California s Common Core Content Standards An Introduction for Teachers

1. California�s Common Core Content StandardsAn Introduction for Teachers Mathematics 1 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview HANDOUT REQUIRED FOR THIS INTRODUCTION: Provide copies of the standards for all participants. This is going to be an introduction of the Common Core Standards for California so that you will be familiar with the organization and content of the standards. HANDOUT REQUIRED FOR THIS INTRODUCTION: Provide copies of the standards for all participants. This is going to be an introduction of the Common Core Standards for California so that you will be familiar with the organization and content of the standards.

2. General Overview Focus and Coherence Mathematical Proficiency Objectives 2 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Session will be in two parts: 1) General overview concentrating on the focus and coherence of the standards and mathematical proficiency as defined by Common Core and 2) the structure and organization of the standards with examples It is clear that there are more similarities than differences with CA standards. However, we will show some shifts and provide some time to explore the standards and some of the differences. Instructor notes: Session will be in two parts: 1) General overview concentrating on the focus and coherence of the standards and mathematical proficiency as defined by Common Core and 2) the structure and organization of the standards with examples It is clear that there are more similarities than differences with CA standards. However, we will show some shifts and provide some time to explore the standards and some of the differences.

3. Common Core Standards Overview:Toward Greater Focus and Coherence 3 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor Notes: Many have heard of the complaint that our curriculum is �a mile wide and an inch deep.� These Standards are a substantial answer to that problem. It is important to note that �fewer� is no substitute for focused standards. Fewer standards would be easy to do by resorting to broad general statements. Instead, these standards aim for clarity and coherence. Instructor Notes: Many have heard of the complaint that our curriculum is �a mile wide and an inch deep.� These Standards are a substantial answer to that problem. It is important to note that �fewer� is no substitute for focused standards. Fewer standards would be easy to do by resorting to broad general statements. Instead, these standards aim for clarity and coherence.

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 Coherence Design 4 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor Notes: Schmidt and Houang* (2002) say that standards are coherent if they are: Articulated over time Reflect the hierarchical nature of the content Evolve from particulars to deeper structures Development of the standards began with research-based learning progressions detailing what is known today about how students� math knowledge, skill and understanding develop over time. Citation: Schmidt, W.H., and Houang, R.t., �Lack of Focus in the Intended Mathematics Curriculum: Symptom or Cause?� in Loveless (ed.), Lessons Learned: What International Assessments Tell Us About Math Achievement. Washington, D.C.: Brookings Institution Press, 2007 Instructor Notes: Schmidt and Houang* (2002) say that standards are coherent if they are: Articulated over time Reflect the hierarchical nature of the content Evolve from particulars to deeper structures Development of the standards began with research-based learning progressions detailing what is known today about how students� math knowledge, skill and understanding develop over time. Citation: Schmidt, W.H., and Houang, R.t., �Lack of Focus in the Intended Mathematics Curriculum: Symptom or Cause?� in Loveless (ed.), Lessons Learned: What International Assessments Tell Us About Math Achievement. Washington, D.C.: Brookings Institution Press, 2007

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 Common Core Standards 5 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor Notes: Emphasis in the Common Core introduction is placed on understanding mathematics. While it is sometimes difficult to assess this, one hallmark of understanding mentioned is the ability to justify, in a way appropriate to the student�s math maturity, why a particular statement is true or where a rule comes from. It is clear from the document, page 4 of Common Core, that both understanding and procedural skill are equally important and can be assessed with rich tasks. International benchmarking was done using a number of top-performing countries: Hong Kong, Korea, Singapore, Finland, etc. Three characteristics of standards found in top-performing countries are rigor, focus and coherence. For further information on international benchmarking see: Benchmarking for Success: Ensuring U.S. Students Receive a World-Class Education. National Governors Association, Council of Chief State School Officers, and Achieve, Inc., 2008 Gingsburg, A., Leinwand, S., and Decker, K., �Informing Grades 1-6 Standards Development: What Can Be Learned from High-Performing Hong Kong, Korea and Singapore:� American Institutes for Research, 2005. Instructor Notes: Emphasis in the Common Core introduction is placed on understanding mathematics. While it is sometimes difficult to assess this, one hallmark of understanding mentioned is the ability to justify, in a way appropriate to the student�s math maturity, why a particular statement is true or where a rule comes from. It is clear from the document, page 4 of Common Core, that both understanding and procedural skill are equally important and can be assessed with rich tasks. International benchmarking was done using a number of top-performing countries: Hong Kong, Korea, Singapore, Finland, etc. Three characteristics of standards found in top-performing countries are rigor, focus and coherence. For further information on international benchmarking see: Benchmarking for Success: Ensuring U.S. Students Receive a World-Class Education. National Governors Association, Council of Chief State School Officers, and Achieve, Inc., 2008 Gingsburg, A., Leinwand, S., and Decker, K., �Informing Grades 1-6 Standards Development: What Can Be Learned from High-Performing Hong Kong, Korea and Singapore:� American Institutes for Research, 2005.

