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OELCS 2005 Math Module 3 Speaker Notes Focus, Coherence, and Rigor

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1 OELCS 2005 Math Module 3 Speaker Notes Focus, Coherence, and Rigor
Mathematical Shifts of the Common Core State Standards Focus, Coherence, and Rigor May 2013 Common Core Training for Administrators Middle Grades Mathematics Division of Academics, Accountability, and School Improvement

2 OELCS 2005 Math Module 3 Speaker Notes
Mathematical Shifts of the Common Core State Standards: Focus, Coherence and Rigor AGENDA Purpose and Vision of CCSSM Implementation Timeline Six Shifts in Mathematics Design and Organization Instructional Implications: Classroom Look-fors Expectations of Student Performance CCSSM Resources: Websites Reflections / Questions and Answers 2 minutes

3 OELCS 2005 Math Module 3 Speaker Notes
Community Norms We are all learners today We are respectful of each other We welcome questions We share discussion time We turn off all electronic devices __________________ 1 minute Review the norms – helps the group be mre thoughtful and productive. Ask the participants if there is any norm they would like to include

4 OELCS 2005 Math Module 3 Speaker Notes
Purpose and Vision of the CCSSM

5 Why Common Standards? Consistency Equity Opportunity Clarity
Previously, every state had its own set of academic standards and different expectations of student performance. Consistency Common standards can help create more equal access to an excellent education. Equity Students need the knowledge and skills that will prepare them for college and career in our global economy. Opportunity 3 minutes 4 reasons for common standards: Consistency among states Equity for all children to have access to an excellent education Opportunity to compete in a global economy Clarity of standards for all stakeholders. Clarity is critical so that the expectations are attained at every grade level so that students are prepared for the upcoming grade level given the hierarchal nature of mathematics. These new standards are clear and coherent in order to help students, parents, and teachers understand what is expected. Clarity

6 College and Career Readiness: Anchor for the Common Core
The Common Core State Standards were back-mapped from the anchor of college and career readiness because governors and state school chiefs realized there was a significant gap between high school expectations for students and what students are expected to do in college/career. Among high school graduates, about only half are academically prepared for postsecondary education. 2 minutes- Misalignment between our expectations (high school diploma) and what college demands—David Coleman/Pres of College Board RESEARCH BASE (Greene & Winters, 2005) Among high school juniors and seniors taking the ACT college entrance exam, half of the students were ready for college-level reading assignments in core subjects like math, history, science, and English (ACT, 2006). Math Students are expected to have deep knowledge of core mathematical concepts (such as ratio and proportion) that are gatekeepers for success in advanced mathematics courses in college or advanced technical training in a career. Students are expected to apply their mathematical knowledge to new and novel situations. ELA Students are expected to read more non-fiction (periodicals, technical manuals, etc.) and more complex texts Students are expected to read with greater independence Quality of students’ writing in college isn’t meeting professors’ expectations

7 College Remediation and Graduation Rates
Remediation rates and costs are staggering As much as 40% of all students entering 4-year colleges need remediation in one or more courses As much as 63% in 2-year colleges Degree attainment rates are disappointing Fewer than 42% of adults aged hold college degrees 1 minute-Misalignment between our expectations (high school diploma) and what college demands—David Coleman/Pres of College Board As much as 40% of all students entering 4-year colleges need remediation in one or more courses. There is a gap in the expectations of colleges and careers and student preparation upon graduation from high school. The “architect” of Common Core, also president if the College Board, speaks to this gap Source: The College Completion Agenda 2010 Progress Report, The College Board

8 Are We Mathematically Ready for College and Careers?
OELCS 2005 Math Module 3 Speaker Notes Are We Mathematically Ready for College and Careers? Video: 3:22/ with discussion 6 minutes Verizon video-click in outer blue area -Do we have a sense of urgency for change? -Ask administrators to self-reflect: Is the instruction occurring in your classrooms going to lead to this level of mathematical understanding for the next generation? -Can someone volunteer to explain the mathematical misconception? 1:38:00 College remediation; David Coleman’s Testimony to the Florida Legislature’s K-12 Subcommittee

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Common Core State Standards Mission The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. 1 minute The CCSS provides consistent and clear understanding of what students are expected to learn. They are designed to be robust and applicable to the real world and reflect the skills needed for career and college success and to effectively compete in the global economy. NAEP and TIMMS (Trends in International Mathematics and Science Study) have always shown a need for change toward how math instruction is delivered.

10 Florida’s Common Core State Standards Implementation Timeline
M-DCPS Year / Grade level K 1 2 3 – 8 9 – 12 F L L CCSS fully implemented B L CCSS fully implemented and assessed F L F L 2 minutes Slide has fly-in. Show first and mention that this timeline is the state of FL implementation timeline. Hit enter for a MDCPS implementation. Miami-Dade County is ahead of the game. Unlike the State timeline, next year’s ( ) 3rd graders will not be seeing CCSSM for the first time. They have been learning it the last two years. We were able to give them a running start since grades 1 and 2 do not take FCAT 2.0. The current first graders are our common core babies. These students will be taking the PARCC assessment in grade 3. They should be better prepared because they have only been instructed using CCSSM. FOR ELEM ONLY – Are all K-2 teachers fully implementing CCSS? Next year, other grades will be using a blended model; the Mathematics pacing guide will guide teachers in this blended model. F – Full Implementation of CCSSM L – Full implementation of content area literacy standards including: text complexity, quality and range in all grades (K-12) B – Blended instruction of CCSS with NGSSS; last year of NGSSS assessed on FCAT 2.0 (Grades 3-8); 4th quarter will focus on NGSSS/CCSSM grade level content gaps 10 10

