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Mississippi Department of Education Common Core State Standards and Assessments

This document provides training for trainers on the design, organization, and implementation of the mathematics standards in grades 3-5. It includes information on the structure of the standards, referencing the standards, and connecting the standards for mathematical practice to mathematical content.

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Mississippi Department of Education Common Core State Standards and Assessments

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  1. Mississippi Department of EducationCommon Core State Standards and Assessments 3 - 5 Training of Trainers Mathematics October - November 2011

  2. *All page references are from this document unless otherwise noted.

  3. Design and Organization of the Mathematics Standards • Introduction • Standards of Mathematical Practice • Standards of Mathematical Content • Glossary

  4. Structure Sample from CCSS – Mathematics (See Page 5)

  5. How to read the grade level 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. Note that standards from different domains may sometimes be closely related.

  6. Domains for grades 3-5 OA = Operations and Algebraic Thinking NBT= Number and Operations in Base Ten NF = Number and Operations-Fractions MD = Measurement and Data G = Geometry

  7. Referencing the CCSS for Mathematics 4.G.1 4.G.1 - Grade level 4.G.1 - Geometry Domain 4.G.1- Standard 5.NF.4b 5.NF.4b - Grade level 5.NF.4b - Number and Operations – Fractions Domain 5.NF.4b - Standard

  8. Work Session 1Activity 1a: Referencing the CCSS Directions: See Work Session 1 Activity page. • Locate Grade 3 of the CCSS for Mathematics. • Indicate the correct reference for this standard: “Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole”. 3. Under which domain does this standard belong?

  9. Work Session 1Activity 1b: Referencing the CCSS Directions: See Work Session 1 Activity page. • Refer to the CCSS for Mathematics. • Indicate the page number for standard 4.OA.4. • Read the standard to your group. • Indicate the page number for standard 5.NF.7c. • Read the standard to your group.

  10. Work Session 1Activity 1c: Referencing the CCSS Directions: See Work Session 1 Activity page. • Locate Grade 5 of the CCSS for Mathematics. • Indicate the correct reference for this standard: “Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.” 3. Under which domain does this standard belong?

  11. Connecting the Standards for Mathematical Practice to Mathematical Content CCSS page 8 The 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 school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

  12. Connecting the Standards for Mathematical Practice to Mathematical Content CCSS page 8 The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word understand are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

  13. Mathematical Practices CCSS page 6-8 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. • Look for and express regularity in repeated reasoning. Note: The mathematical practices should not be used as a checklist because they interact and overlay with each other.

  14. Work session 2Mathematical Practices Directions: See Work Session 2 Activity page. 1. Refer to Mathematical Practice number 1. 2. Highlight key words that help with understanding. 3. Discuss key words with your group.

  15. 1. Make sense of problems and persevere in solving them. CCSS p.6

  16. 2. Reason abstractly and quantitatively. CCSS p.6

  17. 3. Construct viable arguments and critique the reasoning of others CCSS p.6-7

  18. 4. Model with Mathematics CCSS p.7

  19. 5. Use appropriate tools strategically CCSS p.7

  20. 6. Attend to Precision CCSS p.7

  21. 7. Look for and make use of structure CCSS p.8

  22. 8. Look for and express regularity in repeated reasoning CCSS p.8

  23. Understanding the Glossary • Locate pages 88-89 of the CCSS for Mathematics. • See the following tables: Table 1: Common addition and subtraction situations Table 2: Common multiplication and division situations

  24. CCSS p.88

  25. CCSS p.89

  26. Understanding the Glossary Continued • Locate page 90 of the CCSS for Mathematics. • Note the following Tables: Table 3: Properties of operations Table 4: Properties of equality Table 5: Properties of inequality

  27. CCSS p. 90

  28. Work Session 3: Correlation between CCSS and MS Math Framework (MMF) Directions: See Work Session 3 Activity page. • Locate the MMF and reference each objective for Grade 3, Competencies 1 and 2. • If you find a MS 3rd grade objective that matches a Grade 3 CCSS, write the competency/objective beside the Grade 3 CCSS indicated in the table on the activity page. • If you do not find a match in the Grade 3CCSS for Math for an objective in the MMF for Grade 3, Competencies 1 and 2, highlight the MS objective. • Discuss findings at your table.

  29. Work Session 4Focusing on a grade 3 standard Directions: See Work Session 4 Activity page. • Read standard 3.MD.7 . • Briefly describe in writing what this standard means to you. 3.MD.7 Relate area to the operations of multiplication and addition. Recognize area as additive. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

  30. Work Session 4 ContinuedFocusing on a 3rd grade CCSS Video Directions: 1. View the video for the grade 3 CCSS indicated below. 2. Video will be paused and participants will model the activity in the video. 3.MD.7 Relate area to the operations of multiplication and addition. Recognize area as additive. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

  31. Directions: See Work Session 4 Activity page.3. Indicate the mathematical practice(s) that are addressed in the teaching strategy in the video. Discuss at table.______________________________________________________________________________________________________________________________________________________________________________________________ Work Session 4 ContinuedFocusing on a grade 3 standard

  32. Work Session 4 ContinuedFocusing on a grade 3 standard Directions: See Work Session 4 Activity page.4. Explain in writing how the video assisted with your understanding of 3.MD.7. Discuss at table.______________________________________________________________________________________________________________________________________________________________________________________________

  33. Work Session 4 ContinuedFocusing on a grade 3 standard Directions: See Work Session 4 Activity page.5. Indicate any aspect of the standard that you still don’t understand. Discuss at table.______________________________________________________________________________________________________________________________________________________________________________________________

  34. Work Session 5Focusing on a grade 4 standard Directions: See Work Session 5 Activity page. • Read standard 4.NF.1. • Briefly describe in writing what this standard means to you. 4.NF.1Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the nu7mber 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.

