instructional materials evaluation toolkit (imet)...instructional materials to the shifts and the...

Instructional Materials Evaluation Toolkit (IMET) Mathematics, Grades K–8

Grades 6–8

The IMET was developed by Student Achievement Partners. Educators may use or Download this tool at 2

CONTENTS 4 Introduction to the IMET: What, When, Who Non-Negotiable 1: Freedom from Obstacles to Focus Materials must reflect the content architecture of the Standards by not assessing the specific topics named in Metric 1* before the grade level where they first appear in the Standards. 5 NN Metric 1A: Materials reflect the basic architecture of the Standards by not assessing the topics listed below* before the grade level indicated.

Non-Negotiable 2: Focus and Coherence Materials must focus coherently on the Major Work of the grade in a way that is consistent with the progressions in the Standards. 7 NN Metric 2A: In each grade K–8, students and teachers using the materials as designed devote the large majority of time to the Major Work of the grade. 10 NN Metric 2B: Supporting Work enhances focus and coherence simultaneously by also engaging students in the Major Work of the grade. 12 NN Metric 2C: Materials base content progressions on the grade-by-grade progressions in the Standards. Content from previous or future grades does not unduly interfere with or displace

on-grade-level content. 13 NN Metric 2D: Materials are designed to support all students in doing grade-level mathematics. 15 NN Metric 2E: Materials relate on-grade-level concepts explicitly to prior knowledge from earlier grades. 19 NN Metric 2F: Review of material from previous grades is clearly identified as such to the teacher, and teacher and students can see what their specific responsibility is for the current year. 21 NN Metric 2G: Materials include learning objectives that are visibly shaped by CCSSM cluster headings. 22 NN Metric 2H: Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are

natural and important.

Alignment Criterion 1: Rigor and Balance Materials must reflect the balances in the Standards and help students meet the Standards’ rigorous expectations. 24 AC Metric 1A: The materials support the development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content Standards or

cluster headings. 34 AC Metric 1B: The materials are designed so that students attain the fluencies and procedural skills required by the Standards. 39 AC Metric 1C: The materials are designed so that teachers and students spend sufficient time working with applications, without losing focus on the Major Work of each grade.

Alignment Criterion 2: Standards for Mathematical Practice Materials must authentically connect content Standards and practice Standards. 47 AC Metric 2A: Materials address the practice Standards in such a way as to enrich the Major Work of the grade; practice Standards strengthen the focus on Major Work instead of detracting

from it, in both teacher and student materials. 50 AC Metric 2B: Materials attend to the full meaning of each practice Standard. 55 AC Metric 2C: Materials support the Standards’ emphasis on mathematical reasoning. Alignment Criterion 3: Access to the Standards for All Students Materials must provide supports for English Language Learners and other special populations. 62 AC Metric 3A: Support for English Language Learners and other special populations is thoughtful and helps those students meet the same Standards as all other students. The language in

which problems are posed is carefully considered. 63 AC Metric 3B: Materials provide appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners with gradual removal of supports, when

needed, to allow students to demonstrate their mathematical understanding independently. 67 AC Metric 3C: Design of lessons recommends and facilitates a mix of instructional approaches for a variety of learners (e.g., using multiple representations, asking a range of questions,

checking for understanding, flexible grouping, pair-share, deconstructing/reconstructing the language of problems). Indicators of Quality 69 Indicator 1: Lessons are thoughtfully structured and support the teacher in leading the class through the learning paths at hand, with active participation by all students in their own learning

and in the learning of their classmates. 72 Indicator 2: The underlying design of the materials includes both problems and exercises. (In solving problems, students learn new mathematics, whereas in working exercises, students

apply what they have already learned to build mastery.) Each problem or exercise has a purpose. NOTE: This indicator does not require that the problems and exercises be labeled as such. 73 Indicator 3: There are separate teacher materials that support and reward teacher study including, but not limited to: discussion of the mathematics of the units and the mathematical point of

each lesson as it relates to the organizing concepts of the unit, discussion on student ways of thinking and anticipating a variety of students’ responses, guidance on lesson flow, guidance on questions that prompt students’ thinking, and discussion of desired mathematical behaviors being elicited from students.

74 Indicator 4: Manipulatives suggested in the materials are faithful representations of the mathematical objects they represent and are connected to written methods.


75 Indicator 5: Materials include a variety of curriculum-embedded assessments. Examples include pre-, formative, summative, and self-assessment resources. 77 Indicator 6: Assessments contain aligned rubrics, answer keys, and scoring guidelines that provide sufficient guidance for interpreting student performance. 78 Indicator 7: Materials assess student proficiency using methods that are accessible and unbiased, including the use of grade-level language in student prompts. 79 Indicator 8: Materials are carefully evaluated by qualified individuals, whose names are listed, in an effort to ensure freedom from mathematical errors and grade-level appropriateness. 81 Indicator 9: The visual design supports students in engaging thoughtfully with the subject. Navigation through the text is clear. 83 Indicator 10: The materials engage parents in appropriate ways. For example, homework assignments in elementary grades, consist of routine problems, practice with getting answers, and

fluency-building exercises that parents can easily support. 84 Appendix: Scope and Sequence for enVisionmath2.0 (Grades 6–8)


Instructional Materials Evaluation Tool - Mathematics, Grades K–8 (This page is taken directly from the tool found at http://achievethecore.org/IMET) What Are the Purposes of the IMET? This Math IMET is designed to help educators determine whether instructional materials are aligned to the Shifts and major features of the Common Core State Standards (CCSS). The substantial instructional Shifts (http://www.corestandards.org/other-resources/key-shifts-in- mathematics/) at the heart of the Common Core State Standards are:

• Focus strongly where the Standards focus

• Coherence: Think across grades and link to major topics within

the grade

• Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application with equal intensity.

The IMET draws directly from the following documents:

• Common Core State Standards for Mathematics

(www.corestandards.org/Math)

• Publishers’ Criteria for the Common Core State Standards in Mathematics grade K–8 (http://www.corestandards.org/wp-content/uploads/Math_ Publishers_Criteria_K-8_Spring_2013_FINAL1.pdf)

When to Use the IMET

1. Purchasing materials: Many factors go into local purchasing decisions. Alignment to the Standards is a critical factor to consider. This tool is designed to evaluate alignment of instructional materials to the Shifts and the major features of the CCSS. It also provides suggestions of additional indicators to consider in the materials evaluation and purchasing process.

2. Evaluating materials currently in use: The IMET can be used to analyze the degree of alignment of existing materials and help to highlight specific, concrete flaws in alignment. Even where materials and tools currently in use fail to meet one or more of these criteria, the pattern of failure is likely to be informative. States and districts can use the evaluation to create a thoughtful plan to modify or combine existing resources in such a way that students’ actual learning experiences approach the focus, coherence, and rigor of the Standards.

3. Developing programs: Those developing new programs can use

this tool as guidance for creating aligned curricula. Please note this tool was designed for evaluating comprehensive curricula (including any supplemental or ancillary materials), but it was not designed for the evaluation of standalone supplemental materials.

Who Uses the IMET? Evaluating instructional materials requires both subject-matter and pedagogical expertise. Evaluators should be well versed in the Standards (www.corestandards.org/Math) for all grades in which materials are being evaluated. This includes understanding the Major Work of the grade (www.achievethecore/focus), the Supporting and Additional work, how the content fits into the progressions in the Standards (www.achievethecore.org/progressions), and the expectations of the Standards with respect to conceptual understanding, procedural skill and fluency, and application. Evaluators also should be familiar with the substantial instructional Shifts (http:// www.corestandards.org/other-resources/key-shifts-in-mathematics/) of Focus, Coherence and Rigor that are listed above.

The IMET was developed by Student Achievement Partners. Educators may use or Download this tool at

InstructionalMaterialsEvaluationToolkit(IMET)|Mathematics,GradesK–8

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Non-Negotiable 1 Freedom from Obstacles to Focus Non-Negotiable 1: Materials must reflect the content architecture of the Standards by not assessing the specific topics named in Metric 1A* before the grade level where they first appear in the Standards.

Metric

NN Metric 1A:

Materials reflect the basic architecture of the Standards by not assessing the topics listed below before the grade level indicated. • Probability, including chance, likely outcomes, probability models. (Introduced in the CCSSM in grade 7) • Statistical distributions, including center, variation, clumping, outliers, mean, median, mode, range, quartiles; and statistical association or

trends, including two-way tables, bivariate measurement data, scatter plots, trend line, line of best fit, correlation. (Introduced in the CCSSM in grade 6)

• Coordinate transformations or formal definition of congruence or similarity. (Introduced in the CCSSM in grade 8) • Symmetry of shapes, including line/reflection symmetry, rotational symmetry. (Introduced in the CCSSM in grade 4)

Evidence

enVisionmath2.0 (Grades K–8) does not assess any of these topics prior to their introduction by the Common Core.

• Probability, including chance, likely outcomes, and probability models are not assessed in enVisionmath2.0 until Grade 7 Topic 7 when the focus is on cluster 7.SP.C.

• Statistical distributions, including center, variation, clumping, outliers, mean, median, mode, range, and quartiles are not assessed in enVisionmath2.0 until Grade 6 Topic 8 when the focus is on clusters 6.SP.A and 6.SP.B. Statistical association or trends are not assessed in enVisionmath2.0 until Grade 7 Topic 6 when the focus is on clusters 7.SP.A and 7.SP.B. Two-ways tables, bivariate measurement data, scatter plots, trend line, line of best fit, and correlation are not assessed in enVisionmath2.0 until Grade 8 Topic 4 when the focus is on cluster 8.SP.A.

• Coordinate transformations or formal definition of congruence or similarity are not assessed in enVisionmath2.0 until Grade 8 Topic 6 when the focus is on cluster 8.G.A.



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• Symmetry of shapes, including line/reflection symmetry and rotational symmetry are not assessed in enVisionmath2.0 until Grade 4 Topic 16 when the focus is on cluster 4.G.A.

Non-Negotiable 1: Materials must reflect the content architecture of the Standards by not assessing the specific topics named in Metric 1A* before the grade level where they first appear in the Standards. Strengths / Weaknesses:

enVisionmath2.0 (Grades K–8) reflects the content architecture by:

• not assessing topics before the grade level where they first appear in the CCSSM.

• being organized by content Clusters.



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Non-Negotiable 2 Focus and Coherence Non-Negotiable 2: Materials must focus coherently on the Major Work of the grade in a way that is consistent with the progressions in the Standards. Metric

NN Metric 2A:

In each grade K–8, students and teachers using the materials as designed devote the large majority of time to the Major Work of the grade.

Evidence

There are five key ways enVisionmath2.0 (Grades K–8) focuses coherently on the Major Work of the grade:

1. enVisionmath2.0 (Grades K–8) is organized by CCSSM Clusters providing clear content focus. Each Cluster is developed through one or more content topics aligned with the Standards in that Cluster.

2. Each Cluster is identified as Major, Supporting, or Additional; Clusters are color-coded.

Major Clusters – green Supporting Clusters – blue Additional Clusters - yellow

3. The content Clusters/Topics are sequenced to provide content coherence.

• Major Clusters dominate the content developed at each grade.

• Major Clusters are sequenced so they generally appear early in the year.

• Topics that develop standards from the same Cluster generally follow one another.

• Related Clusters are located together to promote connections.



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• A Supporting Cluster at times is placed near the Major Cluster it supports.

• Additional Clusters are located where most appropriate for the content.

4. Standards from some Clusters are integrated into other Clusters where natural connections occur.

5. No extraneous content is taught in any Cluster at any grade in enVisionmath2.0 (Grades K–8).



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For example, Grade 6 Contents reflect the following organization and support of the content emphasis of the CCSSM. Please refer to the Contents for each grade in the Student’s Editions or in each grades’ Teacher’s Edition Program Overview.



