1. Mathematics EQuIP Rubric & Quality Review Training Washington, DC May 19, 2014 1

2. 2 Session Goals Use the EQuIP quality review process to determine the quality and alignment of lessons and units to the Common Core State Standards (CCSS) in mathematics During this session, reviewers will: Develop their ability to use EQuIP criteria to provide observations about CCSS-aligned instructional materials and make suggestions for improvement. Develop a common understanding of the EQuIP quality review process. Develop a common understanding of the rating scale and descriptors for the four rubric dimensions and the rating categories and descriptors for overall ratings. Develop their abilities to use EQuIP criteria, rating scales and rating descriptors to accurately rate instructional materials.

3. 1.CCSS: Before beginning a review, all members of a review team are familiar with the CCSS. 2.Inquiry: Review processes emphasize inquiry rather than advocacy and are organized in steps around a set of guiding questions. 3.Respect & Commitment: Each member of a review team is respected as a valued colleague and contributor who makes a commitment to the EQuIP process. 4.Criteria & Evidence: All observations, judgments, discussions and recommendations are criterion and evidence based. 5.Constructive: Lessons/units to be reviewed are seen as “works in progress.” Reviewers are respectful of contributors’ work and make constructive observations and suggestions based on evidence from the work. 6.Individual to Collective: Each member of a review team independently records his/her observations prior to discussion. Discussions focus on understanding all reviewers’ interpretations of the criteria and the evidence they have found. 7.Understanding & Agreement: The goal of the process is to compare and eventually calibrate judgments to move toward agreement about quality with respect to the CCSS. EQuIP Quality Review: Principles & Agreements 3

4. EQuIP Quality Review Process The EQuIP quality review process is a collegial process that centers on the use of criteria-based rubrics for English language arts (ELA)/literacy and mathematics. The criteria are organized into four dimensions: The Four Dimensions As educators examine instructional materials against the criteria in each dimension, they are able to use common standards for quality and generate evidence-based commentary and ratings on the quality and alignment of materials. EQuIP Quality Review: Process & Dimensions 4 Alignment to the depth of the CCSS Key shifts in the CCSS Instructional supports Assessment

5. Five Quality Review Steps Step 1. Review Materials Record the grade and title of the lesson/unit on the Quality Review Rubric PDF Scan to see what the lesson/unit contains and how it is organized Read key materials related to instruction, assessment and teacher guidance Study and work the tasks that serve as the centerpiece for the lesson/unit, analyzing the content and mathematical practices the tasks require Step 2. Apply Criteria in Dimension I: Alignment to the Depth of the CCSS Identify the grade-level CCSS that the lesson/unit targets Closely examine the materials through the “lens” of each criterion Indicate each criterion for which clear and substantial evidence is found Record input on specific improvements needed to meet criteria or strengthen alignment Enter a rating of 0–3 for Dimension I Note: Dimension I is non-negotiable. For the review to continue, a rating of 2 or 3 is required. If the review is discontinued, consider giving general feedback that might help developers/teachers make decisions regarding next steps. 5

6. Five Quality Review Steps Step 3. Apply Criteria in Dimensions II–IV Examine the lesson/unit through the “lens” of each criterion Indicate each criterion met, record observations and feedback, and then rate 0–3 Step 4. Apply an Overall Rating and Provide Summary Comments Individually review ratings for Dimensions I–IV, adding/clarifying comments as needed Total dimension ratings and record an overall rating (E, E/I, R, N) based on total score Individually write summary comments for the overall rating on the Quality Review Rubric PDF Step 5. Compare Overall Ratings and Determine Next Steps Note the evidence cited to arrive at final ratings, summary comments, and similarities and differences among raters. Recommend next steps for the lesson/unit and provide recommendations for improvement to developers/teachers. 6

7. 7 Review: The 4 Dimensions Alignment to the depth of the CCSS Key shifts in the CCSS Instructional supports Assessment

8. 8 Step 1. Review Materials Step 2. Apply Criteria in Dimension I: Alignment to the Depth of the CCSS Step 3. Apply Criteria in Dimensions II–IV Step 4. Apply an Overall Rating and Provide Summary Comments Step 5. Compare Overall Ratings and Determine Next Steps Review: The 5 Steps

9. Five Quality Review Steps When working with a review team: Have a review plan that considers the experience and expertise of all team members. Team members may choose to compare individual ratings after each dimension or wait until each person has individually rated and recorded all input for Dimensions II–IV before beginning discussion. Individuals should record their overall rating prior to discussion. Adjustments to ratings and/or commentary should take place as a part of the group discussion. 9