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 Common Core Standards 6 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: The following two slides deal with what the standards do and don�t do. They set standards for grades K-8. Remind them that in CA there are no Grade 8 standards and the default set is Algebra 1. In the CA CCS (CCSS) there is a set for grade 8. Since the CCS did not include a set of standards identified for a course, Algebra 1, the commission created one. Therefore in eighth grade a student would have the option of taking either grade 8 or the Algebra 1 set of standards. This insured that there would be no lowering the bar for CA students. Explain that this will discussed later on in the presentation in greater detail. The high school standards are organized by conceptual clusters as opposed to courses. So in the Algebra cluster, there will be standards for both Algebra 1 and 2. Explain that how these standards become courses will be determined at a later date as part of the CA Common Core implementation plan. This also will be discussed later in more detail. As with ELA, the Math CCS provide clear signposts toward the goal of college and career readiness. Instructor notes: The following two slides deal with what the standards do and don�t do. They set standards for grades K-8. Remind them that in CA there are no Grade 8 standards and the default set is Algebra 1. In the CA CCS (CCSS) there is a set for grade 8. Since the CCS did not include a set of standards identified for a course, Algebra 1, the commission created one. Therefore in eighth grade a student would have the option of taking either grade 8 or the Algebra 1 set of standards. This insured that there would be no lowering the bar for CA students. Explain that this will discussed later on in the presentation in greater detail. The high school standards are organized by conceptual clusters as opposed to courses. So in the Algebra cluster, there will be standards for both Algebra 1 and 2. Explain that how these standards become courses will be determined at a later date as part of the CA Common Core implementation plan. This also will be discussed later in more detail. As with ELA, the Math CCS provide clear signposts toward the goal of college and career readiness.

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 Common Core Standards 7 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: These bullets come directly from the Common Core introduction-pages 1-5 (this piece was not adopted by CA but may show up in the framework). Emphasize that these are content not pedagogy standards. Instructor notes: These bullets come directly from the Common Core introduction-pages 1-5 (this piece was not adopted by CA but may show up in the framework). Emphasize that these are content not pedagogy standards.

8. Common Core Standards for Mathematics: Two Types Mathematical Practice (recurring throughout the grades) Mathematical Content (different at each grade level) 8 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: There are two types of standards found in the Common Core Standards for mathematics. One set is for Mathematical Practice. This set is the same across all grade levels and conceptual clusters. The other set is the Mathematical Content standards and these are different for each grade and conceptual cluster. Instructor notes: There are two types of standards found in the Common Core Standards for mathematics. One set is for Mathematical Practice. This set is the same across all grade levels and conceptual clusters. The other set is the Mathematical Content standards and these are different for each grade and conceptual cluster.

9. Common Core Standards: Mathematical Proficiency 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 Teacher Overview Instructor notes: One of the biggest changes to our standards is the Common Core Standards for Mathematical Practice. These are a set of eight practices which describe the varieties of expertise that educators should seek to develop in their students. These also carry across all grade levels. These also relate to the idea of balanced as defined by the CA Mathematics Framework. The next three slides will analyze the practices through the lens of the three components of a balanced math program. Instructor notes: One of the biggest changes to our standards is the Common Core Standards for Mathematical Practice. These are a set of eight practices which describe the varieties of expertise that educators should seek to develop in their students. These also carry across all grade levels. These also relate to the idea of balanced as defined by the CA Mathematics Framework. The next three slides will analyze the practices through the lens of the three components of a balanced math program.