11 (4th 9 weeks Common Core Lockdown)
FCAT 2.0 PARCC K 1 2 3 4 5 6 7 (4th 9 weeks Common Core Lockdown) 8 9 10 11 12 NGSSS CCSSM 3 minutes Will be a blended year (NGSSS /CCSSM) however the NGSSS standards will be assessed, therefore it is essential for schools to ensure the Common Core Standards referenced in the pacing guides are covered fully. The 4th nine weeks for grades 3-8 should be on Common Core Lockdown.

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Six Shifts in Mathematics Activity begins once shifts are introduced

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Mathematical Shifts FOCUS deeply on what is emphasized in the Standards COHERENCE: Think across grades, and link to major topics within grades RIGOR: Require- Fluency Deep Understanding 1 minute There a six major mathematical shifts of the common core state standards. The last four (Fluency, Deep Understanding, Modeling and Dual Intensity) are components of Rigor. These shifts are not independent of each other. In order for one to occur, it is dependent on another. – for example, in order to ensure students have DEEP UNDERSTANDING of the content, there must be a narrower and more specific FOCUS on the priority concepts (not mile wide and inch deep) and the teacher must understand the COHERENCE of the math concepts and how topics progress from year to year in order to design lessons where their students can connect math concepts to each other (fragmented knowledge vs connected knowledge)– a point emphasized by The Structure is the Standards- Essay by CCSSM writers Jason Zimba, Phil Daro, and Bill McCallum. Model/Apply Dual Intensity

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15 OELCS 2005 Math Module 3 Speaker Notes
Name that Shift… At your tables, use the provided cards to place under the appropriate shift listed on the handout: focus, coherence, fluency, deep understanding, application, and dual intensity. After this is complete, your table will receive one large card identifying what the student does and the teacher does for a particular shift. Discuss with your table the shift that the card describes and tape it to the corresponding chart paper. Be ready to explain your reasoning. 15 minutes Slides following are solutions to the activity.

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Shift 1: Focus Teachers use the power of the eraser and significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades. 1 minute It is important to recognize that “fewer standards” are no substitute for focused standards. Achieving “fewer standards” would be easy to do by resorting to broad, general statements. Instead, these Standards aim for clarity and specificity->focus. Concepts are prioritized to narrow focus, be specific and delve deeper. Common Core State Standards document (p. 3) Teachers focus on big concepts throughout the school year. Example: In middle grades, there is a major emphasis on the concept of ratio and proportion. Students are able to transfer mathematical skills and understanding across concepts and grades. Spend more time on Fewer Concepts Achievethecore.org

17 Mathematics Shift 1: Focus
OELCS 2005 Math Module 3 Speaker Notes Mathematics Shift 1: Focus What the Student Does… What the Teacher Does… Spend more time on fewer concepts Extract content from the curriculum Focus instructional time on priority concepts Give students the gift of time 3 minutes to go over all solution slides (30 seconds per solution slide) SOLUTION TO ACTIVITY SLIDE Concepts are prioritized to narrow focus, be specific and delve deeper. Spend more time on Fewer Concepts

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“Teach Less, Learn More…” - Ministry of Education, Singapore Quote by David Coleman, who quoted THE MINISTRY OF EDUCATION from The Singapore Teaching and Learning website

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K-8 Priorities in Math Priorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction, measurement using whole number quantities 3–5 Multiplication and division of whole numbers and fractions 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 1 minute The foundation for students’ success in Algebra 1 is the progression from numbers & fractions ratios and proportional reasoning linear relationships. Note the priorities above. 8 Linear algebra Achievethecore.org

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Shift 2: Coherence Principals and teachers carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. A student’s understanding of learning progressions can help them recognize if they are on track. 2 minutes Each standard is not a new event, but an extension of previous learning. The CCSS Math standards are built on a progression of knowledge from one year to the next. Standards build on previous knowledge, all standards are extensions of what was learning in the past. We need to know our grade-level standards, and at least one grade below and above the classes we teach, keeping in mind what students have learned, what we will teach them, and how will they build on this next year! Common Core provides a three dimensional spiral approach to the curriculum. Teachers build concepts on prior knowledge. Mid: In 7th grade students solve real‐world and mathematical problems involving area, volume and surface area of two‐ and three‐dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms and in 8th grade students solve real‐world and mathematical problems involving volume of cylinders, cones, and spheres. Keep Building on learning year after year Achievethecore.org

21 Mathematics Shift 2: Coherence
OELCS 2005 Math Module 3 Speaker Notes Mathematics Shift 2: Coherence What the Student Does… What the Teacher Does… Build on knowledge from year to year, in a coherent learning progression Connect the threads of critical areas across grade levels Connect to the way content was taught the year before and will be taught the following years Focus on priority progressions SOLUTION TO ACTIVITY SLIDE Student – Build on Knowledge, Teachers Connect focus areas across grades and focus on grade level priority progressions Keep Building on learning year after year

