1. Common Core State Standards K-5 Mathematics

2. Where Are We Going? We can create a process in which all district stakeholders will be ready to implement the Common Core/Essential Standards with fidelity by Fall 2012-2013. Summer Institute Administrator & Lead Teacher Overview Lead Teacher Training Teacher Overview Teacher Training

3. North Carolina Professional Teaching Standards Standard III: Teachers know the content they teach. Standard IV: Teachers facilitate learning for their students. Standard V: Teachers reflect on their practice.

4. Our Goals For Today Look at Standards for Mathematical Practice Understand how to read grade level standards Look at support documents from DPI See a sample flow of progression on fractions from K-5 Grade level sharing of Common Core Sample Lessons Teacher planning time for the implementation of the Common Core Standards

5. Standards Are A Platform For Instructional Systems This is a new platform for better instructional systems and better ways of managing instruction. –Builds on achievements of last 2 decades –Builds on lessons learned in last 2 decades –Lessons about time and teachers Phil Daro, NCCTM 2010

6. Intentional Design Limitations The Standards do NOT define: How teachers should teach. All that can or should be taught. The nature of advanced work beyond the Core. The interventions needed for students well below grade level. The full range of support for English Language Learners and students with special needs.

7. Research By third grade nearly half the students still do not “get” this concept. 16

8. By fourth grade over half the students still do not “get” this concept. A number contains 18 tens, 2 hundreds, and 4 ones. What is that number? 1824 2824 218.4 384 Common Core State Standards begin to specifically address these misunderstandings in Kindergarten and First Grade.

9. 8 + 4 = [ ] + 5 Think for a minute about your answer to this problem, and what students in first through sixth grades might think the answer is.

10. 8 + 4 = [ ] + 5 Percent Responding with Answers Grade71217 12 & 17 1 st -2 nd 5%58%13%8% 3 rd -4 th 9%49%25%10% 5 th -6 th 2%76%21%2%

11. Kindergarten Number and Operations in Base Ten K.NBT Work with numbers 11–19 to gain foundations for place value. First Grade Number and Operations in Base Ten 1.NBT Understand place value Use place value understanding and properties of operations to add and subtract.

12. Number and Operations in Base Ten 1.NBT Understand Place Value Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: a. a. 10 can be thought of as a bundle of ten ones —called a ten. b. b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. c. c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

13. Understanding and Foundation The K-5 standards emphasize building a strong understanding and foundation in: whole numbers additionsubtractionmultiplicationdivision fractions and decimals Building this foundation will help to apply more demanding math concepts and procedures, and move into applications. The students will also explore geometry, measurement, and data, including: graphs (e.g. bar charts, line plots) geometric measurement (e.g. area, volume, angles)

14. Common Core State Standards Standards for Mathematical Practice –Carry across all grade levels (K-12) –Describe habits of mind of a mathematically expert student Standards for Mathematical Content –K-8 standards presented by grade level –Organized into domains that progress over several grades –Grade introductions give 2–4 focal points at each grade level

15. Standards for Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

16. Taking a Closer Look (1) Think about your grade level. (2) Make a chart (with all grades represented) showing each person’s interpretation of the standard. What does this standard mean to you?

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

18. 2. Reason abstractly and quantitatively.

19. 3. Construct viable arguments and critique the reasoning of others.

20. 4. Model with mathematics.

21. 5. Use appropriate tools strategically

22. 6. Attend to precision.

23. 7. Look for and make use of structure.

24. 8. Look for and express regularity in repeated reasoning.

25. Standards for Mathematical Practices What do teachers do in their classrooms to cause these effects on students? Using the cause and effect chart, think of you as the cause, and the student as the effect. Describe things that you do in your classroom to get these desired effects.