10. Mathematical Proficiency as defined by the California Framework 10 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: The three components of a balanced math program are: Computational/Procedural skills, Conceptual Understanding and Problem Solving. Students must be able to do all three to be truly mathematically proficient. Since this has been the model for proficiency for many years it seemed best to review it in order to compare it to the Practices.Instructor notes: The three components of a balanced math program are: Computational/Procedural skills, Conceptual Understanding and Problem Solving. Students must be able to do all three to be truly mathematically proficient. Since this has been the model for proficiency for many years it seemed best to review it in order to compare it to the Practices.

11. Standards for Mathematical Practice� � �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 Teacher Overview Instructor notes: This is an animated slide with particular words being highlighted. It is important to point out the use of the word �practitioners� to describe the math student. The expectation is that students will be �doing� mathematics. Another focus is on the engagement with the tasks. The practices are asking for a deeper interaction with the content and this should be noted. The last emphasis underscores the idea that all students from elementary through high school can demonstrate these skills but at the appropriate maturity level.Instructor notes: This is an animated slide with particular words being highlighted. It is important to point out the use of the word �practitioners� to describe the math student. The expectation is that students will be �doing� mathematics. Another focus is on the engagement with the tasks. The practices are asking for a deeper interaction with the content and this should be noted. The last emphasis underscores the idea that all students from elementary through high school can demonstrate these skills but at the appropriate maturity level.

12. Standards for Mathematical PracticeMathematically proficient students: 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 Teacher Overview Instructor notes: The mathematical practices are on the next two slides. Underneath is a short statement highlighting some of the information from the paragraph that describes each practice. Explain that there is a strong emphasis on student problem solving, reasoning and �practicing� mathematics. Instructor notes: The mathematical practices are on the next two slides. Underneath is a short statement highlighting some of the information from the paragraph that describes each practice. Explain that there is a strong emphasis on student problem solving, reasoning and �practicing� mathematics.

13. 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 PracticeMathematically proficient students: 13 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: See notes on slide 12 It might be important to note that the tools listed under # 5 practice include: paper and pencil, concrete models, ruler, protractor, calculator, spreadsheet, computer algebra system, statistical package, dynamic geometry software and digital content located on a website. If time, have participants discuss the implications of these practices for both teaching and assessment. Instructor notes: See notes on slide 12 It might be important to note that the tools listed under # 5 practice include: paper and pencil, concrete models, ruler, protractor, calculator, spreadsheet, computer algebra system, statistical package, dynamic geometry software and digital content located on a website. If time, have participants discuss the implications of these practices for both teaching and assessment.

14. Locate the Mathematical Practices With a partner compare the eight mathematical practices to the three components of a balanced math program as defined by the California Framework Which practices align best to Conceptual Understanding? Computation and Procedures? Problem Solving? Be ready to share out with the entire group. Try It!Mathematical Practices 14 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Distribute a copy of the CCSS to the participants. Have them find the eight mathematical practices. If time is available give them a few minutes to read the entire description for each practice. Have the participants compare the practices to the three components of the Venn Diagram by discussing where each practice might sit in the diagram. It is important to point out that the practices go much deeper in defining mathematical proficiency but that most fit into some part of the Venn Diagram descriptions of balance. Have a few groups share their findings. Follow-up question What implications do these practices have on your current instruction? Instructor notes: Distribute a copy of the CCSS to the participants. Have them find the eight mathematical practices. If time is available give them a few minutes to read the entire description for each practice. Have the participants compare the practices to the three components of the Venn Diagram by discussing where each practice might sit in the diagram. It is important to point out that the practices go much deeper in defining mathematical proficiency but that most fit into some part of the Venn Diagram descriptions of balance. Have a few groups share their findings. Follow-up question What implications do these practices have on your current instruction?

