CCSS within the Teaching Framework: Mathematics 3-6

1. CCSSwithin the Teaching Framework:Mathematics 3-6 Rochester City School District August 9, 2012 August 10, 2012 August 20, 2012 August 27, 2012

2. Today’s Agenda • 8:30-9:30 (Full Group) • Overview and Change of Direction • Overview of Danielson Rubric • 9:45-10:45 • CCSS Sample Questions from NYSED – June 2012 • 10:45 – 11:30 • Modules • Correlation charts • 11:30-12:30 Lunch • 12:30-2:30 (Grade Level Groups) • Emphases in Common Core Standards for Mathematical Content • NBT / Tasks • 2:30–3:00 Accessing Resources on Rochester Curriculum and the RCSD Math Department’s web page • 3:00-3:30 Evaluation and Feedback

3. When not knowing Math can cost you $15,000… http://www.youtube.com/watch?v=BbX44YSsQ2I

4. Math Sample Questions

5. CCSS Sample Questions from NYSEDJune 2012 • Answer each of the sample questions. • How do you think students would interpret these questions? • Match each question to one of the standards. • Discuss the following: • What are your impressions of the questions? • What would your students think of these questions?

6. NYS Common Core Standards addressed in the sample questions: 3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷3, 6 × 6 = ? 3.OA.5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6= 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2= 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. 4.OA.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 4.NF.4c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 5.NBT.3a.Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). 5.NBT.2. 2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10. 6.RP.3c. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

7. So… • What areyour impressions of the questions? • What would your students think of these questions? • How do you think students would interpret these questions? Now let’s see how well we matched the grade level standards to each question…

8. NYS Common Core Standards addressed in the sample questions: • 3.OA.4, 3.OA.5, 3.OA.6 • 4.OA.3, 4.NF.4c • 5.NBT.3a and 5.NBT.2 • 6.RP.3c Commentary: This question aligns with CCLS 3.OA.4, 3.OA.5, and 3.OA.6 and assesses the student’s ability to determine the unknown whole numbers in multiplication and division equations. The question also assesses the student’s ability to apply properties of operations as strategies to multiply or divide and to understand division as an unknown factor problem. Commentary: This question is aligned with CCLS 4.NF.4c and 4.OA.3. It assesses a student’s ability to multiply a fraction by a whole number and to solve a multi-step word problem using addition and subtraction of whole numbers. Commentary: This question aligns with CCLS 5.NBT.3a and 5.NBT.2 because it requires students to write decimals in equivalent forms and apply understanding of place value. Commentary: This question aligns to CCLS 6.RP.3c because it assesses a student’s ability to work with percents, namely, finding a percent of a quantity in a contextual situation.

9. How to Use Different Aspects of the Sample Questions • Interpret the way the standards are conceptualized in each question. • Note the multiple ways the standard is assessed throughout the sample questions. • Take note of numbers (e.g., fractions instead of whole numbers) used in the samples. • Pay attention to the strong distractors in each multiple‐choice question. • Don’t consider these questions to be the only way the standard will be assessed. • Don’t assume that the sample questions represent a mini‐version of future state assessments.

10. Understanding Math Sample Questions Multiple Choice • Sample multiple‐choice math questions are designed to assess CCSS math standards and incorporate both standards and math practices in real‐world applications. Math multiple‐choice questions assess procedural and conceptual standards. Unlike questions on past math assessments, many require the use of multiple skills and concepts. Answer choices are also different from those on past assessments. Within the sample questions, all distractors will be based on plausible missteps.

11. Understanding Math Sample Questions Short Constructed Response • Math short constructed response questions are similar to past 2‐point questions, asking students to complete a task and show their work. Like multiple‐choice questions, short constructed response questions will often require multiple steps, the application of multiple math skills, and real‐world applications. Many of the short constructed response questions will cover conceptual and application standards.

12. Understanding MathSample Questions Extended Constructed Response • Math extended constructed response questions are similar to past 3‐point questions, asking students to show their work in completing two or more tasks or one more extensive problem. Extended constructed response questions allow students to show their understanding of math procedures, conceptual understanding, and application.

