instructional materials evaluation toolkit (imet)

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

Grades K–5

The IMET was developed by Student Achievement Partners. Educators may use or adapt. http://creativecommons.org/licenses/by/3.0/ Download this tool at http://achievethecore.org/IMET

CONTENTS Introduction to Instructional Materials Evaluation Tool—Mathematics, Grades K–8 Non-Negotiable 1: Freedom from Obstacles to Focus Materials must reflect the content architecture of the Standards by not assessing the topics named before the grade level where they first appear in the Standards. 1 NN Metric 1A: Materials reflect the basic architecture of the Standards by not assessing the listed topics 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. 3 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. 6 NN Metric 2B: Supporting Work, where present, enhances focus and coherence simultaneously by also engaging students in the Major Work of the grade. 11 NN Metric 2C: Materials follow 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. 12 NN Metric 2D: Materials give all students extensive work with on-grade-level problems. 14 NN Metric 2E: Materials relate on-grade-level concepts explicitly to prior knowledge from earlier grades. 17 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. 19 NN Metric 2G: Materials include learning objectives that are visibly shaped by CCSSM cluster headings. 20 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. 35 AC Metric 1B: The materials are designed so that students attain the fluencies and procedural skills required by the Standards. 41 AC Metric 1C: The materials are designed so that teachers and students spend sufficient time working with engaging applications, without losing focus on the Major Work of each grade.

Alignment Criterion 2: Standards for Mathematical Practice Materials must demonstrate authentic connections between content Standards and practice Standards. 54 AC Metric 2A: Materials address the practice Standards in such a way as to enrich the Major Work of the grade; practices strengthen the focus on Major Work instead of detracting from it, in

both teacher and student materials. 56 AC Metric 2B: Materials attend to the full meaning of each practice Standard. 62 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. 68 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. 70 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. 73 AC Metric 3C: Design of lessons recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (e.g., including models, using a

range of questions, checking for understanding, flexible grouping, pair-share). Indicators of Quality 76 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. 80 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 Criterion does not require that the problems and exercises be labeled as such. 81 Indicator 3: Design of assignments is not haphazard: exercises are given in intentional sequences in order to strengthen students’ mathematical understanding. 82 Indicator 4: 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 among students.

http://creativecommons.org/licenses/by/3.0/

http://achievethecore.org/IMET


84 Indicator 5: Manipulatives suggested in the materials are faithful representations of the mathematical objects they represent and are connected to written methods. 86 Indicator 6: Materials include a variety of curriculum-embedded assessments. Examples include pre-, formative, summative, and self-assessment resources. 88 Indicator 7: Assessments contain aligned rubrics, answer keys, and scoring guidelines that provide sufficient guidance for interpreting student performance. 89 Indicator 8: Materials assess student proficiency using methods that are accessible and unbiased, including the use of grade-level language in student prompts. 90 Indicator 9: Materials are carefully evaluated by qualified individuals, whose names are listed, in an effort to ensure freedom from mathematical errors and grade-level appropriateness. 93 Indicator 10: The visual design supports students in engaging thoughtfully with the subject. Navigation through the text is clear. 97 Indicator 11: 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. 98 Appendix: Scope and Sequence for enVision® Mathematics




Introduction to Instructional Materials Evaluation Tool—Mathematics, Grades K–8 (This information is taken from the tool found at https://achievethecore.org/aligned/wp-content/uploads/2015/04/IMET_Version2.pdf) 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 grades K–8 (Spring 2013) (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 (https://achievethecore.org/category/774/mathematics-focus-by-grade-level), 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.



https://achievethecore.org/aligned/wp-content/uploads/2015/04/IMET_Version2.pdf

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http://www.corestandards.org/other-resources/key-shifts-in-

http://www.corestandards.org/Math)

http://www.corestandards.org/Math)

https://achievethecore.org/category/774/mathematics-focus-by-grade-level

https://achievethecore.org/category/774/mathematics-focus-by-grade-level

http://www.achievethecore.org/progressions)

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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 topics named 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 listed topics 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)

• Similarity, congruence, or geometric transformations. (Introduced in the CCSSM in grade 8) • Symmetry of shapes, including line/reflection symmetry, rotational symmetry. (Introduced in the CCSSM in grade 4) Evidence

enVision® Mathematics does not assess any of these topics prior to their introduction by the Common Core.

• Probability, including chance, likely outcomes, and probability models, is not assessed in enVision® Mathematics Grades K–5.

• 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 are not assessed in enVision® Mathematics Grades K–5. This content is introduced later (e.g., in Grade 6 when Clusters 6.SP.A and 6.SP.B are the focus of instruction).

• Similarity, congruence, and geometric transformations are not assessed in enVision® Mathematics Grades K–5.

• Symmetry of shapes, including line/reflection symmetry and rotational symmetry, is not assessed in enVision® Mathematics until Grade 4 Topic 16 when the focus is on Cluster 4.G.A.




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

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

enVision® Mathematics reflects the content architecture by:

• not assessing the topics before the grade level where they first appear in the CCSSM. Topics are consistently assessed at the grade level where they first appear in the Standards.





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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 enVision® Mathematics focuses coherently on the Major Work of the grade:

1. enVision® Mathematics is organized by CCSSM Clusters providing clear content focus. Each CCSSM 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.

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.

• A Supporting Cluster is placed, at times, near the Major Cluster it supports. Standards from Major Clusters are often revisited through applications within Topics and Lessons that focus on Supporting Clusters.





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• Additional Clusters are located where most appropriate for the content. Standards from Major Clusters are often revisited through applications within Topics and Lessons that focus on Additional Clusters.

4. Standards from some Clusters are integrated into other Clusters where natural connections occur. Cross-cluster connections reinforce the Major Work of the grade.

5. No extraneous content is taught in any Cluster at any grade in enVision® Mathematics.





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For example, Grade 4 Contents reflect the following organization and support of the content emphasis of the CCSSM. Please refer to the Content Overview pages (Grade 4 sample shown below) and the Table of Contents for each grade in the Teacher’s Editions.

From Grade 4 Teacher’s Edition pp. F3–F4





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Metric

NN Metric 2B: Supporting Work, where present, enhances focus and coherence simultaneously by also engaging students in the Major Work of the grade.

Evidence

Supporting Work at each grade of the enVision® Mathematics program enhances focus and coherence simultaneously by engaging students in the Major Work of the grade. This is noted by the Standards listed for each lesson in the Table of Contents for each grade.

These tables show how Supporting Work enhances the Major Work of the grade in Grades K–5.

GRADE K

Supporting Cluster Major Cluster(s) Connections

K.MD.B supports K.CC.B and K.CC.C in:

• Topic 5 Classify and Count Data

K.CC.B and K.CC.C are developed in:

• Topic 1 Numbers 0 to 5

• Topic 2 Compare Numbers 0 to 5



• In Lessons 5-2 and 5-3, students apply their understandings of numbers from 0 to 10 developed in Topics 1 and 3 to counting objects in data categories.

• In Lesson 5-3, students count the number of objects in two data categories and then compare the numbers for each category using their understandings developed in Topics 2 and 4.

• In Lesson 5-4, students analyze other students’ reasoning about counting and comparing the number of objects in data categories.

K.G.B supports K.CC.B, K.CC.C, K.OA.A, and K.G.A

K.CC.B, K.CC.C, K.OA.A, and K.G.A are developed in:





• The work in Topic 13 involves counting shapes and comparing various attributes of two-dimensional and three-dimensional shapes. For this work, students apply their understanding of counting, comparing numbers, addition, and subtraction developed in earlier Topics.

• Every lesson in Topic 13 builds on the work students did in Topic 12 identifying and describing two-dimensional and three-dimensional





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• Topic 6 Understand Addition

• Topic 7 Understand Subtraction

• Topic 8 More Addition and Subtraction

• Topic 9 Count Numbers to 20

• Topic 11 Count Numbers to 100

• Topic 12 Identify and Describe Shapes

shapes.

GRADE 1


1.MD.C supports 1.OA.A in:

• Topic 6 Represent and Interpret Data

1.OA.A is developed in:

• Topic 1 Understand Addition and Subtraction

• Every lesson in Topic 6 involves work with addition and subtraction facts developed in Topics 1 through 5.

• In Lessons 6-4 and 6-5, equations are explicitly used as in Topic 5.

Topic 6 also supports 1.OA.B, 1.OA.C, and 1.OA.D.

1.OA.C is developed in:

• Topic 2 Fluently Add and Subtract Within 10

• Topic 3 Addition Facts to 20: Use Strategies

1.OA.B is developed in:

• Topic 4 Subtraction Facts to 20: Use Strategies

1.OA.D is developed in:

• Topic 5 Work with Addition and Subtraction Equations

GRADE 2


2.OA.C supports 2.OA.B in:

2.OA.B is developed in: • Every lesson in Topic 2 involves work with addition facts developed in Topic 1. In Lessons 2-1 and 2-2, addition facts are connected to





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• Topic 2 Work with Equal Groups

• Topic 1 Fluently Add and Subtract Within 20 odd and even arrays to show the total. In Lessons 2-3, 2-4, 2-5, equal groups are added.

2.MD.C supports 2.NBT.A in:

• Topic 8 Work with Time and Money

2.NBT.A is developed in:

• Topic 9 Numbers to 1,000

• Topic 8 work that involves skip counting within 100 helps prepares students for the Topic 9 work on skip counting to 1,000.

• In most Topic 8 lessons where students find the values of collections of coins, adding or subtracting within 100 can be used as developed in Topics 3 through 6.

• In Lesson 8-4, students use addition and subtraction within 100 to solve one- and two-step word problems as developed in Topic 7. Topic 8 also

supports 2.OA.A. 2.OA.A is developed in:

• Topic 7 More Solving Problems Involving Addition and Subtraction

2.MD.D supports 2.OA.A in:

• Topic 15 Graphs and Data


• Topic 7 More Solving Problems Involving Addition and Subtraction

• The Topic 15 problems that involve putting together, taking apart, and comparing data apply the work with addition and subtraction word problems in Topic 7.

• Lessons 15-1 and 15-2 apply the work with length developed in Topics 12 (and 14).

Topic 15 also supports 2.MD.A.

2.MD.A is developed in:

• Topic 12 Measuring Length

GRADE 3


3.MD.B supports 3.OA.A in:



• Topic 1 Understand Multiplication and Division of Whole Numbers

• Topic 2 Multiplication Facts: Use Patterns

• In Topic 7, students interpret and create picture graphs where scales represent equal groups. This applies the multiplication concepts and facts developed in Topics 1 and 2.