15. 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 Connecting Practices to Content 15 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: It is important to point out that despite the emphasis on reasoning and problem solving the practices maintain a balanced combination of procedure and understanding. This should not be perceived as another swing of the pendulum but a focus on both skills and understanding. Explain that the expectations in the standards that begin with the word �understand� are considered opportunities to connect practice to content. Students who lack understanding may rely too heavily on procedures. According to the CCS document, page 2, �[The Understand expectations] are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development and student achievement in mathematics.� As stated earlier the standards build in complexity and provide more clarity for expected performance. An example of providing more clarity is found on the next slide. Instructor notes: It is important to point out that despite the emphasis on reasoning and problem solving the practices maintain a balanced combination of procedure and understanding. This should not be perceived as another swing of the pendulum but a focus on both skills and understanding. Explain that the expectations in the standards that begin with the word �understand� are considered opportunities to connect practice to content. Students who lack understanding may rely too heavily on procedures. According to the CCS document, page 2, �[The Understand expectations] are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development and student achievement in mathematics.� As stated earlier the standards build in complexity and provide more clarity for expected performance. An example of providing more clarity is found on the next slide.

16. 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 Connecting Practices to Content: Example 16 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is an example from first grade that demonstrates the clarity for the understanding place value grade one performance. Explain that there is a heavy emphasis on number in the primary grades. Because of this focus on clarity there may be more standards in certain grades than in the CA standards to support the details in the topic. For example, in Kindergarten there were 18 standards and in the CA CCS (CCSS) there are 26. However, with the CA standards 33% have to do with number, while in the CCSS over 57% are number related. Instructor notes: This is an example from first grade that demonstrates the clarity for the understanding place value grade one performance. Explain that there is a heavy emphasis on number in the primary grades. Because of this focus on clarity there may be more standards in certain grades than in the CA standards to support the details in the topic. For example, in Kindergarten there were 18 standards and in the CA CCS (CCSS) there are 26. However, with the CA standards 33% have to do with number, while in the CCSS over 57% are number related.

17. 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 Grade K-8 Standards 17 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This slide begins the discussion on the organization of the standards The format for each grade level, K-8, is the same. The overview page lists the domains, clusters and mathematical practices The standards are listed by grade level. Standards that relate form clusters. Clusters that relate form domains. The additional standards or language is bolded and underlined. Instructor notes: This slide begins the discussion on the organization of the standards The format for each grade level, K-8, is the same. The overview page lists the domains, clusters and mathematical practices The standards are listed by grade level. Standards that relate form clusters. Clusters that relate form domains. The additional standards or language is bolded and underlined.

18. 18 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is a screen shot that shows what the domains, clusters and standards look like on the actual page. If participants have the document explain that they will have a chance to investigate a grade level or levels with the next few slides. Instructor notes: This is a screen shot that shows what the domains, clusters and standards look like on the actual page. If participants have the document explain that they will have a chance to investigate a grade level or levels with the next few slides.

19. K-8 Grade Section Overview Page 19 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is a screen shot of the grade 3 overview page. Explain that this page lists only the domains, clusters and practices. The practices are the same for each grade level and the high school clusters. If they have the document have them turn to this page as you explain the organization. Instructor notes: This is a screen shot of the grade 3 overview page. Explain that this page lists only the domains, clusters and practices. The practices are the same for each grade level and the high school clusters. If they have the document have them turn to this page as you explain the organization.

20. Grade 2 Example 20 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is a screen shot of the first standards page for grade 2. Point out the domain name and its symbol: Operations and Algebraic Thinking 2.OA; the cluster names and the standards. Also draw their attention to the two added standards which are bolded and underlined. Instructor notes: This is a screen shot of the first standards page for grade 2. Point out the domain name and its symbol: Operations and Algebraic Thinking 2.OA; the cluster names and the standards. Also draw their attention to the two added standards which are bolded and underlined.

21. Locate one grade level in your standards handout Find Introduction Domains Clusters Standards At your table, pick a grade level and have each person or partner group choose a domain and read. Share out reactions to language, content and structure. Share out with the large group. Try It!Introducing the Standards 21 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Have participants pick a grade level from K-8 and spend a few minutes looking through the entire grade. Then, in table groups, either individually or in pairs, have them take a domain and read. Have them share out reactions to language, content, structure, etc. at their table. Have a few of the table groups share out to the large group.Instructor notes: Have participants pick a grade level from K-8 and spend a few minutes looking through the entire grade. Then, in table groups, either individually or in pairs, have them take a domain and read. Have them share out reactions to language, content, structure, etc. at their table. Have a few of the table groups share out to the large group.