10. 10 Using Dimension Ratings and Descriptive Scales To Synthesize Judgment Rating Scale for Dimensions I–IV: 3: Meets most to all of the criteria in the dimension 2: Meets many of the criteria in the dimension 1: Meets some of the criteria in the dimension 0: Does not meet the criteria in the dimension

11. 11 Using Dimension Ratings and Descriptive Scales To Synthesize Judgment Rating Scale for Dimensions I–IV: 3: Meets most to all of the criteria in the dimension 2: Meets many of the criteria in the dimension 1: Meets some of the criteria in the dimension 0: Does not meet the criteria in the dimension Descriptors for Dimensions I–IV: 3: Exemplifies CCSS Quality — meets the standard described by criteria in the dimension, as explained in criterion-based observations 2: Approaching CCSS Quality — meets many criteria but will benefit from revision in others, as suggested in criterion-based observations 1: Developing toward CCSS Quality — needs significant revision, as suggested in criterion-based observations 0: Not representing CCSS Quality — does not address the criteria in the dimension

12. 12 Using Dimension Ratings and Descriptive Scales To Synthesize Judgment Rating Scale for Dimensions I–IV: 3: Meets most to all of the criteria in the dimension 2: Meets many of the criteria in the dimension 1: Meets some of the criteria in the dimension 0: Does not meet the criteria in the dimension Descriptors for Dimensions I–IV: 3: Exemplifies CCSS Quality — meets the standard described by criteria in the dimension, as explained in criterion-based observations 2: Approaching CCSS Quality — meets many criteria but will benefit from revision in others, as suggested in criterion-based observations 1: Developing toward CCSS Quality — needs significant revision, as suggested in criterion-based observations 0: Not representing CCSS Quality — does not address the criteria in the dimension

13. 13 Using Overall Ratings to Summarize Judgment Overall Rating for the Lesson/Unit: E: Exemplar — Aligned and meets most to all of the criteria in Dimensions II–IV (total 11–12) E/I: Exemplar if Improved — Aligned and needs some improvement in one or more dimensions (total 8–10) R: Revision Needed — Aligned partially and needs significant revision in one or more dimensions (total 3–7) N: Not Ready to Review — Not aligned and does not meet criteria (total 0–2)

14. 14 Using Overall Ratings to Summarize Judgment Overall Rating for the Lesson/Unit: E: Exemplar — Aligned and meets most to all of the criteria in Dimensions II–IV (total 11–12) E/I: Exemplar if Improved — Aligned and needs some improvement in one or more dimensions (total 8–10) R: Revision Needed — Aligned partially and needs significant revision in one or more dimensions (total 3–7) N: Not Ready to Review — Not aligned and does not meet criteria (total 0–2)

15. 15 Using Overall Ratings to Summarize Judgment Overall Rating for the Lesson/Unit: E: Exemplar — Aligned and meets most to all of the criteria in Dimensions II–IV (total 11–12) E/I: Exemplar if Improved — Aligned and needs some improvement in one or more dimensions (total 8–10) R: Revision Needed — Aligned partially and needs significant revision in one or more dimensions (total 3–7) N: Not Ready to Review — Not aligned and does not meet criteria (total 0–2)

16. 16 The EQuIP Rubric for Mathematics

17. 17 Grade 7 — Draft Unit Plan 7.NS.1-3: Operations with Rational Numbers This lesson is part of a unit plan titled “Operations with Rational Numbers.” The state that developed this unit plan uses the CCSS and the EQuIP rubrics to guide development of instructional materials and has charged its developers with creating overall conceptual unit plans, based on the CCSS domains, with one standards-based lesson, plus one or more lesson seeds, based on the cluster. Teachers are able to use these as models in their planning as they further develop the unit. Step 1. Review Materials Sample Unit: Operations with Rational Numbers

18. Step 1. Review Materials – 10 minutes Sample Unit: Operations with Rational Numbers Record the grade and title of the lesson/unit on the Quality Review Rubric Grade 7, “Draft Unit Plan – 7.NS.1-3: Operations with Rational Numbers” Read key materials related to instruction, assessment and teacher guidance −Unit overview narratives (pp. 1–3) −Unit alignment/standards (pp. 4–11) −Common misconceptions (p. 11) −Vocabulary (pp. 12–14) −Lesson plan and materials – 7.NS.1 (pp. 15–29) −Lesson seeds – 7.NS.1b, 7.NS.1d (pp. 30–32) Activities: −Four Corners (relating additive inverses to students’ lives) −Traveling Around Maryland Gallery Walk (adding and subtraction whole and decimal numbers) −Rational Flow Chart (combining positive and negative decimals) −The Zero Circle and Block Party (Lesson Seeds : using ‘opposites’ to find zero) 18