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Shift 3: Fluency Teachers help students to study algorithms as “general procedures” so they can gain insights to the structure of mathematics (e.g. organization, patterns, predictability). Students are expected to have speed and accuracy with simple calculations and procedures so that they are more able to understand and manipulate more complex concepts. Students are able to apply a variety of appropriate procedures flexibly as they solve problems. 2 minutes 1st component of rigor---Fluent in the Standards means “fast and accurate.” It might also help to think of fluency as meaning the same thing as when we say that somebody is fluent in a foreign language: when you’re fluent, you flow. Fluent isn’t halting, stumbling, or reversing oneself. Assessing fluency requires attending to issues of time (and even perhaps rhythm, which could be achieved with technology). The word fluency was used judiciously in the Standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity. In fact, the rarity of the word itself might easily lead to fluency becoming invisible in the Standards—one more among 25 things in a grade, easily overlooked. Assessing fluency could remedy this, and at the same time allow data collection that could eventually shed light on whether the progressions toward fluency in the Standards are realistic and appropriate. Common Core requires specific sets of fluencies per grade levels in order to achieve a deep conceptual understanding of related concepts. Mid: For example in third grade, students need to multiply/divide fluently within 100. This fluency is an essential prerequisite to completing the calculations for surface area and volume in the middle grades. Spend time Practicing (First Component of Rigor) Achievethecore.org

23 Mathematics Shift 3: Fluency
OELCS 2005 Math Module 3 Speaker Notes Mathematics Shift 3: Fluency What the Student Does… What the Teacher Does… Spend time practicing, with intensity, skills (in high volume) Push students to know basic skills at a greater level of fluency Focus on the listed fluencies by grade level Uses high quality problem sets, in high volume SOLUTION TO ACTIVITY SLIDE Student – Practice time, Teachers- focus on grade level fluencies utilizing high quality problem sets which provides opportunities for ample practice Spend time Practicing

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K-8 Key Fluencies Grade Required Fluency K Add/subtract within 5 1 Add/subtract within 10 2 Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 100 Add/subtract within 1000 4 Add/subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7 Solve px + q = r, p(x + q) = r 8 Solve simple 22 systems by inspection 2 minutes Please note that there are algorithmic fluencies in middle grades. Associated with particular standards. Procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), Grade 5 does include division with 2 digit divisor and 4 digit dividend. Grade 6 it becomes multi-digit division. Grade 5 has +,-,x,/ with decimals up to the hundredths. Grade 6 it becomes multi-digit decimals. MACC.6.NS.2.2 Multi-digit division MACC.6.NS.2.3 Multi-digit decimal operations MACC.7.EE.2.4a Solve px + q = r, p(x + q) = r MACC.8.EE.3.8b Solve simple 22 systems by inspection Achievethecore.org

25 OELCS 2005 Math Module 3 Speaker Notes
Shift 4: Deep Conceptual Understanding Teachers teach more than “how to get the answer;” they support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding. 1 minute The instructional approach emphasizes conceptual understanding rather than memorization of procedures. Students articulate mathematical reasoning and understand why the algorithms work. Mid: For example, students in 6th grade find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. Students are able to transfer prior knowledge to derive the formulas. Understand Math, Do Math, and Prove it (Second Component of Rigor) Achievethecore.org

26 Mathematics Shift 4: Deep Understanding
OELCS 2005 Math Module 3 Speaker Notes Mathematics Shift 4: Deep Understanding What the Student Does… What the Teacher Does… Show mastery of material at a meaningful level Articulate mathematical reasoning Demonstrate deep conceptual understanding of priority concepts Explain and justify their thinking Create opportunities for students to understand the “answer” from a variety of access points Ensure that students understand WHY they are doing what they’re doing-ASK PROBING QUESTIONS Guide student thinking instead of telling the next step Continuously self reflect and build knowledge of concepts being taught SOLUTION TO ACTIVITY SLIDE Last bullet- teachers should continue to build their own knowledge of the mathematical content and connections in a collaborative setting Student – demonstrate mastery of material and articulates mathematical reasoning, Teachers-create opportunities for students to understand the answers at varying access points, ask probing questions and guide students thinking opposed to telling and continuously self-reflect to build capacity. Understand Math, Do Math, and Prove it

27 OELCS 2005 Math Module 3 Speaker Notes
Shift 5: Applications (Modeling) Teachers provide opportunities to apply math concepts in “real world” situations. Teachers in content areas outside of math ensure that students are using math to make meaning of and access content. Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. 1 minute Students are able to choose and apply various concepts when solving real-world problems. Mid: For example, students in 6th grade solve real‐world and mathematical problems involving area, surface area, and volume. It is critical that students understand the meaning of the concepts to be able to apply in real world situations. For example, if students are presented with a scenario where they must determine the amount of paint needed to paint the walls of a room, there may not be the word “area” used in the word problem to clue the students as to which formula to apply. They must know the meaning of the term to know when to apply the formula. Apply math in Real World situations (Third Component of Rigor)