22. California Comparison 22 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This chart shows the differences between the strands of the California Standards and the domains of the California Common Core Standards. A more graphic display of CCSS across grade levels is on the next slide. Instructor notes: This chart shows the differences between the strands of the California Standards and the domains of the California Common Core Standards. A more graphic display of CCSS across grade levels is on the next slide.

23. 23 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This chart illustrates the domains included at each grade level in kindergarten through grade eight. The domain Counting and Cardinality is only included in kindergarten while Number and Operations-Fractions is included in grades three through five. The remaining domains: Operations and Algebraic Thinking, Number and Operations in Base Ten, Measurement and Data, and Geometry are all included in kindergarten through grade five. Notice that Ratios and Proportional Relationships is only included in grades six and seven while Functions is only in grade eight. Instructor notes: This chart illustrates the domains included at each grade level in kindergarten through grade eight. The domain Counting and Cardinality is only included in kindergarten while Number and Operations-Fractions is included in grades three through five. The remaining domains: Operations and Algebraic Thinking, Number and Operations in Base Ten, Measurement and Data, and Geometry are all included in kindergarten through grade five. Notice that Ratios and Proportional Relationships is only included in grades six and seven while Functions is only in grade eight.

24. Ties Between Domains: Example 24 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This chart shows how a number of domains connect across grade levels. Have participants look at the chart and describe the connections they see. What you would like them to notice is how properties of operations is woven into each standard and that this is an example of how key ideas are articulated over time. It is also important to point out that the verbs move from understand to use to apply thus connecting back to the mathematical practices. Instructor notes: This chart shows how a number of domains connect across grade levels. Have participants look at the chart and describe the connections they see. What you would like them to notice is how properties of operations is woven into each standard and that this is an example of how key ideas are articulated over time. It is also important to point out that the verbs move from understand to use to apply thus connecting back to the mathematical practices.

25. Choose a domain that covers at least two grade levels and read the standards at each grade level. What do you notice? What big ideas are repeated Be prepared to share key findings with the group. Try It!Trace across Domains 25 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Direct participants to choose a domain that covers at least two grade levels. Facilitate the selection of different domains to trace by encouraging selection from different grade spans. Have them discuss the questions and be ready to share out.Instructor notes: Direct participants to choose a domain that covers at least two grade levels. Facilitate the selection of different domains to trace by encouraging selection from different grade spans. Have them discuss the questions and be ready to share out.

26. 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) 26 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: It is important to reiterate the CCSS focus on arithmetic and fluency with whole numbers in the early grades. The kindergarten through grade five standards provide students with a solid foundation in whole numbers arithmetic (addition, subtraction, multiplication and division), fractions, and decimals. Mastery of these skills prepares students for learning more advanced concepts and procedures in later grades. Here are two standards that explicitly call for the use of concrete models or drawings. Instructor notes: It is important to reiterate the CCSS focus on arithmetic and fluency with whole numbers in the early grades. The kindergarten through grade five standards provide students with a solid foundation in whole numbers arithmetic (addition, subtraction, multiplication and division), fractions, and decimals. Mastery of these skills prepares students for learning more advanced concepts and procedures in later grades. Here are two standards that explicitly call for the use of concrete models or drawings.

27. 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) 27 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: There is also a clear direction for numerical fluency in the CCSS. Here are selected standards for grades 3 and 5. By the time students exit grade 5, they should be using algorithms to manipulate numbers fluently. The CCSS build upon practices of countries with high achievement in mathematics.Instructor notes: There is also a clear direction for numerical fluency in the CCSS. Here are selected standards for grades 3 and 5. By the time students exit grade 5, they should be using algorithms to manipulate numbers fluently. The CCSS build upon practices of countries with high achievement in mathematics.