19. 19 Step 2: Criteria for Dimension I: Alignment to the Depth of the CCSS The lesson/unit aligns with the letter and spirit of the CCSS:  Targets a set of grade-level CCSS mathematics standard(s) to the full depth of the standards for teaching and learning  Standards for Mathematical Practice that are central to the lesson are identified, handled in a grade-appropriate way and well connected to the content being addressed  Presents a balance of mathematical procedures and deeper conceptual understanding inherent in the CCSS

20. 20 Dimension I

21. 21 Step 2: Apply Criteria in Dimension I Try it now – on your own at first.

22. 22 The lesson/unit aligns with the letter and spirit of the CCSS:  Targets a set of grade-level CCSS mathematics standard(s) to the full depth of the standards for teaching and learning  Standards for Mathematical Practice that are central to the lesson are identified, handled in a grade-appropriate way and well connected to the content being addressed  Presents a balance of mathematical procedures and deeper conceptual understanding inherent in the CCSS Step 2. Apply Criteria in Dimension I: Alignment Normed Response: Operations with Rational Numbers

23. 23 Criterion:  1. Targets a set of grade-level CCSS mathematics standard(s) to the full depth of the standards for teaching and learning Observations and suggestions: The lesson clearly targets 7.NS.1a, b and d as the content standard for the lesson. The lesson does not list 7.NS.1c, although the lesson includes discussion of additive inverse. Developers should re-evaluate the omission of 7.NS.1c. It is assumed that further development of lessons beyond the two lesson seeds provided will address the requirements of 7.NS.2 and 7.NS.3. Step 2. Apply Criteria in Dimension I: Alignment Normed Response: Operations with Rational Numbers

24. 24 Criterion:  2. Standards for Mathematical Practice that are central to the lesson are identified, handled in a grade-appropriate way and well connected to the content being addressed Observations and suggestions: The lesson plan identifies all eight of the Standards for Mathematical Practice as being addressed in the lesson/unit. And although references to the Practices are listed throughout the lesson/unit, it is not clear how central practices connect to specific activities and tasks. Those that are most central should be identified and their connection clearly defined. Those central Practices should serve as a focal point for the lesson and those that are not central should either be eliminated or noted as serving in a supporting role. Step 2. Apply Criteria in Dimension I: Alignment Normed Response: Operations with Rational Numbers

25. 25 The lesson/unit aligns with the letter and spirit of the CCSS:  3. Presents a balance of mathematical procedures and deeper conceptual understanding inherent in the CCSS Observations and suggestions: The balance between mathematical procedures and conceptual understanding is strong in the unit. As refinements are made, and as the Lesson Seeds are more fully developed, care needs to be taken to ensure that this balance is maintained. Step 2. Apply Criteria in Dimension I: Alignment Normed Response: Operations with Rational Numbers Individually consider the rating for this dimension.

26. 26 Observations/Feedback and Rating The lesson clearly targets 7.NS.1a, b and d as the content standard for the lesson. The lesson does not list 7.NS.1c, although the lesson includes discussion of additive inverse. Developers should re-evaluate the omission of 7.NS.1c. As the unit is further developed, attention should be given to lessons that address 7.NS.2 and 7.NS.3, if that is the unit’s intent. The lesson plan identifies all eight of the Standards for Mathematical Practice as being addressed. Those that are most central should be identified and serve as a focal point for the lesson. Those that are not central should either be eliminated or noted as serving in a supporting role. In addition, references should be made throughout the lesson as to how central practices relate to specific activities and tasks. The balance between mathematical procedures and conceptual understanding is strong in the unit. Rating: 2 (Missing clear connections between the Practices and the activities. Will benefit from some revision in this area.) Approaching CCSS Quality — meets many criteria but will benefit from revision in others, as suggested in criterion-based observations Step 2. Apply Criteria in Dimension I: Alignment Normed Response: Operations with Rational Numbers

27. 27 Compare Criterion-Based Checks, Observations/Feedback and Rating What is the pattern within our team in terms of the criteria we have checked? Do our observations and feedback reference the criteria and evidence (or lack of evidence) in the instructional materials? Do our ratings correspond to the rating and descriptors in the rubric? Step 2. Apply Criteria in Dimension I: Alignment Sample Unit: Operations with Rational Numbers