28 Mathematics Shift 5: Application (Modeling)
OELCS 2005 Math Module 3 Speaker Notes Mathematics Shift 5: Application (Modeling) What the Student Does… What the Teacher Does… Utilize math in other content areas and situations, as relevant Choose the right math concept to solve a problem when not necessarily prompted to do so Apply math including areas/ courses where it is not directly required (i.e. in science) Provide students with real world experiences and opportunities to apply what they have learned SOLUTION TO ACTIVITY SLIDE Student will have the ability to apply math in other content areas, and make appropriate decisions about which strategies can be used to solve problems without being prompted. Teachers incorporate cross content references when applicable and provides real world experiences to allow students the opportunity to apply what they learned. Apply math in Real World situations

29 OELCS 2005 Math Module 3 Speaker Notes
Shift 6: Dual Intensity There is a balance between practice and understanding; both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. 1 minute There is a balance between fluency and problem solving applications. Mathematical understanding and procedural skill are equally important. For example, teachers provide time to practice while giving opportunities for students to apply learned concepts and model real life situations with mathematics. Mid: For example in 8th grade, students not only need to understand conceptually the derivation of the formula for the volume of a cylinder, but they must also get sufficient practice to achieve a combination of procedural fluency and conceptual understanding. Think fast and Solve problems (Fourth Component of Rigor) Achievethecore.org

30 Mathematics Shift 6: Dual Intensity
What the Student Does… What the Teacher Does… Practice math skills with an intensity that results in fluency Practice math concepts with an intensity that forces application in novel situations Find the balance between conceptual understanding and practice within different periods or different units Be ambitious in demands for fluency and practice, as well as the range of application SOLUTION TO ACTIVITY SLIDE Students and Teachers understand the importance of both Fluency and Application. Think fast and Solve problems

31 OELCS 2005 Math Module 3 Speaker Notes
Design and Organization

32 Design and Organization
Standards for Mathematical Content K-8 standards presented by grade level Organized into domains that progress over several grades Grade introductions give 2–4 focal points at each grade level Standards for Mathematical Practice Carry across all grade levels Describe habits of mind of a mathematically expert student 4 minutes Stress that in math, there are TWO components. What will drive the transformation in mathematics instruction are the MATH PRACTICES. This will become the focus of the Look-fors later in the presentation. The Common Core Mathematics Standards are composed of the 8 Mathematical Practices that carries across all grade levels and focuses on the habits of mind that students must develop to demonstrate mastery of common core standards. The second component of the CCSSM are the Mathematical Content Standards which are grade level standards that are organized by domains which progress over several grades fork-8 and organized by conceptual categories for grades 9-12. 32 32

33 OELCS 2005 Math Module 3 Speaker Notes
Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Far fewer things done well– go deeper

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Standards for Mathematical Practice How we teach is just as important as what we teach…. 2 minutes The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Common Core State Standards document (p. 9) “The Standards for Mathematical Practice are unique in that they describe how teachers need to teach to ensure their students become mathematically proficient. We were purposeful in calling them standards because then they won’t be ignored.” Quote by Bill McCallum- CCSSM writer

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Learning Experiences It matters how students learn Learning mathematics is more than just learning concepts and skills. Equally important are the cognitive and metacognitive process skills. These processes are learned through carefully constructed learning experiences. For example, to encourage students to be inquisitive and have a deeper understanding of mathematics, the learning experiences must include carefully structured opportunities where students discover mathematical relationships and principles on their own. 2 minutes It matters how the students are learning the math content, not just that they can get an answer to a procedure. From: Ministry of Education, Singapore (quoted as a key resource for Common Core writers such as Jason Zimba, Bill McCallum, & the “architect” David Coleman) ■2013 O & N(A)-Level Mathematics Teaching and Learning Syllabus See link above---NOTE how the Singapore syllabi made for teachers (from which this slide is developed) emphasizes metacognition and cognitive processes just as much, if not more, than the math content. Students need to be taught to be intelligent thinkers-analytical, inquisitive, in-depth thinkers.

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What does mathematical understanding look like? 4 minutes Leading into HABITS OF MIND slide One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain where the rule comes from understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). -Common Core State Standards document (p. 4)

37 OELCS 2005 Math Module 3 Speaker Notes
Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Reasoning and Explaining Modeling and Using Tools Seeing Structure and Generalizing 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning 2 minutes The practices can be further organized into 4 Categories (Habits of Mind, Reasoning and Explaining, Modeling and Using Tools, and Seeing Structure and generalizing.

38 OELCS 2005 Math Module 3 Speaker Notes
Overarching Habits of Mind of a Productive Mathematical Thinker MP 1: Make sense of problems and persevere in solving them. MP 6: Attend to precision Mathematically proficient students can… Mathematically proficient students can… use mathematical vocabulary to communicate reasoning and formulate precise explanations calculate accurately and efficiently and specify units of measure and labels within the context of the situation explain the meaning of the problem and look for entry points to its solution monitor and evaluate their progress and change course if necessary use a variety of strategies to solve problems 3 minutes MP 1 A worthwhile mathematical taks is one that is open-ended does not have a solution path that is immediately obvious (or implied) requires students to think and not just rely on memorized procedures requires students to connect mathematical skill, understanding, and reason requires students to interpret and communicate results. Provide students with problems that require planning (before solving) and evaluation (after solving). Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and They plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Elementary: Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Secondary: Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. MP 6 Mathematically proficient students: try to communicate precisely to others. try to use clear definitions in discussion with others and in their own reasoning. state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. carefully specify units of measure and label axes to clarify the correspondence with quantities in a problem. calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Students give carefully formulated explanations to each other. Students have learned to examine claims and make explicit use of definitions. Gather Information Make a plan Anticipate possible solutions Continuously evaluate progress Check results Question sense of solutions