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Metric

NN Metric 2B:

Supporting Work enhances focus and coherence simultaneously by also engaging students in the Major Work of the grade.

Evidence

Supporting Work at each grade of the enVisionmath2.0 (Grades 6–8) program enhances focus and coherence simultaneously by engaging students in the Major Work of the grade. This is explicitly noted in the Table of Contents for each grade.

GRADE 6

Supporting Cluster Major Cluster(s) Connections

6.G.A supports 6.EE.A and 6.EE.B in:

• Topic 7 Solve Area, Surface Area, and Volume Problems

6.EE.A and 6.EE.B are developed in:

• Topic 3 Numeric and Algebraic Expressions

• Topic 4 Represent and Solve Equations and Inequalities

• In Topic 7, formulas are developed and used that apply students’ understandings of numerical and algebraic expressions developed in Topics 3 and 4.



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GRADE 7


7.SP.A and 7.SP.B support 7.RP.A and 7.EE.B in:

• Topic 6 Use Sampling to Draw Inferences About Populations

7.RP.A and 7.EE.B are developed in:

• Topic 2 Analyze and Use Proportional Relationships

• Topic 3 Analyze and Solve Percent Problems

• Topic 5 Solve Problems Using Equations and Inequalities

• In Topic 6, students draw comparative inferences about populations while applying their understandings of numerical and algebraic expressions and equations developed in Topics 2, 3, and 5. Additionally, students use their understanding of proportional relationships, developed in Topic 2, to make estimates from sample data.

• In Topic 7, students find probabilities using their understandings of writing and solving equations developed in Topic 5.

7.SP.C supports 7.EE.B in:

• Topic 7 Probability

7.EE.B is developed in:


GRADE 8


8.NS.A supports 8.EE.A in:

• Topic 1 Real Numbers

8.NS.A is developed in:

• Topic 1 Real Numbers

• In Topic 1, students use their understandings of rational and irrational numbers as they work with radicals and integer exponents.

• In Topic 4, students apply their understandings of linear functions and graphs from Topic 3 to analyze linear associations of bivariate data and make predictions.

8.SP.A supports 8.F.A and 8.F.B in:

• Topic 4 Investigate Bivariate Data

8.F.A and 8.F.B are developed in:

• Topic 3 Use Functions to Model Relationships



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Metric

NN Metric 2C:

Materials base content progressions on the grade-by-grade progressions in the Standards. Content from previous or future grades does not unduly interfere with or displace on-grade-level content.

Evidence

enVisionmath2.0 (Grades K–8) offers a unique organization to focus on the Common Core clusters in a way that is consistent with the progressions in the Standards. Each Cluster is developed through one or more content Topics where lessons are aligned with the Standards in that Cluster. Content from previous or future grades does not unduly interfere with or displace on-grade-level content.

(Please see the Contents section in the Grades 6–8 Teacher’s Edition Program Overviews starting on p. 6. Also please reference the program Scope and Sequence in the Grades 6–8 Teacher’s Edition Program Overviews on p. 84. The Grades 6–8 Scope and Sequence is provided in an Appendix to this document.)



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Metric

NN Metric 2D:

Materials are designed to support all students in doing grade-level mathematics.

Evidence

enVisionmath2.0 (Grades 6–8) lessons give students extensive work with on-grade-level problems.

IN EVERY LESSON

• Solve & Discuss It!, Explain It!, and Explore It! introduce lesson content through problem solving.

• Visual Learning Bridge and its digital counterpart, the Visual Learning Animation Plus and other interactive examples, provide stepped-out visual examples and can be used as direct instruction for the important math in the lesson.

• Additional Examples, available digitally, provide even more directed instruction and practice on the important math concepts in the lesson.

• Try It! and Convince Me! ask students to practice skills and construct arguments related to understanding lesson content.

• Do You Understand? is formative assessment of lesson conceptual understanding.

• Do You Know How? builds proficiency of lesson skills.

• Practice & Problem Solving and Additional Practice focus on lesson skills and problem solving that include reinforcement of math practices, vocabulary questions to build understanding, higher order thinking problems, and CCSSM Assessment items.

• MathXL for School provides Practice and Problem Solving, as well as Additional Practice for every lesson.

• Virtual Nerd App provides instructional tutorials for every lesson.

LESSON DIFFERENTIATION AND ASSESSMENT

• Reteach to Build Understanding provides scaffolded reteaching for key lesson concepts.

• Additional Vocabulary Support helps students develop and reinforce understanding of key terms and concepts.



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• Build Mathematical Literacy provides support for struggling readers to build mathematical literacy.

• Enrichment presents engaging problems and activities that extend the lesson concepts.

• Lesson Quiz assesses the important math in the lesson.

• Technology Center includes Digital Math Tools Activities, Today’s Challenge, and Online Math Games.



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Metric

NN Metric 2E:

Materials relate on-grade-level concepts explicitly to prior knowledge from earlier grades.

Evidence

enVisionmath2.0 (Grades K–8) was developed so content coherence is maintained within and across grades. Here are some key aspects of the Grades 6–8 program that speak to this.

CLUSTER OVERVIEWS RELATE ON-GRADE-LEVEL CONCEPTS TO PRIOR KNOWLEDGE

In the Teacher’s Edition, each Math Background contains a section on Coherence that describes how new on-grade-level content is connected to prior knowledge in an earlier grade and, if appropriate, to prior knowledge earlier in the year. Here is an excerpt from the Coherence section of the Grade 7 Topic 1 Math Background. See page 2C for the complete Coherence section.



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PRIOR KNOWLEDGE FROM GRADE 6 NEW ON-GRADE-LEVEL CONTENT IN GRADE 7



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LESSON OVERVIEWS RELATE ON-GRADE-LEVEL CONCEPTS TO PRIOR KNOWLEDGE

In the Teacher’s Edition, each Lesson Overview contains a section on Coherence that describes how new on-grade-level content is connected to prior knowledge. Here is an excerpt from the Coherence section of the Lesson Overview for the first two lessons on integers and rational numbers in Grade 7.

FROM THE LESSON 1-1 LESSON OVERVIEW FROM THE LESSON 1-2 LESSON OVERVIEW

Lesson 1-1 Relate Integers and Their Opposites, p. 7A

Lesson 1-2 Understand Rational Numbers, p. 13A CCSSM Clusters that Call for Extending Prior Knowledge In Grade 6, Topic 1 focuses on Cluster 6.NS.A, Topic 2 focuses on Cluster 6.NS.C, and Topic 3 focuses on Cluster 6.EE.A. In Grade 7, Topic 1 focuses on Cluster 7.NS.A. Each of these clusters call for extending prior knowledge:

• Cluster 6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• Cluster 6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers.

• Cluster 6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions.

• Cluster 7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

The examples given above from Grade 6 Topics 1–3 and Grade 7 Topic 1 are evidence of how on-grade-level content in enVisionmath2.0 (Grades 6–8) is developed by applying and extending prior knowledge.



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LESSON INSTRUCTION RELATES ON-GRADE-LEVEL CONCEPTS TO PRIOR KNOWLEDGE

Step 1 in the instructional model for every lesson is Problem-Based Learning called Solve & Discuss It!, Explain It!, or Explore It! where students use prior knowledge, sometimes from earlier grades, to solve a problem that contains ideas from the lesson. The problem-based learning process connects new ideas to prior knowledge.

Step 2 in the instructional model for each lesson is Visual Learning where teachers use the Visual Learning Bridge and/or the Visual Learning Animation Plus and other interactive examples to explicitly connect students’ thinking in the Solve & Discuss It!, Explain It!, or Explore It! (which may use prior knowledge from earlier grades) to important math ideas in the lesson.



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Metric

NN Metric 2F:

Review of material from previous grades is clearly identified as such to the teacher, and teacher and students can see what their specific responsibility is for the current year.

Evidence

I Can… Statements This feature appears on the first page of each lesson in the Student’s Edition. Students’ responsibilities for each lesson are articulated in this statement.

CCSM Standards Codes The featured content and process Standards codes appear on the first page of each lesson in both the Student’s and Teacher’s Editions. Students can use the codes to locate their responsibilities for each lesson by utilizing the CCSM Standards documents located in the front of their Student’s Editions.

Key Concept This feature appears in each lesson in the Student’s Edition. The Key Concept presents a summary of the main math concepts presented in the lesson.

Fluency Practice Activity This feature is at the end of each topic. For any topic before a topic that teaches an on grade level fluency standard, this feature provides practice on a fluency standard from a previous grade and the standard from the previous grade is identified on the Student’s Edition page.

Today’s Challenge The Today’s Challenge problems to use with Topic 1 can be used anytime in Topic 1. So those problems use content taught in a previous grade or topic.

Beginning-of-Year Assessment This is a readiness assessment at the start of the year that assesses content taught in the previous grade. The Item Analysis chart in the Teacher’s Edition identifies standards from the previous grade.

Topic Readiness Assessment This is a readiness assessment at the start of each topic that assesses content related to the topic that is taught in the previous grade and/or previous topics. The Item Analysis chart in the Teacher’s Edition identifies standards from the previous grade and/or topics.



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Get Ready! Review What You Know! This feature is at the start of each topic and reviews content related to the topic that is taught in the previous grade and/or previous topics. The Item Analysis chart in the Teacher’s Edition identifies standards from the previous grade and/or topics.

Math Background, Look Back This feature is at the start of each topic in the Teacher’s Edition and describes connections to content in the previous grades and/or topics. This feature also includes a description of the content within the topic and related content to come.



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Metric

NN Metric 2G:

Materials include learning objectives that are visibly shaped by CCSSM cluster headings.

Evidence

enVisionmath2.0 (Grades K–8) is organized to help students focus on clusters of Common Core standards within a grade. Each Common Core cluster is the focus of one or more topics to promote in-depth development.

The Lesson Overview and Mathematics Overview begins every lesson in the Teacher’s Edition, outlining the Domain, Cluster, Content Standard, and Mathematical Practices covered in the lesson, as well as a Lesson Objective that is clearly shaped by CCSSM cluster headings. Student’s Edition pages feature prominent learning objectives in the form of “I can” statements that state the content objective in accessible language. Student pages also provide the Content Standard and Mathematical Practices on the page, giving students the opportunity to make a direct connection to the specific Standard and Practice as well.

The following chart illustrates the relationship among some CCSSM cluster headings and enVisionmath2.0 (Grades 6–8) topic titles, lesson titles, and student “I can” objectives for lessons. Also refer to the Contents for each grade in the Teacher’s Edition Program Overview.

SOME TITLES AND LEARNING OBJECTIVES SHAPED BY CCSSM CLUSTER HEADINGS

CCSSM Cluster Heading Topic Title Lesson Title Lesson Objective (“I can”)

Grade 6 Cluster 6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.

Topic 5 Understand and Use Ratio and Rate

Lesson 5-1 Understand Ratios

I can use a ratio to describe the relationship between two quantities.

Grade 7 Cluster 7.EE.A Use properties of operations to generate equivalent expressions.

Topic 4 Generate Equivalent Expressions

Lesson 4-2 Generate Equivalent Expressions

I can write equivalent expressions for given expressions.

Grade 8 Cluster 8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.

Topic 2 Analyze and Solve Linear Equations

Lesson 2-7 Analyze Linear Equations: y = mx

I can write equations to describe linear relationships.