28. 28 Consider the rating for Dimension I: Is the overall rating for alignment a 3 or 2? Does the quality of the alignment to the CCSS warrant continuing with the review? If yes, continue with Dimensions II – IV. Step 2. Apply Criteria in Dimension I: Alignment Sample Unit: Operations with Rational Numbers

29. 29 Step 3. Apply Criteria for Dimensions II – IV Criteria for Dimension II: Key Shifts in the CCSS The lesson/unit addresses key shifts in the CCSS:  Focus: Lessons and units targeting the major work of the grade provide an especially in-depth treatment, with especially high expectations. Lessons and units targeting supporting clusters have visible connection to the major work of the grade and are sufficiently brief. Lessons and units do not hold students responsible for material from later grades.  Coherence: The content develops through reasoning about the new concepts on the basis of previous understandings and provides opportunities for students to transfer knowledge and skills within and across domains and learning progressions.

30. 30  Rigor: Requires students to engage with and demonstrate challenging mathematics with appropriate balance among the following: * o Application: Provides opportunities for students to independently apply mathematical concepts in real-world situations and problem solve with persistence, choosing and applying an appropriate model or strategy to new situations o Conceptual Understanding: Develops students’ conceptual understanding through tasks, brief problems, questions, multiple representations and opportunities for students to write and speak about their understanding. o Procedural Skill and Fluency: Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately. * Note: The three components of rigor are not considered/checked separately but only as required by the targeted content standards. Key words: APPROPRIATE BALANCE Step 3. Apply Criteria for Dimensions II – IV Criteria for Dimension II: Key Shifts in the CCSS

31. 31 Step 3: Apply Criteria in Dimension II Try it now – on your own at first.

32. 32 Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Normed Response: Operations with Rational Numbers The lesson/unit addresses reflects evidence of key shifts that are reflected in the CCSS:  Focus: Lessons and units targeting the major work of the grade provide an especially in-depth treatment, with especially high expectations. Lessons and units targeting supporting clusters have visible connection to the major work of the grade and are sufficiently brief. Lessons and units do not hold students responsible for material from later grades.  Coherence: The content develops through reasoning about the new concepts on the basis of previous understandings and provides opportunities for students to transfer knowledge and skills within and across domains and learning progressions.  Rigor : Requires students to engage with and demonstrate challenging mathematics with appropriate balance among the following o Application: Provides opportunities for students to independently apply mathematical concepts in real- world situations and problem solve with persistence, choosing and applying an appropriate model or strategy to new situations o Conceptual Understanding: Develops students’ conceptual understanding through tasks, brief problems, questions, multiple representations and opportunities for students to write and speak about their understanding.  Procedural Skill and Fluency : Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately

33. 33 The lesson/unit addresses reflects evidence of key shifts that are reflected in the CCSS:  Focus: There is strong focus on 7.NS.1 in the unit plan, including the model lesson plan and lesson seeds—particularly on 7.NS.1a, 7.NS.1b, and 7.NS.1d. The concepts and level of rigor prioritized in the standards are evident in the lessons/lesson seeds. 7.NS.1c is not clearly evident, and may be addressed in future lesson seeds.  Coherence: There are clear connections made to prior learning in the section on Vertical Alignment. There are other references among the UDL Components, the Overview and Teacher Notes. In the Enduring Understandings there is a hint at how the concepts of this unit will be carried forward. However, more information might be provided for the teacher in how the real number system will evolve from the learning in this unit. Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Normed Response: Operations with Rational Numbers

34. 34  Rigor: Requires students to engage with and demonstrate challenging mathematics with appropriate balance among the following: *  Application: There is evidence of opportunity for application of some skills with the activities involving real-world context (card sort, gallery walk). However, many activities provide the opportunity for students to work in small groups or with partners, so there are few clear opportunities for independent application of concepts and skills to real-world contexts. Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Normed Response: Operations with Rational Numbers * NOTE: Norming for this lesson began early in the life of the EQuIP review process. Earlier versions of the rubric allowed for separate checks for the three components of rigor.

35. 35  Rigor: Requires students to engage with and demonstrate challenging mathematics with appropriate balance among the following: *  Application:  Conceptual Understanding: The guiding questions and other question prompts provide tools for teachers to probe for understanding. It is not clear, however, if all students will be expected to respond to questions verbally or in writing or if questions will be used solely to guide student discussion. Students are not presented with contextual problems that require deeper thinking and persistence and that can provide an indication of deeper conceptual understanding. Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Normed Response: Operations with Rational Numbers * NOTE: Norming for this lesson began early in the life of the EQuIP review process. Earlier versions of the rubric allowed for separate checks for the three components of rigor.