39 OELCS 2005 Math Module 3 Speaker Notes
Reasoning and Explaining MP 2: Reason abstractly and quantitatively. MP 3: Construct viable arguments and critique the reasoning of others Mathematically proficient students can… Mathematically proficient students can… have the ability to contextualize and decontextualize problems involving quantitative relationships: decontextualize - to abstract a given situation and represent it symbolically and manipulate the representing symbols contextualize - to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. make a mathematical statement (conjecture) and justify it listen, compare, and critique conjectures and statements 4 minutes MP 2: Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Elementary: This particular Standard for Mathematical Practice incorporates an elementary student’s ability to translate a word problem into an equivalent number sentence. Additionally, students recognize that all relevant information within the problem has a quantitative value that is expressed with a numeric symbol. After successfully solving the problem, students are able to explain how the numeric representations are related to the original word problem. Secondary: Teachers who are developing students’ capacity to "reason abstractly and quantitatively" help their learners understand the relationships between problem scenarios and mathematical representation, as well as how the symbols represent strategies for solution. A middle childhood teacher might ask her students to reflect on what each number in a fraction represents as parts of a whole. A different middle childhood teacher might ask his students to discuss different sample operational strategies for a patterning problem, evaluating which is the most efficient and accurate means of finding a solution. MP 3: Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Children begin to reason from their own experiences and create arguments, even mathematical ones, before they enter the classroom as students. As students begin a more formal of study of mathematics, teachers should seek to build on the understandings that students bring to the classroom and move students toward commonly employed techniques for mathematical arguments and more formal language. Students’ early mathematical reasoning often relies on pattern recognition, generalization from examples, and classification; therefore, teachers should provide students with ample opportunities for students to use these skills as they make conjectures, search for evidence, and explain and justify ideas. For example, the teacher may ask, “If we skip count by twos, will we ever get a number that ends in 5?” Students can offer conjectures about the answer and study the pattern of numbers created when skip counting by twos. When they reach conclusions, the teacher can ask them to explain their thinking and critique the explanations of their classmates. Students need such opportunities on a regular basis in order to develop the ability to create and examine mathematical arguments. Students learn to determine areas to which an argument applies. listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve arguments. For example, a teacher can ask students to relate the domain of the function to its graph and, where applicable, to the quantitative relationship it describes. Students should be able to explain why a domain is appropriate for a given situation.

40 OELCS 2005 Math Module 3 Speaker Notes
Modeling and Using Tools MP 4: Model with Mathematics. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. MP 5: Use appropriate tools strategically Mathematically proficient students can… Mathematically proficient students can… apply mathematics to solve problems that arise in everyday life reflect and make revisions to improve their model as necessary map mathematical relationships using tools such as diagrams, two-way tables, graphs, flowcharts and formulas. consider the available tools when solving a problem (i.e. ruler, calculator, protractor, manipulatives, software) use technological tools to explore and deepen their understanding of concepts 5 minutes MP 4: Mathematically proficient students: • apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. • apply what they know to make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. • identify important quantities in a practical situation. • map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. • analyze those relationships mathematically to draw conclusions. • routinely interpret their mathematical results in the context of the situation. • reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Elementary: In early grades, this might be as simple as writing an addition equation to describe a situation. Secondary: In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. MP 5: Mathematically proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Being able to choose the appropriate tool for a real world situation when the tool is not identified. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. For example, middle school students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 6 might use physical objects or applets to construct nets and calculate the surface area of three-dimensional figures. High school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to use technological tools to explore and deepen their understanding of concepts.

41 OELCS 2005 Math Module 3 Speaker Notes
Seeing Structure and Generalizing MP 7: Look for and make use of structure MP 8: Look for and express regularity in repeated reasoning Mathematically proficient students can… Mathematically proficient students can… look closely to determine possible patterns and structure (properties) within a problem analyze a complex problem by breaking it down into smaller parts notice repeating calculations and look for efficient methods/ representations to solve a problem generalize the process to create a shortcut which may lead to developing rules or creating a formula 3 minutes MP 7: Mathematically proficient students: look closely to discern a pattern or structure. Elementary: Young students might notice that three and seven more is the same amount as seven and three more. They may sort a collection of shapes according to how many sides the shapes have. Students can recognize the pattern that exists in the teen numbers; every teen number is written with a 1 (representing one ten) and ends with the digit that is first stated. Intermediate students will see 7 x 8 equals the well-remembered 7 x x 3, in preparation for the distributive property. Secondary: Mathematically proficient students, in the expression x2 + 9x + 14, can see 14 as 2 x 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. The can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. (e.g., They see 5 - 3(x - y)2 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 numbers x and y.) MP 8: notice if calculations are repeated. look both for general methods and for shortcuts Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. i.e. 2/11= …. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Students can write the ratio of the circumference of a circle to the diameter of the circle; discuss, develop and justify this ratio for several circles; and determine that this ratio is constant for all circles. As high school students work to solve a problem, derive formulas or make generalizations, they maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

42 GROUP ACTIVITY Each group will receive:
An envelope containing 8 small cards and a handout of the listed mathematical practices. Instructions: Match each of the 8 small cards with its mathematical practice (using the handout of the listed mathematical practices). 20 minutes 2 minutes to explain and distribute materials 10 minutes for participants to do activity 8 minutes to review solutions Here are the standards of mathematical practice laid out in a way that classifies them as much as possible. Each group has 8 cards. Each card matches one of the standards of mathematical practice as it appears in the classroom. Try to match the card with the standard. Each group will also get a set of 8 large cards – one for each practice. When we review each one, hold up the correct card for the practice!