• In Topic 7, students solve word problems involving data given in a graph. This prepares them for the more extensive work in Topic 11 in





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Topic 7 also supports 3.OA.D.

3.OA.D is developed in:

• Topic 11 Use Operations with Whole Numbers to Solve Problems

which they use all four operations to solve one- and two-step problems.

3.G.A supports 3.MD.C and 3.NF.A in:

• Topic 15 Attributes of Two-Dimensional Shapes

3.MD.C is developed in:

• Topic 6 Connect Area to Multiplication and Addition

3.NF.A is developed in:

• Topic 12 Understand Fractions as Numbers

• Topic 13 Fraction Equivalence and Comparison

• In Topic 15 students learn the attributes of shapes that can be covered exactly by unit squares—a foundation that connects area to multiplication and addition.

• The work in Topic 15 connects the concepts of Topics 12 and 13 to the use of a fraction to describe a part of a shape or a part of a set of shapes.

GRADE 4


4.OA.B supports 4.NBT.B in:

• Topic 7 Factors and Multiples

4.NBT.B is developed in:

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

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

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

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

• In Topic 7, students apply the multiplication concepts developed in Topics 3 and 4 as they work with factors, multiples, primes, and composites.

4.MD.B supports 4.NF.A and 4.NF.B in:

• Topic 11 Represent and

4.NF.A and 4.NF.B are developed in:

• Topic 8 Extend Understanding of Fraction Equivalence and Ordering

• Topic 9 Understand Addition and Subtraction of Fractions

• In Topic 11, some data come from length measurements represented as fractions and mixed numbers as developed in Topic 8.

• In Topic 11, word problems call for students to apply their knowledge of operations with fractions developed in Topics 9 and 10.





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Interpret Data on Line Plots

• Topic 10 Extend Multiplication Concepts to Fractions

4.MD.A supports 4.OA.A and 4.NF.B in:

• Topic 13 Measurement: Find Equivalence in Units of Measure


• Topic 6 Use Operations with Whole Numbers to Solve Problems

4.NF.B is developed in:

• Topic 9 Understand Addition and Subtraction of Fractions

• Topic 10 Extend Multiplication Concepts to Fractions

• In Topic 13, many of the conversions of units of measure involve using operations with whole numbers to solve problems, as developed in Topic 6. Conversions of customary measurement units involve multiplying fractions and whole numbers developed in Topic 10. Some Topic 13 problems involve adding and subtracting fractions, as developed in Topic 9.

GRADE 5


5.MD.B supports 5.NF.A and 5.NF.B in:


5.NF.A and 5.NF.B are developed in:

• Topic 7 Use Equivalent Fractions to Add and Subtract Fractions

• Topic 8 Apply Understanding of Multiplication to Multiply Fractions

• Topic 9 Apply Understanding of Division to Divide Fractions

• In Topic 10, some data come from measurements represented as fractions and mixed numbers developed in Topic 7.

• In Topic 10, some word problems call for students to apply their knowledge of operations with fractions developed in Topics 8 and 9.

5.MD.A supports 5.NBT.A and 5.NBT.B in:

• Topic 12 Convert Measurements

5.NBT.A is developed in:

• Topic 1 Understand Place Value

5.NBT.B is developed in:

• Topic 2 Use Models and Strategies to Add and Subtract Decimals

• Topic 3 Fluently Multiply Multi-Digit Whole Numbers

• Topic 4 Use Models and Strategies to Multiply Decimals

• In Topic 12, making conversions among measurement units in the metric system uses place-value understandings, as developed in Topic 1.

• In Topic 12, converting units of measure and solving problems involving conversions incorporate using the four operations with both decimals and whole numbers, as developed in Topics 2–6.





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• Topic 5 Use Models and Strategies to Divide Whole Numbers

• Topic 6 Use Models and Strategies to Divide Decimals

Metric

NN Metric 2C: Materials follow 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

enVision® Mathematics offers an organization to focus on the Common Core Clusters in a way that is consistent with the progressions in the Standards. Each Common Core 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 Teacher’s Edition for each grade (K–5) starting on page F4. Also, please reference the Grades K–5 program Scope and Sequence in the Teacher’s Edition Program Overview. This Grades K–5 program Scope and Sequence is also provided in an Appendix to this document, starting on page 98.)





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Metric

NN Metric 2D: Materials give all students extensive work with on-grade-level problems.

Evidence

enVision® Mathematics lessons give students extensive work with on-grade-level problems.

IN EVERY LESSON

On-grade-level problems are infused in the instructional design. Examples: • Daily Review is ongoing review of on-grade-level and previously learned content. Formats include free response and selected response.

• Solve & Share begins the lesson by engaging students with a problem in which new math ideas are embedded. Students solve the problem in ways they choose. Teachers are encouraged to give students time to struggle. Research shows that as students think about ways to solve a problem, conceptual understandings emerge.

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

• Convince Me! is formative assessment that checks for understanding of content related to the lesson objective immediately after the instruction; it asks students to construct arguments related to understanding lesson content.

• Guided Practice is formative assessment of lesson content that checks for understanding before students do Independent Practice.

• Independent Practice builds proficiency in lesson content as students work on their own.

• Problem Solving lesson practice provides work with specific math practices, vocabulary questions to build understanding, higher-order thinking problems, STEM problems, and CCSSM Assessment items.

• Assessment Practice in each lesson helps prepare students for major assessments.

• Additional Practice Workbook provides two pages of additional practice for each lesson; it includes an additional example and focuses on lesson skills and problem solving that reinforces math practices, vocabulary questions to build understanding, higher-order thinking problems,





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and assessment practice items.

• Practice Buddy Online at Grades K–5 provides online interactive practice to use with Additional Practice. At Grades 3–5, Practice Buddy Online is also used with Independent Practice and Problem Solving. At Grades 3–5, it is auto-scored with on-screen help, including Help Me Solve This and View an Example.

LESSON DIFFERENTIATION

• Intervention Activity is on-grade-level, focused, teacher-directed help.

• Reteach to Build Understanding is guided reteaching that helps prepare students for the homework. It also includes vocabulary support.

• Build Mathematical Literacy provides students with reading-comprehension support for an item from the lesson.

• Enrichment Is on-grade-level, focused, higher-order thinking work that helps students develop deeper understandings.

• Activity Centers contain a variety of activities to support and extend lesson content, including Pick a Project, Problem-Solving Leveled Reading Mat Activities, and enVision® STEM Activities.

• Technology Center includes Digital Math Tools, Activities, and Online Math Games (thinking games and fluency games reinforce concepts, procedural skills, and fluency).

• Reteaching Set, at the end of each Topic in the Student Edition, provides lesson examples and practice for Tier 1 Intervention after Guided Practice as needed.





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Metric

NN Metric 2E: Materials relate on-grade-level concepts explicitly to prior knowledge from earlier grades.

Evidence

enVision® Mathematics was developed so that content coherence is maintained within and across grades. Here are some key aspects of the program that speak to this.

MATH BACKGROUND: COHERENCE RELATES ON-GRADE-LEVEL CONCEPTS TO PRIOR KNOWLEDGE

In the Teacher’s Edition, the Math Background at the beginning of each Topic contains a section about Coherence that describes how new on-grade-level content is connected to prior knowledge from an earlier grade and, if appropriate, to prior knowledge from earlier in the year. Here is an example of the Coherence section from Grade 4, Topic 9.

From Grade 4 Teacher’s Edition, p. 329F





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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 fractions in Grade 4.

FROM THE LESSON 8-1 LESSON OVERVIEW FROM THE LESSON 8-2 LESSON OVERVIEW

From Grade 4 Lesson 8-1 Equivalent Fractions: Area Models, p. 293A

From Grade 4 Lesson 8-2 Equivalent Fractions: Number Lines, p. 297A





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CCSSM Clusters that Call for Extending Prior Knowledge In Grade 4, Topic 8 focuses on Cluster 4.NF.A and Topic 9 focuses on 4.NF.B. Both of these clusters call for extending prior knowledge:

• Cluster 4.NF.A Extend understanding of fraction equivalence and ordering.

• Cluster 4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

The examples given on page 15 from Topic 8 (Coherence in Lessons 8-1 and 8-2) and given on page 14 from Topic 9 (Math Background: Coherence) are evidence of how on-grade-level content in enVision® Mathematics is developed by applying and extending prior knowledge.

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 & Share, in which students use prior knowledge, sometimes from earlier grades, to solve a problem that contains new math ideas from the lesson. The problem-based learning process creates new ideas connected to prior knowledge.

STEP 1: PROBLEM-BASED LEARNING

From Grade 4 Lesson 8-1 Equivalent Fractions: Area Models, p. 293





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

STEP 2: VISUAL LEARNING

From Grade 4 Lesson 8-1 Equivalent Fractions: Area Models, p. 294

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

Review of material from previous grades is clearly identified as such to the teacher. For example:

Fluency Practice Activity This feature is at the end of each Topic. For any Topic that precedes a Topic that teaches a fluency standard, this feature provides practice on a fluency standard from a previous grade. The standard from the previous grade is identified on the Student Edition page.





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Daily Review The Daily Review for Lesson 1-1 reviews content from a previous grade. The Standards cited in the Teacher’s Edition for those items are from the previous grade.

Today’s Challenge The Topic 1 Today’s Challenge problems cover content taught in a previous grade and can be used anytime during Topic 1. The Standards cited for these problems in the Today’s Challenge Teacher’s Guide are from the previous grade.

Readiness Test At Grades 1–5, there is a readiness test 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. In Grade K, the readiness test assesses some content taught early in Grade K.

Review What You Know The Topic 1 Review What You Know items at the start of each Topic cover content taught in previous grades. The Item Analysis chart in the Teacher’s Edition identifies Standards from the previous grade. In Grade K Topic 1, the Review What You Know items cover pre-kindergarten skills.

The program includes a wealth of materials to help teachers see what their specific responsibility is for the current year. For example, at the beginning of each Topic, the Topic Planner and the Math Background pages provide teachers with an overview of the mathematical content and focus for the Topic. The Math Background section includes examples to support teachers in understanding the mathematics content and pedagogy. Each lesson begins with the mathematics objective and the Essential Understanding for the lesson. Questions that teachers can ask students are provided in a blue font throughout the lessons to help ensure that teachers have the guidance they need for instructional purposes. The program also includes materials to ensure students understand what their specific responsibility is for the current year. For example, in the Student Edition, each lesson begins with an “I can…” statement that states the math content objective for the lesson in student-friendly language; listed under the “I can…” statement are the content Standards and the Mathematical Practices for the lesson. At Grades 3–5, each lesson in the Student Edition begins with an Essential Question to help students understand the math objective for the lesson. Further, as students work through a lesson, they are given multiple opportunities to gauge their own understanding of lesson content. For example, they share and analyze student work during the Solve & Share. Convince Me! offers students an opportunity to check their understanding of the lesson content. Guided Practice steps students through some problems before they work independently. All of these features help student understand their specific responsibility. At the end of each lesson, there is Assessment Practice that students can use to check their understanding of the lesson content. There are many other self-assessment opportunities presented throughout the program; see page 87 of this document for additional examples.