36. 36  Rigor: Requires students to engage with and demonstrate challenging mathematics with appropriate balance among the following: *  Application:  Conceptual Understanding:  Procedural Skill and Fluency: The unit emphasizes procedural skill and there are specific references in the unit to the development of fluency with operations with rational numbers as a goal of grade 7. However, it is not clear how much calculator usage is allowed, causing some concern as to how quickly and accurately students might perform calculations and procedures without this tool. Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Normed Response: Operations with Rational Numbers

37. 37 Observations/Feedback and Rating The lesson is exemplary in its focus on a reasonable portion of the major work of the grade. However, many activities provide the opportunity for students to work in small groups or with partners, so there are few clear opportunities for independent application of concepts and skills to real-world contexts. The guiding questions and other question prompts provide tools for teachers to probe for understanding. It is not clear, however, if all students will be expected to respond to questions verbally or in writing or if questions will be used solely to guide student discussion. The unit emphasizes procedural skill and there are specific references in the unit to the development of fluency with operations with rational numbers as a goal of grade 7. However, it is not clear how much calculator usage is allowed, causing some concern as to how quickly and accurately students might perform calculations and procedures without technological support. Rating: 2 (Missing rigorous application problems and opportunities for deep conceptual understanding. Some revision required to fill these gaps.) Approaching CCSS Quality — meets many criteria but will benefit from revision in others, as suggested in criterion-based observations Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Normed Response: Operations with Rational Numbers

38. 38 Compare Criterion-Based Checks, Observations/Feedback and Rating What is the pattern within our team in terms of the criteria we have checked? Do our observations and feedback reference the criteria and evidence (or lack of evidence) in the instructional materials? Do our ratings correspond to the rating and descriptors in the rubric? Step 3. Apply Criteria in Dimension II: Key Shifts in the CCSS Sample Unit: Operations with Rational Numbers

39. 39 Criteria for Dimension III: Instructional Supports The lesson/unit is responsive to varied student learning needs:  Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media.  Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g., pictures, symbols, expressions, equations, graphics, models) in the discipline.  Engages students in productive struggle through relevant, thought-provoking questions, problems and tasks that stimulate interest and elicit mathematical thinking.  Addresses instructional expectations and is easy to understand and use.

40. 40 Criteria for Dimension III: Instructional Supports  Provides appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners. *  Supports diverse cultural and linguistic backgrounds, interests and styles.  Provides extra supports for students working below grade level.  Provides extensions for students with high interest or working above grade level. * Note: All three of these components are required in a high quality lesson or unit.

41. 41 Criteria for Dimension III: Instructional Supports A unit or longer lesson should: longer lessons:  Recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (including models), using a range of questions, checking for understanding, flexible grouping, pair-share, etc.  Gradually removes supports, requiring students to demonstrate their mathematical understanding independently.  Demonstrates an effective sequence and a progression of learning where the concepts or skills advance and deepen over time.  Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately.

42. 42 Step 3: Apply Criteria in Dimension III Try it now – on your own at first.

43. 43 The lesson/unit is responsive to varied student learning needs:  Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media  Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g., pictures, symbols, expressions, equations, graphics, models) in the discipline  Engages students in productive struggle through relevant, thought-provoking questions, problems and tasks that stimulate interest and elicit mathematical thinking  Addresses instructional expectations and is easy to understand and use  Provides appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners  Supports diverse cultural and linguistic backgrounds, interests and styles  Provides extra supports for students working below grade level  Provides extensions for students with high interest or working above grade level A longer unit or lesson should:  Recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (including models), using a range of questions, checking for understanding, flexible grouping, pair-share, etc.  Gradually removes supports, requiring students to demonstrate their mathematical understanding independently  Demonstrates an effective sequence and a progression of learning where the concepts or skills advance and deepen over time  Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

44. 44 Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers The lesson/unit is responsive to varied student learning needs:  Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media  Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g., pictures, symbols, expressions, equations, graphics, models) in the discipline  Engages students in productive struggle through relevant, thought-provoking questions, problems and tasks that stimulate interest and elicit mathematical thinking  Addresses instructional expectations and is easy to understand and use

45. 45  Provides appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners  Supports diverse cultural and linguistic backgrounds, interests and styles  Provides extra supports for students working below grade level  Provides extensions for students with high interest or working above grade level Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