43 OELCS 2005 Math Module 3 Speaker Notes
SOLUTIONS TO ACTIVITY

44 OELCS 2005 Math Module 3 Speaker Notes
SOLUTIONS TO ACTIVITY

45 OELCS 2005 Math Module 3 Speaker Notes
SOLUTIONS TO ACTIVITY

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SOLUTIONS TO ACTIVITY

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SOLUTIONS TO ACTIVITY

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SOLUTIONS TO ACTIVITY

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SOLUTIONS TO ACTIVITY

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SOLUTIONS TO ACTIVITY

51 OELCS 2005 Math Module 3 Speaker Notes
Connecting the Math Practices to Look-Fors Activity begins once shifts are introduced

52 OELCS 2005 Math Module 3 Speaker Notes
4 minutes Provide this look-fors document for administrators to use as they watch the video of classroom instruction on the following slide. NOTE: First Priority- OBSERVE THE LEARNING in the next video– Use the “Students:” column of the look-fors 1st. When we note that students i.e. communicated and defended their reasoning, then we can check what the teacher did to accomplish this with the students. There are many more check boxes that can exist—an infinite number- so do not assume that because a part of the video is not exact to a checkbox that it is not indicating the practice. There are many more ways to demonstrate each of these practices, so much so that they cannot all be listed. Use this as a GUIDE.

53 OELCS 2005 Math Module 3 Speaker Notes
Observe the learning, not the teaching… Focal point is the student learning…

54 OELCS 2005 Math Module 3 Speaker Notes
Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Reasoning and Explaining Modeling and Using Tools Seeing Structure and Generalizing 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning 30 seconds In most cases, when look-fors can be grouped as above. If I see students modeling and using tools, it is likely they are engaged in both practices 4 and 5. If students are reasoning and explaining, it is likely they are engaged in practices 2 and 3. And so forth…

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Video Clip: Classroom Instruction

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15 minutes Don’t click on picture!! Click on the black to play!! I DO and WE DO slide– use this video to pause intermittently and review the demonstration of practices. Side note: Notice her classroom environment—it is obvious she is not a traditional teacher with the rich activities posted Very beginning 1st question asked- MP# 6 Attend to Precision -Notice how from the very beginning, the teacher probes the thinking of the student to be more precise with his definition of a variable. This is important to ensure there are no misconceptions or false generalizations in the minds of ALL the students in the class. It also ensures a deeper understanding of the term. Asks WHY would you use a variable?—purpose of the symbol not just what it is-Goes Deeper (Math Shift) 1:26 minutes MP#4 Model with Mathematics- multiple representations 2:03 minutes MP#7 Look for and Make Use of Structure- analyze patterns and apply them in context 2:13 min MP#8 Look for and express regularity in repeated reasoning – write a rule to show the relationship (generalize the process to develop a rule) 2:13 min MP#4 Model with Mathematics & MP#5 Use appropriate tools strategically- use the tool pattern blocks to model growth patterns 3:40 min MP#3 Construct Viable Arguments -#3 Diego justified his reasoning- uses the meaning of indep and dep variables to justify his answer Note: He was not prompted to justify yet he did so regardless-habit of mind. If he had not, the teacher should ask for the rationale. Can stop at 3:50 min Increase in rigor discussed at 6:25-7:05~ Focus, Deeper Understanding, Application

57 OELCS 2005 Math Module 3 Speaker Notes
Content Standards and Progressions

58 Clusters are groups of related standards
Domains are larger groups/categories of standards that progress across grades Clusters are groups of related standards Content standards define what students should understand and be able to do Domain 1 minute Domains are large groups of related standards. Clusters are groups of related standards. Standards define what students should be able to understand and be able to do- part of a cluster- Have the participants locate Grade 7 Expressions and Equations Domain on page 49 Cluster Standard 58 58

59 2 minutes Explain the different components of the document. Have participants follow along in their common core binder. 1st page of each grade level provides the “critical areas” of that grade level. (grade 6 on page 39-40) 2nd page is an overview (or snapshot) of the domains and clusters of that grade level. Also includes the 8 Mathematical Practices. (grade 6 right after page 40-should be numbered page 41 but is not numbered in the document) 3rd and following pages are the standards. (grade 6 starts on page 42)

60 OELCS 2005 Math Module 3 Speaker Notes
New Florida Coding for CCSSM Note: In the state of Florida, clusters will be numbered. MACC.7.EE.1.2 MACC.2.OA.1.1 Domain Common Core 1 minute MA-stands for math CC stands for Common Core 7 is the grade level EE is the domain- Expressions and Equations 1 is the cluster # 2 is the standard # Math Standard Cluster Grade Level

61 Florida Coding Scheme for Common Core State Standards
MACC.8.EE.3.7 Identify the cluster Identify the grade level Identify the standard Identify the domain Find the standard in your CCSSM binder 2 minutes- p.54 ANS: Cluster 3 Grade 8 Standard 7 Domain EE- Expressions and Equations Located on page 54 of the Common Core State Standards- call on someone to read aloud to ensure the participants understand how to navigate through the document.

