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Day 1 Common Core State Standards Session 3 6-12 Mathematics • List words that begin with each letter of the alphabet that identify aspects of the Common Core State Standards for Mathematics. WELCOME Agenda • The Common Core State Standards for Mathematics from A to Z • Exploring the Standards for Mathematical Practice • Exploring the Common Core State Standards for Mathematics Critical Areas • Exploring the Mathematics Progressions “We live in a time of vast changes that include accelerating globalization, mounting quantities of information, the dominating influence of science and technology, and the clash of civilizations. Those changes call for new ways of learning and thinking in school, business, and the professions.” -Howard Gardner Five Minds for the Future (2007) Expected Outcomes 1. Enhance knowledge base of the Common Core Standards for Mathematics. 2. Become familiar with the structure of the Common Core State Standards for Mathematics. 3. Enhance knowledge of the Common Core Standards for Mathematical Practice. 4. Understand how the critical areas bring focus to key mathematical concepts for students to learn at each grade level. 5. Consider how the learning progressions can be used to inform curriculum and guide instruction. Day 1 Digital Checkpoint Session 3 6-12 Mathematics Digital Checkpoint • Poll Everywhere • http://www.polleverywhere.com/ • To participate in the polls you will need a phone with text capability or a computer with Internet access. • If using your phones, please remember to leave them on silent. • This is a standard rate text message, so it may be free for you, or up to twenty cents on some carriers if you do not have a text messaging plan. • The service we are using is serious about privacy. We cannot see your phone numbers, and you’ll never receive follow-up text messages outside this presentation. How to Vote via Web Response The texting process uses the example shown on the slide. • The URL for Internet Polling is http://www.polleverywhere.com/vote • In the Box, type in 37607 – this is our session “phone number.” • Then enter the “Code” for the poll. • Remember to leave a space between each entry. • Remember if using this Presentation, you will need to create your own Polling Account. Poll Everywhere is free for educators to use with up to 40 participants. Schools may subscribe and create their own account for unlimited usage. Digital Resources for the Common Core • Apple • http://itunes.apple.com/us/app/common-core-standards/id 439424555 • Android • https://play.google.com/store/apps/details?id=com.edutat er.corestandards • Common Core State Standards for Mathematics • http://www.corestandards.org/ Domains for K-12 This table lists the domains for Kindergarten through grade 12. For each domain, the shaded areas indicate the grade levels where it is addressed. Notice that most of the domains span multiple grades level. Notice the abbreviation for each of the Domains. Counting and Cardinality (CC) – unique to Kindergarten Operations and Algebraic Thinking (OA) Number and Operations in Base Ten (NBT) Number and Operations – Fractions (NF) Ratios and Proportional Relationships (RP) The Number System (NS) Expressions and Equations (EE) Functions (F) Geometry (G) Measurement and Data (MD) Statistics and Probability (SP) Number and Quantity (N) – unique to high school Algebra (A) – unique to high school The Common Core Standards for Mathematics structure includes Domains, Clusters and Standards and in High School there are Conceptual Categories. The domains progress over several grades and standards from different domains may sometimes be closely related. High School Conceptual Categories • Number and Quantity (N) • Algebra (A) • Functions (F) • Modeling (*) • Geometry (G) • Statistics and Probability (S) Grade 7 Overview This page provides an overview of the standards, first organized by domains. Domains describe large groups of related standards. For grade 7, the domains are: • Ratios and Proportional Relationships • The Number System • Expressions and Equations • Geometry • Statistics and Probability Within each domain, you’ll find cluster headings, which describe smaller groups of related standards. For example, within the Ratios and Proportional Relationships domain, there is only one cluster heading: • Analyze proportional relationships and use them to solve real-world and mathematical problems. Also note the Statistics and Probability domain has 3 cluster headings: • Use random sampling to draw inferences about a population. • Draw informal comparative inferences about two populations. • Investigate chance processes and develop, use, and evaluate probability models. Number and Quantity Overview Each of the Conceptual Categories has the same format as the elementary and middle school, except that the categories are not by grade