46. 46 A unit or longer lesson should: longer lessons:  Recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (including models), using a range of questions, checking for understanding, flexible grouping, pair-share, etc.  Gradually removes supports, requiring students to demonstrate their mathematical understanding independently  Demonstrates an effective sequence and a progression of learning where the concepts or skills advance and deepen over time  Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

47. 47 The lesson/unit is responsive to varied student learning needs:  Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media There is clear and sufficient guidance in this Draft Unit Plan to provide teachers the support they need to help students reach the target standards. The layout, format, and introduction are generally clear and helpful.  Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g., pictures, symbols, expressions, equations, graphics, models) in the discipline The language used in the Draft Unit Plan is generally accurate and grade- appropriate. The section on Vocabulary/Terminology/Concepts is a nice addition and will help teachers understand the content as well as provide a tool for them to ensure the use of precise language in their classrooms. Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

48. 48 The lesson/unit is responsive to varied student learning needs:  Engages students in productive struggle through relevant, thought-provoking questions, problems and tasks that stimulate interest and elicit mathematical thinking The guiding questions posed in the unit are thought provoking and are likely to stimulate student interest and some mathematical thinking. However the level of rigor required does not indicate that they are likely to engage students in a productive struggle.  Addresses instructional expectations and is easy to understand and use The Possible Student Outcomes defined in the Draft Unit Plan help clearly define the instructional expectations. The varied questions and activities offered in the Model Lesson Plan provide teachers with a range of ways to address instructional expectations. The materials are user-friendly and generally easy to understand. Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

49. 49  Provides appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners  Supports diverse cultural and linguistic backgrounds, interests and styles  Provides extra supports for students working below grade level  Provides extensions for students with high interest or working above grade level There is not enough support for students working below grade level in the unit or those with language difficulties. The concepts presented in this unit might prove to be challenging for English-language learners, who would benefit from the use of manipulatives and visual supports to demonstrate their mathematical understanding independently. There are places in the unit where activities are listed for intervention (a video), extension (writing a rap song), and for ELL (using the Frayer method of writing definitions). It is not clear how these activities would support those students needing more support from this lesson. Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

50. 50 A unit or longer lesson should:  Recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (including models), using a range of questions, checking for understanding, flexible grouping, pair-share, etc. The Model Lesson Plan presents a variety of instructional approaches including multiple representations, using a range of questions, checking for understanding, flexible grouping, and think-pair-share.  Gradually removes supports, requiring students to demonstrate their mathematical understanding independently With little support provided students with particular needs, it is not clear that there is a gradual move to independent learning. As students become more adept, over the course of the unit, supports can be removed strategically from some activities, requiring students to demonstrate their mathematical understanding independently. Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

51. 51 A unit or longer lesson should:  Demonstrates an effective sequence and a progression of learning where the concepts or skills advance and deepen over time The unit appears to present an effective sequence and progression of learning.  Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately Procedural skill and fluency are clearly addressed in the unit. If the Lesson Seeds are developed in the same way as the Model Lesson Plan then the unit-level instructional supports should be sound. Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

52. 52 The lesson/unit is responsive to varied student learning needs:  Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media  Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g., pictures, symbols, expressions, equations, graphics, models) in the discipline  Engages students in productive struggle through relevant, thought-provoking questions, problems and tasks that stimulate interest and elicit mathematical thinking  Addresses instructional expectations and is easy to understand and use  Provides appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners  Supports diverse cultural and linguistic backgrounds, interests and styles  Provides extra supports for students working below grade level  Provides extensions for students with high interest or working above grade level A longer unit or lesson should:  Recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (including models), using a range of questions, checking for understanding, flexible grouping, pair-share, etc.  Gradually removes supports, requiring students to demonstrate their mathematical understanding independently  Demonstrates an effective sequence and a progression of learning where the concepts or skills advance and deepen over time  Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

53. 53 The lesson/unit is responsive to varied student learning needs:  Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media  Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g., pictures, symbols, expressions, equations, graphics, models) in the discipline  Engages students in productive struggle through relevant, thought-provoking questions, problems and tasks that stimulate interest and elicit mathematical thinking  Addresses instructional expectations and is easy to understand and use  Provides appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners  Supports diverse cultural and linguistic backgrounds, interests and styles  Provides extra supports for students working below grade level  Provides extensions for students with high interest or working above grade level A longer unit or lesson should:  Recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (including models), using a range of questions, checking for understanding, flexible grouping, pair-share, etc.  Gradually removes supports, requiring students to demonstrate their mathematical understanding independently  Demonstrates an effective sequence and a progression of learning where the concepts or skills advance and deepen over time  Expects, supports and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