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Metric

NN Metric 2H:

Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

Evidence

In enVisionmath2.0 (Grades K–8), each Common Core cluster is the focus of one or more topics. But topics also contain connections to other clusters and other domains where the connections are valuable and sensible. Below are examples of some examples of key cross-cluster and cross-domain connections within Grades 6–8.

GRADE 6 Connected Clusters/Domains Connections Examples

6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.B Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers.

6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions.

• Topic 1 Use Positive Rational Numbers

Topic 2 Integers and Rational Numbers

Topic 3 Numeric and Algebraic Expressions

Topics 1 and 2 focus on operations with fractions and decimals, integers, and rational numbers (6.NS.A–6.NS.C). Topic 3 connects to the content in Topics 1 and 2 in that it applies that knowledge to work with numeric and algebraic expressions (6.EE.A).



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GRADE 7 7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.


Topic 8 Solve Problems Involving Geometry

Topic 8 focuses on solving real-world problems about area, surface area, and other geometric concepts (7.G.B) but connects to the concepts learned in Topic 5 in that students must write and solve equations (7.EE.B) to solve the problems.

GRADE 8 8.EE.B Understand the connections between proportional relationships, lines, and linear equations.

8.F.A Define, evaluate, and compare functions.

8.F.B Use functions to model relationships between quantities.

8.SP.A Investigate patterns of association in bivariate data.

• Topic 2 Analyze and Solve Linear Equations

Topic 3 Use Functions to Model Relationships

Topic 4 Investigate Bivariate Data

Topic 4 analyzes linear associations on graphs (8.SP.A) but connects to and supports the concepts learned in Topic 2 (8.EE.B) and Topic 3 (8.F.A, 8.F.B) by analyzing linear equations and identifying linear and nonlinear functions.

Non-Negotiable 2: Materials must focus coherently on the Major Work of the grade in a way that is consistent with the progressions in the Standards.

Strengths/Weaknesses

The Common Core Cluster organization of enVisionmath2.0 (Grades K–8) builds understanding and best promotes mathematical content connections. The program explicitly aligns to the Major, Supporting, and Additional Work at every grade. Teachers will find this very helpful in understanding and being confident about how enVisionmath2.0 (Grades K–8) supports Focus and Coherence.



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Alignment Criterion 1 Rigor and Balance

AlignmentCriterion1:MaterialsmustreflectthebalancesintheStandardsandhelpstudentsmeettheStandards’rigorousexpectations.

Metric

AC Metric 1A:

The materials support the development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content Standards or cluster headings.

Evidence

Is conceptual understanding attended to thoroughly where the Standards set explicit expectations for understanding or interpreting?

“Understanding” is explicitly called out in many of the Grades 6–8 Standards in CCSSM. The following are descriptions of how “understanding” is given special attention in 6.RP.A, 7.NS.A, and 8.G.A in Grades 6–8 in enVisionmath2.0.

6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.

The enVisionmath2.0 highly-effective approach to ratio concepts and ratio reasoning embraces: (a) an understanding of equivalent ratios in models, tables, and graphs, (b) an understanding and use of unit rate to compare and convert measures, and (c) an understanding of percent as a rate.

• Ratios are written to describe models that represent the relationship between two quantities in Topic 5 Lesson 1 while tables of equivalent ratios are used to compare ratios and solve problems in Topic 5 Lessons 2 and 3.

• The understanding that straight lines on graphs show that equivalent ratios are points on the same line is explored in Topic 5 Lesson 4.

• The understanding that unit rates are a special type of ratio that can be used to compare quantities with different units of measure is used to solve problems in Topic 5 Lessons 6 and 7.



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• The use of dimensional analysis helps students to understand the use of ratio reasoning to convert between measurement units in Topic 5 Lessons 8–10.

• Finding percent of a quantity as a rate per 100 in Topic 6 further develops students’ understanding of ratio concepts and ratio reasoning.

7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

The enVisionmath2.0 highly-effective approach to understanding operations with rational numbers embraces an application of students’ understanding of operations with fractions.

• In earlier grades, students learned the processes to add, subtract, multiply, and divide whole numbers and fractions. In Grade 6, students are introduced to the term rational number, which includes whole numbers and integers, as they perform operations with fractions and decimals and explore opposites. In Grade 7 Topic 1, students are introduced to operations with integers. Having the understanding that rational numbers consist of any number (positive or negative) that can be written as a ratio, students extend that understanding to operations with rational numbers.

• The inverse relationship between integers and their opposites is explored in Topic 1 Lesson 1 which launches students into adding and subtracting integers in Topic 1 Lessons 3 and 4 and sets the stage for developing understanding with adding and subtracting rational numbers with different signs in Topic 1 Lesson 5.

• The examination of multiplication with different pairings of positive and negative integers in Topic 1 Lesson 6 prepares students for understanding the multiplication of rational numbers in Topic 1 Lesson 7.

• The extension of the rules for multiplying rational numbers prepares students for understanding the division of rational numbers in Topic 1 Lesson 9. The processes and understanding for adding, subtracting, multiplying, and dividing rational numbers are developed purposefully throughout the lessons in Topic 1.

8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software.

The enVisionmath2.0 highly-effective approach to understanding congruence and similarity embraces a focus on analyzing transformations including translations, reflections, rotations, and dilations to determine if one figure maps onto another.

• The properties of translations are utilized to perform and describe a translation of a two-dimensional figure. In Topic 6 Lesson 1, students understand the concept of translating a figure and how the resulting image is related to the preimage.

• The properties of reflections are utilized to create images that have the same size and shape but a different orientation. In Topic 6 Lesson 2, students understand the concept of reflecting figures and recognize how the image is a flip of the preimage and that the two figures are an equal distance from the line of reflection.



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• Rotations are described in terms of the center of rotation and the number of degrees the figure has been rotated. In Topic 6 Lesson 3, students understand that the image and preimage of a rotated figure have the same size and shape.

• Dilations are identified as a transformation that changes the size of a figure but not its shape, angle measure, or proportional size. The properties of dilations are used to dilate or enlarge a figure in a coordinate plane.

• After developing an understanding of transformations, the concepts of congruence and similarity are purposefully introduced. The understanding of transformations is used to develop a deeper understanding of congruent and similar figures. In Topic 6 Lesson 5, students use transformations to identify congruent figures that have the same shape and size. In Topic 6 Lesson 7, students understand that figures are similar if a sequence of transformations maps one figure onto another.

Do the materials feature high-quality conceptual problems and conceptual discussion questions?

The core instructional model used for every lesson in enVisionmath2.0 (Grades K–8) is one shown through research to be the most effective for developing conceptual understanding. Here is a recent research article that describes the instructional model used in enVisionmath2.0.

“Researchers have identified a three-phase classroom activity structure [for high-quality mathematics instruction]. In well executed lessons…the teacher poses a problem and ensures that all students understand the context and expectations, students develop strategies and solutions (typically in collaboration with each other), and, through reflection and sharing, the teacher and students work together to explicate the mathematical concepts underlying the lesson’s problem.” [Munter, Charles. “Developing Visions of High-Quality Mathematics Instruction.” Journal for Research in Mathematics Education, Volume 45, Number 5, November 2014, pp. 584–635.]

The instructional approach described above has also been called Teaching Mathematics Through Problem Solving. Here is a reference for an NCTM publication that describes this model and provides additional research as to its effectiveness.

Lester, F.K. & Charles, R.I. (EDs). (2003). Teaching Mathematics Through Problem Solving, Grades PreK–6. Reston, VA: NCTM.

The instructional model described above and used for every lesson in enVisionmath2.0 (Grades 6–8) begins with an experience titled Solve & Discuss It!, Explore It!, or Explain It!. These experiences give students a non-routine problem to solve. A problem is non-routine if students have not already learned an efficient procedure for solving it. For example, “Find 0.28 x 4.” is a non-routine problem for a sixth grader who has not yet developed an efficient strategy for finding the product of a whole number and a decimal. Students might use decimal grids to find the solution to this non-routine problem.

Non-routine problems used to introduce new ideas in mathematics (teaching mathematics through problem solving) means that “high quality conceptual problems” are at the core of enVisionmath2.0 (Grades K–8). Furthermore, the teaching actions shown in the Teacher’s Edition show that “conceptual discussion questions” are at the core of enVisionmath2.0.



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Below is a sample Solve & Discuss It! problem from enVisionmath2.0 (Grades 6–8).

In the following Grade 6 lesson, students are working with a “non-routine” problem to explore patterns in the number of folds and the number of sections when folding paper. Being able to model the math in the problem is foundational for understanding the patterns. Multiple strategies can be used to explore and describe the patterns.



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The Teacher’s Edition provides Before, During, and After questions to support best practices and promote discourse. In addition, sample student work is provided, as shown at the right, to ensure a quality classroom discussion to analyze student work. In this particular example, both students successfully demonstrate different strategies to solve the problem. See Grade 6, Topic 3 Teacher’s Edition p. 117.



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Below, the Visual Learning Bridge for this lesson makes the concepts from the Solve & Discuss It! become more explicit and provides blue guiding questions in the Teacher’s Edition that ask students, “What?” and “How?” to promote conceptual understanding.



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Additional Teacher Support Related to Conceptual Understanding In the Teacher’s Edition, each Math Background and each Lesson Overview has a section called “Rigor” which includes information about conceptual understanding.

From Grade 6 Topic 3 Math Background, p. 112D

From Grade 6 Topic 3 Lesson 1 Lesson Overview, p. 117A



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Do the materials feature opportunities to identify correspondences across mathematical representations?

Students are prompted to use different graphic models as they explore concepts and are encouraged to represent their thinking with concrete, pictorial, and abstract representations as shown at the right. (As an example, see Grade 6 Teacher’s Edition pages 252A–252D.)



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In Grade 6 Lesson 5-1, students use a bar diagram as a model to solve a ratio problem. (6.RP.A.1, 6.RP.A.3)

Also in Grade 6 Lesson 5-1, students use a double number line diagram as a model to solve a ratio problem. (6.RP.A.1, 6.RP.A.3)

In Grade 6 Lesson 5-4, students graph ratios to visually see that equivalent ratios form a straight line when graphed on a coordinate plane. All points on the line represent equivalent ratios. (6.RP.A.3a)



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In Grade 7 Lessons 4-1, 4-4 through 4-6, and 4-8, students use bar diagrams to write variable expressions that represent given problem-solving situations. (7.EE.A)

In Grade 8 Lessons 2-1 through 2-4, students use bar diagrams to represent equations. They decompose and reorder bar diagrams to solve for x. (8.EE.C.7a, 8.EE.C.7b)



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Metric

AC Metric 1B:

The materials are designed so that students attain the fluencies and procedural skills required by the Standards.

Evidence

Is progress toward fluency and procedural skill interwoven with students’ developing conceptual understanding of the operations in question?

The following table provides the CCSSM Fluency Standards for each grade, the associated enVisionmath2.0 (Grades K–5) topics that provide the development with understanding and foundation for fluency across and within grades, and the corresponding “capstone” topic at each grade that ensures fluency for the required CCSSM.

CCSSM Fluency Standard

enVisionmath2.0 Topics that Provide the Foundation for Fluency “Capstone” enVisionmath2.0 Topic

Grade K Fluently add and subtract within 5. (K.OA.A.5)

• Grade K, Topic 6 Understand Addition

• Grade K, Topic 7 Understand Subtraction

• Grade K, Topic 8 More Addition and Subtraction (Provides the development with understanding that students need to fluently add and subtract within 5.)