26. Math Talks: Using Classroom Discussions Math Talks help students develop their mathematical understanding and conceptual knowledge. Students are asked to solve problems, explain their solutions, answer questions, and justify their answers. “ Helping children verbally work through math problems and articulate problem solving is an important aspect of their understanding.” Dr. Paola Sztajn, Professor of Mathematics at NC State University

27. Elements of Gradual Transformation: Beginning to Use Math Talks 1. 1. A shift from teacher as sole questioner to both children and teacher as questioners. 2. 2. Children increasingly explaining and articulating their ideas. 3. 3. A shift from teacher as the source of all math ideas to children’s ideas also influencing the direction of lessons. 4. 4. Children increasingly taking responsibility for learning and for the evaluation of themselves and others. 5. 5. Increasing amounts of child-to-child talks with teacher guidance as needed.

28. Standards for Mathematical Content K-8 standards presented by grade level Organized into domains that progress over several grades Grade introductions give 2–4 focal points at each grade level

29. How To Read the State Standards Standards define what students should understand and be able to do. Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Domains are larger groups of related standards. Standards from different domains may sometimes be closely related.

30. Design Domains Counting and Cardinality (Kindergarten CC) Operations and Algebraic Thinking (OA) Number and Operations in Base Ten (NBT) Number and Operations Fractions (3 rd -5 th NF) Measurement and Data (MD) Geometry (G)

31. How to read the grade level standards Domains are larger groups of related standards. Standards from different domains may sometimes be closely related. Standards define what students should understand and be able to do. Domain Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Number and Operations in Base Ten 3.NBT Domain Use place value understanding and properties of operations to perform multi-digit arithmetic. Bold Heading for Standards 1. Use place value understanding to round whole numbers to the nearest 10 or 100. 2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Cluster of Standards

32. Support Documents Unpacking/Unpacked Content Common Core Crosswalk Document Common Core Standards By Grade Level

33. Unpacking/Unpacked Content A response for each standard to the question ―What does this standard mean? The unpacked content describes carefully and specifically what the standards mean a child will understand and be able to do. It explains the different knowledge or skills that constitute that standard.

34. Unpacking http://www.dpi.state.nc.us/acre/standards/support-tools/

35. CONTENT APPEARING TO BE THE SAME MAY ACTUALLY BE DIFFERENT!! The CCSS Requires CLOSE Reading!!! Standards may seem similar but have very different expectations.

36. Crosswalk

37. The Crosswalk is meant to provide a starting point from which educators can organize and begin the necessary discussions about their local curricula. Score Points and Descriptors 3 The concepts and skills of the North Carolina Math Standard Course of Study (NC ELA SCOS) are strongly aligned to the concepts and skills in the Math Common Core State Standards. 2 The concepts and skills of the NC Math SCOS are reasonably aligned to the concepts and skills in the Math CCSS. 1 The concepts and skills of the NC Math SCOS are minimally aligned to the concepts and skills in the Math CCSS. NE It is a new expectation found in the Math CCSS. Crosswalk Rubric

38. Grade Specific Standards K−12 Standards: Are grade-specific end-of-year expectations. Are developmentally appropriate. mathematical understanding through the grade levels. Are a cumulative progression of mathematical understanding through the grade levels.

39. Critical Areas The Common Core identifies 3 – 5 Critical Areas of Focus for each grade level. These appear at the beginning of each grade level.

40. Third Grade Critical Area Instructional time should focus on four critical areas: (1)developing understanding of multiplication and division and strategies for multiplication and division within 100 (2)developing understanding of fractions, especially unit fractions (fractions with numerator 1) (3)developing understanding of the structure of rectangular arrays and of area; and (4)describing and analyzing two-dimensional shapes.

41. Don’t Forget the Examples Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. K.OA.1. Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. K.OA.2. 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.3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). K.OA.4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. K.OA.5. Fluently add and subtract within 5.