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Metric

NN Metric 2G: Materials include learning objectives that are visibly shaped by CCSSM cluster headings.

Evidence

enVision® Mathematics 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 begins every lesson in the Teacher’s Edition, outlining the content Standard and Mathematical Practices covered in the lesson, as well as a Mathematics Objective for the lesson that is clearly shaped by CCSSM Cluster headings. Student Edition pages feature prominent learning objectives in the form of “I can” statements that state the math content objective in student-accessible language. Student pages also provide the content Standard and Mathematical Practices, giving students the opportunity to make a direct connection to the specific Standard(s) and Practice(s) as well.

The following chart illustrates the relationship among some CCSSM Cluster headings and enVision® Mathematics topic titles, lesson titles, and lesson objectives (student “I can” statements) for lessons. Also refer to the Contents for each grade in the Teacher’s Edition (beginning on page F4) or in the Teacher’s Edition Program Overview (beginning on page 6).

SOME TITLES AND LEARNING OBJECTIVES SHAPED BY CCSSM CLUSTER HEADINGS

CCSSM Cluster Heading Topic Title Lesson Title Lesson Objective (“I can”) Grade K Cluster K.G.B Analyze, compare, create, and compose shapes.

Topic 13 Analyze, Compare, and Create Shapes

Lesson 13-1 Analyze and Compare Two-Dimensional (2-D) Shapes

I can analyze and compare 2-D shapes.

Grade 1 Cluster 1.NBT.A Extend the counting sequence.

Topic 7 Extend the Counting Sequence

Lesson 7-1 Count by 10s to 120

I can count by 10s to 120.

Grade 2 Cluster 2.OA.C Work with equal groups of objects to gain foundations for multiplication.

Topic 2 Work with Equal Groups

Lesson 2-3 Use Arrays to Find Totals

I can find the total number of objects in a set of rows and columns.





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

Grade 3 Cluster 3.MD.C Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Topic 6 Connect Area to Multiplication and Addition

Lesson 6-4 Areas of Squares and Rectangles

I can find the areas of squares and rectangles by multiplying.

Grade 4 Cluster 4.NF.A Extend understanding of fraction equivalence and ordering.

Topic 8 Extend Understanding of Fraction Equivalence and Ordering

Lesson 8-1 Equivalent Fractions: Area Models

I can recognize and generate equivalent fractions.

Grade 5 Cluster 5.NF.B Apply and extend previous understandings of multiplication and division.

Topic 9 Apply Understanding of Division to Divide Fractions

Lesson 9-6 Divide Whole Numbers and Unit Fractions

I can divide with unit fractions.

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 enVision® Mathematics, 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 key cross-cluster and cross-domain connections within Grade 1 and Grade 4.

GRADE 1 Connected Clusters/Domains Connections Examples 1.OA.A Represent and solve problems using addition and subtraction.

1.OA.B Understand and apply properties of

• Topic 1 Understand Addition and Subtraction focuses on representing and solving addition and subtraction word problems (1.OA.A) and also uses addition and subtraction equations as representations (1.OA.D).





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operations and the relationship between addition and subtraction.

1.OA.C Add and subtract within 20.

1.OA.D Work with addition and subtraction equations.

• Topic 2 Fluently Add and Subtract Within 10 focuses on developing the relationship between counting on and counting back from a number to add and subtract (1.OA.C) and also includes experiences solving word problems using basic facts (1.OA.A), and applying properties of addition (1.OA.B).

• Topic 3 Addition Facts to 20: Use Strategies focuses on developing basic facts (1.OA.C) and also includes experiences solving word problems using basic facts (1.OA.A), as well as properties of operations (1.OA.B).

• Topic 4 Subtraction Facts to 20: Use Strategies focuses on developing operation properties and relationships (1.OA.B) and also develops subtraction facts (1.OA.C) and includes real-world problems solved using basic facts (1.OA.A).

• Topic 5 Work with Addition and Subtraction Equations develops a deeper understanding of equations (1.OA.D) and also uses properties of operations (1.OA.B), develops fluency with addition and subtraction facts (1.OA.C), and includes real-world problems solved using basic facts (1.OA.A).

1.OA.A Represent and solve problems using addition and subtraction.

1.MD.C Represent and interpret data.

• Topic 6 Represent and Interpret Data focuses on organizing, collecting, representing, and interpreting data (1.MD.C) and also includes real-world problems solved using basic facts (1.OA.A).

1.NBT.A Extend the counting sequence.

1.NBT.B Understand place value.

• Topic 7 Extend the Counting Sequence focuses on counting, reading, and writing for numbers to 120 (1.NBT.A) and also develops an understanding of place value for these numbers. (1.NBT.B).

1.NBT.B Understand place value.

1.NBT.C Use place value understanding and properties of operations to add and subtract.

• Topic 9 Compare Two-Digit Numbers focuses comparing 2 two-digit numbers based on meanings of the tens and ones digits (1.NBT.B) and also develops the concept of mentally finding 10 more or 10 less than a given two-digit number without having to count. (1.NBT.C).

GRADE 4 4.OA.A Use the four operations with whole numbers to solve problems.

4.NBT.B Use place-value understanding and properties of operations to perform multi-digit arithmetic.

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

Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers

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

Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers

Topics 2 through 5 focus on developing operations with multi-digit numbers (4.NBT.B) but also infuse real-world problems solved using those operations (4.OA.A).





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• Topic 6 Use Operations with Whole Numbers to Solve Problems focuses on solving comparison problems and multi-step problems (4.OA.A) but also involves operations with multi-digit numbers (4.NBT.B).

4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

• Topic 10 Extend Multiplication Concepts to Fractions focuses on multiplying fractions or mixed numbers by whole numbers (4.NF.B) but also uses this to solve problems involving time conversions (4.MD.A).

• Topic 13 Measurement: Find Equivalence in Units of Measure focuses on finding equivalent measures involving customary units (4.MD.A) and also includes some conversions involving multiplying whole numbers by fractions or mixed numbers (4.NF.B).

4.MD.B Represent and interpret data.

4.NF.A Extend understanding of fraction equivalence and ordering.

4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

• Topic 11 Represent and Interpret Data on Line Plots focuses on reading and making line plots (4.MD.B) and also includes data and word problems that involve understanding equivalent fractions as well as adding and subtracting fractions and multiplying a fraction by a whole number. (4.NF.A, 4.NF.B).

4.NF.C Understand decimal notation for fractions, and compare decimal fractions.

4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

• Topic 12 Understand and Compare Decimals focuses on fraction and decimal understandings (4.NF.C) but also uses these to solve problems involving operations with money (4.MD.A).

4.MD.C Geometric measurement: understand concepts of angle and measure angles.

4.NBT.B Use place-value understanding and properties of operations to perform multi-digit arithmetic.

4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

• Topic 15 Geometric Measurement: Understand Concepts of Angles and Angle Measurement focuses on measuring angles (4.MD.C) and also develops an understanding of lines, rays, and angles at the start of the Topic and uses that through the rest of the Topic (4.G.A) as angles measures are added and subtracted (4.NBT.B).





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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 enVision® Mathematics 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 enVision® Mathematics supports Focus and Coherence.





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Alignment Criterion 1 Rigor and Balance

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

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? In enVision® Mathematics, conceptual understanding is attended to thoroughly in those places where the Standards set specific expectations.

For example, there are several instances where “understand” or “understanding” are explicitly called out in the CCSSM at Grade 1. The following is a description of how two of these instances is given special attention in enVision® Mathematics Grade 1.

1.OA.B Understand and apply properties of operations and the relationship between addition and subtraction.

The enVision® Mathematics highly effective approach to developing basic addition and subtraction facts embraces: (a) basic fact strategies, (b) the use of properties of operations at appropriate places in the development of facts, and (c) the relationship between addition and subtraction for developing subtraction basic facts.

• The strategies of counting on (including the use of the additive identity—adding 0), doubles, and near doubles can be used to develop the basic addition facts. The Commutative Property for Addition is introduced after those fact strategies. The make a ten strategy is then introduced to develop the remaining facts and their commutative partners. This development is handled in Topics 2 and 3.





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• The Associative Property of Addition is introduced (informally) later in Topic 5 when student use strategies (e.g., make a ten) to add three addends, connecting basic facts to work with solving addition equations and solving word problems with three addends.

• After developing counting strategies for finding subtraction basic facts, another approach to developing subtraction facts in enVision® Mathematics is “think addition.” In other words, the relationship between addition and subtraction is used to develop basic subtraction facts. See Grade 1 page 178 where students are given 13 – 8 = ?. They use the relationship between addition and subtraction to think of the related addition equation 8 + ? = 13. They use their knowledge of the addition fact to find the value of 13 – 8.

From Grade 1 Lesson 4-5: Use Addition to Subtract, p. 178

• The understanding of properties justifies many of the strategies used to develop basic addition and subtraction facts in enVision® Mathematics (even though these are not made formal and explicit to students at Grade 1). For example, the informal thinking used for the near doubles strategy (introduced in Grade 1 Lesson 2-3) is grounded in the Associative Property of Addition.

From Grade 1 Lesson 2-3: Near Doubles, p. 66





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1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2

• The concept of equality is developed intuitively and purposefully throughout Topics 1–4. For example, the Visual Learning Bridge at the top of page 14 (shown below) uses an equation to show the whole and the parts. This shows how the equal sign is used to show that two quantities are the same. The last box of the Visual Learning Bridge develops the idea that different numerical expressions can name (or be equal to) the same quantity. These are essential understandings about equality.

From Grade 1 Lesson 1-3: Both Addends Unknown, p. 14

• Topic 5 then focuses on 1.OA.D.7. The following understandings related to equality are developed here:

– An unknown quantity in an addition or subtraction equation can be replaced with a specific number so the value on the left of the equal sign is the same as the value on the right.

– Numerical expressions in an equation can be on the right or left of the equal sign. – An equation is true if the value on the left of the equal sign is the same as the value on the right of the equal sign. – The value of a numerical expression with three addends can be found in different ways and an equation can be written to show the total.

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

The core instructional model used for every lesson in enVision® Mathematics consistently features high-quality conceptual problems and conceptual discussion questions. This model is shown through research to be the most effective for developing conceptual understanding. A recent research article describes the instructional model used in enVision® Mathematics.