Grade 1 Fluently add and subtract within 10. (1.OA.C.6)

• Grade K, Topic 8 More Addition and Subtraction

• Grade 1, Topic 1 Solve Addition and Subtraction Problems to 10

• Grade 1, Topic 2 Fluently Add and Subtract Within 10

Grade 2 Fluently add and subtract within 20. (2.OA.B.2)


• Grade 1, Topic 3 Addition Facts to 20: Use Strategies

• Grade 1, Topic 4 Subtraction Facts to 20: Use Strategies




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Grade 2 Fluently add and subtract within 100. (2.NBT.B.5)

• Grade 1, Topic 10 Use Models and Strategies to Add with Tens and Ones

• Grade 1, Topic 11 Use Models and Strategies to Subtract Tens


• Grade 2, Topic 3 Add Within 100 Using Strategies

• Grade 2, Topic 5 Subtract Within 100 Using Strategies

• Grade 2, Topic 4 Fluently Add Within 100

• Grade 2, Topic 6 Fluently Subtract Within 100

Grade 3 Fluently add and subtract within 1,000. (3.NBT.A.2)

• Grade 2, Topic 4 Fluently Add Within 100

• Grade 2, Topic 6 Fluently Subtract Within 100

• Grade 2, Topic 10 Add Within 1,000 Using Models and Strategies

• Grade 2, Topic 11 Subtract Within 1,000 Using Models and Strategies

• Grade 3, Topic 8 Use Strategies and Properties to Add and Subtract

• Grade 3, Topic 9 Fluently Add and Subtract Within 1,000

Grade 3 Fluently multiply and divide within 100. (3.OA.C.7)

• Grade 2, Topic 2 Work with Equal Groups

• Grade 3, Topic 1 Understand Multiplication and Division of Whole Numbers

• Grade 3, Topic 2 Multiplication Facts for 0, 1, 2, 5, 9, 10

• Grade 3, Topic 3 Apply Properties: Multiplication Facts for 3, 4, 6, 7, and 8

• Grade 3, Topic 4 Use Multiplication to Divide: Division Facts

• Grade 3, Topic 5 Fluently Multiply and Divide Within 100

Grade 4 Fluently add and subtract multi-digit whole numbers. (4.NBT.B.4)

• Grade 3, Topic 9 Fluently Add and Subtract Within 1,000

• Grade 4, Topic 2 Fluently Add and Subtract Multi-Digit Whole Numbers

Grade 5 Fluently multiply multi-digit whole numbers. (5.NBT.B.5)

• Grade 3, Topic 5 Fluently Multiply and Divide Within 100

• Grade 3, Topic 10 Multiply by Multiples of 10

• Grade 4, Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers

• Grade 4, Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers

• Grade 5, Topic 3 Fluently Multiply Multi-Digit Whole Numbers



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In the elementary years, the fluency Standards focus on operation or computational fluency so that by the end of Grade 6, students have operational fluency with whole and rational numbers. While students in Grade 6 are finalizing operational fluency, the focus of instruction in middle grades shifts from operational fluency to ratio reasoning and proportionality and to algebraic concepts and linearity. Middle grades students are expected to apply operational fluency as they begin study of algebraic concepts and develop proficiency with applying ratio and proportional reasoning and solving algebraic equations. The following table provides the CCSSM fluency Standards for Grades 6–8, the associated enVisionmath2.0 (Grades 6–8) topics that provide the development with understanding and foundation for fluency across and within grades, and the corresponding “capstone” topic at each grade.



Grade 6 Fluently divide multi-digit whole numbers. (6.NS.B.2)

• Grade 4, Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers

• Grade 5, Topic 5 Use Models and Strategies to Divide Whole Numbers

• Grade 6, Topic 1 Use Positive Rational Numbers

Grade 6 Fluently add, subtract, multiply, and divide multi-digit decimals. (6.NS.B.3)

• Grade 5, Topic 2 Add and Subtract Decimals to Hundredths

• Grade 5, Topic 4 Use Models and Strategies to Multiply Decimals

• Grade 5, Topic 6 Use Models and Strategies to Divide Decimals


Grade 6 Apply and extend previous understandings of arithmetic to algebraic expressions. (6.EE.A)

• Grade 5, Topic 1 Understand Place Value


• Grade 6, Topic 3 Numeric and Algebraic Expressions

Grade 7 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (7.NS.A)


• Grade 6, Topic 2 Integers and Rational Numbers

• Grade 7, Topic 1 Integers and Rational Numbers

Grade 7 Apply properties of operations to generate equivalent expressions. (7.EE.A.1)

• Grade 6, Topic 3 Numeric and Algebraic Expressions

• Grade 7, Topic 4 Generate Equivalent Expressions



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Grade 7 Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. (7.EE.B.4a)

• Grade 6, Topic 4 Represent and Solve Equations and Inequalities


• Grade 7, Topic 5 Solve Problems Using Equations and Inequalities

Grade 8 Solve linear equations in one variable. (8.EE.C.7)

• Grade 7, Topic 2 Analyze and Use Proportional Relationships


• Grade 7, Topic 5 Solve Problems Using Equations and Inequalities

• Grade 8, Topic 2 Analyze and Solve Linear Equations

Grade 8 Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. (8.EE.C.8b)


• Grade 8, Topic 2 Analyze and Solve Linear Equations

• Grade 8, Topic 5 Analyze and Solve Systems of Linear Equations

Are purely procedural problems and exercises present that include cases in which opportunistic strategies are valuable and generic cases that require efficient algorithms present?

There are opportunities throughout enVisionmath2.0 (Grades K–8) for students to use strategies that make sense to them and may be more non-traditional, as well as opportunities that require more efficient algorithms. For example, see Grade 6 Student’s Edition Lesson 1-1 pp. 10–12. On p. 7, Problem-Based Learning (Solve & Discuss It!) opens the lesson with a rich problem for students to solve any way they choose and then discuss and share solution strategies. During Visual Learning Bridge on pp. 8–9, students see how they can use different ways to solve addition problems with decimals. During Convince Me! students explain how adding 0.267 and 50.9 is different from adding 0.26 and 50.9. Do You Understand? and Do You Know How? offers opportunities for students to find sums, differences, and products and describe the procedures they use.



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Do the materials in grades K–6 provide repeated practice toward attainment of fluency Standards?

For each fluency Standard, there is an enVisionmath2.0 topic that develops fluency with understanding. That topic is the culmination of foundations for fluency taught previously.

For example, in addition to foundational topics in Grade 4 and Grade 5, Topic 1 in Grade 6 provides the development with understanding that students need to fluently add, subtract, multiply, and divide whole numbers and decimals.

In addition,

• Fluency Practice Activities found at the end of each of topic support Fluency Standards. Students can collaborate on these activities to help each other as needed.

• Fluency Practice/Assessment Worksheets Activities (enVisionmath2.0 Grades K–5) provide ongoing practice/assessment of fluency subskills.

• Online Practice Buddy (enVisionmath2.0 Grades 3–5) or Online MathXL for School (enVisionmath2.0 Grades 6–8) provides auto-scored items on subskills of a fluency standard. If students are uncertain how to complete an item, or if they answer incorrectly, intervention is available through on-screen help. This includes “Help Me Solve This” which guides the student through a solution, and “View an Example” which shows a worked-out solution for a similar item.

• Game Center at PearsonRealize.com provides opportunities for practicing fluency skills.

• Math Diagnosis and Intervention System 2.0 lessons provide remediation for students who struggle with a fluency standard.

• “My Fluency Progress” Form (enVisionmath2.0 Grades K–5) in the Teaching Tools is a resource that can be used to help students become personally responsible for monitoring their progress on the fluency subskills.



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Metric

AC Metric 1C

The materials are designed so that teachers and students spend sufficient time working with applications, without losing focus on the Major Work of each grade.

Evidence

Are there single- and multi-step contextual problems that develop the mathematics of the grade, afford opportunities for practice, and engage students in problem solving? Do the problems attend thoroughly to those places in the content Standards where expectations for multi-step and real-world problems are explicit?

Students begin every enVisionmath2.0 (Grades 6–8) lesson with Problem-Based Learning, Solve & Discuss It!, Explain It!, or Explore It!, which introduces new math in the context of solving a real problem, and a standard solution procedure may not be readily available to students. Students make connections to prior knowledge and determine their own methods for solving the problem. But students may not have learned a procedure/technique for solving the problem in the most efficient manner. Classroom discussion exposes them to alternate methods for solving the problem and deepens their understanding. This non-routine, daily engagement with the Mathematical Practices is the best way for students to develop understanding.

For example, in Grade 7 Lesson 5-1, p. 253, students explore different ways to represent a two-step problem. In Grade 8 Lesson 5-1, p. 257, students identify three ways pairs of lines can intersect or not intersect in preparation for solving systems of linear equations by inspection (one solution, no solution, or infinitely many solutions).

Many of the lesson’s Visual Learning Bridges start with real-world applications. These real-world applications provide a context and motivate students to see why the concept or skill is important and also reinforces operation meanings and relationships for use in solving problems.

Single- and multi-step contextual problems, including non-routine problems, that afford opportunities for students to practice and engage in problem solving are also found in every lesson’s Do You Understand?, Do You Know How?, and Practice & Problem Solving. For example, in Grade 6 Lesson 1-7, pp. 52–54, students use strategies such as deciding the steps to use to solve the problem, choosing the correct operations, identifying the information needed from the problem, correctly using the information, calculating accurately, interpreting solutions, and checking that the solution is reasonable to help solve multi-step problems and communicate their reasoning. Lessons like, Grade 7 Lesson 2-6, pp. 124–126 provide additional examples.



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Students have a variety of opportunities to apply their understanding in STEM Activities, 3-Act Mathematical Modeling Activities, Digital Math Tools Activities, and Online Math Games. (See Grade 7 Teacher’s Edition, pp. 352 and pp. 381A to 383–384 for examples.)

Grade 7 Lesson 5-7, p. 299, Item 9 and p. 300, Item 12 are just a couple examples of multi-step applications provided in the Practice & Problem Solving portion of the daily lesson. In addition to multi-step contextual problems being available to students in daily lessons, enVisionmath2.0 (Grades 6–8) provides full lessons devoted to supporting the CCSSM where expectations for multi-step and real-world problems are explicit. As an example, the chart below lists the appropriate Standard and identifies the enVisionmath2.0 topic and lessons within those topics that were written to address those Standards.

SOME LESSONS ON STANDARDS THAT EXPLICITLY EXPECT MULTI-STEP AND REAL-WORLD PROBLEMS

Grade CCSSM Standards enVisionmath2.0 Topics enVisionmath2.0 Lessons

Grade 7

7.RP.A.3

Use proportional relationships to solve multi-step ratio and percent problems.

Topic 2 Analyze and Use Proportional Relationships

Lesson 2-1 Connect Ratios, Rates, and Unit Rates

Lesson 2-2 Determine Unit Rates with Ratios of Fractions

Lesson 2-6 Apply Proportional Reasoning to Solve Problems

Topic 3 Analyze and Solve Percent Problems

Lesson 3-1 Analyze Percents of Numbers

Lesson 3-2 Connect Percent and Proportion

Lesson 3-3 Represent and Use the Percent Equation

Lesson 3-4 Solve Percent Change and Percent Error Problems

Lesson 3-5 Solve Markup and Markdown Problems

Lesson 3-6 Solve Simple Interest Problems



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SOME LESSONS ON STANDARDS THAT EXPLICITLY EXPECT MULTI-STEP AND REAL-WORLD PROBLEMS

Grade CCSSM Standards enVisionmath2.0 Topics enVisionmath2.0 Lessons

Grade 7 7.EE.B.3

Solve multi-step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

Topic 1 Integers and Rational Numbers

Lesson 1-10 Solve Problems with Rational Numbers

Topic 4 Generate Equivalent Expressions Lesson 4-1 Write and Evaluate Algebraic Expressions

Topic 5 Solve Problems Using Equations and Inequalities

Lesson 5-2 Solve Two-Step Equations

Lesson 5-3 Solve Equations Using the Distributive Property

Topic 7 Probability

Lesson 7-1 Understand Likelihood and Probability

7-4 Use Probability Models


Lesson 8-6 Solve Problems Involving Area of a Circle

Lesson 8-8 Solve Problems Involving Surface Area

Lesson 8-9 Solve Problems Involving Volume

7.G.B.5

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.