28. 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) 28 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Included in the CCSS is a clear and articulated sequence for the development of fractions. Student mastery of the conceptual and procedural knowledge about fractions are essential to success in algebra. In grade three, students begin to develop an understanding of fractions as numbers and represent fractions on a number line diagram. Addition and subtraction of fractions are introduced in grade four, and multiplication and division in grade five. Instructor notes: Included in the CCSS is a clear and articulated sequence for the development of fractions. Student mastery of the conceptual and procedural knowledge about fractions are essential to success in algebra. In grade three, students begin to develop an understanding of fractions as numbers and represent fractions on a number line diagram. Addition and subtraction of fractions are introduced in grade four, and multiplication and division in grade five.

29. 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) 29 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor note: Additional audience participation (as time allows) Let�s take a few minutes to look at one of the CCSS on fractions. Working with the fractions pairs: 7/9 and 4/9 (same denominator) 4/9 and 4/7 (same numerator) Take a few minutes to review the standard listed here. Discuss with a neighbor your thoughts about how to compare these pairs of fractions using a visual model. Instructor note: Answer 7/9 > 4/9 and 4/7 > 4/9. May use a rectangle divided in fractional parts (same denominator); a number line from 0 to 1 (same numerator � 3.NF. 2a-b) Instructor note: Additional audience participation (as time allows) Let�s take a few minutes to look at one of the CCSS on fractions. Working with the fractions pairs: 7/9 and 4/9 (same denominator) 4/9 and 4/7 (same numerator) Take a few minutes to review the standard listed here. Discuss with a neighbor your thoughts about how to compare these pairs of fractions using a visual model. Instructor note: Answer 7/9 > 4/9 and 4/7 > 4/9. May use a rectangle divided in fractional parts (same denominator); a number line from 0 to 1 (same numerator � 3.NF. 2a-b)

30. Fraction Concepts 30 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: The next two slides are optional depending on the time and audience discussion and provide an example of how to use a number line to compare these fractions. This is an image of 4/9 on the number line. Notice the � mark on the slide; 4/9 is smaller than �. Instructor notes: The next two slides are optional depending on the time and audience discussion and provide an example of how to use a number line to compare these fractions. This is an image of 4/9 on the number line. Notice the � mark on the slide; 4/9 is smaller than �.

31. Fraction Concepts 31 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is an image of 4/7 on the number line. Notice the � mark on the slide; 4/7 is greater than �. If time, you might illicit other ways of determining which fraction is bigger. This model compares the two fraction to � as a way to determine which is bigger. Instructor notes: This is an image of 4/7 on the number line. Notice the � mark on the slide; 4/7 is greater than �. If time, you might illicit other ways of determining which fraction is bigger. This model compares the two fraction to � as a way to determine which is bigger.

32. Goal for 8th 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 8th grade Common Core, high school Algebra content cluster and CA Algebra standards 8th 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 California Grade 8 Options 32 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: The CCSS are consistent with the goal that all students succeed in Algebra 1. Students who master the content and skills through grade seven will be well-prepared for algebra in grade eight. Recognizing that all students must continue their study of mathematics, the CCSS move students forward with grade eight standards that prepare them for higher mathematics, including Algebra 1. Instructor notes: The CCSS are consistent with the goal that all students succeed in Algebra 1. Students who master the content and skills through grade seven will be well-prepared for algebra in grade eight. Recognizing that all students must continue their study of mathematics, the CCSS move students forward with grade eight standards that prepare them for higher mathematics, including Algebra 1.

33. California Algebra 1 33 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is a screen shot of CCCS Algebra 1. Since this course was not identified in the original CCS document, it was created by the commission using standards from: Grade 8 Common Core CA Algebra 1 Common Core Algebra Conceptual Cluster Since it was considered an addition, all standards are bolded and underlined.Instructor notes: This is a screen shot of CCCS Algebra 1. Since this course was not identified in the original CCS document, it was created by the commission using standards from: Grade 8 Common Core CA Algebra 1 Common Core Algebra Conceptual Cluster Since it was considered an addition, all standards are bolded and underlined.