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“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, pages 584-635.]

The instructional approach described above has also been called Teaching Mathematics Through Problem Solving. The following NCTM publication describes this model and provides additional research as to its effectiveness:

Lester, F.K. & Charles, R.I., eds. Teaching Mathematics Through Problem Solving: Prekindergarten–Grade 6. Reston, VA: NCTM, 2003

The instructional model described above and used for every lesson in enVision® Mathematics begins with an experience titled Solve & Share. 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 9 + 6.” is a non-routine problem for a first grader who has not yet developed an efficient strategy for finding the sum. Students might use counting strategies and objects 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 enVision® Mathematics. Furthermore, the teaching actions shown in the Teacher’s Edition show that “conceptual discussion questions” are at the core of enVision® Mathematics.

On page 28 is a sample Solve & Share problem from enVision® Mathematics. In this Grade 1 lesson, students are working with a non-routine problem to demonstrate that some addition facts can be solved by changing them to an equivalent fact with 10. Being able to use benchmark numbers such as 5 and 10 is an important idea foundational for understanding place value, number, and operations. Multiple strategies can be used to solve the following problem.





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From Grade 1 Lesson 3-5: Make 10 to Add, p. 125

“Conceptual discussion questions” are at the core of enVision® Mathematics. The Teacher’s Edition provides Before, During, and After questions for teachers to use with the Solve & Share to support best practices and promote discourse.





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Sample student work is provided with every lesson to ensure a quality class discussion that involves analyzing sample student work. In this particular example shown at the right, one student successfully explained how to use counters and think about ten to solve the addition fact. The other student used cubes but made a mistake in how he used ten to solve the addition fact.

From Grade 1 Teacher’s Edition Lesson 3-5: Make 10 to Add, p. 125





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The Visual Learning Bridge (which is Step 2) for this lesson makes the concepts from the Solve & Share more explicit and provides teachers with guiding questions (in blue font in the Teacher’s Edition) that ask students, “Why?” and “How?” to promote conceptual understanding.

From Grade 1 Teacher’s Edition Lesson 3-5: Make 10 to Add, p. 126





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Additional Teacher Support Related to Conceptual Understanding In the Teacher’s Edition, the Math Background at the beginning of each Topic and each Lesson Overview has a section called “Rigor” which includes information about how conceptual understanding is developed at both the topic level and the lesson level.

From Grade 4 Topic 9 Math Background: Rigor, p. 329G

From Grade 4 Topic 9, Lesson 9-1 Lesson Overview, p. 333A





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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. See the examples from Grade 1 shown below. In Lesson 4-1, students use a number line as a model to solve subtraction by counting on or counting back to find the difference. When students count on to subtract, they are thinking addition to subtract and connecting operations. (1.OA.C.5, 1.OA.C.6)

From Grade 1 Lesson 4-1: Count to Subtract, p. 162





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In Lesson 4-2, students explore how to use 10 to subtract within 20 using ten-frames and counters. (1.OA.C.6)

From Grade 1 Lesson 4-2: Make 10 to Subtract, p. 166





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In Lesson 4-4, students use part-part-whole mats (bar diagrams) to visually see the connection between addition and subtraction. The addition-subtraction relationship becomes even more visible than before as students work with fact families. (1.OA.B.4, 1.OA.C.6)

From Grade 1 Lesson 4-4: Fact Families, p. 174





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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? For each Fluency Standard, there is an enVision® Mathematics Topic that develops fluency with understanding. That Topic is the culmination of foundations for fluency taught previously.

The following table provides the CCSSM Fluency Standards for each grade, the associated enVision® Mathematics 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. For example, in addition to foundational Topics in Grade K and Grade 1, Topic 2 in Grade 1 provides the development with understanding that students need to fluently add and subtract within 10.

CCSSM Fluency Standard

enVision® Mathematics Topics that Provide the Foundation for Fluency

“Capstone” enVision® Mathematics Topic(s)

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 (Develops the conceptual understanding needed to fluently add and subtract within 5.)

Grade 1 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. (1.OA.C.6)

• Grade K, Topic 8 More Addition and Subtraction

• Grade 1, Topic 1 Understand Addition and Subtraction

• Grade 1, Topic 3 Addition Facts to 20: Use Strategies • Grade 1, Topic 4 Subtraction Facts to 20: Use Strategies

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





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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


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: Use Patterns

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

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

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





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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

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 enVision® Mathematics 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 the Grade 1 Student Edition Lesson 4-7, pages 185–188. On page 185, Problem-Based Learning (Solve & Share) opens the lesson with a rich problem for students to solve any way they choose, after which students discuss and share solution strategies. During Visual Learning on page 186, students see how they can use different ways to solve subtraction facts. During Convince Me!, students explain how one strategy can be used to solve 10 – 3. Guided Practice offers opportunities for students to find each difference and explain the strategy they used.





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From Grade 1 Lesson 4-7: Explain Subtraction Strategies, p. 185





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From Grade 1 Lesson 4-7: Explain Subtraction Strategies, p. 186





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

In addition to the fluency progress detailed in the table above (on pages 35–37) that shows the lessons for the CCSSM Fluency Standards, enVision® Mathematics (Grades K–5) also provides the following:

• A Fluency Review or a Fluency Practice Activity is found in every Topic in Grades 1–5. These provide either review of fluency Standards from the previous grade or practice of fluency Standards from the current grade. At Kindergarten, Fluency Practice Activities are found in Topics 8–14. Grade 1 Topic 1 reviews the Kindergarten fluency standard (K.OA.A.5), and Grade 1 Topics 2–15 support Fluency Standard 1.OA.C.6.

• Fluency Practice/Assessment Worksheets, found in the Teacher’s Resource Masters, provide ongoing practice/assessment of fluency subskills.

• Practice Buddy Online at Grades K–5, provides online interactive practice for use with Additional Practice. At Grades 3–5, Practice Buddy Online is also used with Independent Practice and Problem Solving. Practice Buddy Online at Grades 3–5 is autoscored with on-screen help, including 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.

• Fluency Math Games at PearsonRealize.com provide opportunities for practicing fluency skills.

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

• “My Fluency Progress” Form 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. For an example, see Grade 1 Teacher’s Resource Masters, Teaching Tool 41, page 41.





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Metric

AC Metric 1C The materials are designed so that teachers and students spend sufficient time working with engaging applications, without losing focus on the Major Work of each grade.

Evidence

Are there single- and multi-step contextual problems, including non-routine 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 enVision® Mathematics lesson with Problem-Based Learning, the Solve & Share, which introduces new math in the context of solving a real problem for which a standard solution procedure may not be readily available to students. Students apply their prior knowledge to engaging problem situations and determine their own methods for solving the problem. Students may not yet have learned a procedure/technique for solving the problem in the most efficient manner. Classroom discussion and analyzing sample student work exposes students to alternate methods for solving the problem and deepens their conceptual understanding of the Major Work of the grade. Research suggests that this non-routine, daily engagement with the Mathematical Practices is the best way for students to develop understanding. (Kapur, 2010; Lester and Charles, 2003; Scott, 2014).

For example in Grade 1 Lesson 4-7, page 185 (pictured previously on page 38), students explore different ways to solve a subtraction problem and communicate their thinking. In Grade 4 Lesson 9-6, page 353, students extend their understanding of adding and subtracting fractions with like denominators by using diagrams, including a number line, to model these operations.

The Visual Learning Bridge starts with a real-world application in many lessons. These real-world applications provide a context and motivate students to see why the concept or skill is important; the real-world applications also reinforce operation meanings and relationships to help students develop their mathematical problem-solving skills.

Single- and multi-step contextual problems, including non-routine problems that afford opportunities for students to practice and engage in problem solving, are found in every lesson (e.g., in Guided Practice, Independent Practice, Problem Solving, and/or Additional Practice). For example, in Grade 1 Lesson 4-7, pages 185–188, as well as Additional Practice pages 67–68, students use strategies such as count, make 10, and think addition to help them solve subtraction problems and communicate their reasoning. Lessons like Grade 4 Lesson 9-6, pages 353–356, as well as Additional Practice pages 123–124, provide more examples.





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Students have a variety of opportunities to apply their understanding of the important grade-level content via engaging applications in the enVision® STEM Project and Pick a Project activities found at the beginning of each Topic, and the 3-Act Math activities found at the beginning of each odd-numbered Topic. Additional opportunities are provided in Digital Math Games, Digital Math Tools Activities, and Problem-Solving Reading Mat Activities, shown on the last two pages of each lesson in the Teacher's Edition.

In Grade 4, page 336, Items 19 and 21; page 340, Items 11 and 14; and; page 344 Items 22 and 23 are just a few examples of multi-step applications provided in the Problem Solving section of the daily lesson. In addition to multi-step contextual problems being available to students in daily lessons, each grade in enVision® Mathematics provides full “Problem Solving” lessons devoted to supporting the CCSSM where expectations for multi-step and real-world problems are explicit. The chart below lists some Standards that explicitly focus on multi-step and real-world problems; it also identifies the enVision® Mathematics Topics and lessons within those Topics that were written to address those Standards.

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

Grade CCSSM Standards enVision® Mathematics Topics

enVision® Mathematics

Lessons Grade 2

2.OA.A.1

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Topic 3 Add Within 100 Using Strategies Lesson 3-6 Solve One-Step and Two-Step Problems

Topic 4 Fluently Add Within 100 Lesson 4-8 Solve One-Step and Two-Step Problems

Topic 5 Subtract Within 100 Using Strategies Lesson 5-7 Solve One-Step and Two-Step Problems

Topic 6 Fluently Subtract Within 100 Lesson 6-6 Solve One-Step and Two-Step Problems

Topic 7 More Solving Problems Involving Addition and Subtraction

Lesson 7-4 Solve Two-Step Problems

Lesson 7-5 Continue to Solve Two-Step Problems

Grade 3

3.OA.D.8

Solve two-step word problems using the four operations. Represent these

Topic 11 Use Operations with Whole Numbers to Solve Problems

Lesson 11-1 Solve 2-Step Word Problems: Addition and Subtraction

Lesson 11-2 Solve 2-Step Word





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problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Problems: Multiplication and Division

Lesson 11-3 Solve 2-Step Word Problems: All Operations

Grade 3

3.MD.B.3

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Topic 7 Represent and Interpret Data

7-4 Solve Word Problems Using Information in Graphs [includes two-step problems]

Grade 4

4.OA.A.3

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Topic 6 Use Operations with Whole Numbers to Solve Problems

Lesson 6-3 Model Multi-Step Problems

Lesson 6-4 More Model Multi-Step Problems

Lesson 6-5 Solve Multi-Step Problems

Grade 5 5.MD.A.1

Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Topic 12 Convert Measurements Lesson 12-1 Convert Customary Units of Length

Lesson 12-2 Convert Customary Units of Capacity

Lesson 12-3 Convert Customary Units of Weight





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Lesson 12-4 Convert Metric Units of Length

Lesson 12-5 Convert Metric Units of Capacity

Lesson 12-6 Convert Metric Units of Mass

Lesson 12-7 Convert Units of Time

Lesson 12-8 Solve Word Problems Using Measurement Conversions





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Multi-Step Problems in Lessons In Grades 3–5, in the Teacher’s Edition, at the bottom of the fourth page in a lesson, there is a list of the multi-step problems in the lesson, (See the example shown at the right.)

