relationship Ability to recognize multiplicative situations and distinguish it from additive situations Ability to recognize multiplicative situations and distinguish it from additive situations Ability to recognize and explain the difference between proportional (y=kx) and non-proportional (y=kx+b) situations Ability to recognize and explain the difference between proportional (y=kx) and non-proportional (y=kx+b) situations Ability to recognize and distinguish different types of proportionality: direct and inverse Ability to recognize and distinguish different types of proportionality: direct and inverse

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Ability to use proportionality as a mathematical model in real- world contexts Ability to use proportionality as a mathematical model in real- world contexts Knowing and use of the language of proportionality Knowing and use of the language of proportionality Understanding the concept of function to express the co- variation Understanding the concept of function to express the co- variation Knowing that the graph of a direct proportional situation is a straight line that passes through the origin Knowing that the graph of a direct proportional situation is a straight line that passes through the origin Knowing that the graph of non-proportional situation is a straight line intersecting the y axis b units from the origin Knowing that the graph of non-proportional situation is a straight line intersecting the y axis b units from the origin Knowing that the graph of an inversely proportional situation is a hyperbola Knowing that the graph of an inversely proportional situation is a hyperbola Understanding that k is the constant ratio in a direct proportional situations Understanding that k is the constant ratio in a direct proportional situations Understanding that k is the constant product in an inversely proportional situations Understanding that k is the constant product in an inversely proportional situations Key Characteristics of a Proportional Thinker (cont.)

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Metacognition and Self-Monitoring Restate Principle Three in your own words. What does it mean to you? Restate Principle Three in your own words. What does it mean to you? How does the sequence of Mission activities address Principle Three? List at least two examples. How does the sequence of Mission activities address Principle Three? List at least two examples. In using metacognition, who is doing the work? What is the teacher doing? What is the student doing? In using metacognition, who is doing the work? What is the teacher doing? What is the student doing?

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Post Test Work on test items individually. Work on test items individually. Write your participant number on the test. Write your participant number on the test.

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Reflection Describe one significant event, theme or idea from your summer professional development. How is the idea you selected connected to your teaching practice? How are you planning to implement in your classroom these ideas from your professional development experience for your own benefit and the benefit of your students?

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Reflection Questions Where does this unit fit in my curriculum? Where does this unit fit in my curriculum? What support will I need to implement this unit? What support will I need to implement this unit? What will I do differently as a result of my experience in the institute? What will I do differently as a result of my experience in the institute?

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