level, but rather by content. Overarching “big ideas” that connect topics across the grades/courses. Descriptions of the mathematical content to be learned, elaborated through clusters and standards Clusters indicate WHAT students should know and be able to do: • May appear in multiple grade levels/courses with increasing developmental standards as the grade levels progress • Reflect both mathematical understandings and skills, which are equally important Florida’s Numbering of the Common Core State Standards MACC.7.EE.1.1 MACC Mathematics Common Core = Subject 7 = Grade EE Expressions and Equations = Domain 1 = Cluster (Use properties of operations to generate equivalent expressions) 1 = Standard (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.) Florida employs a Statewide Course Numbering System. The Florida Common Core State Standards basically follows the Common Core State Standards for Mathematics numbering system. In the Florida numbering system a number is assigned to the Cluster Headings. The numbering of the individual standards continues throughout the Clusters. That is the numbering of the standards do not begin again with each new cluster, they continue from the previous cluster’s standards. Florida’s Numbering of the Common Core State Standards MACC.912.N-RN.2.3 This slide shows the difference in the high school standards – there is the addition of the Conceptual Category. There are six Conceptual Categories: Number and Quantity (N) Algebra (A) Functions (F) Modeling (*) Geometry (G) Statistics and Probability (S) Standards for Mathematical Practice “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.” - Bill McCallum 8 Standards for Mathematical Practice 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 sense of structure 8. Look for and express regularity in repeated reasoning Florida’s Common Core State Standards Implementation Timeline 2011-2012: K = FL, 1 – 12 = L 2012 – 2013: K - 1 = FL, 2 – 12 = L 2013-2014: K -2 = FL, 3 – 12= BL 2014-2015 CCSS fully implemented and assessed: FL Across the board F - full implementation of CCSS for all content areas L – begin full implementation of content area literacy standards including: (1) use of informational text, text complexity, quality and range in all grades (K-12), and (2) CCSS Literacy Standards in History/Social Studies, Science, and Technical Subjects (6-12) B - blended instruction of CCSS with Next Generation Sunshine State Standards (NGSSS); last year of NGSSS assessed on FCAT 2.0 Day 1 Standards for Mathematical Practice Session 3 6-12 Mathematics Poll Question Please take out your phones or use your computers to respond to this Poll: How many Common Core Standards for Mathematical Practices are there? Text the Code #129153 and your response to 37607 or submit your response to http://PollEv.com/vote Standards for Mathematical Practice 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 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others Modeling and Using Tools 4. Model with mathematics 5. Use appropriate tools strategically Seeing Structure and Generalizing 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning http://www.corestandards.org/the-standards/mathematics/introduction/stand ards-for-mathematical-practice Standards for Mathematical Practice Video Clip • Develops dispositions and habits of mind “Characteristic of an educated person” • Precision in thought • Precision in the use of language and terms • Precision of argument • Sense making happens through conversations The Standards for Mathematical Practice • Please locate the Common Core State Standards for Mathematics. • Take a moment to examine the first three words of the narrative description for each of the 8 mathematical practices. • What do you notice? Mathematically Proficient Students… Task Your Task is to: •At your table, count off 1-8. •Read your assigned Mathematical Practice. •Identify and underline the words (verbs) that illustrate the student actions for this practice. Consider the Learners • Over 240,000 ELLs in Florida • Almost every district has ELLs • 300 languages are spoken among ELLs • 79% of ELLs are in Mainstream/Inclusion model classrooms • ELLs are learning in the same classrooms as non-ELLs Making the Content Comprehensible • Use the standards vocabulary as a teaching tool. “Generalize, develop, describe, analyze, apply, measure,” etc. are all words ELLs will hear in the classroom and need to understand. • ELLs may know how to perform the skill using their language, they just may not yet have the English vocabulary. • Use pictures, graphs, and charts whenever possible. • Make use of root words and cognates. Classroom Strategies • Group ELLs with non-ELLs to work together. • Allow more wait time for ELLs to respond. -Silence does not necessarily mean the student does not know the answer, the ELL may be translating the answer and needs more time. • Remember that ELLs from different countries may display mathematical functions in different ways. Critical Areas and Mathematics Progressions Session 3 6-12 Mathematics K-12 Domains and Critical Areas It is important that teachers are fully aware of the critical areas for their grade level as well as the critical areas from the prior grade level and the next grade level. 