From Grade 4 Teacher’s Edition, p. 343-344





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

APPLICATIONS IN PROGRAM ELEMENTS

Here are some Grade 4 examples of how the following elements of enVision® Mathematics address this metric in the many Topics that focus on Major Clusters as well as in other Topics with connections to Major Clusters.

Solve & Share in Lessons Every lesson begins with students solving a problem (problem-based instruction) in an experience called Solve & Share. 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 4 examples are: Lesson 1-3, page 13; Lesson 2-1, page 37; Lesson 3-2, page 85; Lesson 4-5, page 145; Lesson 5-1, page 169.

Visual Learning Bridge in Lessons Every lesson includes a Visual Learning Bridge (VLB) that builds on the Solve & Share. The VLB is the important mathematics of the lesson made explicit. Most of these begin with a real-world 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 4, page 150, Box A.

• To reinforce operation meanings using bar diagrams to show relationships between quantities. See, for example, Grade 4, page 186 Box A.

• To provide a real-world visual context that helps students make sense of the concept or procedure developed. See, for example, Grade 4, page 154 Box A and page 490 Box A.

Problem Solving Items in Lessons This section of every lesson includes content-appropriate applications that:

• focus on the content of the lesson and revisit Major Content as appropriate. See, for example, all exercises on Grade 4, page 400.

• involve the use of multiple Mathematical Practices and provide a teaching opportunity to discuss a particular Mathematical Practice. See, for example, Grade 4, Item 11, page 392.

• provide an opportunity to use higher-cognitive levels of reasoning. See, for example, Grade 4, Item 14, page 496.

• apply previously learned concepts and procedures in a task representative of those from common important assessments. See, for example, Grade 4, Items 22 and 23, page 351.

“Problem Solving” Lessons Mathematical Practices are experienced in every lesson. For example, the Solve & Share experience that begins every lesson is a particularly rich area for developing students’ abilities with Mathematical Practices. Every Solve & Share task requires students to select, coordinate, and implement multiple Mathematical Practices. However, the enVision® Mathematics authors believe it is important to stop occasionally throughout the year to have in-depth discussions about each Mathematical Practice. Therefore, each Topic in





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enVision® Mathematics includes a lesson that focuses on a particular Mathematical Practice. Almost all of these lessons are driven in the Solve & Share and Visual Learning Bridge by applications that can be solved by employing the Mathematical Practice of focus for that lesson together with other Mathematical Practices as appropriate. Some important aspects of these “Problem Solving” lessons are:

• An emphasis on “finding the hidden question” when solving multi-step problems (e.g., Grade 4, page 155).

• The extensive use of bar diagrams to show how quantities in problems are related and the operations that can be used (e.g., Grade 4, page 110).

• The use of common assessment-type items that make explicit the role and use of multiple Mathematical Practices in finding solutions (e.g., Grade 4, page 112).

• The use of visual representations as a vehicle for introducing data needed to solve problems (e.g., page 247 Items 4 and 5; page 248 Items 6–8) or to promote an understanding of a problem (e.g., Grade 4, page 371 Items 1 and 4).

Assessments and Applications Applications are infused in assessments.

• Assessment Practice items in daily lessons are included at the end of the Problem Solving practice in each lesson. Many of these items are applications of concepts and skills developed in the lesson. Some examples in Grade 4 are page 88, Items 16 and 17; page 176, Item 33.

• Topic Assessment items are typically driven by real-world applications rather than just skill assessment. Some Grade 4 examples are: Grade 4 Teacher’s Edition, Topic 3 Assessment and Topic 3 Performance Task, pages 119–124A and Grade 4 Teacher’s Edition, Topic 5 Assessment and Topic 5 Performance Task, pages 215–220A.

REPRESENTING AND SOLVING THE FULL-RANGE OF ONE-STEP ADDITION AND SUBTRACTION WORD PROBLEMS

The charts on the next page show the plan used in enVision® Mathematics to introduce various word-problem types for addition and subtraction in Grade 1. Problems with these structures are then used throughout the remainder of Grade 1, where appropriate, and then continue to be used in subsequent grades.





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The types in italics are review from Grade K.

The types in bold are new to Grade 1 and should be mastered in Grade 1.

The types in bold and underlined are new to Grade 1 and should be mastered in Grade 2.

PROBLEM TYPES IN GRADE 1 TOPIC 1: UNDERSTAND ADDITION AND SUBTRACTION

Lessons Problem Types 1-1 Add To Add to: result unknown

1-2 Put Together Put together: total unknown

1-3 Both Addends Unknown Put together/take apart: addends unknown

1-4 Take From Take from: result unknown

1-5 Compare Situations Compare: difference unknown (more)

1-6 More Compare Situations Compare: difference unknown (fewer)

1-7 Change Unknown Add to: change unknown

1-8 Practice Adding and Subtracting Put together/take apart: addend unknown

1-9 Problem Solving: Construct Arguments All types from above

PROBLEM TYPES IN GRADE 1 TOPICS 2, 3, 4, 5 Topics Problem Types Topic 2 Fluently Add and Subtract Within 10 Any types from above plus:

Take from: change unknown

Topic 3 Addition Facts to 20: Use Strategies

Any types from above plus: Compare: bigger unknown (more) and Compare: bigger unknown (fewer)

Topic 4 Subtraction Facts to 20: Use Strategies

Any types from above plus: Add to: start unknown and Take from: start unknown

Topic 5 Work with Addition and Subtraction Equations

Any types from above plus Compare: smaller unknown (more) and Compare: smaller unknown (fewer)





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Does modeling build slowly across Grades K–8, with applications that are relatively simple in earlier grades and when students are encountering new content?

BAR DIAGRAMS

Many problems involve two known quantities and one unknown quantity students need to find. 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 enVision® Mathematics demonstrates how students apply understanding in simple situations in early grades and then slowly build across the grades. See Teacher’s Edition Program Overview pages 60–63 for a complete overview of the Bar Diagram progression across grades in enVision® Mathematics. Below are just a few examples from the program to demonstrate how this modeling is built slowly across grades.

As shown at the right, conceptual development in Grades K–2 moves from concrete to pictorial to symbolic.

From Teacher’s Edition Program Overview, p. 60

At the right is an example of how Kindergarten students learn to use two-color counters in a bar diagram to model an addition equation. They connect the model to a story about monkeys to develop a conceptual understanding of addition equations.

From Kindergarten Lesson 8-3, p. 302





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At the right is an example of a Grade 1 “Put Together: Total Unknown” problem situation. Students use connecting cubes in a bar diagram to model the situation.

From Grade 1 Lesson 1-2, p. 10

At the right is a Grade 2 example of a “Compare: Difference Unknown” problem situation. Students use numbers in a bar diagram to model the situation.






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At the right is a Grade 3 “Equal Groups: Product Unknown” problem situation. Students use counters to model the situation; then they write numbers in a bar diagram to model the situation.






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At the right is a Grade 4 example of “Compare with Multiplication” and “Compare with Addition.” Notice how the quantities are represented proportionately.

From Grade 4 Lesson 6-1, p 226





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At the right is a Grade 5 example of using bar diagrams to help solve a multi-step problem that involves “Add To: Result Unknown” and then “Take From: Result Unknown.”






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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

enVision® Mathematics requires students to engage with and demonstrate challenging mathematics with appropriate balance among application, conceptual understanding, and procedural skill and fluency.

Alignment Criterion 2 Standards for Mathematical Practice Alignment Criterion 2: Materials must demonstrate authentic connections between content Standards and practice Standards. Metric

AC Metric 2A: Materials address the practice Standards in such a way as to enrich the Major Work of the grade; practices strengthen the focus on Major Work instead of detracting from it, in both teacher and student materials.

Evidence

enVision® Mathematics is organized to focus on CCSSM Clusters, and Clusters are identified as Major, Supporting, and Additional. Another layer in organizing enVision® Mathematics 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.





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The goal and promise of enVision® Mathematics is to develop understanding, so the Mathematical Practices are infused in all aspects of the program. For example, see the Math Practices and Problem Solving Handbook shown in the Teacher’s Edition in every grade (e.g., Grade 1 Teacher’s Edition, pages F27–F45). See also each Topic’s planning pages (e.g., Grade 1 Topic 4 Teacher's Edition, pages 157A–157C, and 157H) for an overview of how the Mathematical Practices are integrated throughout the program in ways the enrich the important work of the Topic.

IN EVERY LESSON

Everyday engagement of the Standards for Mathematical Practice enables learners to develop conceptual 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. The Solve & Share problem-based learning section, found in every lesson in enVision® Mathematics, requires the student to select, implement, and apply Mathematical Practices. The Standards for Mathematical Practice are reflected in the questions (in blue font) provided for teachers for each Solve & Share.

For each daily Visual Learning Bridge, the avatar comments 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.

Convince Me!, which immediately follows the Visual Learning Bridge in each lesson, is a formative assessment opportunity which focuses on a specific Mathematical Practice.

In each lesson’s Problem Solving section, as well as in the Additional Practice Workbook, specific items are labeled with a Mathematical Practice. These items may involve applying more than one Mathematical Practice—providing an opportunity for rich classroom conversations and teaching moments.

LESSONS THAT FOCUS ON MATH PRACTICES

In every Topic in every grade, there is a comprehensive “Problem Solving” lesson that gives special focus to one of the eight Mathematical Practices and is completely devoted to helping students develop the “habits of mind” for the Standards for Mathematical Practice. Features of the “Problem Solving” lessons include the following. (For reference as an example, see Grade 1 Teacher’s Edition Lesson 4-9, pages 193A-196B and Grade 4 Teacher’s Edition, Lesson 9-10, pages 369A–372B.)

“I can…” statement The “I can” statement, on the first page of each “Problem Solving” lesson, highlights specific features of the math practice.