54. 54 Observations/Feedback and Rating The lesson/unit is well organized, provides an effective sequence, and includes many support features for teachers that are easy to understand and use. It encourages a mix of instructional approaches, including providing a range of questions, checking for understanding, flexible grouping, and think-pair-share. However there is too little support for diverse learning styles and no suggestions for engaging students in the productive struggle that would accompany rigorous application problems. Most activities ask for students to be paired or grouped, with little opportunity for them to demonstrate independent learning. Since supports are missing there can be no gradual removal of support over time. Revisions required for this unit to be considered exemplary, would be more substantial than a “quick fix.” Rating: 1 (Missing strategies for engaging students in productive struggle, support for diverse learners, and gradual removal of support with an emphasis on independent understanding. Including all of these supports would require significant revision of the unit.) Developing toward CCSS Quality — needs significant revision, as suggested in criterion- based observations Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

55. 55 Observations/Feedback and Rating The lesson/unit is well organized, provides an effective sequence, and includes many support features for teachers that are easy to understand and use. It encourages a mix of instructional approaches, including providing a range of questions, checking for understanding, flexible grouping, and think-pair-share. However there is too little support for diverse learning styles and no suggestions for engaging students in the productive struggle that would accompany rigorous application problems. Most activities ask for students to be paired or grouped, with little opportunity for them to demonstrate independent learning. Since supports are missing there can be no gradual removal of support over time. Revisions required for this unit to be considered exemplary, would be more substantial than a “quick fix.” Rating: 1 (Missing strategies for engaging students in productive struggle, support for diverse learners, and gradual removal of support with an emphasis on independent understanding. Including all of these supports would require significant revision of the unit.) Developing toward CCSS Quality — needs significant revision, as suggested in criterion- based observations Step 3. Apply Criteria in Dimension III: Instructional Supports Normed Response: Operations with Rational Numbers

56. 56 Step 3. Apply Criteria in Dimension III: Instructional Supports Compare Criterion-Based Checks, Observations/Feedback and Rating What is the pattern within our team in terms of the criteria we have checked? Do our observations and feedback reference the criteria and evidence (or lack of evidence) in the instructional materials? Do our ratings correspond to the rating and descriptors in the rubric?

57. 57 Criteria for Dimension IV: Assessment The lesson/unit regularly assesses whether students are mastering standards-based content and skills:  Is designed to elicit direct, observable evidence of the degree to which a student can independently demonstrate the targeted CCSS.  Assesses student proficiency using methods that are accessible and unbiased, including the use of grade-level language in student prompts.  Includes aligned rubrics, answer keys and scoring guidelines that provide sufficient guidance for interpreting student performance. In addition, for units and longer lessons: and longer lessons:  Uses varied modes of curriculum-embedded assessments that may include pre-, formative, summative and self-assessment measures.

58. 58 Step 3: Apply Criteria in Dimension IV Try it now – on your own at first.

59. 59 The lesson/unit regularly assesses whether students are mastering standards-based content and skills:  Is designed to elicit direct, observable evidence of the degree to which a student can independently demonstrate the targeted CCSS  Assesses student proficiency using methods that are accessible and unbiased, including the use of grade-level language in student prompts  Includes aligned rubrics, answer keys and scoring guidelines that provide sufficient guidance for interpreting student performance In addition, for units and longer lessons: and longer lessons:  Uses varied modes of curriculum-embedded assessments that may include pre-, formative, summative and self-assessment measures Step 3. Apply Criteria in Dimension IV: Assessment Sample Unit: Operations with Rational Numbers

60. 60 The lesson/unit regularly assesses whether students are mastering standards-based content and skills:  Is designed to elicit direct, observable evidence of the degree to which a student can independently demonstrate the targeted CCSS This lesson includes a readiness assessment but no formative or end-of-lesson assessment to determine the degree to which students can independently demonstrate the targeted CCSS. An observation checklist for the teacher to use when observing students working collaboratively with others would be helpful, as would adding opportunities for independent demonstration of understanding and skill.  Assesses student proficiency using methods that are accessible and unbiased, including the use of grade-level language in student prompts Since there aren’t actual assessment tasks, there is no evidence for this criterion. Step 3. Apply Criteria in Dimension IV: Assessment Normed Response: Operations with Rational Numbers