6 Grade Domains: Ratios and Proportional Relationships The Number System Expressions and Equations Geometry Statistics and Probability 6 Grade Critical Areas: Connecting ratio and rate to whole number multiplication and division using concepts of ratio and rate to solve problems. Completing understanding of division of fractions and extending the notation of numbers to the system of rational numbers which includes negative numbers. Writing, interpreting, and using expressions and equations. Developing understanding of statistical thinking. 7 Grade Domains: Ratios and Proportional Relationships The Number System Expressions and Equations Geometry Statistics and Probability 7 Grade Critical Areas: Developing understanding of and applying proportional relationships. Developing understanding of operations with rational numbers and working with expressions and linear equations. Solving problems involving scale drawings and informal geometric constructions, and working with two-and three-dimensional shapes to solve problems involving area, surface area, and volume. Drawing inferences about populations based on samples. 8 Grade Domains: The Number System Expressions and Equations Functions Geometry Statistics and Probability 8 Grade Critical Areas: Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations. Grasping the concept of a function and using functions to describe quantitative relationships. Analyzing two-and-three dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying Pythagorean Theorem. Algebra 1 Domains: The Real Number System Quantities Seeing Structure in Expressions Arithmetic with Polynomials and Rational Expressions Creating Equations Reasoning with Equations and Inequalities Interpreting Functions Building Functions Linear, Quadratic, and Exponential Models Interpreting Categorical and Quantitative Data Algebra 1 Critical Areas: Relationships between Quantities and Reasoning with Equations; Analyze and explain the process of solving an equation. Linear and Exponential Relationships; Learn function notation and develop the concept of domain and range. Descriptive Statistics; Learn formal means of assessing how a model fits (data regression techniques, graphical representations and goodness of fit.) Expressions and Equations; Create and solve equations, inequalities, and systems of equations involving quadratic expressions. Quadratic Functions and Modeling; Comparing key characteristics. Four critical areas in 6th Grade 1. Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems 2. Completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers 3. Writing, interpreting, and using expressions and equations 4. Developing understanding of statistical thinking Identify the 6th Grade Critical Area • 1 - Ratio: Understand ratio concepts and use ratio reasoning to solve problems. • 4 - Statistics: Summarize and describe distributions. • 2 - Rational #: Compute fluently with multi-digit numbers and find common factors and multiples. • 3 - Equations: Reason about and solve one-variable equations and inequalities. • 3 - Expressions: Apply and extend previous understandings of arithmetic to algebraic expressions. • 2 - Rational #: Apply and extend previous understandings of numbers to the system of rational numbers. • 3 - Equations: Represent and analyze quantitative relationships between dependent and independent variables. • 4 - Statistics: Develop understanding of statistical variability. • 2 - Rational #: Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Solve real-world and mathematical problems involving area, surface area, and volume was not included in the above list. It is included as one of the areas to build on their work in elementary school. Four critical areas in 7th Grade 1. Developing understanding of and applying proportional relationships 2. Developing understanding of operations with rational numbers and working with expressions and linear equations 3. Solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume 4. Drawing inferences about populations based on samples Identify the 7th Grade Critical Areas 1-Proportions, 2-Rational: Analyze proportional relationships and use them to solve real-world and mathematical problems. 3-Geometry: Draw, construct and describe geometrical figures and describe the relationships between them. 2-Rational #: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 4-Statistics: Use random sampling to draw inferences about a population. 