Solve & Share The problem posed can be analyzed and solved by engaging the math practice featured in the lesson. The Thinking Habits, shown in the cloud on page 1 of the “Problem Solving” lesson, provide questions that help guide students’ thinking in a way that supports the math practice.





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In Look Back! (Grades 3–5), students use one or more of the math practices featured in the lesson to interpret their solution or analyze another aspect of the problem.

Visual Learning Bridge Students are provided a stepped-out example to model the kind of thinking that helps them engage the math practice.

Convince Me! Students show how they can engage the same math practice to solve a problem related to the one in the Visual Learning Bridge. Teachers can use this feature to check for students’ understanding of the math practice.

Guided Practice Students solve a problem by enlisting the math practice featured in the lesson.

Independent Practice Students do work on their own to engage the math practice.

Problem Solving Performance Task Assessment Students engage multiple math practices to solve a problem like ones they will encounter in performance task assessments.

In the Additional Practice Workbook, Another Look! provides another example that models how to apply the math practice (“habits of mind”). Students do work involving the specific math practice that is the focus of the lesson. Then they engage various math practices to solve problems like ones they will encounter in common performance task assessments.

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?

Math practices and content Standards are connected throughout all Grades K–5 Topics and lessons including the special “Problem Solving” lessons that specifically focus on math practices. The following are examples within just one grade and Topic (Grade 4, Topic 9) of how math practices are meaningfully present in the form of assignments, activities, or problems that stimulate students to develop the “habits of mind” described in the practice Standard. Similar examples can be found in any grade for all Topics and lessons; refer to the Math Practices and ETP





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page at the beginning of each Topic for additional examples.

From Grade 4 Teacher’s Edition, page 329H

MATH PRACTICES

MP.1 Make sense of problems and persevere in solving them. Students persevere as they try to understand problems involving fractions, plan how to solve them, and determine if their solutions make sense. (e.g., p. 334, Convince Me!)

MP.2 Reason abstractly and quantitatively. Students use quantitative reasoning as they solve subtraction problems involving fractions. (e.g., p. 352, Item 20)

MP.3 Construct viable arguments and critique the reasoning of others. Students critique strategies for computing with fractions, and they construct arguments to justify their results. (e.g., p. 342, Visual Learning Bridge)

MP.4 Model with mathematics. Students model with math when they apply bar models and equations to represent problems involving fraction operations. (e.g., p. 369, Look Back!)

MP.5 Use appropriate tools strategically. Students use tools such as fraction strips to represent fraction operations and solve problems. (e.g., p. 333, Look Back!)

MP.6 Attend to precision. Students attend to precision when they use and explain fraction computations and when they choose the correct units in their solutions. (e.g., p. 345, Look Back!)

MP.7 Look for and make use of structure. Students look for structure when they examine relationships in fraction computations. (e.g., p. 341, Look Back!)

MP.8 Look for and express regularity in repeated reasoning. Students use repeated reasoning when they generalize about fraction operations. (e.g., p. 361, Solve & Share)





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“Problem Solving” Lessons In every Topic, at Grades K–5, a comprehensive “Problem Solving” lesson is completely devoted and gives special focus to one of the eight math practices. To develop proficiency with a math practice, the lesson instruction offers thinking habits questions for that math practice and models thinking related to that math practice when solving a rich problem. (For an example of a “Problem Solving” lesson, see Grade 4 Teacher’s Edition Lesson 9-10 Model with Math pages 369A–372B.)

The “Problem Solving” lessons provide practice with the kinds of rich problems students will encounter on high-stakes performance-based assessments. (For an example, see Grade 4 Teacher’s Edition page 371-372.)

Math practices with content Standards are also assessed in Topic Assessments and Topic Performance Tasks, both of which include rubrics in the Teacher’s Edition and in the Assessment Sourcebook. (For examples, see the Grade 4 Teacher’s Edition, Topic 9 Topic Assessment on pages 377-378 and 378A and the Topic 9 Performance Task on pages 379-380 and 380A.)

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?

The goal and promise of enVision® Mathematics is to develop understanding, so the Mathematical Practices are infused in all aspects of the program, especially in daily lessons and in “Problem Solving” lessons. Developmentally appropriate problem solving materials offer rich problems and tasks that demonstrate growing sophistication across the grades. These problem solving materials incorporate grade-appropriate language, grade-appropriate math content, and grade-appropriate thinking skills.

On the next page are two examples (one each from Grade 1 and Grade 4) of a Visual Learning Bridge from a “Problem Solving” lesson that addresses MP.4 Model with Math. In addition to addressing MP.4, the Grade 1 lesson also addresses Grade 1 content Standards 1.NBT.C.6 and 1.NBT.C.5. The Grade 4 lesson also addresses Grade 4 content Standards 4.NF.B.3d, 4.NF.B.3a, and 4.NF.B.3c. In each lesson, the language and math concepts are appropriate to the grade. These examples show how MP.4 progresses from an appropriately simple approach at Grade 1 to a more sophisticated approach at Grade 4. In the Grade 1 lesson, MP.4 involves drawing a picture and writing an equation to represent a subtraction situation. In the Grade 4 lesson, MP.4 involves using a bar diagram and an equation with a letter variable to represent a fraction subtraction situation.





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From Grade 1 Lesson 11-7 Problem Solving: Model with Math, p. 478

From Grade 4 Lesson 9-10 Problem Solving: Model with Math, p. 370





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The chart below cites the “Problem Solving” lessons as one way in which practice Standards are developed across Grades K–5.

PROBLEM SOLVING LESSONS

CCSSM Grade K Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 MP.1 Make sense of problems and persevere in solving them.

Lesson 13-4

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

Lesson 11-6 Lesson 4-9; Lesson 12-8

Lesson 4-7; Lesson 6-6


MP.2 Reason abstractly and quantitatively.



Lesson 6-7; Lesson 7-8; Lesson 8-5; Lesson 15-6


Lesson 2-8


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

Lesson 1-10 Construct Arguments Lesson 5-4 Critique Reasoning

Lesson 1-9 Construct Arguments Lesson 3-9 Critique Reasoning

Lesson 1-10 Construct Arguments Lesson 3-7 Construct Arguments Lesson 5-8 Critique Reasoning

Lesson 9-7 Construct Arguments Lesson 11-4 Critique Reasoning Lesson 13-8 Construct Arguments

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

Lesson 3-9 Critique Reasoning Lesson 10-4 Critique Reasoning Lesson 12-4 Critique Reasoning Lesson 16-4 Construct Arguments

MP.4 Model with mathematics.





Lesson 3-8; Lesson 5-10; Lesson 9-10; Lesson 10-5


MP.5 Use appropriate tools strategically.

Lesson 7-7

Lesson 12-4

Lesson 14-5 Lesson 1-6

Lesson 15-6

Lesson 11-5

MP.6 Attend to precision.


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

Lesson 13-7 Lesson 12-9

MP.7 Look for and make use of structure.



Lesson 9-10 Lesson 5-6; Lesson 6-7; Lesson 10-4


Lesson 1-7

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

Lesson 4-5

Lesson 7-7


Lesson 3-7

Lesson 7-3

Lesson 9-8





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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 available at PearsonRealize.com and at the front of the Teacher’s Edition has two parts: Math Practices provides a page for each math practice as a developmentally appropriate resource for students and teachers to use throughout the year when discussing math practices. Problem Solving provides resources for students and teachers that facilitate problem solving experiences on a daily basis. (For example, see Grade 4 Teacher’s Edition, pages F27–F51.)

Math Practices pages in the Teacher’s Edition include:

1. Develop the math practice, which discusses how to use the student page (i.e., What does the math practice mean? Sample uses of the math practice and Thinking Habits for the math practice)

2. Connect the math practice, which discusses connecting the math practice to content Standards and to other math practices

3. Assess the math practice, which lists behaviors related to the math practice and gives a scoring rubric for evaluating students’ proficiency with the math practice

Problem Solving Guide is a resource teachers can use to facilitate students’ work solving problems. The guide incorporates math practices and includes questions to guide students’ thinking.

Problem Solving Recording Sheet can be made available at any time to help students use the Problem Solving Guide and provide a structure for recording their work as they solve a problem.

Bar Diagrams At Grades 3–5, the Math Practices and Problem Solving Handbook (pages 13–18 of the handbook) presents bar diagrams for different situations along with examples of how they can be used to solve problems. In the Teacher's Edition, pages F46–F51 provide comprehensive notes that support understanding of bar diagrams. For a general discussion of bar diagrams in Grades K–5, see the Teacher’s Edition Program Overview, pages 60–63.

A comprehensive “Problem Solving” lesson in every Topic is completely devoted and gives special focus to one of the eight math practices. Features of the “Problem Solving” lessons include extensive teaching notes and practice Performance Assessments.

All enVision® Mathematics lessons include extensive teaching notes and explanations, including accurate alignments and support for the practice Standards.





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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 below are from Grade 4, Topics 5 and 9 Teacher’s Edition, but any grade offers comparable opportunities for students.

• The Solve & Share in every lesson supports MP.3 by calling for students to explain their thinking (construct viable arguments) and critique the arguments of others (analyze student work) during class discussions.

• The Visual Learning Bridge has guiding questions in the Teacher’s Edition that often support MP.3 (Grade 4 Teacher’s Edition, pages 178, 182, 342).

• Convince Me!, which occurs immediately after the Visual Learning Bridge, often supports MP.3 (Grade 4 Teacher’s Edition, pages 174, 178, 182, 342).

• Problem Solving exercises often support MP.3 (Grade 4 Teacher’s Edition, pages 176, 184).

• Additional Practice often supports MP.3 (Grade 4 Additional Practice Workbook, pages 114, 128).

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?

Throughout enVision® Mathematics, students are encouraged to use reasoning, to explain their strategies, and to use diagrams, models, and tools appropriately. For example, students are prompted to use visual models for fraction addition and subtraction, such as fraction strips, number lines, and partitioned regions. In Grade 4 Topic 9, students are encouraged to represent their thinking with concrete and abstract





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representations throughout. (For an overview, see Grade 4 Teacher’s Edition, pages 329E–329H.)

Bar diagrams help students understand and represent the relationship between known and unknown quantities in order to solve problems—a direct application of MP.2 Reason abstractly and quantitatively. (See Teacher's Edition Program Overview, pages 60–63, for an overview of bar diagrams.)

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 statement about what good math thinkers do when they engage MP.2, as shown at the right on page 4 of the Grade 1 handbook.

• They are provided with a sample problem that lends itself to engaging MP.2.

• Students are also provided with Thinking Habits questions that help them engage MP.2 when solving problems.

