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

2.  General Content Overview ◦ Focus and Coherence ◦ Similarities ◦ Shifts ◦ Structure/Organization ◦ Activity Objectives 2  Mathematical Practices Overview ◦ Purpose ◦ Structure ◦ 8 Practices ◦ Standards Alignment ◦ Activity

3. 3 Avoid the problem of “mile wide and an inch deep” Recognize that “fewer standards” are no substitute for focused standards Aim for clarity and specificity © 2011 California County Superintendents Educational Services Association Mathematics General Overview

4.  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

5.  Define what students should understand and be able to do in their study of mathematics ◦ Is the ability to justify appropriate to student’s 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

6.  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

7.  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

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

9. Mathematical Content 9

10. 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 Mathematical Content Comparison 10 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

11. 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. 11 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

12. 12 © 2013 Actions Learning Systems, Inc. K12345678 Counting & Cardinality x Operations & Algebraic Thinking xxxxxX Number & Base Ten xxxxXx Numbers & Operations- Fractions xxX Ratios & Proportional Relationships xx The Number system xxx Expressions & Equation xxX Functions x Geometry xxxxxxxxX Measurement & Data xxxxxX Statistics & Probability xxx

13. 13 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

14. 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 14 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

15.  Teaching Probability of Independent and Dependent Events https://www.teachingchannel.org/videos/tea ching-dependent-and-independent- events?fd=1 Example of Common Core Math in a 6 th Grade Classroom © 2011 California County Superintendents Educational Services Association Mathematics General Overview 15

16.  Overview page ◦ Lists domains, clusters and mathematical practices  Domains ◦ Larger groups of related standards. Standards from different domains may be closely related.  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 16 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

17. 17 Grade 6 Overview Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. The Number System Apply and extend previous understandings of Multiplication and division to divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and multiples. Apply and extend previous understandings of numbers to The system of rational numbers. Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables. Geometry Solve real-world and mathematical problems involving area, surface area, and volume. Statistics and Probability Develop understanding of statistical variability. Summarize and describe distributions. Mathematical Practices 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 use of structure. 8. Look for and express regularity in repeated reasoning.

18. Reading the Standards 18 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. Ratios and Proportional Relationships 6.RP Domain Understand ratio concepts and use ratio reasoning to solve problems. 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1 Standard CLUSTERCLUSTER

19.  Label the Domain, Standards, and Clusters for Grade 6 Common Core Mathematics.  Circle, underline or highlight key academic vocabulary within the standards.  Go to U Drive, Curriculum, Common Core State Standards, Common Core Unpacked Math, k-12_math_crosswalk pgs. 97-128.  Choose a standard and compare similarities and differences.  Be prepared to discuss. Mathematical Content Activity 19

20. Mathematical Practice 20

21.  Standards for Mathematical Practice ◦ Describe habits of mind of a mathematically expert student ◦ Relate to mathematical proficiency as defined by the California Framework ◦ Consistent throughout grade levels 21 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

22. Problem Solving Computational & Procedural Skills DOING MATH Conceptual Understanding “WHERE” THE MATHEMATICS WORK “HOW” THE MATHEMATICS WORK “WHY” THE MATHEMATICS WORK 22 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

23. “ …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.” 23 © 2011 California County Superintendents Educational Services Association Mathematics General Overview

24. Structuring the Standards for Mat hematical Practice 24 1. Make sense of problems and persevere in solving them. 6. Attend to precision 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of other Reasoning and explaining 4.Model with mathematics 5.Use appropriate tools strategies Modeling and using tools 7.Look for and make use of structure 8.Look for and express regularity in repeated reasoning Seeing structure and generalizing

25. 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning 4. Model with mathematics 25

26. 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for an express regularity in repeated reasoning Standards for Mathematical Practice Mathematically proficient students: 26

27.  Example of MP 3, 6, 7, and 8 in an intermediate classroom Inside Mathematics Video © 2011 California County Superintendents Educational Services Association Mathematics General Overview 27 http://www.insidemathematics.org/index.php/exemplary-lessons- integrating-practice-standards

28.  Activity ◦ Label the Mathematical Practices that align with specific standards. ◦ Choose a standard and its Mathematical Practice(s) ◦ Go to U Drive, Curriculum, Common Core State Standards, Common Core Unpacked Math, Find your grade level ◦ Brain storm how you can integrate the Mathematical Practice with the Common Core Standard you chose. ◦ Be prepared to share. Mathematical Practices Aligned to Common Core Standards 28

29. SBAC Math Test http://www.smarterbalanced.org/practice-test/ SCROLL DOWN AND CLICK ON PRACTICE TEST 29