61. 61  Includes aligned rubrics, answer keys and scoring guidelines that provide sufficient guidance for interpreting student performance Answer keys are provided for some, but not all, of the student activities. This feature will be very helpful for teachers but reviewers suggested that in some cases an answer key could be more fully developed. In addition, for units and longer lessons:  Uses varied modes of curriculum-embedded assessments that may include pre-, formative, summative and self-assessment measures This Unit Plan includes reference to a variety of types of curriculum embedded assessment, including a readiness or pre-assessment, formative assessments, and an Exit Ticket assessment as a closure activity. At this point, no culminating summative assessment is identified for the end of the unit, although such may be intended for the Sample Assessment Items section. Attention to varied modes of assessment is a strength of this Unit Plan but the unit could be stronger still by adding a rigorous summative assessment. Step 3. Apply Criteria in Dimension IV: Assessment Normed Response: Operations with Rational Numbers

62. 62 Observations/Feedback and Rating Opportunities for informal assessment using a variety of modes are abundant in the unit, including group discussion, exit tickets, and a “Gallery Walk.” An observation checklist for the teacher to use when observing students working collaboratively with others would be helpful. Other than a readiness assessment, this unit does not include other independent and rigorous assessments of the individual student’s understanding. Missing is a rigorous summative assessment that would elicit direct, observable evidence of independent understanding and that is supported by assessments of similar rigor throughout the lesson/unit. Rating: 1 (Missing a rigorous summative assessment that is supported by other assessment opportunities throughout the lesson/unit.) Developing toward CCSS Quality — needs significant revision, as suggested in criterion-based observations Step 3. Apply Criteria in Dimension IV: Assessment Normed Response: Operations with Rational Numbers

63. 63 Observations/Feedback and Rating Opportunities for informal assessment using a variety of modes are abundant in the unit, including group discussion, exit tickets, and a “Gallery Walk.” An observation checklist for the teacher to use when observing students working collaboratively with others would be helpful. Other than a readiness assessment, this unit does not include other independent and rigorous assessments of the individual student’s understanding. Missing is a rigorous summative assessment that would elicit direct, observable evidence of independent understanding and that is supported by assessments of similar rigor throughout the lesson/unit. Rating: 1 (Missing a rigorous summative assessment that is supported by other assessment opportunities throughout the lesson/unit.) Developing toward CCSS Quality — needs significant revision, as suggested in criterion-based observations Step 3. Apply Criteria in Dimension IV: Assessment Normed Response: Operations with Rational Numbers

64. 64 Compare Criterion-Based Checks, Observations/Feedback and Rating What is the pattern within our team in terms of the criteria we have checked? Do our observations and feedback reference the criteria and evidence (or lack of evidence) in the instructional materials? Do our ratings correspond to the rating and descriptors in the rubric? Step 3. Apply Criteria in Dimension IV: Assessment Sample Unit: Operations with Rational Numbers

65. 65 Step 4. Apply an Overall Rating and Provide Summary Comments Overall Rating for the Lesson/Unit: E: Exemplar — Aligned and meets most to all of the criteria in dimensions II–IV (total 11–12) E/I: Exemplar if Improved — Aligned and needs some improvement in one or more dimensions (total 8–10) R: Revision Needed — Aligned partially and needs significant revision in one or more dimensions (total 3–7) N: Not Ready to Review — Not aligned and does not meet criteria (total 0–2) Try it now – on your own at first.

66. 66 Step 4. Provide Summary Comments Normed Response: Operations with Rational Numbers 66 RATING: 2211 R – Revision Needed While this lesson plan needs improvement, it has great potential. The lesson plan offers numerous and varied opportunities for the skillful teacher to engage students in content, but the less-skilled teacher will need more guidance. All teachers would benefit from an instructional resource that demonstrates stronger alignment (including clear identification of the Standards for Mathematics Practice that are most central to the activities) and the addition of assessments that will provide clear evidence as to whether individual students grasp the knowledge, skills and conceptual understandings that are addressed in the unit. Supports for students with disabilities and English language learners will strengthen the lesson.

67. 67 Step 5. Compare Summary Comments and Determine Next Steps Guiding questions to compare overall ratings and synthesize criterion-based observations and suggestions: How do our overall ratings compare? Does this example serve as a model of CCSS instruction? What are its strengths? Areas for improvement?

68. 68 Reflection on Session Goals Did we … Develop your ability to use EQuIP criteria to provide observations about CCSS-aligned instructional materials and make suggestions for improvement? Develop a common understanding of the EQuIP quality review process? Develop a common understanding of the rating scale and descriptors for the four rubric dimensions and the rating categories and descriptors for overall ratings? Develop your ability to use EQuIP criteria, rating scales and rating descriptors to accurately rate instructional materials?

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