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Overview 1 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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General Overview Focus and Coherence Mathematical Proficiency Objectives 2 Structure Organization Grade 8 Options Similarities Shifts Next Steps © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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3 Avoid the problem of mile wide and an inch deep Aim for clarity and specificity

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Topics and performances are logical over time Based on learning progressions research on how students learn Reflect hierarchical nature of the content Evolve from particulars to deeper structures 4 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Define what students should understand and be able to do in their study of mathematics Is the ability to justify appropriate to students math maturity Understanding and procedural skill are equally important and can be assessed using tasks of sufficient richness Are internationally benchmarked Reflect rigor, focus and coherence of standards in top-performing countries 5 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Do: Set grade-level standards K-8 Identify standards for Algebra 1 Provide conceptual cluster standards in high school Provide clear signposts along the way toward the goal of college and career readiness for all students 6 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Do not: Define intervention methods or materials Define the full range of supports for English learners, students with special needs and students who are well above or below grade level expectations Dictate curriculum or teaching methods 7 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Mathematical Practice (recurring throughout the grades) Mathematical Content (different at each grade level) 8 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Standards for Mathematical Practice Describe habits of mind of a mathematically expert student Relate to mathematical proficiency as defined by the California Framework 9 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Problem Solving Computational & Procedural Skills DOING MATH Conceptual Understanding WHERE THE MATHEMATICS WORK HOW THE MATHEMATICS WORK WHY THE MATHEMATICS WORK 10 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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…describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high schools years. 11 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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1. Make sense of problems and persevere in solving them …start by explaining to themselves the meaning of a problem and looking for entry points to its solution 2. Reason abstractly and quantitatively …make sense of quantities and their relationships to problem situations 3. Construct viable arguments and critique the reasoning of others …understand and use stated assumptions, definitions, and previously established results in constructing arguments 4. Model with mathematics …can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace 12 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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5. Use appropriate tools strategically …consider the available tools when solving a mathematical problem 6. Attend to precision …calculate accurately and efficiently 7. Look for and make use of structure …look closely to discern a pattern or structure 8. Look for and express regularity in repeated reasoning …notice if calculations are repeated, and look for both general methods and for shortcuts Standards for Mathematical Practice Mat hematically proficient students: 13 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Balanced combination of procedure and understanding Understand expectations connect practice to content. Lack of understanding prevents students from engaging in the mathematical practices Weighted toward central and generative concepts that most merit the time, resources, innovative energies and focus Build in complexity and provide more clarity for expected performance 14 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Grade One Understand Place Value The two digits of a two-digit number represent amounts of tens and ones 10 can be thought of as a bundle of ten ones – called a ten. The numbers from 11 to 19 are composed of a ten and one, two, etc. The numbers 10, 20, 30, … refer to one, two, three, …tens and zero ones 15 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Overview page Lists domains, clusters and mathematical practices Standards-by grade level Defines what students should understand and be able to do Clusters Groups of related standards. Standards from different clusters may be closely related Domains Larger groups of related standards. Standards from different domains may be closely related. Additional standard language or whole standards Bolded and underlined Added to maintain rigor of California expectations 16 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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17 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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18 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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19 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Common Core State Standards for CA DOMAINS California Standards Grades K-7 STRANDS K-5 Counting and Cardinality (K only) Operations and Algebraic Thinking Number and Operations in Base 10 Number and Operations-Fractions Measurement and Data 6-8 Ratio and Proportional Relationships (grade 6-7) The Number System Expressions and Equations Functions (Grade 8) Geometry Statistics and probability Number Sense Algebra and Functions Measurement and Geometry Statistics, Data Analysis and Probability Mathematical Reasoning California Comparison 20 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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21 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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22 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Develop Conceptual Understandings Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. (K.OA.2) Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. (2NBT.7) 23 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Emphasis on Fluency Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers. (3.OA.7) Fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.5) 24 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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A Strong Focus on Fractions Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. (3.NF.2.a) Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g. by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5+ 