62 Note that the Common Core Standards can be found in CPALMS

63 2 minutes Ask participants: Which domain is K-8? Geometry Which domain is solely in grade 8? Functions Note the different naming conventions in K-5 vs middle grades, with the exception of Geometry. Note that it is not a 1 to 1 horizontal progression across the domains.

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Common Core Progressions Common Core Progressions

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Think across grades and link to major topics within grades The Standards are designed around coherent progressions from grade to grade. Teachers carefully connect the learning across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension to previous learning. 1 minute Definition: Describes a topic across a number of grade levels based on conceptual development and the logical structure of mathematics. The Standards are designed around coherent progressions from grade to grade. Teachers need to be knowledgeable of developmental pathways and key learning trajectories, so that all students at all levels have the opportunity to learn the mathematics they need to progress. Each standard is not a new event, but an extension to previous learning. Siemon, Breed & Virgona 2006

66 OELCS 2005 Math Module 3 Speaker Notes
Geomet ry 3 Understand that shapes in different categories may share attributes. 6 Find the area of right triangles, other triangles, special quadrilaterals, and polygons. 6 3 minutes Participants will be provided samples of grade 3 and grade 6 Geometry progressions standards. Provide participants with copies of the Geometry Progression handout and explain that key components of the standards have been extracted to focus on the main point of the standard. Use this slide to provide participants with Third and Sixth grades Geometry standards in preparation for the progression activity.

67 Progression- Activity
OELCS 2005 Math Module 3 Speaker Notes Progression- Activity In your groups, use the geometry progression handout to identify the grade level corresponding to each bullet. 10 minutes Provide geometry progression handout to participants In groups, participants will match each bullet to the corresponding grade level Participants will continue with the remaining grade levels (K, 1, 2, 4, 5, 7, 8, HS) Summarize findings as a whole group with next slide

68 OELCS 2005 Math Module 3 Speaker Notes
Correctly name shapes regardless of their orientations and overall size. Geomet ry 1 Distinguish between defining attributes versus non‐defining attributes . 2 Recognize and draw shapes having special attributes. 3 Understand that shapes in different categories may share attributes. 4 Classify two-dimensional figures based on the presence or absence of parallel and perpendicular lines. 5 Understand that attributes belonging to a category of two‐dimensional figures also belong to all subcategories of that category. 6 Find the area of right triangles, other triangles, special quadrilaterals, and polygons. 6 7 Solve real‐world and mathematical problems involving area, volume and surface area. 4 minutes Use this slide to conduct a whole class discussion of Geometry progression activity findings. 8 Know the formulas for the volumes of cones, cylinders, and spheres. HS Prove theorems about parallelograms.

69 OELCS 2005 Math Module 3 Speaker Notes
The Key Progression of Middle Grades Math Common Core Progressions 1 minute There is one progression that is KEY to students’ readiness to high school Algebra- “the cornerstone”– as David Coleman explains: Elem: Fractions Middle Grades: Ratios and Proportional Reasoning These are the predictors of a child’s readiness and success in Algebra. THE MATH THAT MATTERS MOST

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Elementary grades work with part to whole relationships in fraction form. In 6th grade, the concept of ratio is introduced, and students look at tables of equivalent ratios and their representations on a graph. In 7th grade, students focus more on the use of multiple representations to solve proportion problems, including ratio tables, graphs, and equations, with a special focus on the constant of proportionality or unit rate. Finally in 8th grade, students extend the proportional reasoning developed in 6th and 7th grade to examine linear relationships with multiple representations. Grades 3-5 Grade 6 2 minutes Explain how the content of each grade level is dependent on the understanding of the prior content. Teachers also need to be aware of how the concept will be developed in the upcoming grade levels. The foundation for students’ success in Algebra 1 (the cornerstone) is the progression from numbers & fractions ratios and proportional reasoning linear relationships. Grade 7 Grade 8 Ratio and Proportionality Grade Level Progression

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Expectations of Student Performance PARCC Sample Items vs. FCAT 2.0 Sample Items Now we will compare and contract the expectations of student performance from PARCC versus FCAT 2.0 utilizing PARCC prototypes and FCAT 2.0 Sample Items.

72 The Standards & The Assessment
OELCS 2005 Math Module 3 Speaker Notes The Standards & The Assessment Define what students should understand and be able to do in their study of mathematics These standards are “focused” and “coherent” (i.e., conceptually DEEP) 1 minute The CCSSM define what students should understand and be able to do –focused, coherent—must teach for deeper understanding. PARCC assesses at a deeper level (i.e. performance assessment). Will determine whether students are on track to be college and career ready. Next generation assessment system Technology-based Assesses at a conceptually DEEP level