34. Locate the 8th grade and Algebra 1 standards in your standards handout. Identify the domains in each set of standards Which are the same? Different? Read the ends of the Algebra 1 standards? Where did these standards come from? Why are they all underlined and bolded? What implications does this choice for grade 8 mathematics have on your school/district? Try It!Options for Grade 8 34 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Have participants find both the Algebra 1 and 8th grade standards in their handout. Have them compare the domains in both sets of standards. Since there will be some 8th grade standards in Algebra 1 it is important to point out that there will be some similarities and differences in the domain names. Make sure that they understand that the large titles in Algebra 1, i.e. Number and Quantity, Algebra, Functions, Geometry and Statistics and Probability are not domains but titles taken from different parts of the CCS. Make sure that they understand that Algebra 1 standards come from different places: 8th grade Common Core Algebra Content Cluster in the High School Common Core CA Algebra 1 standards Because Algebra 1 was configured by the CA commission the standards are all bolded and underlined. Instructor notes: Have participants find both the Algebra 1 and 8th grade standards in their handout. Have them compare the domains in both sets of standards. Since there will be some 8th grade standards in Algebra 1 it is important to point out that there will be some similarities and differences in the domain names. Make sure that they understand that the large titles in Algebra 1, i.e. Number and Quantity, Algebra, Functions, Geometry and Statistics and Probability are not domains but titles taken from different parts of the CCS. Make sure that they understand that Algebra 1 standards come from different places: 8th grade Common Core Algebra Content Cluster in the High School Common Core CA Algebra 1 standards Because Algebra 1 was configured by the CA commission the standards are all bolded and underlined.

35. Mathematics Standards for High School 35 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: The High School standards are listed by conceptual cluster, not by course. The structure of domain, cluster and standard is the same as in K-8. Instructor notes: The High School standards are listed by conceptual cluster, not by course. The structure of domain, cluster and standard is the same as in K-8.

36. 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 Mathematics Standards for High School 36 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Explain that the high school standards specify the math that all students should study to be college and career ready. The standards also identify additional standards that students should learn in order to take advanced courses such as calculus, advanced statistics and discrete mathematics. These are indicated by (+). The standards commission added two California courses, Calculus and Advanced Placement Statistics and Probability, to the CCS. Development of suggested course descriptions will be done by CDE as part of their long-range implementation plan. It is expected to include pathways for both traditional and integrated courses. Instructor notes: Explain that the high school standards specify the math that all students should study to be college and career ready. The standards also identify additional standards that students should learn in order to take advanced courses such as calculus, advanced statistics and discrete mathematics. These are indicated by (+). The standards commission added two California courses, Calculus and Advanced Placement Statistics and Probability, to the CCS. Development of suggested course descriptions will be done by CDE as part of their long-range implementation plan. It is expected to include pathways for both traditional and integrated courses.

37. 37 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This is a screen shot of a page from the Geometry content cluster. Notice that under the cluster: Translate between the geometric description and the equation for a conic section, all students would be expected to master standards 1 and 2. Standard 3 reflects mathematics that students pursuing advanced courses in mathematics should study, as indicated by the (+) symbol. More information about model course pathways for high school mathematics can be found at www.corestandards.org. This information, however, was not adopted as part of the CCCS document. Instructor notes: This is a screen shot of a page from the Geometry content cluster. Notice that under the cluster: Translate between the geometric description and the equation for a conic section, all students would be expected to master standards 1 and 2. Standard 3 reflects mathematics that students pursuing advanced courses in mathematics should study, as indicated by the (+) symbol. More information about model course pathways for high school mathematics can be found at www.corestandards.org. This information, however, was not adopted as part of the CCCS document.

38. 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 (?) Mathematics Standards for High School 38 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: The example above is from the Geometry standards. Point out the star (?) at the end of each standard. More information about this can be found on page 45 which is the overview page for the Mathematics Standards for High School and pages 59-60 which are the overview pages for the Modeling Cluster.Instructor notes: The example above is from the Geometry standards. Point out the star (?) at the end of each standard. More information about this can be found on page 45 which is the overview page for the Mathematics Standards for High School and pages 59-60 which are the overview pages for the Modeling Cluster.