1/2 = 3/7, by observing that 3/7 < 1/2. (5.NF.2) 25 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Fraction Concepts Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. ( 3.NF.3d) Discuss how you might compare pairs of fractions using a visual fraction model. For discussion purposes, use the following two fraction pairs: 7/9 and 4/9 (same denominator) 4/9 and 4/7 (same numerator) 26 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Fraction Concepts 27 Source: www.mathisfun.com/numbers/fraction-number-line.html © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Fraction Concepts 28 Source: www.mathisfun.com/numbers/fraction-number-line.html © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Goal for 8 th grade students is Algebra 1 Not all students have the necessary prerequisite skills for Algebra 1 Two sets of standards for grade 8 Each set will prepare students for college and career Standards for Algebra 1 Taken from 8 th grade Common Core, high school Algebra content cluster and CA Algebra standards 8 th grade Common Core Goal of grade 8 Common Core is to finalize preparation for students in high school K-7 standards as augmented prepare students for either set of standards 29 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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30 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Arranged by conceptual cluster (NOT by course): Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability Same K-8 structure of domain, cluster and standard 31 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Specify the math that all students should study to be college and career ready Identify additional math standards that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics. These are indicated by (+). Include the addition of two courses from California: Calculus Advanced Placement Statistics and Probability Development of suggested course descriptions will be done by CDE as part of their long-range implementation plan Traditional vs. Integrated 32 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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High School Example-Geometry Content Cluster 33 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Modeling Cluster Not a collection of topics but viewed in relation to other standards A Standard of Mathematical Practice Specific modeling standards appear throughout the high school standards and are indicated by a star symbol ( ) 34 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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GradeCalifornia StandardCommon Core KindergartenUse concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10). Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. FirstCount, read, and write whole numbers to 100. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. ThirdMemorize to automaticity the multiplication table for numbers between 1 and 10. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division and the properties of operations. 35 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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GradeCalifornia StandardCommon Core FifthUnderstand the concept of multiplication and division of fractions. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (A unit fraction is one with a numerator of 1 and the denominator is a positive integer) SixthInterpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ). Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. SeventhUse variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three is less than a number, half as large as area A). Use variables to represent quantities in real- world and mathematical problems and construct simple equations and inequalities to solve problems about the quantities. 36 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Grade/C ourse California StandardCommon Core SeventhConstruct and read drawings and models made to scale. Solve problems involving scale drawings of geometric figures, including actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. AlgebraAlgebra 1: Solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Algebra Content Cluster: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. GeometryGeometry: Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Geometry Content Cluster: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 37 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Grade Shifts: Examples Developed by SCFIRD Concept1997 StandardsCCCS Compose simple shapes to form larger shapes (e.g., 2 triangles to form a rectangle) Grade 2 K Introduction to Probability Grade 3 Grade 7 Introduction of fractions as numbers Grade 2 Grade 3 Add and subtract simple fractions Grade 3 Grade 4 Introduction of integersGrade 4 Grade 6 38 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Based on the following central questions: What K-12 CA Mathematics standards were not reflected in the CCS document? Which (of those) standards would substantively enhance and improve the CCS? Which would maintain the rigor of Californias standards? Californias Additional 15% 39 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Added standards to develop ideas not included in CCS Grade 2-Operations and Algebraic Thinking Grade 5-Operations and Algebraic Thinking High School Geometry-Geometric Measurement and Dimension Examples of Additional 15%: 40 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Added language to existing standard Grade 2-Measurement and Data Grade 4-Geometry Examples of Additional 15%: 41 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Added a substantial section to an existing cluster Grade 6-The Number System Examples of Additional 15%: 42 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Added two courses from California Standards: Calculus Advanced Placement Probability and Statistics Examples of Additional 15%: 43 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Stay the Course! More similarities than differences in the standards Implement a truly balanced math program as this will support the mathematical practices Continue to use quality assessments to inform and drive effective instruction Provide opportunities for teachers to collaborate and plan 44 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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Websites Common Core Standards: www.corestandards.org www.corestandards.org California Common Core Standards: Visit the California Department of Educations Common Core State Standards Web page at: http://www.cde.ca.gov/be/st/cc/index.asp The standards Frequently asked questions Informational flyers Additional resources 45 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

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