Lesson 8-4 Solve Problems Using Angle Relationships



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Do application problems particularly stress applying the Major Work of the grade?

APPLICATIONS IN PROGRAM ELEMENTS

Here are some examples of how the following elements of enVisionmath2.0 (Grades 6–8) address this metric in the many topics that focus on major clusters as well as in other topics with connections to major clusters.

Solve & Discuss It!, Explain It!, Explore It! in Lessons Every lesson begins with students solving a problem (problem-based instruction) in an experience called Solve & Discuss It!, Explain It!, or Explore It!. Introducing concepts and procedures through problem solving develops understanding. Many of these problems are applications. Students apply previously-learned concepts and procedures to solve problems. Some Grade 6 examples are: Lesson 1-4, p. 31; Lesson 1-7, p. 49; Lesson 2-3, p. 77; Lesson 5-2, p. 263; Lesson 5-4, p. 275. Some Grade 7 examples are: Lesson 1-1, p. 7; Lesson 1-8; p. 51; Lesson 3-1, p. 137; and Lesson 4-3, p. 201. Some Grade 8 examples include: Lesson 1-5, p. 31; Lesson 2-5, p. 117; and Lesson 4-1, p. 211.

Visual Learning Bridge in Lessons Every lesson includes a Visual Learning Bridge (VLB) as a follow-up to the Solve & Discuss It!, Explain It!, or Explore It!. The VLB is the important mathematics of the lesson made explicit. Most VLBs begin with a real-world, highly visual problem. These are included for the following reasons:

• To show an application of the concept or procedure developed in that lesson. See, for example, Grade 6 Topic 5, p. 284.

• To reinforce operation meanings often using bar diagrams, number lines, or other models to show relationships between quantities. See, for example, Grade 7 Topic 1, p. 26 or Grade 7 Topic 2, p. 87.

• To provide a real-world visual context that helps students make sense of the concept or procedure developed. See, for example, Grade 8 Topic 2, p. 130.

Practice & Problem Solving Items in Lessons This is the final section of every lesson. Content-appropriate applications are included that:

• Focus on the content of the lesson but revisit Major Content as appropriate. See, for example, all exercises on Grade 6 Topic 8, p. 465.

• Involve the use of multiple Mathematical Practices but provide a teaching opportunity to discuss a particular Mathematical Practice. See, for example, Grade 8 Topic 3, Item #12, p. 175.

• Provide an opportunity to use higher-cognitive levels of reasoning. See, for example, Grade 7 Topic 4, Item #15, p. 211.

• Apply previously-learned concepts and procedures in a task representative of those from Common Core Assessments. See, for example, Grade 6 Topic 8, Item #18, p. 478.



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3-Act Mathematical Modeling Lessons Mathematical Practices are applied in every lesson. For example, the Solve & Discuss It!, Explain It! or Explore It! experience that begins every lesson is a particularly rich area for developing students’ abilities with Mathematical Practices. Every Solve & Discuss It!, Explain It! or Explore It! task requires students to select, coordinate, and implement multiple Mathematical Practices. However, the enVisionmath2.0 (Grades K–8) authors believe it is important to stop occasionally throughout the year to have in-depth applications of Mathematical Practices in real-world problems that also apply previously-learned mathematics and the mathematics in the topic of study. Therefore, each topic in enVisionmath2.0 (Grades 6–8) includes a lesson that focuses on mathematical modeling and specifically applies Mathematical Practices. Some aspects of these lessons of particular note are:

• An emphasis on addressing the “Main Question” (e.g., Grade 6 Topic 7, p. 419).

• The use of number lines to show how quantities in problems are related (e.g., Grade 7 Topic 7, p. 381).

• The use of multiple Mathematical Practices in finding solutions (e.g., Grade 6 Topic 6, p. 374).

• The use systematic questioning to promote an understanding of a problem (e.g., Grade 8 Topic 8, p. 444).

Assessments and Applications Applications are infused in assessments.

• Common Core assessment items in lessons are included at the end of the Practice & Problem Solving section. Many of these items are applications of concepts and skills developed in the lesson. Some examples in Grade 8 are p. 44, Item 27 and p. 140, Items 13 and 14.

• Common Core assessment items with real-world application are also included in the Mid-Topic Checkpoint and Mid-Topic Performance Task. Some examples in Grade 6 are p. 137, Items 8–9 and p. 210, Part C.

• Common Core lesson and topic assessments items are often driven by real-world applications rather than just skill assessment. Some Grade 7 examples are: Topic 6 Lesson 1 Quiz, p. Q6-1 and Topic 8 Topic Assessment Item 7, p 2 of 2.



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Does modeling build slowly across K–8, with applications that are relatively simple in earlier grades and when students are encountering new content? In Grades 6–8, do the problems begin to provide opportunities for students to make their own assumptions or simplifications in order to model a situation mathematically?

BAR DIAGRAMS (GRADES K–8)

Bar diagrams help students understand and represent the relationship between quantities and solve problems. Bar diagrams help students use the math they know, use a diagram, and write an equation to show the problem. The use of Bar Diagrams in enVisionmath2.0 (Grades K–8) demonstrates how students apply understanding in simple situations in early grades and then slowly build across the grades. See the Grades K–5 Program Sampler and Review Guide pp. 58–61 and the Grades 6–8 Teacher’s Edition Program Overview pp. 48–51 for a complete overview of the Bar Diagram progression across grades in enVisionmath2.0 (Grades K–8). Below are just a few examples from the program to demonstrate how this modeling is builds across Grades 6–8.

Below is a Grade 6 example of bar diagram use with ratios. Below is a Grade 7 example of bar diagram use with percent.

Below is a Grade 7 example of bar diagram use with two-step equations. Below is a Grade 8 example of bar diagram use with functions.



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A Grade 8 example of using bar diagram to solve equations is shown at the right.

MATHEMATICAL MODELING (GRADES 6–8)

Having explored a variety of mathematical models in Grades K–5, each topic in Grades 6–8 includes a 3-Act Mathematical Modeling Lesson which provides opportunities for students to make their own assumptions or simplifications to model mathematically. Students make a prediction, identify information needed to develop a mathematical model, then develop a model that represents the problem. Finally, they use their models to solve the problem. At the right is an example from Grade 7 Topic 7 Teacher’s Edition p. 382.



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Alignment Criterion 1: Materials must reflect the balances in the Standards and help students meet the Standards’ rigorous expectations.

Strengths / Weaknesses

enVisionmath2.0 (Grades K–8) requires students to engage with and demonstrate challenging mathematics with appropriate balance among application, conceptual understanding, and procedural skill and fluency.



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Alignment Criterion 2 Standards for Mathematical Practice

AlignmentCriterion2:MaterialsmustauthenticallyconnectcontentStandardsandpracticeStandards.

Metric

AC Metric 2A:

Materials address the practice Standards in such a way as to enrich the Major Work of the grade; practice Standards strengthen the focus on Major Work instead of detracting from it, in both teacher and student materials.

Evidence

enVisionmath2.0 (Grades K–8) is organized to focus on CCSSM clusters, and clusters are identified as Major, Supporting, and Additional. Another layer in organizing enVisionmath2.0 (Grades K–8) relates to the eight Standards for Mathematical Practice as described in the Common Core Standards. Mathematical practices are the habits of mind, processes, and dispositions that enable a learner to understand mathematics and to use or do mathematics with understanding. Mathematical practices translate to observable verbal and written behaviors by students as they do mathematics.

The goal and promise of enVisionmath2.0 (Grades K–8) is to develop understanding, so the mathematical practices are infused in all aspects of the program. (For reference as an example, see Grade 6 Teacher’s Edition pp. 252E, 257A to 262–262 and Grade 8 Teacher’s Edition pp. 292E, 333A to 337–338.)

IN EVERY LESSON

Everyday engagement of the Standards for Mathematical Practice enables learners to develop understandings and use mathematics with understanding. It is clear from research that problem-based teaching and learning is THE best vehicle for developing expertise with the mathematical practices. Every lesson Solve & Discuss It!, Explain It! or Explore It! problem-based learning in enVisionmath2.0 (Grades 6–8) requires the student to select, implement, and manage multiple mathematical practices. The Standards for Mathematical Practice are reflected in the questions provided for teachers for each Solve & Discuss It!, Explain It! or Explore It!, and particular questions to highlight are noted in the teacher materials and in the student materials in blue Mathematical Practice boxes.



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For each Visual Learning Bridge, the comments and questions in the blue Mathematical Practice boxes in the student materials and the conversation-starter questions in the teacher materials connect to mathematical practices. These comments and questions are embedded in the online Visual Learning Animation Plus and other interactive examples.

The Visual Learning Bridge that follows from Grade 7 Topic 3 Lesson 3 shows examples of comments and questions in blue Mathematical Practice boxes that focus instruction on one or more particular mathematical practices.



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In each lesson’s Do You Understand?, Do You Know How? and Practice & Problem Solving section, specific items are labeled with one mathematical practice, but these items may involve more than one mathematical practice—providing an opportunity for rich classroom conversations and teaching moments.

LESSONS THAT HIGHLIGHT MATH PRACTICES

In Grades 6–8, a comprehensive 3-Act Mathematical Modeling Lesson in every topic gives special focus to the math practices. Features of the 3-Act Mathematical Modeling Lesson include the following. (For reference as an example, see Grade 8 Teacher’s Edition pp. 377A to 379–380.)

Act 1: The Hook The lesson begins with a multimedia presentation of a problem situation where students brainstorm possible questions to answer. Once the main question has been identified, students apply math practices as they propose predictions that answer the main question.

Act 2: The Model Students identify information needed to develop a mathematical model. Students use their models and specific math practices to calculate solutions and compare them to their initial predictions.

Act 3: The Solution and Sequel Students view the solution through a multimedia presentation. They use math practices to explain disparities or differences between their calculated solutions and the actual solutions.

For further information regarding the implementation of the math practices, see the enVisionmath2.0 (Grades K–8) Math Practices and Problem Solving Handbook at the start of each grade’s Teacher’s and Student’s Editions.



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Metric

AC Metric 2B:

Materials attend to the full meaning of each practice Standard.

Evidence

Over the course of any given year of instruction, is each mathematical practice standard meaningfully present in the form of assignments, activities, or problems that stimulate students to develop the habits of mind described in the practice standard?

Each mathematical practice is highlighted in a Math Practices and Problem Solving lesson (Grades K–5) and a 3-Act Mathematical Modeling lesson (Grades 6–8) at least once per grade as shown in the following chart.

CCSSM Grade K Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8

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

Lesson 13-4

Lesson 6-5; Lesson 9-6; Lesson 14-9

Lesson 11-7 Lesson 4-9; Lesson 12-8

Lesson 4-11; Lesson 6-5


Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

MP.2 Reason abstractly and quantitatively.



Lesson 6-9; Lesson 7-6; Lesson 8-5; Lesson 14-6


Lesson 2-6


Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

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

Lesson 1-11 Construct Arguments Lesson 5-4 Critique Reasoning



Lessons 9-8 and 13-8 Construct Arguments Lesson 11-4 Critique Reasoning

Lessons 1-5 and 8-7 Construct Arguments Lessons 11-4 and 16-6 Critique Reasoning

Lessons 3-7 and 12-4 Critique Reasoning Lesson 16-4 Construct Arguments

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8



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CCSSM Grade K Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 MP.4 Model with mathematics.