From Grade 1 Math Practices and Problem Solving Handbook, p. 4





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Teacher notes develop MP.2 through class discussion of the student’s page. Teacher notes also connect MP.2 to content Standards and to other math practices. Teacher notes also provide a rubric for assessing students’ ongoing development of proficiency with MP.2.

From Grade 1 Teacher’s Edition, pp. F30–F31

See also Teacher's Edition Program Overview page 56.

Other Support for an Emphasis on Mathematical Reasoning includes:

• “Problem Solving” lessons, called Reasoning, that focus on MP.2, as well as lessons, called Constructing Arguments and Critique Reasoning, that focus on MP.3

• Topic and Cumulative/Benchmark Assessment items that assess MP.2 and MP.3





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The Bar Diagrams section of the handbook provides comprehensive explanations and examples of the various bar diagrams students will use throughout the year. This collection of examples supports students as they learn how to use mathematical reasoning and model with math to show how the quantities in a problem are related. Page 13 of the Grade 4 Math Practices and Problem Solving Handbook is pictured at the right.

Further, the Teacher's Edition contains comprehensive notes to support teachers as they work through the handbook with students. Example:

From Grade 4 Teacher’s Edition, p. F46

From Grade 4 Math Practices and Problem Solving Handbook, p. 13





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

“Problem Solving” 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 Thinking Habits questions for that math practice and models thinking related to that math practice when solving a rich problem. (See Grade 4 Teacher’s Edition, pages 369A–372B.)

The “Problem Solving” lessons provide practice with the kinds of rich problems students will encounter on high-stakes performance-based assessments. (See Grade 4 Teacher’s Edition, pages 369A–372B.)

The Thinking Habits provided in the Math Practices and Problem Solving Handbook as well as on the Solve & Share page of “Problem Solving” lessons model language and help student engage with the math practice. For example,

• What do the numbers and symbols in the problem mean?

• How are the numbers or quantities related?

• How can I show a word problem using pictures, numbers, or equations?

Build Mathematical Literacy Lesson Masters These masters provide support to help students read and understand a problem from the lesson. The support is given in a variety of ways to enhance a student’s ability to comprehend the kind of text and visual displays in a math problem.

My Word Cards Write-on vocabulary cards are provided online at PearsonRealize.com. Students use information on the front of the card to complete the back of the card. Additional activities are provided with online with the cards.

Vocabulary Activities The Teacher’s Edition provides vocabulary activities at the start of Topics. (See Grade 4 Teacher's Edition, page 330.) These include activities for vocabulary in My Word Cards and/or activities for vocabulary in Review What You Know. Some of the activities use graphic organizers in the Teacher’s Resource Masters.

Vocabulary in Reteach to Build Understanding Vocabulary support is featured in Grades 1–5 at the start of each Reteach to Build Understanding lesson master.

Vocabulary and Writing in Lessons New words are introduced in the Visual Learning Bridge and, in Grades 1–5, are highlighted in yellow. Lesson practice includes questions to reinforce understanding of vocabulary used in the Topic. In addition, students are often asked to do writing in math in lesson practice to explain their thinking.

Vocabulary Review At the end of each Topic is a page of Vocabulary Review. (See Grade 4, page 374.) It includes questions to reinforce





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understanding of the vocabulary used in the Topic and asks students to use vocabulary in writing. Additional activities in the Teacher’s Edition reinforce oral language and writing in math.

Animated Glossary An animated glossary is always available to students and teachers at PearsonRealize.com. Students can click on a word to hear the word and the definition read aloud.

Vocabulary Game Online The Game Center at PearsonRealize.com includes a vocabulary game that students can access anytime.

Problem-Solving Leveled Reading Mat and Activity Another component that helps students understand the language of mathematics is the Problem-Solving Leveled Reading Mat. One mat is provided for each Topic in the Quick-and-Easy Centers Kit for Differentiated Instruction. One side of the mat has on-level reading and the other side has below-level reading. A Problem-Solving Reading Activity master is provided for two lessons in a Topic. The activity has problems that use a context similar to the context on the mat.

Interactive Math Stories, at Grades K–2 Each Topic begins with an Interactive Math Story which exposes students to math concepts and related mathematical language. It is available as an online story, as an animated story, and as a color-in, take-home story in the Teacher’s Resource Masters.

Alignment Criterion 2: Materials must demonstrate authentic connections between content Standards and practice Standards.

Strengths / Weaknesses

enVision® Mathematics 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.

enVision® Mathematics identifies Thinking Habits questions for each math practice. Math practices are infused and discussed on a daily basis starting from Day 1. Also, within each Topic, a special “Problem Solving” lesson is provided as an opportunity to focus on specific math practices.

Specially flagged comments and problems in all lessons focus on specific math practices. Math practices are assessed along with the content Standards, and rubrics are provided for assessing students’ overall proficiency with each math practice.





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Alignment Criterion 3 Access to the Standards for All Students Alignment Criterion 3: Materials must provide supports for English Language Learners and other special populations. 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 includes a Lesson Language Objective, which indicates a way that students can demonstrate their understanding of the math content. Instruction suggestions are given for the Solve & Share and the Visual Learning Bridge (or Convince Me!) to ensure students access the key ideas of the lesson before beginning the Independent Practice and Problem Solving. This leveled instruction includes suggestions for students at Entering, Emerging, Developing, Expanding, and Bridging levels of English language proficiency aligned with WIDA (World-Class Instructional Design and Assessment). Examples:

From Grade 1 Teacher’s Edition, p. 90 From Grade 1 Teacher’s Edition, p. 170





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Language Support Online provides online Academic Vocabulary Activities for 40 words at Grades K–2 and 40 words at Grades 3–5 that are useful for communicating ideas in all content areas. In addition, an online Lesson Self-Assessment (Grades 3–5) encourages students to reflect on their understanding of the language and math goals of the lesson.

Language Support Handbook provides topic- and lesson-level instructional support that promotes both receptive and productive language development for all students. It includes teaching support for Academic Vocabulary, Lesson Self-Assessment Recording Sheets, and more.

Visual Learning Animation Plus provides motion and sound to help lower language barriers to learning and to enhance comprehension. Text for questions that are read aloud appears on-screen to help English language learners connect oral and written language.

Visual Learning Bridge includes visual models to convey math concepts and give meaning to math language. Instruction is stepped out to visually organize important ideas. Teachers may want to use the Student Edition eText to display the Visual Learning Bridge after using the Visual Learning Animation Plus to help ELL students make visual/verbal connections. Each Visual Learning Bridge is followed by a Convince Me! feature that provides an opportunity for students to practice verbalizing their understanding of the concept or skill just learned.

My Word Cards Vocabulary cards with illustrated definitions of key math-related words in each Topic are provided online at Pearsonrealize.com.

Animated Glossary is always available to students and teachers while using digital resources. The glossary (in English and Spanish) contains the math vocabulary highlighted in each Topic.

Pictures with a purpose appear in lesson practice to help communicate information related to math concepts or to real-world problems. Teachers may want to display the Interactive Student Edition pages so they can point to specific pictures or words on the pages when discussing the practice.

ELL Consultants, Janice Corona from Dallas, Texas, and Jim Cummins from Toronto, Canada, have ensured quality ELL instruction.





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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

enVision® Mathematics meets the needs of all students and provides Error Intervention and/or Response to Intervention during each lesson, at the end of each lesson, at the end of each Topic, and anytime as indicated.

ONGOING INTERVENTION, DURING THE LESSON

• Prevent Misconceptions during the Visual Learning Bridge includes a remediation strategy to address a common misconception about the lesson concept.

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

• Reteaching Set is provided before Independent Practice to ensure understanding prior to practice.

• Learning Aids Online in Practice Buddy Interactive online practice is provided at Grades K–5. At Grades 3–5, Practice Buddy Online includes online personalized practice for Independent Practice and Problem Solving. Practice Buddy Online at Grades 3–5 is autoscored with on-screen help including 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.

• Higher Order Thinking problems in lesson practice challenge students’ thinking.

STRATEGIC INTERVENTION, AT THE END OF THE LESSON

• Quick Check items in each lesson (identified by a pink checkmark in the Teacher’s Edition) can be used to help prescribe differentiated instruction during Step 3 of the lesson. Also, Online Quick Check assesses the lesson using five online, machine-scored items.

• Intervention Activity supports teachers working with small groups of struggling students.

• Reteach to Build Understanding provides guided reteaching of the lesson.





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• Build Math Literacy provides support to help students read and understand a problem from the lesson.

• Enrichment provides activities and extensions that challenge on-level and advanced learners.

• Pick a Project offers students the opportunity to pick from a choice of three or four projects related to the Topic. Early finishers might complete more than one project. Extension suggestions are included for each project.

• enVision® STEM Activity for on-level and advanced learners relates to the Topic’s STEM theme.

• Problem-Solving Leveled Reading Mat and Activity includes a big, beautiful, data-filled mat for each Topic and a Problem-Solving Reading Activity with lesson-related problems to solve using the data on the mat. One side of the mat has on-level reading and the other side has below-level reading. The accompanying Activity Guide contains suggestions for using the mat.

• 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.

• Leveled Additional Practice offers two pages of practice for every lesson. It begins with Another Look!, an example to remind students and parents about important ideas in the lesson. Practice items focus on skills (often with scaffolding) and on problem solving that includes reinforcement of math practices, vocabulary, Higher Order Thinking, and Assessment Practice. Leveled assignments for Additional Practice are given in the Teacher’s Edition. They provide for Intervention, On-level, and Advanced students.

• Learning Aids Online in Practice Buddy, at Grades K–5, provides online interactive practice for use with Additional Practice for every lesson. At Grades 3–5, Practice Buddy Online is also used with Independent Practice and Problem Solving. Practice Buddy Online, at Grades 3–5, is autoscored with on-screen help, including 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.

INTENSIVE INTERVENTION, AS NEEDED ANYTIME

• Steps to Fluency Success offer a variety of print and digital resources to help ensure all students achieve fluency success.

• Math Diagnosis and Intervention System includes two-page lessons that provide guided instruction and practice. (See more about the Math Diagnosis and Intervention System on the next page.)

• Visual Learning Animation Plus is a step-by-step visual learning animation that is used in each lesson to connect the Problem-Based Learning Solve & Share to the Guided Practice in a visual way. This animation can be used anytime to refresh understanding.





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

• Learning Aids Online Practice Buddy (described earlier on page 70)

• ExamView® provides unlimited exercises for additional practice.

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

• Diagnosis is provided by the diagnostic tests in the system. Diagnosis is also provided by the Item Analysis charts given with program assessments at the start of a grade or Topic, or at the end of a Topic, a group of Topics, or the year.

• Intervention Lessons include guided instruction followed by practice. Lessons that are below grade level can be assigned, if needed.