42. Fractions Fractions are a rich part of mathematics, but we tend to manipulate fractions by rote rather than try to make sense of the concepts and procedures. Researchers have concluded that this complex topic causes more trouble for students than any other area of mathematics. Why???? Bezuk and Bieck 1993

43. Progression of Fractions K-5

44. Kindergarten Fractions Kindergarten- What do they do with fractions? Identify, describe, analyze, compare, create and compose shapes. (In later grades this leads to understanding that 3 triangles can make a trapezoid. One triangle is 1/3 of a trapezoid.) Place value and the understanding of a whole will provide a basis for the understanding of fractions.

45. First Grade 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Second Grade 2.G.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Third Grade 3.G.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. 3.NF.1-3 Number and Operations— Fractions Develop understanding of fractions as numbers.

46. Fourth Grade Fractions 4.NF. 1-7 Fourth- What do they do with fractions? 1.Explain why a fraction a/b is equivalent to a fraction (n ×a)/(n ×b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 3.Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

47. Fifth Grade Fractions 5.NF. 1-7 Fifth- What do they do with fractions? 1. Use equivalent fractions as a strategy to add and subtract fractions. 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. Making sense 2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? Video help Video help

48. Fifth Continued 4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5. Use equivalent fractions as a strategy to add and subtract fractions. 6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g. by using visual fraction models or equations. 7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Solve real world problems involving division of unit fractions by non- zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? Multiplying Fractions Multiplying Fractions Dividing Fractions Dividing Fractions

49. Before, During and After REMEMBER Math lessons need a before, during and after too!! Just like guided reading!! 1. Activate prior knowledge 2. What is the task?? 3. Have a discussion that is mostly lead by students

50. What are the features of a good math task?

51. What are Features of a Good Math Task? It begins where the students are; accessible to a wide range of learners. It is seen as something to make sense of. It requires justifications and explanations for answers and methods. The focus is on making sense of the mathematics involved and thereby increasing understanding. It challenges the learners to think for themselves. It offers different levels of challenge. It encourages collaboration and discussion. It has the potential for revealing patterns or leading to generalizations. It invites children to make decisions. John Van de Walle, Elementary & Middle School Mathematics, Teaching Developmentally NRICH Project @ University of Cambridge, Nrich.maths.org/5662

52. Math Tasks Break into grade level groups to look at and discuss good math tasks from your notebook. (WRESA workshop- Getting to the Core of Elementary Mathematics.) Think about sharing resources, internet sites, etc. with others.

53. National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html DPI Mathematics Site http://math.ncwiseowl.org Common Core State Standards www.corestandards.org ACCOUNTABILITY AND CURRICULUM REFORM EFFORT http://www.ncpublicschools.org/acre/ NCCTM INACTIVE MATH SITE http://illuminations.nctm.org/ActivityDetail.aspx?ID=18 In the near future, we will have other support documents from the state (Week by Week Essentials) as well as the county (pacing guides). Resources National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html DPI Mathematics Site http://math.ncwiseowl.org Common Core State Standards www.corestandards.org ACCOUNTABILITY AND CURRICULUM REFORM EFFORT http://www.ncpublicschools.org/acre/ NCCTM INACTIVE MATH SITE http://illuminations.nctm.org/ActivityDetail.aspx?ID=18 In the near future, we will have other support documents from the state (Week by Week Essentials) as well as the county (pacing guides).

54. PLANNING TIME!! 1. Look at the Crosswalks. 2. Use the Rubric to evaluate the CCSS in comparison to the NCSCS. 3. Make a list of NEW things you will be teaching at your grade level. 4. Use Math Inventory Sheet (if time permits) to begin planning.

55. As you are planning, remember… Mathematical practices describe the habits of mathematically proficient students… Who is doing the talking? Who is doing the thinking? Who is doing the thinking? Who is doing the math?

56. Things to Think About: What do you want students to do? How will you know if they get it? What will you do if they don’t? What will you do if they did?

58. Old Boxes WE are the next step If we JUST swap out the old standards and put the new CCSS: in the old boxes into old systems and procedures into the old relationships Into old instructional materials formats Into old assessment tools… …then nothing will change, and perhaps nothing will. Phil Daro, NCCTM 2010