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PARCC Assessments WHAT HOW Transformative Formats PROBLEMS WORTH DOING Multi-step problems, conceptual questions, applications, and substantial procedures will be common, as in an excellent classroom. BETTER STANDARDS DEMAND BETTER QUESTIONS Instead of reusing existing items, PARCC will develop custom items to the Standards. FOCUS PARCC assessments will focus strongly on where the Standards focus. Students will have more time to master concepts at a deeper level. DRAG & DROP FILL-IN RESPONSES COMPARISONS RADIO BUTTONS / MC CHECK BOXES WRITTEN RESPONSES FOCUS PARCC assessments will focus strongly on where the Standards focus. Students will have more time to master concepts at a deeper level. PROBLEMS WORTH DOING Multi-step problems, conceptual questions, applications, and substantial procedures will be common, as in an excellent classroom. BETTER STANDARDS DEMAND BETTER QUESTIONS Instead of reusing existing items, PARCC will develop custom items to the Standards. 3 minutes Transformative formats making possible what can not be done with traditional paper-pencil assessments (e.g., simulations to improve a model, game-like environments, drawing/constructing diagrams or visual models, etc.) Review each of the transformative formats: Radio Buttons/Multiple Choice (one possible answer) Drag & Drop Check Boxes (many possible answers) Comparisons (pull down menu to select choice- i.e.: <,>, etc) Fill-in Responses (type in the response) Written Responses (infuses writing to explain)

74 Overview of PARCC Mathematics Task Types
Description of Task Type I. Tasks assessing concepts, skills and procedures Balance of conceptual understanding, fluency, and application Can involve any or all mathematical practice standards Machine scorable including innovative, computer-based formats Will appear on the End of Year and Performance Based Assessment components II. Tasks assessing expressing mathematical reasoning Each task calls for written arguments / justifications, critique of reasoning, or precision in mathematical statements (MP.3, 6). Can involve other mathematical practice standards May include a mix of machine scored and hand scored responses Included on the Performance Based Assessment component III. Tasks assessing modeling / applications Each task calls for modeling/application in a real-world context or scenario (MP.4)

75 Design of PARCC Math Summative Assessments
Performance Based Assessment (PBA) Type I items (Machine-scorable) Type II items (Mathematical Reasoning/Hand-Scored – scoring rubrics are drafted) Type III items (Mathematical Modeling/Hand-Scored and/or Machine-scored - scoring rubrics are drafted) End-of-Year Assessment (EOY) Type I items only (All Machine-scorable)

76 Grade 6 Example Numbering / Ordering Numbers / Absolute Value
OELCS 2005 Math Module 3 Speaker Notes Grade 6 Example Numbering / Ordering Numbers / Absolute Value

77 OELCS 2005 Math Module 3 Speaker Notes
FCAT 2.0 – Grade 6 30 seconds Here, students are asked to find the least value. MA.6.A.5.1 Use equivalent forms of fractions, decimals, and percents to solve problems - Moderate DOK

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PARCC – Grade 6 Part a 4 minutes Note-they must read the entire word problem to proceed with the following slides. Use pointer to show locations on number line Discuss: In terms of what the item requires of students, compare/contrast: -Format -Depth and Rigor Note: Drag & Drop Format. Students do not calculate here; they must understand number sense to appropriately place the negative decimals on the number line.

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PARCC – Grade 6 Part b 1 minute 1st answer is correct: Cake F weighs less than Cake G because -5 < -3. Discuss: In terms of what the item requires of students, compare/contrast: -Format -Depth and Rigor Note: Students do not just select the inequality but the choice with the appropriate rationale.

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PARCC – Grade 6 Part c 2 minutes Last answer is correct: The cake that is furthest from the target weight. Discuss: In terms of what the item requires of students, compare/contrast: -Format -Depth and Rigor Note that the ability to answer this question revolves on students’ conceptual understanding of absolute value. There is no calculation/procedure with this problem.

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PARCC – Grade 6 Part d 2 minutes Select all that apply. There are two solutions that should be check marked. The two on the right: A cake with a reading of -5.3 oz. A cake with a reading of 3.1 oz. Discuss: In terms of what the item requires of students, compare/contrast: -Format -Depth and Rigor Note: Students must select all that apply; therefore, simply guessing will not have the same probability of being correct as with FCAT’s multiple choice. Note: CPALMs- states that Florida will use all 4 levels of DOK with PARCC. The 4th level is Extended Thinking & Complex Reasoning. Since PARCC has performance assessment/tasks, this level which requires extended time can be utilized, whereas, it was not for FCAT (information as of 4/5/13)

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CCSSM Resources: Websites

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1 minute The following sites will be uploaded on the commoncore.dadeschools.net website. There are CCSSM powerpoints already there from presentations with department leaders. Council of the Great City Schools: Common Core Works The Partnership for Assessment of Readiness for College and Careers (PARCC) Florida Department of Education: COUNTDOWN TO COMMON CORE Implementing the Common Core State Standards The Teaching Channel Videos Illustrative Mathematics Achieve Tools for the Common Core Standards commoncoretools.me Sense Making in Mathematics math.serpmedia.org/tools_specialized.html Inside Mathematics

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Reflections How will the implementation of the Common Core Mathematical Practices shape future classroom instruction? Provide opportunities for reflections

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Handout to take with them: take home reading. Principals from multiple states share how they met the challenges of shifting to Common Core. Some of these states are in full implementation already so they can provide valuable reflection for where we are headed in terms of best practices as an administrator.

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Office of Academics and Transformation Division of Academics, Accountability, & School Improvement Questions/Concerns: Department of Mathematics and Science Middle Grades Mathematics 1501 N.E. 2nd Avenue, Suite 326 Miami, FL Office: Fax: Thank the participants for their time and efforts.


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