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**California Common Core Standards for Mathematics**

Overview Instructor notes: The audience for this power point is administrators and teachers and serves as a beginning overview for educators. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Objectives General Overview Structure Similarities Shifts Next Steps**

Focus and Coherence Mathematical Proficiency Structure Organization Grade 8 Options Similarities Shifts Next Steps Instructor notes: This is going to be a brief overview of the Common Core State Mathematics Standards for California (CCSS) so that you will be familiar with the organization and content of the standards. 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Common Core Standards Overview: Toward Greater Focus and Coherence**

Avoid the problem of “mile wide and an inch deep” Avoid the problem of “mile wide and an inch deep” Aim for clarity and specificity 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Coherence Design 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 Instructor Notes: Schmidt and Houang* (2002) say that standards are coherent if they are: Articulated over time Reflect he 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 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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Common Core Standards 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 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Common Core Standards 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 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Common Core Standards 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 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Common Core Standards for Mathematics: Two Types**

Mathematical Practice (recurring throughout the grades) Mathematical Content (different at each grade level) Instructor notes: One of the biggest changes to our standards is the Common Core Standards for Mathematical Proficiency 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**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 Instructor notes: One of the biggest changes to our standards is the Common Core Standards for Mathematical Proficiency 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Mathematical Proficiency as defined by the California Framework**

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

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**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.” 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**1. Make sense of problems and persevere in solving them **

Standards for Mathematical Practice Mathematically 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 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. Have the participants compare the practices to the three components of the Venn Diagram by discussing where each practice might sit in the diagram. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**5. Use appropriate tools strategically **

Standards for Mathematical Practice Mathematically proficient students: 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 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Connecting Practices to Content**

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

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**Connecting Practices to Content: Example**

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

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**Grade K-8 Standards Overview page Standards-by grade level Clusters**

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

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

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**K-8 Grade Section Overview Page**

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

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**Grade 2 Example 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. Have participants pick a grade level from K-8 and spend a few minutes looking through the entire grade. Ask them to be ready to share out their reactions to language, content, structure, etc. to the whole group. If you want, you may direct them to a specific grade level. For example, in third grade, the initial development on fractions is very interesting to participants and brings up many points about content knowledge of teachers, instructional materials, assessment, etc. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**California Comparison**

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

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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 are only included in grades six and seven while Functions is only in grade eight. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Ties Between Domains: Example**

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

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

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

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

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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) 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) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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Fraction Concepts 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 ½ mark on the slide; 4/9 is smaller than ½. Source: © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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Fraction Concepts Instructor notes: This is an image of 4/7 on the number line. Notice the ½ mark on the slide; 4/7 is greater than ½. If time, you might illicit other ways of determining which fraction is bigger. This model compares the two fraction to ½ as a way to determine which is bigger. Source: © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**California Grade 8 Options**

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

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

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**Mathematics Standards for High School**

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

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**Mathematics Standards for High School**

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 Instructor notes: Explain that the high school standards specify the math the 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**High School Example-Geometry Content Cluster**

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 This information, however, was not adopted as part of the CCCS document. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Mathematics Standards for High School**

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 (★) 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 which are the overview pages for the Modeling Cluster. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Some comparison examples**

Grade California Standard Common Core Kindergarten Use 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. First Count, read, and write whole numbers to 100. Count to 120, starting at any number less than In this range, read and write numerals and represent a number of objects with a written numeral. Third Memorize 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. Instructor notes: Now that the structure of the K-12 standards has been explained, it is important to point out that there are many more similarities than differences between the CCS and California standards. The following three slides are a small sampling of the similarities starting with Kindergarten and moving through to high school. Give participants some time to review. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Some comparison examples**

Grade California Standard Common Core Fifth Understand 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) Sixth Interpret 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. Seventh Use 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. Instructor notes: See notes from slide 35. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Some comparison examples**

Grade/Course California Standard Common Core Seventh Construct 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. Algebra Algebra 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. Geometry Geometry: 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. Instructor notes: See notes from slide 35. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Grade Shifts: Examples**

Concept 1997 Standards CCCS Compose simple shapes to form larger shapes (e.g., 2 triangles to form a rectangle) Grade 2 K Introduction to Probability 3 7 Introduction of fractions as numbers Add and subtract simple fractions 4 Introduction of integers 6 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 will move 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview Developed by SCFIRD © 2011 California County Superintendents Educational Services Association

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**California’s Additional 15%**

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

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**Examples of Additional 15%:**

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 Instructor notes: One addition was to add standards. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Examples of Additional 15%:**

Added language to existing standard Grade 2-Measurement and Data Grade 4-Geometry Instructor notes: Some additions were simply added language to clarify. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Examples of Additional 15%:**

Added a substantial section to an existing cluster Grade 6-The Number System Instructor notes: In some grade level/courses a substantial section was added to an existing cluster. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Examples of Additional 15%:**

Added two courses from California Standards: Calculus 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**What Now? 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 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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**Wrap-Up and Questions Websites**

Common Core Standards: California Common Core Standards: Visit the California Department of Education’s Common Core State Standards Web page at: The standards Frequently asked questions Informational flyers Additional resources 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. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview © 2011 California County Superintendents Educational Services Association

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