3-Geometry: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 4-Statistics: Draw informal comparative inferences about two populations. 2-Rational #: Use properties of operations to generate equivalent expressions. 2-Rational #: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Additional: Investigate chance process and develop, use, and evaluate probability models. Three critical areas in 8th Grade 1. Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations 2. Grasping the concept of a function and using functions to describe quantitative relationships 3. Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem Five critical areas (units) in Algebra 1 1. Relationships Between Quantities and Reasoning with Equations 2. Linear and Exponential Relationships 3. Descriptive Statistics 4. Expressions and Equations 5. Quadratic Functions and Modeling Six critical areas (units) in Geometry 1. Congruence, Proof, and Constructions 2. Similarity, Proof, and Trigonometry 3. Extending to Three Dimensions 4. Connecting Algebra and Geometry Through Coordinates 5. Circles With and Without Coordinates 6. Applications of Probability Four critical areas (units) in Algebra 2 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions 4. Inferences and Conclusions from Data Building Fluency • What is meant by fluency? The Common Core State Standards for Mathematics explicitly sets a demand for students to attain a variety of fluencies at specific grade levels. The Common Core State Standards for Mathematics document uses the phrase “fast and accurate” to describe the notion of fluency. To further clarify the idea they use an analogy of being fluent in a foreign language – the ability to communicate and understand with automaticity. By adhering to the Common Core State Standards for Mathematics, students will receive instruction through a progression of stages surrounding a target concept that ultimately provides them with the knowledge and practice needed to acquire the endpoint goal of fluency. When it comes to measuring the full range of the Standards, usually the first things that come to mind are the mathematical practices, or perhaps the content standards that call for conceptual understanding. However, the Standards also address another aspect of mathematical attainment that is seldom measured at scale either: namely, whether students can perform calculations and solve problems quickly and accurately. At each grade level in the Standards, one or two fluencies are expected:. 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. Key Fluencies K - MACC.K.OA.1.5 - Add/subtract within 5 1 - MACC.1.OA.3.6 - Add/subtract within 10 2 - MACC.2.OA.2.2 - Add/subtract within 20 MACC.2.NBT.2.5 - Add/subtract within 100 (pencil and paper) 3 - MACC.3.NBT.1.2 - Add/subtract within 1,000 MACC.3.OA.3.7 - Multiply/divide within 100 4 - MACC.4.2.4 - Add/subtract within 1,000,000 Critical Area #1 - Develop fluency with efficient procedures for multiplying whole numbers 5 - MACC.5.NBT.2.5 - Multi-digit multiplication Critical Area #1 - Developing fluency with addition and subtraction of fractions 6 - MACC.6.NS.2.2 - Multi-digit division MACC.6.NS.2.3 - Multi-digit decimal operations 7 - MACC.7.EE.2.4a - Solve px + q = r, p(x + q) = r 8 - MACC.8.EE.3.8b - Solve simple 22 systems by inspection Mathematics Progressions • The Importance of Mathematics Progressions VIDEO The mathematics progressions are a narrative description of how a particular domain plays out over a grade level. They build from grade to grade and topic to topic, connecting topics logically and sequentially - providing K-12 focus and coherence. The mathematics progressions are research-based learning detailing what is known about students’ mathematical knowledge, skill, and understanding. The mathematics progression offer explanations for the sequence of the standards, potential cognitive difficulties, and pedagogical solutions which may be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks. Mathematics Progressions Project • Kindergarten Counting and Cardinality • K-5 Number and Operations in Base Ten • 3-5 Number and Operations—Fractions • K–5 Operations and Algebraic Thinking • K-5 Measurement and Data (Data part) • Geometry Progression (coming soon!) • 6-8 Expressions and Equations • 6-7 Ratios and Proportional Relationships • 6-8 Statistics and Probability • High school Statistics and Probability • http://commoncoretools.me/category/progressions/ Reflective Thoughts 1. How will you use the Common Core Standards for Mathematical Practices to inform your curriculum and guide your instruction? 2. How will the Critical Areas and the Cluster headings help to inform your curriculum and guide your instruction? 3. How will you use the Learning Progressions to inform your curriculum and guide your instruction?

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posted: | 8/31/2012 |

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