Lesson 3-10; Lesson 5-10; Lesson 9-11; Lesson 10-6


Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

MP.5 Use appropriate tools strategically.

Lesson 7-9

Lesson 12-5


Lesson 1-7

Lesson 15-6

Lesson 10-6

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

Topics 1, 2, 3, 4, 5, 6, 7, 8

MP.6 Attend to precision.


Lesson 5-7 Lesson 12-9 Lesson 7-5; Lesson 15-4

Lesson 13-7 Lesson 11-8 Topics 4, 6, 8

Topics 3, 7

Topics 2, 3, 6

MP.7 Look for and make use of structure.



Lesson 9-10 Lesson 5-8; Lesson 6-7; Lesson 10-9


Lesson 1-7

Topics 5, 7

Topics 2, 8

Topics 1, 2, 3

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

Lesson 4-6

Lesson 7-7


Lesson 3-8

Lesson 7-3

Lesson 9-8

Topics 3, 5

Topics 4, 7

Topics 1, 5, 8

Additionally, math practices and content standards are connected within all topics and lessons. The following are examples within just one grade and topic (Grade 7, Topic 3) of how math practices connect to content standards. Similar examples can be found for all topics and lessons. Also refer to the Math Background page on Math Practices, i.e., Grade 7 Teacher’s Edition p. 132E.

MP.1 Make sense of problems and persevere in solving them. Students make sense of a problem as they describe the equivalent ratios they can use to solve the problem. (e.g., p. 142, Item 217)

MP.2 Reason abstractly and quantitatively. Students use reasoning to analyze relationships between quantities greater than 100% or less than 1%. (e.g., p. 140, Item 2)

MP.3 Construct viable arguments and critique the reasoning of others. Students construct arguments to justify that finding 25% of a number is not the same as dividing it by 1/4. (e.g., p. 140, Item 3)

MP.4 Model with mathematics. Students model with math when they use ratios to help compare the quantities in the problem. (e.g., p. 149, Solve & Discuss It!)



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MP.5 Use appropriate tools strategically. Students select and use appropriate tools to gather the information they need. (e.g., p. 164, Item 7)

MP.6 Attend to precision. Students attend to precision when they use correct terminology. (e.g., p. 151, Example 3)

MP.7 Look for and make use of structure. Students look for structure when they examine relationships in percents and proportions. (e.g., p. 143, Solve & Discuss It!)

MP.8 Look for and express regularity in repeated reasoning. Students generalize about the mathematics of markups and markdowns. (e.g., p. 170, Item 3)

Do the materials treat the practice standards as developing across grades or grade bands? Are the practice standards in early grades appropriately simple? Do they display an arc of growing sophistication across the grades?

As stated previously, the goal and promise of enVisionmath2.0 (Grades K–8) is to develop understanding, so the mathematical practices are infused in all aspects of the program throughout all daily lessons (Grades K–8), all Math Practices and Problem Solving lessons (Grades K–5), and all 3-Act Mathematical Modeling lessons (Grades 6–8). Developmentally appropriate student lessons offer rich problems and tasks that demonstrate growing sophistication across the grades. One example of how a practice standard is developed across Grades K–8 follows.

MP.4 Model with Mathematics In primary grades, students write simple expressions or equations to represent and describe a problem situation. As students become more proficient in modeling with math, they progress to more sophisticated representations such as equations with one or more variables, application of proportional reasoning, or functions.

Grade 1 Topic 10 Lesson 5



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Are there teacher-directed materials that explain the role of the practice standards in the classroom and in students’ mathematical development? Are alignments to practice standards accurate?

The Math Practices and Problem Solving Handbook at the front of the Student’s and Teacher’s Editions is a developmentally appropriate resource for students and teachers to use throughout the year when discussing math practices. It provides resources for students and teachers that facilitate problem-solving experiences on a daily basis. (See the Grades 6–8 Teacher’s Edition Program Overview p. 44.)

Math Practices and Problem Solving Handbook pages in the Teacher’s Edition include:

1. Building students’ proficiency with the math practice, which discusses what the mathematical practice means and provides sample uses of the mathematical practice.

2. Math practice behaviors, which describe what to listen and look for when monitoring students’ ongoing development of proficiency with each math practice.

3. Math practices proficiency rubric, which gives a scoring rubric for evaluating students’ proficiency with the math practice.

Bar Diagrams in the Grades 6–8 Math Practices and Problem Solving Handbook present bar diagrams and give examples of how they can be used to solve problems. For a discussion of bar diagrams in Grades 6–8, see the Teacher’s Edition Program Overview pp. 48–51.



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Metric

AC Metric 2C:

Materials support the Standards’ emphasis on mathematical reasoning.

Evidence

Do the materials support students in constructing viable arguments and critiquing the arguments of others concerning grade-level mathematics that is detailed in the content Standards?

Students demonstrate proficiency with MP.3 Construct viable arguments and critique the reasoning of others throughout lessons and topics connected directly to grade-level content Standards in the following ways. Examples cited are from Grade 8, Topic 1 Teacher’s Edition, but any grade and topic offers comparable opportunities for students.

• Lesson 1-2 Explain It! supports MP.3 by inherently calling for students to explain their reasoning and thinking (construct viable arguments) and critique the arguments of others during class discussions.

• Focus on Math Practices at the end of Solve & Discuss It!, Explore It!, or Explain It! often supports MP.3 (p. 13).

• The Visual Learning Bridge has guiding questions in the Teacher’s Edition that often support MP.3 (pp. 14–15, 20–21, 26–27, 46–47, 58–59).

• Do You Understand? often supports MP.3 (pp. 16, 22, 28, 34, 42, 54, 60).

• Practice & Problem Solving often supports MP.3 (pp. 17, 24, 30, 36, 44, 49, 56, 71).

Do the materials support students in producing not only answers and solutions, but also, in a grade-appropriate way, arguments, explanations, diagrams, mathematical models, etc., especially in the Major Work of the grade?

Students are encouraged to use reasoning and explain their strategies often throughout lessons as well as use diagrams, models, and tools appropriately. For example, in Grade 6, Topic 1 students are prompted to use visual models for fraction division, such as number lines and area models. Students are encouraged to represent their thinking with concrete and abstract representations throughout. (For an overview, see Grade 6 Teacher Edition pages 2B–2E.)

Bar diagrams help students understand and represent the relationship between known and unknown quantities to solve problems—a direct application of both MP.2 Reason abstractly and quantitatively and MP.4 Model with mathematics. (See the Teacher’s Edition Program Overview pp. 48–51 for an overview.)



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The Math Practices and Problem Solving Handbook provides a page for each math practice as a resource for students and teachers to use throughout the year when discussing math practices.

• Students are provided with a clarifying statements about what mathematically proficient students do when they engage MP.2 or MP.4, as shown below on Grade 7, Student’s Edition pp. F14–F15.



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• Sample problems are provided that lend themselves to engaging MP.2 or MP.4 as shown below on Grade 6, Student’s Edition pp. F17 and F19. Questions are also provided that help students engage MP.2 and MP.4 when solving problems.



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Teaching notes help develop MP.2 and MP.4 through class discussion of the student’s book page. Teacher notes also help to identify the behaviors that develop with proficiency with MP.2 and MP.4.



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Teacher notes also provide a rubric for assessing students’ ongoing development of proficiency with the math practices, as shown at the right.

See also Grades 6–8 Teacher’s Edition Program Overview p. 44.



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Other Support for an Emphasis on Mathematical Reasoning Other support includes:

• 3-Act Mathematical Modeling lessons that focus questions on math practices.

• Lesson, topic, and cumulative/benchmark assessment items that assess math practices.

Do materials explicitly attend to the specialized language of mathematics? Is the language of argument, problem solving, and mathematical explanations taught rather than assumed?

3-Act Mathematical Modeling lessons are opportunities to stop and focus on a specific math practice during problem solving. To enhance proficiency with a math practice, the lesson instruction offers questions for that math practice and models thinking and reflecting on thinking related to that math practice when solving a rich, real-world problem. (See Grade 7 Teacher’s Edition pp. 163A to 165–166)

The Math Practices and Problem Solving Handbook models language and helps students engage with the math practices. For example,

• What do the numbers in the problem represent?

• How do the numbers relate?

• Which numbers and operations will you use to represent the problem mathematically?

• What unit will you need to use in the solution?

Topic Vocabulary, Vocabulary section on the Review What You Know! activity, Prepare for Reading Success activity, and the Vocabulary Review section on the Topic Review pages are provided in the Student’s Edition (SE Grade 7 Topic 3, pp. 132, 135, 136, 179). The Teacher’s Edition provides vocabulary review activities at the start and end of topics (TE Grade 7 Topic 3, pp. 135, 179). These include activities for vocabulary in Review What You Know! and the Topic Review pages.

A bilingual glossary is provided at the back of the Student’s Edition. An online, bilingual, glossary with audio builds understanding of math vocabulary.

An online vocabulary game is available in the Game Center.



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Alignment Criterion 2: Materials must authentically connect content Standards and practice Standards.


enVisionmath2.0 (Grades K–8) infuses math practices during instruction, practice, and assessment and provides opportunities to focus on specific math practices. Math practices are habits of mind, processes, and dispositions that enable a learner to understand mathematics and to use mathematics with understanding.

enVisionmath2.0 (Grades K–8) identifies questions for each math practice. Math practices are infused and discussed on a daily basis starting from Day 1.

Specially flagged comments and problems in all lessons focus on specific math practices. Math practices are assessed with the content standards, and a rubric is provided for assessing students’ overall proficiency with the math practices.



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Alignment Criterion 3 Access to the Standards for All Students

AlignmentCriterion3:MaterialsmustprovidesupportsforEnglishLanguageLearnersandotherspecialpopulations.

Metric

AC Metric 3A:

Support for English Language Learners and other special populations is thoughtful and helps those students meet the same Standards as all other students. The language in which problems are posed is carefully considered.

Evidence

Daily ELL instruction support is provided in the Teacher’s Edition for every lesson. This support is used with a specified part of the lesson such as Solve & Discuss It!, Explore It!, or Explain It!, Examples in the Visual Learning Bridge, Try It! or Convince Me!. The support is leveled instruction that includes suggestions for students at Beginning, Intermediate, and Advanced levels of English language proficiency.

ELL Toolkit provides additional support for English Language Learners.

Visual Learning Animation Plus and Other Interactive Examples provides motion and sound to help lower language barriers to learning. Questions that are read aloud also appear on screen to help English Language Learners connect oral and written language.

Visual Learning Bridge often has visual models to help give meaning to math language. Instruction is stepped out to visually organize important ideas. Teachers may want to use the Student’s Edition eText to display the Visual Learning Bridge after using the Visual Learning Animation Plus and other interactive examples.

Pictures that appear in lesson practice have a purpose, to provide comprehensible input and help communicate information related to math concepts or to real-world problems.

Additional Vocabulary Support masters in each lesson help students develop and reinforce understanding of key terms and concepts.



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Metric

AC Metric 3B:

Materials provide appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners with gradual removal of supports, when needed, to allow students to demonstrate their mathematical understanding independently.

Evidence

enVisionmath2.0 (Grades K–8) meets the needs of all students and provides Response to Intervention at the start of the year, at the start of each topic, during each lesson, at the end of each lesson, at the end of each topic, and anytime as indicated. Specifically for Grades 6–8:

ONGOING INTERVENTION, AT THE START OF THE YEAR

• Beginning-of-the-Year Assessment available both print and digitally. This assessment diagnoses students’ areas of strengths and weaknesses. The results can be used to prescribe differentiated intervention.