13. For additional information and sample questions, follow this link: http://www.p12.nysed.gov/apda/common-core-sample-questions/

14. NYS Curriculum Modules

15. Common Core State Standards NYS Curriculum Modules Rochester Curriculum

16. NYS Common Core Curriculum Maps in Mathematics

17. Rochester Curriculum Main Page

18. Grade 3 Correlation Chart

19. Grade 4 Correlation Chart

20. Grade 5 Correlation Chart

21. Grade 6 • As of today, the state has not released the Common Core Curriculum map for Grade 6, so no correlation chart is available at this time. • The Rochester Curriculum will provide the framework for Grade 6 to begin the school year. • As new information becomes available, Administrators and Grade 6 teachers will be informed.

22. Lunch 11:30 – 12:30

23. Afternoon Agenda • 12:30-2:30 (Grade Level Groups) • Emphases in Common Core Standards for Mathematical Content • NBT / Tasks • 2:30–3:00 • Accessing Resources on Rochester Curriculum and the RCSD Math Department’s web page • 3:00-3:30 Evaluation and Feedback

24. Emphases in Common Core Standards for Mathematical Content

25. Emphases in Common Core Standards for Mathematical Content Kindergarten – High School March 12, 2012 • Not all of the content in a given grade is emphasized equally in the standards. • The list of content standards for each grade is not a flat, one‐dimensional checklist; this is by design. • Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness. • In addition, an intense focus on the most critical material at each grade allows depth in learning, which is carried out through the Standards for Mathematical Practice.

26. Emphases in Common Core Standards for Mathematical Content Kindergarten – High School March 12, 2012 Explanation of terms used: Major clusters– areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting clusters– rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters– expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).

27. Grade 3 1 Work should be positioned in support of area measurement and understanding of fractions. 2 Students multiple and divide to solve problems using information presented in scaled bar graphs. Pictographs and scaled bar graphs are a visually appealing context for one‐ and two‐step word problems.

28. Grade 4 3 Work in this cluster supports students’ work with multi‐digit arithmetic as well as their work with fraction equivalence. 4 The standard in this cluster requires students to use a line plot to display measurements in fractions of a unit and to solve problems involving addition and subtraction of fractions, connecting it directly to the Number and Operations – Fractions clusters.

29. Grade 5 5 The standard in this cluster provides an opportunity for solving real‐world problems with operations on fractions, connecting directly to both number and Operations – Fractions clusters. 6 Work in these standards supports computation with decimals. For example, converting 5 cm to .05 m involves computation with decimals to hundredths.

30. Grade 6 7 In this cluster, students work on problems with areas of triangles and volumes of right rectangular prisms, which connects to work in the Expressions and Equations domain. In addition, another standard within this cluster asks students to draw polygons in the coordinate plane, which supports work with the coordinate plane in the Number System domain.

31. Grade Level Tasks

32. NBT • Please read the Overview in the K-5, Number and Operations In Base Ten learning progression document. • Then please read the section for your grade level: • Grade 3-4 – pages 11-14 • Grade 5-6 – pages 16-19 • Discussion to follow...

33. Task Analysis • Complete the tasks for your grade level. • Note the standards that are addressed in your task. • Be prepared to share one of your tasks with the larger group, considering the following questions: • What are your impressions of the task? • What would your students think of this task? • How do you think students would interpret this task? • How does the task address the standards? • What mathematical content was needed to complete the task?

34. Grade 3 CCSS Domain:Number and Operations in Base Ten

35. Grade 4 CCSS Domain:Number and Operations in Base Ten

36. Grade 5 CCSS Domain:Number and Operations in Base Ten

37. Grade 6 CCSS Domain:Number System

38. Accessing Rochester Curriculum

39. Accessing Rochester Curriculum

40. Accessing Rochester Curriculum

41. Accessing Rochester Curriculum

42. Accessing Rochester CurriculumUnit Page 1

43. Accessing Rochester CurriculumUnit Page 2

44. How to:NYS P-12 Common Core Resources • www.rcsdk12.org • Click Departments • Click Mathematics • Click Common Core Math Resources

45. Thank you for your time and attention today… Have a wonderful year!!