39. Turn to the High School Conceptual Clusters Choose one of the clusters and identify the (+) and (?) standards What do you notice about them? Discuss: How does this organization of standards work in the context of your existing high school structure? What implications do these standards have on your instruction? Try It!Explore the High School Conceptual Clusters 39 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Have participants turn to the High School mathematics section. Have each table group choose a cluster and identify both the (+) and (?) standards. Each group should discuss their own cluster first then be ready to share out with the large group. Because the standards are not written in courses, it is important for participants to discuss how these standards and their organizations might be taught in their own school setting. Instructor notes: Have participants turn to the High School mathematics section. Have each table group choose a cluster and identify both the (+) and (?) standards. Each group should discuss their own cluster first then be ready to share out with the large group. Because the standards are not written in courses, it is important for participants to discuss how these standards and their organizations might be taught in their own school setting.

40. Some comparison examples 40 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: There are many more similarities than differences between the CCS and California standards. The following three slides are a small sampling of the similarities. Give participants some time to review.Instructor notes: There are many more similarities than differences between the CCS and California standards. The following three slides are a small sampling of the similarities. Give participants some time to review.

41. Some comparison examples 41 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: See notes from slide 35.Instructor notes: See notes from slide 35.

42. Some comparison examples 42 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: See notes from slide 35.Instructor notes: See notes from slide 35.

43. Grade Shifts: Examples 43 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Although the two sets of standards are very similar, there are some topics that will be taught at different grades. Here are some examples of topics moving both up and down one or more grade levels. Notice that the introduction to the probability of chance has moved from grade 3 in the 1997 standards to grade 7 in the CCSS. The introduction of fractions as numbers moves from grade two to grade three. Although introduced later, the CCSS addresses the development of fractions in a very focused and coherent manner. Instructor notes: Although the two sets of standards are very similar, there are some topics that will be taught at different grades. Here are some examples of topics moving both up and down one or more grade levels. Notice that the introduction to the probability of chance has moved from grade 3 in the 1997 standards to grade 7 in the CCSS. The introduction of fractions as numbers moves from grade two to grade three. Although introduced later, the CCSS addresses the development of fractions in a very focused and coherent manner.

44. 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 California�s standards? California�s Additional 15% 44 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: This slide shows the central questions used in determining California�s additional 15%. These are similar to ELA. The following slides are a sampling of how the standards were included.Instructor notes: This slide shows the central questions used in determining California�s additional 15%. These are similar to ELA. The following slides are a sampling of how the standards were included.

45. 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%: 45 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: One addition was to add standards. Instructor notes: One addition was to add standards.

46. Added language to existing standard Grade 2-Measurement and Data Grade 4-Geometry Examples of Additional 15%: 46 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Some additions were simply added language to clarify. Instructor notes: Some additions were simply added language to clarify.

47. Added a substantial section to an existing cluster Grade 6-The Number System Examples of Additional 15%: 47 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: In some grade level/conceptual clusters a substantial section was added to an existing cluster.Instructor notes: In some grade level/conceptual clusters a substantial section was added to an existing cluster.

48. Added two courses from California Standards: Calculus Advanced Placement Probability and Statistics Examples of Additional 15%: 48 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: Another addition was the inclusion of two courses from the CA standards: Calculus and Advanced Placement Probability and Statistics. Instructor notes: Another addition was the inclusion of two courses from the CA standards: Calculus and Advanced Placement Probability and Statistics.

49. 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 What Now? 49 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: It is important that until there is further direction from CDE, districts should stay the course. The current work of teaching a balanced math program, using quality formative assessments and providing opportunities for teachers to collaboratively plan should continue and be supported. Instructor notes: It is important that until there is further direction from CDE, districts should stay the course. The current work of teaching a balanced math program, using quality formative assessments and providing opportunities for teachers to collaboratively plan should continue and be supported.

50. Websites Common Core Standards: www.corestandards.org California Common Core Standards: Visit the California Department of Education�s 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 Wrap-Up and Questions 50 � 2011 California County Superintendents Educational Services Association � Mathematics Teacher Overview Instructor notes: These are the current URLs for both the California Common Core State Standards and the original Common Core Standards. Instructor notes: These are the current URLs for both the California Common Core State Standards and the original Common Core Standards.