ONGOING INTERVENTION, AT THE START THE TOPIC

• Topic Readiness Assessment available both print and digitally. An Adaptive Study Plan with Intervention Lessons is automatically generated if the assessment is taken online.

• Review What You Know is a feature at the start of each topic that reviews essential vocabulary and checks students’ understanding of key math concepts needed for the topic.

ONGOING INTERVENTION, DURING THE LESSON

• Prevent Misconceptions during Do You Understand? and Do You Know How? includes a remediation strategy to address a common misconception about the lesson concept.

• Error Intervention (If… Then…) during Practice & Problem Solving identifies a common error and provides remediation strategy.

• Challenge during Practice & Problem Solving provides an extension of a problem’s concept to enrich students’ understanding.

• Response to Intervention during the Visual Learning Bridge provides additional assistance with one or more of the Examples.



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• Enrichment during the Visual Learning Bridge extends one or more of the Examples to challenge students.

• Try It! and Convince Me! Assess students’ understanding of concepts and skills presented in each example. The results can be used to modify instruction as needed.

• Do You Understand? Do You Know How? Assesses students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.

• MathXL for School supports personalized practice for Practice & Problem Solving for every lesson. MathXL is autoscored and provides on-screen help including Help Me Solve This and View an Example.

• Higher Order Thinking problems in Practice & Problem Solving challenge student thinking.

STRATEGIC INTERVENTION, AT THE END OF THE LESSON

• Lesson Quiz assesses students’ conceptual understanding and procedural fluency with lesson content. The results can be used to prescribe differentiated instruction.

• Reteach to Build Understanding provides guided reteaching as a follow-up to the lesson.

• Additional Vocabulary Support helps students develop and reinforce understanding of key terms and concepts.

• Build Mathematical Literacy provides support for struggling readers to build mathematical literacy.

• Digital Math Tools Activities reinforce lesson content or previously taught content.

• Online Math Games reinforce lesson content or previously taught content and include thinking games rather than just drill games. Students learn concepts through the games.

• Practice & Problem Solving offers two (or 3) pages of practice in the Student’s Edition at the end of every lesson. Items focus on skills (often with leveled practice/scaffolding for the first few exercises) and on problem solving that includes reinforcement of math practices, vocabulary, Higher Order Thinking, and Common Core Assessment.

• Additional Practice available both print and digitally. Provides an additional set of practice for students’ needing reinforcement.

• Enrichment presents engaging problems and activities that extend the lesson concepts.

• Learning Aids Online in MathXL for School during Practice & Problem Solving include personalized practice for every lesson. MathXL is autoscored with on-screen help to include Help Me Solve This and View an Example.



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STRATEGIC INTERVENTION, MID-TOPIC

• Mid-Topic Checkpoint (or Assessment)/Mid-Topic Performance Task assesses students’ conceptual understanding and procedural fluency with several lesson’s content. The results can be used to prescribe differentiated instruction.

INTENSIVE INTERVENTION, AS NEEDED ANYTIME

• Visual Learning Animation Plus is a step-by-step visual learning animation that is used in each lesson to connect Example 1 to the Practice & Problem Solving in a very visual way. This animation and the other interactive examples can be used anytime to refresh understanding.

• Online Math Games can be used anytime for more reinforcement.

• Learning Aids Online MathXL for School (described earlier)

Math Diagnosis and Intervention System 2.0 (MDIS) Helps teachers diagnose students’ needs and provide effective intervention that’s on or below grade level.

• Diagnosis Use the diagnostic tests in the system. Also, use the item analysis charts provided in the TE with program assessments at the start of a grade or topic, or at the end of a topic, group of topics, or the year.

• Intervention Lessons These two-page lessons include guided instruction followed by practice. You can assign lessons that are below grade level if needed.

• Teacher Support Teacher Notes provide the support needed to conduct a short lesson. The lesson focuses on vocabulary, concept development, and practice. The Teacher’s Guide contains individual and class record forms and correlations to Student’s Edition lessons.

AUTO-ASSIGNED DIFFERENTIATION AT PEARSONREALIZE.COM

• Differentiation Before a Topic is based on results of the Topic Readiness Assessment. If students take the assessment online, an Adaptive Study Plan with Intervention Lessons will be automatically generated.

• Differentiation After a Lesson is based on results of the Lesson Quiz. If given online, activities from the Differentiation Library can be automatically assigned.



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• Differentiation After a Topic or Group of Topics is based on results of online Topic Assessments and online Cumulative/Benchmark Assessments. Students are assigned remediation or enrichment which can include the following:

– Visual Learning Animation Plus – Online Math Game – Digital Math Tools Activity – Reteach to Build Understanding – Intervention Lessons in the Math Diagnosis and Intervention System 2.0 – MathXL for School Online Practice

• Assignment Reports show the status of assigned resources.

• Usage Data lets teachers know how much time students are spending in the online resources.

• Today’s Challenge Online shows 5 problems using the same data on 5 different days. Problems apply prior knowledge and reinforce the kind of thinking students need for success on high-stakes tests. Problems increase in difficulty within a set. Today’s Challenge Teacher’s Guide includes teaching actions organized under Before, During, and After in addition to Vocabulary Review, ELL Support, and Extension.



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Metric

AC Metric 3C:

Design of lessons recommends and facilitates a mix of instructional approaches for a variety of learners (e.g., using multiple representations, asking a range of questions, checking for understanding, flexible grouping, pair-share, deconstructing/reconstructing the language of problems).

Evidence

Students frequently use models to represent their thinking and solutions in the Problem-Based Learning Part 1 of each lesson, Solve & Discuss It!, Explain It!, or Explore It!.

Teachers are provided with guiding questions to facilitate this discussion and Before, During, and After teaching actions model classroom discourse.

• Before uses Teaching Action #1 (Implement Tasks that Promote Reasoning and Problem Solving) and #2 (Build Understanding) to start understanding. This is whole-class discussion.

• During uses Teaching Action #3 (Support Productive Struggle in Learning Mathematics) when students are stumped and are working together as the teacher facilitates.

• After represents another whole-class discussion. Use Teaching Action #4 (Facilitate Meaningful Mathematical Discourse) and #5 (Transition to Visual Learning) to discuss students’ thinking and work, and to make math ideas explicit. Use #6 (Extension for Early Finishers) as needed.

Blue guiding questions are provided throughout each lesson to support classroom discourse.

Focus on Math Practices checks for understanding right after the instruction and connects to the essential understanding of the lesson and to math practices.

A variety of activities are provided for differentiation where students may be working with the teacher in small groups or in pairs on a digital device or in print. Different learning modalities offer greater access for all students. Pair-share is also used in the games for On-level and Advanced Learners. Deconstructing/reconstructing the language of the problems is specifically addressed in the English Language Leaners section of each lesson’s Teacher’s Guide. The Teacher’s Guide for each lesson also includes a section on Response to Intervention and Enrichment that correlates to the Examples in the lesson.



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Alignment Criterion 3: Materials must provide supports for English Language Learners and other special populations.


enVisionmath2.0 (Grades K–8) provides appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners. Support for English Language Learners helps all students demonstrate mathematical proficiency with the standards.

enVisionmath2.0 (Grades 6–8) includes:

• Daily ELL instruction in the Teacher’s Edition which is used during the instructional part of the lesson.

• Leveled instruction with suggestions for students at Beginning, Intermediate, and Advanced levels of English language proficiency.

• Additional Vocabulary Support masters to help students develop and reinforce understanding of key terms and concepts.

• ELL Teacher Support with Today’s Challenge problems (online only).

• ELL Toolkit with additional support for English Language Learners (online only).



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Indicators of Quality

Indicator 1

Lessons are thoughtfully structured and support the teacher in leading the class through the learning paths at hand, with active participation by all students in their own learning and in the learning of their classmates.

Evidence

There are two key steps to the core instructional model in enVisionmath2.0.

STEP 1 DEVELOP: PROBLEM-BASED LEARNING: Introduce concepts and procedures with a problem-solving experience.

Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014). Conceptual understanding results because the process of solving a problem that involves a new concept or procedure requires students to make connections of prior knowledge to the new concept or procedure. The process of making connections between ideas builds understanding. In enVisionmath2.0 (Grades 6–8), this problem-solving experience is called Solve & Discuss It!, Explain It! or Explore It!.



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STEP 2 DEVELOP: VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1.

The important mathematics is the new concept or procedure students should understand (Hiebert, 2003; Rasmussen, Yackel, and King, 2003). Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions for the Solve & Discuss It!, Explain It!, or Explore It! task. Regardless of whether the important mathematics comes from discussing students’ thinking and work, understanding the important mathematics is further enhanced when teachers use a high-cognitive-level classroom conversation to explicitly present and discuss an additional problem related to the new concept or procedure.

The second step in the core instructional model is called enhanced direct instruction because making the important math explicit through an additional problem or problems should always begin by connecting the students’ thinking and solutions for the Solve & Discuss It!, Explain It!, or Explore It! to the new ideas of the lesson.

Every lesson in enVisionmath2.0 (Grades 6–8) provides a worked-out problem in the student materials that introduces the important mathematics of the lesson through a series of visual illustrations that promote understanding, followed by additional examples that further develop understanding of the concepts. This is called the Visual Learning Bridge. It is a bridge because it connects the students’ thinking and solutions for the Solve & Discuss It!, Explain It!, or Explore It! to the new mathematical ideas of the lesson. In other words, it is not left to chance that students will take away the important mathematics from their work with the the Solve & Discuss It!, Explain It!, or Explore It!; the important mathematics is made explicit in the student materials.



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Making the important mathematics explicit through the Visual Learning Bridge has the added benefit of providing a resource for teachers, students, and parents. Example 1 of every Visual Learning Bridge in enVisionmath2.0 (Grades 6–8) is available as an online Visual Learning Animation Plus, thus facilitating teacher-led classroom conversations. All subsequent examples are available digitally as well in various, interactive forms. Also, illustrations inside each Visual Learning Bridge are accompanied in the teacher materials by high-cognitive-level questions for teachers to use with students.



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Indicator 2

The underlying design of the materials includes both problems and exercises. (In solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery.) Each problem or exercise has a purpose. NOTE: This indicator does not require that the problems and exercises be labeled as such.

Evidence

The following illustrate how students are engaged in solving problems and working exercises in enVisionmath2.0 (Grades 6–8), using Grade 6 as an example.

The Lesson Step 1 Problem-Based Learning activity for each lesson, Solve & Discuss It!, Explain It! or Explore It!, is designed to engage students with a problem in which new math ideas are embedded. Coherence is facilitated as students connect prior knowledge to the new math ideas. Students solve the problem in any way they choose; are given time to struggle; and as students think, conceptual understandings emerge. Solve & Discuss It!, Explain It! or Explore It!, online utilizes the DrawPad to have students write and share their solutions on screen. (For example, Teacher’s Edition Lesson 1-4, pp. 31)

Visual Learning Bridge in Step 2 of each lesson, including the online Visual Learning Animation Plus and other interactive examples, increases the cognitive level of instruction by connecting concrete and pictorial representations to abstract symbols. The questions provided in the Teacher’s Edition have students reflect on the work that is shown, make connections among ideas, and justify the steps. The Try It! and Convince Me! features in each lesson foster communication during visual learning. (For example, Teacher’s Edition Lesson 1-4, pp. 32)

Practice & Problem Solving builds proficiency as students work on their own. (For example, Teacher’s Edition Lesson 1-4, pp. 35–36)

Students communicate often about their thinking and work, using reasoning during Assess & Differentiate in Part 3 of each lesson through Build Mathematical Literacy Activities, Additional Vocabulary Support, Additional Practice, Digital Math Tools Activities, and Online Math Games. (For example, Teacher’s Edition Lesson 1-4, pp. 36A–36B)



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Indicator 3

There are separate teacher materials that support and reward teacher study including, but not limited to: discussion of the mathematics of the units and the mathematical point of each lesson as it relates to the organizing concepts of the unit, discussion of student ways of thinking and anticipating a variety of students’ responses, guidance on lesson flow, guidance on questions that prompt students thinking, and discussion of desired mathematical behaviors being elicited among students.