  35. Work Session 5 ContinuedFocusing on a grade 4 standard video Directions:See Work Session 5 Activity Page. 1. View the video for the Grade 4 CCSS indicated below.2. Video will be paused and participants will model the activity in the video. 4.NF.1Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the nu7mber 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.

  36. Directions:See Work Session 5 Activity page. 3. Indicate the mathematical practice(s) that are addressed in the teaching strategy in the video._____________________________________________________________________________________________________________________________________________________________________________________________________________ Work Session 5 ContinuedFocusing on a grade 4 standard

  37. Work Session 5 ContinuedFocusing on a grade 4 standard Directions:See Work Session 5 Activity page. 4. Explain in writing how the video assisted with your understanding of 4.NF.1. Discuss at table._____________________________________________________________________________________________________________________________________________________________________________________________________________

  38. Work Session 5 ContinuedFocusing on a grade 4 standard Directions: See Work Session 5 Activity page. 5. Indicate any aspect of the standard that you still don’t understand. Discuss at your table. _____________________________________________________________________________________________________________________________________________________________________________________________________________

  39. Work Session 6Focusing on a grade 5 standard Directions: See Work Session 6 Activity page. Read standard 5.NF.5. Briefly describe what this standard means to you. • 5.NF.5 • Interpret multiplication as scaling (resizing) by: • comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. • b. Explaining why multiplying a given number by a fraction greater then 1 results in a product greater then the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence alb = (nxa)/(nxb) to the effect of multiplying a/b by 1.

  40. Work Session 6 ContinuedFocusing on a grade 5 standard video Directions: See Work Session 6 Activity page. • View the video for the Grade 5CCSS indicated below. • Video will be paused and participants will model the activity in the video. • 5.NF.5 • Interpret multiplication as scaling (resizing) by: • comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. • b. Explaining why multiplying a given number by a fraction greater then 1 results in a product greater then the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence alb = (nxa)/(nxb) to the effect of multiplying a/b by 1.

  41. Directions: See Work Session 6 Activity page.3. Indicate the mathematical practice(s) that are addressed in the teaching strategy. Discuss at table._____________________________________________________________________________________________________________________________________________________________________________________________________________ Work Session 6 ContinuedFocusing on a grade 5 standard

  42. Work Session 6 ContinuedFocusing on a grade 5 standard Directions: See Work Session 6 Activity page. • Explain in writing how the video assisted with your understanding of 5.NF.5. Discuss at table. _____________________________________________________________________________________________________________________________________________________________________________________________________________

  43. Work Session 6 ContinuedFocusing on a grade 5 standard Directions: See Work Session 6 Activity page. 5. Indicate any aspect of the standard that you still don’t understand. Discuss at your table. _____________________________________________________________________________________________________________________________________________________________________________________________________________

  44. PARCC Draft Model Content Frameworks for Mathematics • Purpose: • Identify the big ideas in the CCSS for each grade level • Help determine the focus for the various PARCC assessment components • Support the development of the assessment blueprints

  45. PARCC Draft Model Content Frameworks for Mathematics • Structure: • Examples of key advances from the previous grade • Fluency expectations and examples of culminating standards • Examples of major with-in grade dependencies • Examples of opportunities for connections among standards, clusters, or domains • Examples of opportunities for connecting mathematical content and mathematical practices • Instructional emphasis by cluster

  46. PARCC Draft Model Content Frameworks for Mathematics • Examples of key advances from the previous grade: • Highlight major grade-to-grade steps as knowledge and skills increase and progress • Stress the need to treat topics in ways that take into account where students have been in previous grades

  47. PARCC Draft Model Content Frameworks for Mathematics • Fluency expectations and examples of culminating standards: • Highlight individual standards that set expectations for fluency • Stress the need to provide supports and opportunities for practice

  48. PARCC Draft Model Content Frameworks for Mathematics • Fluent: • Means fast and accurate • Means to flow without halting, stumbling, or reversing • Marks the endpoints of progressions of learning • Does not happen all at once in a single grade • Requires attention to student understanding along the way (is not meant to come at the expense of understanding)

  49. PARCC Draft Model Content Frameworks for Mathematics • Examples of major within-grade dependencies: • Highlight cases in which a body of content in a given grade depends conceptually or logically upon another body of content within that same grade • Stress the need to organize material coherently within each grade level • Focus only on the large-scale dependencies (coherence is also important for small-scale dependencies)

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