Evidence

The Listen and Look For Lesson Videos provide information about what the teacher should look for in student work and discussions.

A professional development Topic Overview Video for each topic is provided. In the video, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic.

Extensive teaching notes support every part of every daily lesson, all lesson and topic student resources, and assessments in the topic. These include blue guiding questions to stimulate classroom discourse, incorporating math practices, preventing misconceptions, supporting coherence, error intervention, and analyzing student work. Each lesson also includes comprehensive print and digital resources to assess and differentiate to keep all students engaged and on track. (For example, Grade 8 Teacher’s Edition Lesson 6-1, pp. 302A–302B)

Comprehensive teaching support is also provided in the Math Background in Topic Overview pages, with a “deep dive” into Focus, Coherence, Rigor, and Math Practices and how these are implemented in enVisionmath2.0 (Grades 6–8) for every topic. See the Grades 6–8 Teacher’s Edition Program Overview pp. 52–55.



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Indicator 4

Manipulatives suggested in the materials are faithful representations of the mathematical objects they represent and are connected to written methods.

Evidence

In Grades 6–8, manipulatives are used to support student learning and correspond with the student lessons. They are available online as virtual manipulatives called Math Tools. The connections of mathematical objects to written methods appear throughout enVisionmath2.0 (Grades 6–8).



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Indicator 5Materials include a variety of curriculum-embedded assessments. Examples include pre-, formative, summative, and self-assessment resources. Evidence

enVisionmath2.0 (Grades 6–8) provides students with multiple ways to show what they have learned.

DIAGNOSTIC ASSESSMENT

• At the start of the YEAR

– Beginning-of-Year Assessment Online in TestNav™ – Beginning-of-Year Assessment Masters in Assessment Sourcebook – Diagnostic Test Masters in Math Diagnosis and Intervention System 2.0

• At the start of a TOPIC

– Diagnostic Test Masters in Math Diagnosis and Intervention System 2.0 – Topic Readiness Assessment Online in TestNav™ – Topic Readiness Assessment Masters in Assessment Sourcebook – Review What You Know in Student’s Edition or eText

FORMATIVE ASSESSMENT

• During a LESSON

– Questions in the Visual Learning Animation Plus and Other Interactive Examples – Questions to use with the Visual Learning Bridge in Teacher’s Edition – Try It! and Convince Me! in Student’s Edition or eText – Do You Understand? or Do You Know How? in Student’s Edition or eText

• At the end of a LESSON

– Lesson Quiz Online in TestNav™ – Lesson Quiz Masters in Assessment Sourcebook

SUMMATIVE ASSESSMENT



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• Mid-Topic (After a Series of Lessons) – Mid-Topic Checkpoint in Student’s Edition or eText – Mid-Topic Assessment Online in TestNav™ – Mid-Topic Assessment Masters in Assessment Sourcebook – Self Assessment Tool in Mid-Topic Assessment

– Mid-Topic Performance Task in Student’s Edition or eText

• At the end of a TOPIC – Topic Assessment Online in TestNav™ – Topic Assessment Masters in Assessment Sourcebook – Topic Assessment by ExamView® CD-ROM – Topic Performance Assessment Masters in Assessment Sourcebook

• After a group of TOPICS

– Cumulative/Benchmark Assessment Online in TestNav™ – Cumulative/Benchmark Assessment Masters in Assessment Sourcebook – Practice Performance Tasks Online in TestNav™ – Practice Performance Tasks in Assessment Sourcebook

• At the end of the YEAR

– End-of-Year Assessment Online in TestNav™ – End-of-Year Assessment Masters in Assessment Sourcebook – Next Generation Practice Test Online in TestNav™ – Next Generation Practice Test Online in Assessment Sourcebook

SELF ASSESSMENT

– Self Assessment Tool in Mid-Topic Assessment Online in TestNav™ – Self Assessment Tool in in Mid-Topic Assessment in Assessment Sourcebook



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Indicator 6

Assessments contain aligned rubrics, answer keys, and scoring guidelines that provide sufficient guidance for interpreting student performance.

Evidence

enVisionmath2.0 provides comprehensive assessment resources for interpreting student performance.

• Online assessments are auto-scored and reports generated include individual and class views of the assessment results, including mastery of CCSSM. Teachers can edit student information and organize students into groups based on assessment data.

• Math Practices Proficiency Rubric is provided in the Teacher’s Edition for the Math Practices and Problem Solving Handbook and can be used at any time.

• Scoring guides/rubrics/answer keys are included for assessments in the Teacher’s Edition. Sample student work is included for some assessments in the Assessment Sourcebook.

• Item Analysis for Diagnosis and Intervention in the Teacher’s Edition provides CCSSM, DOK level, and intervention resources for each item.



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Indicator 7

Materials assess student proficiency using methods that are accessible and unbiased, including the use of grade-level language in student prompts.

Evidence

enVisionmath2.0 (Grades 6–8) assessments utilize the same accessible and unbiased formats as found on state and national high-stakes assessments. The formats of enVisionmath2.0 (Grades 6–8) assessment items, which prepare students for high-stakes Common Core tests, include the following:

• Selected response, e.g., single response, multiple response

• Constructed response, e.g., short or extended responses, sometimes using an on-screen symbols palette

• Technology-enhanced items, e.g., drag and drop, drop-down menus, graphing, on-screen tools

• Performance tasks, hand scored or machine scored

This range of assessment item types is useful in providing reliable data on students’ level of understanding.

A variety of auto-generated assessment reports are available for online assessments.

• Individual and class views of progress are provided in an easy-to-view format.

• Common Core Standards mastery reports show individual students’ mastery or class wide mastery for each standard.

Assessment data can be used to organize students into groups for purposes of making instructional decisions and assigning differentiation resources.

Every lesson includes Common Core assessment practice items in formats that help prepare students for Common Core high-stakes tests. enVisionmath2.0 (Grades 6–8) lessons use grade-level appropriate language in student prompts as shown by the Lexile measure. Assessment items in quizzes and tests are based on content developed in the lessons and student prompts on assessments use the same familiar language as prompts found in the lessons providing access to all students.



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Indicator 8

Materials are carefully evaluated by qualified individuals, whose names are listed, in an effort to ensure freedom from mathematical errors and grade-level appropriateness.

Evidence

A team of widely-recognized authors, some of whom were involved in the writing of the CCSS-M and Progressions, were directly responsible for the creation of enVisionmath2.0 (Grades K–8). They are experienced elementary and middle grades math educators; nationally recognized math education leaders; and are experts in key math education issues. The program authors are listed on p. 1 of the Grades 6–8 Teacher’s Edition Program Overview. These authors are listed and provide a perspective of the program goal and organization, as well as the instructional model on pp. 18–27 of the Grades 6–8 Teacher’s Edition Program Overview.

Mathematician Reviewers for enVisionmath2.0 (Grades 6–8) are:

Karen Edwards, Ph.D. Mathematics Lecturer Arlington, MA

Gary Lippman, Ph.D. Professor Emeritus Mathematics and Computer Science California State University, East Bay Hayward, California

Authors for enVisionmath2.0 (Grades 6–8) are:

Robert Q. Berry, III Associate Professor of Mathematics Education Department of Curriculum, Instruction and Special Education University of Virginia Charlottesville, Virginia



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Zachary Champagne Assistant in Research Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) Jacksonville, Florida

Eric Milou Professor of Mathematics Rowan University Glassboro, New Jersey

Jane F. Schielack Professor Emerita of Mathematics Texas A&M University College Station, Texas Jonathan A. Wray Mathematics Instructional Facilitator Howard County Public Schools Ellicott City, Maryland

Randall I. Charles Professor Emeritus Department of Mathematics San Jose State University San Jose, California Francis (Skip) Fennell L. Stanley Bowlsbey Professor of Education and Graduate and Professional Studies McDaniel College Westminster, Maryland



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Indicator 9

The visual design supports students in engaging thoughtfully with the subject. Navigation through the text is clear.

Evidence

Visual design supports the two key steps in the core instructional model in enVisionmath2.0 (Grades K–8) and is engaging to young learners.

Step 1 Problem-Based Learning

Introduce concepts and procedures with a problem-solving experience. Conceptual understanding results because the process of solving a problem that involves a new concept or procedure requires students to make connections of prior knowledge to the new concept or procedure. The process of making connections between ideas builds understanding. In enVisionmath2.0 (Grades 6–8) this problem-solving experience is called Solve & Discuss It!, Explain It!, or Explore It!.

Step 2 Visual Learning

Make the important mathematics explicit with enhanced direct instruction connected to Step 1. Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions for the Solve & Discuss It!, Explain It!, or Explore It! task. Regardless of whether the important mathematics comes from discussing students’ thinking and work, understanding the important mathematics is further enhanced when teachers use a high-cognitive-level classroom conversation to explicitly present and discuss an additional problem related to the new concept or procedure.

The second step in the core instructional model is called enhanced direct instruction because making the important math explicit through an additional problem should always begin by connecting the students’ thinking and solutions for the Solve & Discuss It!, Explain It!, or Explore It! to the new ideas of the lesson.

Every lesson in enVisionmath2.0 (Grades 6–8) provides worked-out problems in the student materials that introduce the important mathematics of the lesson through a series of visual illustrations, particularly in Example 1, that promote understanding. This is called the Visual Learning Bridge. It is a bridge because it connects the students’ thinking and solutions for the Solve & Discuss It!, Explain It!, or Explore It! to the new mathematical ideas of the lesson. In other words, it is not left to chance that students will take away the important mathematics from their work with the Solve & Discuss It!, Explain It!, or Explore It!; the important mathematics is made explicit in the student materials.

Making the important mathematics explicit through the Visual Learning Bridge, in addition to further developing understanding, has the added benefit of providing a resource for teachers, students, and parents.

Every Visual Learning Bridge in enVisionmath2.0 (Grades 6–8) is available online as the Visual Learning Animation Plus and other interactive



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examples, thus facilitating teacher-led classroom conversations. Also, illustrations and worked-out solutions within the Visual Learning Bridge are often accompanied in the student materials by an extensive use of call-outs that supports students by visually highlighting key aspects of the instruction. The photos and other illustrations in the student materials are deliberately chosen to engage students and contribute to their comprehension. In the teacher materials, high-cognitive-level questions are provided for teachers to use with students.



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Indicator 10

The materials engage parents in appropriate ways. For example, homework assignments in elementary grades consist of routine problems, practice with getting answers, and fluency-building exercises that parents can easily support.

Evidence

The Visual Learning Bridge, the Visual Learning Animation Plus, and other interactive examples provide useful resources for parents to understand the concepts as they were taught in the lesson. Student assignments are available online for parents to review. Worked-out Examples, digitally and in print, remind students and parents about important ideas in the lesson. A Home-School Connection letter that summarizes the math concepts is provided (English and Spanish) for each topic. Additionally, online practice powered by MathXL for School includes learning aids such as View an Example and Help Me Solve This.



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APPENDIX: Scope and Sequence for enVisionmath2.0 (Grades 6–8)



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