• 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 Edition lessons.

AUTO-ASSIGNED DIFFERENTIATION AT PEARSONREALIZE.COM

• Differentiation After a Lesson is based on results of online Quick Checks. For remediation, Another Look! Video is assigned at Grades K–2, and Practice Buddy for the Additional Practice is assigned at Grades 3–5. For enrichment, the Enrichment master is assigned.

• Differentiation After a Topic is based on results of online Topic Assessments. For remediation, one or more Visual Learning Animations Plus are assigned. For enrichment, a game is assigned.

• Differentiation After a Group of Topics is based on results of an online Cumulative/Benchmark Assessment. For remediation, one or more Another Look! Videos are assigned. For enrichment, a game is assigned.

• Assignment Reports show the status of assigned resources.

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





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Metric

AC Metric 3C: Design of lessons recommends and facilitates a mix of instructional approaches for a variety of learners such as using multiple representations (e.g., including models, using a range of questions, checking for understanding, flexible grouping, pair-share).

Evidence

Each lesson is carefully designed to use a variety of instructional approaches, grouping arrangements, models and tools, and learning modalities in order to engage a wide range of learners and facilitate their understanding. The Solve & Share activity— the Problem-Based Learning Step 1 of each lesson—exemplifies this design approach. Students frequently use models and tools to represent their thinking. The teacher can use the guiding questions and prompts (provided in blue font) in the Before, During, and After sidebars in order to support and promote mathematical discourse.





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• The Before section helps teachers present the problem that students will solve using their prior knowledge. In a whole-class discussion, the guiding questions ensure that all students understand the problem well enough to begin the task.

• The During section helps teachers focus their observations of students as they work on the problem (typically in pairs or small groups) and offers guiding questions to help teachers prompt students who might be struggling.

• The After section helps teachers lead a whole-class discussion in which students share and discuss their solutions and teachers share and analyze sample student work. An extension activity is also provided for teachers to use with early finishers. Guiding questions transition students to the Visual Learning Bridge, where new math ideas are made explicit.

From Grade 4 Teacher’s Edition, p. 333





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Other enVision® Mathematics program features that show how the design of the lessons facilitates a mix of instructional approaches for a variety of learners:

• Guiding questions in a blue font are provided for the teacher in each lesson throughout the Teacher’s Edition to support classroom discourse.

• The Convince Me! section that follows the Visual Learning Bridge serves as a way for teachers to check for student understanding (formative assessment) and connect to the Essential Understanding of the lesson and to math practices.

• A variety of activities for differentiation are provided. In these activities, 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. Examples of the differentiation options include review, intervention, and enrichment activities; projects; 3-ACT Math activities; Problem-Solving Leveled Reading Mats; and games.

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

enVision® Mathematics provides the appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners. Support for English Language Learners is provided with every lesson to help students demonstrate mathematical proficiency with the Standards.





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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 enVision® Mathematics.

STEP 1 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 between their prior knowledge and the new concept or procedure. The process of making connections between ideas builds understanding. In enVision®

Mathematics this problem-solving experience is called Solve & Share.

During the Solve & Share, all students actively participate in their own learning and in the learning of their classmates. Also, students learn by sharing their work and by analyzing the work of others in the Solve & Share. Guiding questions are provided to help the teacher facilitate a productive discussion among classmates so that learning is a collaborative activity.





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From Grade 1 Teacher’s Edition Lesson 4-7, p. 185





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STEP 2 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 & Share 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 always begins by connecting the students’ thinking and solutions for the Solve & Share to the new ideas of the lesson.

Every lesson in enVision® Mathematics provides a worked-out problem in the student materials. The problem introduces the important mathematics of the lesson through a series of visual illustrations that promote understanding. This is called the Visual Learning Bridge. It is a bridge because it connects the students’ thinking and solutions from the Solve & Share 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 during the Solve & Share; the important mathematics is made explicit in the student materials (i.e., in the Visual Learning Bridge).

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 enVision® Mathematics is available as an online Visual Learning Animation Plus, thus facilitating teacher-led classroom conversations as well as home access to lesson content. Further, every frame within each Visual Learning Bridge is accompanied in the teacher materials by high-cognitive-level questions for teachers to use with students.

Active participation by all students in their own learning and in the learning of their classmates is an integral part of the enVision® Mathematics program. For example, in every lesson, the Visual Learning Bridge is followed by Convince Me! (formative and collaborative learning) and Guided Practice (active learning).





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From Grade 1 Teacher’s Edition Lesson 4-7, p. 186





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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 Criterion does not require that the problems and exercises be labeled as such.

Evidence

Each problem or exercise has a purpose. The following illustrate how students are engaged in solving problems and working exercises in every lesson in enVision® Mathematics, using Grade 4 as an example.

Step 1 Problem-Based Learning for each lesson provides the Solve & Share, which is designed to engage students with a problem in which new math ideas are embedded. For example, see Teacher’s Edition Lesson 9-6, page 353. Coherence is facilitated as students connect prior knowledge to the new math ideas. Students solve the problem in any way they choose. They are given time to struggle and, as students think, conceptual understandings emerge. The Solve & Share online utilizes the DrawPad to enable students to write and share their solutions on-screen.

Step 2 Visual Learning Bridge of each lesson, or the online version—Visual Learning Animation Plus, increases the cognitive level of instruction by connecting concrete and pictorial representations to abstract symbols. The questions provided in the Teacher’s Edition ask students to reflect on the work that is shown, make connections among ideas, and justify the steps. The Convince Me! feature in each lesson fosters communication during visual learning. (For example, see Teacher’s Edition Lesson 9-6, page 354.)

Guided Practice, in every lesson, is an opportunity for students to check their understanding of lesson concepts before they begin Independent Practice. Independent Practice, in every lesson, gives students an opportunity to build proficiency as they work on their own. (For example, see Teacher’s Edition Lesson 9-6, page 355.)

Problem Solving, in every lesson, offers varied rich problems that support students’ development as mathematical problem solvers. (For example, see Teacher’s Edition Lesson 9-6, page 356.)

Students communicate often about their thinking and work, using reasoning, during Step 3 Assess and Differentiate of each lesson. They utilize the Technology Center (including Math Games and Digital Math Tools Activities) as well as Activity Centers (including Problem-Solving Leveled Reading Mats, Pick a Project, enVision® STEM Activities. (For examples, see Teacher’s Edition Lesson 9-2, pages 340A–340B; Lesson 9-6, pages 356A–356B; Lesson 9-4, pages 348A–348B.)





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Indicator 3

Design of assignments is not haphazard: exercises are given in intentional sequences in order to strengthen students’ mathematical understanding.

Evidence

In enVision® Mathematics, the lesson structure is consistent and intentional. The following examples demonstrate how exercises are given in intentional sequences in order to strengthen students’ mathematical understanding.

Guided Practice provides a formative assessment opportunity for students to check their understanding of lesson concepts before they begin Independent Practice. It is accompanied by Error Intervention in the Teacher’s Edition which identifies common errors and provides remediation.

Independent Practice and Problem Solving always follows Guided Practice so that students can build proficiency as they work on their own. Leveled practice with scaffolding is included at times in Grades 3–5. Exercises are varied and rich throughout. They include vocabulary questions to build understanding, number sense problems, and Higher Order Thinking problems for even more rigor. Exercises that allow for a focus on a particular Math Practice are indicated in red font. enVision® STEM problems are tied to the enVision® STEM Topic theme. Assessment Practice items are provided to prepare students for high-stakes tests.

Practice Buddy Online at Grades K–5 provides online interactive practice for use with Additional Practice. At Grades 3-5, Practice Buddy Online is also used with Independent Practice and Problem Solving. Practice Buddy Online at Grades 3–5 is auto-scored with on-screen help including Help Me Solve This and View an Example.

Additional Practice Workbook provides two pages of practice for every lesson. Exercises focus on skills (often with scaffolding) and on problem solving that includes reinforcement of math practices, vocabulary, Higher Order Thinking, and Assessment Practice. For each lesson, it includes Another Look!, which provides an additional example to remind students and parents about important ideas in the lesson. Leveled Assignments in the Teacher’s Edition are provided for Intervention, On-Level, and Advanced students. (For example, see Grade 4 Teacher’s Edition page 356B.)

Another Look! Video provides online help from an engaging video that presents an example similar to the one in Another Look! in the Additional Practice Workbook.

Adaptive Practice at Grades 3–5 delivers just-right math content to each student. Each online assignment is created to fit the student’s learning needs and includes on-level work, as well as instruction and practice of prerequisite skills where the student demonstrates extra help would be beneficial.





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Indicator 4

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 among students.

Evidence

Teachers are provided with a wealth of resources to support their implementation of enVision® Mathematics. Program-, Topic-, and Lesson-level Professional Development support is available 24–7 online at www.pearsonrealize.com.

For example, author-hosted Professional Development Classroom Videos explain and support each part of the lesson. (Teachers can use the online menu, shown at the right, to access those videos.)

Each Topic in each Grade includes a Professional Development Topic Overview Video in which one of the program authors highlights and gives helpful perspectives on important mathematics concepts and skills in the Topic.

Further, each Topic in enVision® Mathematics begins with comprehensive materials to help teachers understand the organizing concepts and the mathematics in the Topic. There are Math Background pages related to Focus, Coherence, and Rigor, as well as a summary of Math Practices and Effective Teaching Practices (ETP) for each Topic. (For example, see Grade 4 Teacher’s Edition, pages 165E–165H.)



http://www.pearsonrealize.com/



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A Listen and Look For Lesson Video, provided for most lessons, discusses the lesson’s essential understanding and then takes an in-depth look at a key problem students will solve using the targeted CCSS. The video tells the teacher what to look for in student work and discussions, and it suggests ways to help students who may struggle. (Look online for the sample icon shown at the top right.)

The Listen and Look For Lesson Videos are also cross-referenced in the Teacher’s Edition, on the first page of a lesson. The icon shown below, which appears on the first page of a lesson in the Teacher’s Edition, reminds teachers to view the video before launching the lesson.

Extensive teaching notes support every part of each daily lesson, all student resources, and all of the assessments in each Topic. Comprehensive teacher notes accompany every page of the student lesson, topic resources, and assessments. These notes include blue guiding questions to help teachers stimulate classroom discourse, as well as suggestions for incorporating math practices, preventing misconceptions, supporting coherence, providing error intervention, and analyzing student work. Each lesson also includes comprehensive print and digital assessment and differentiation resources to help teachers keep all students engaged and on track. (For example, see Grade 1 Teacher’s Edition Lesson 4-7, pages 185A–188B; pages 197–198.)

An online Listen and Look For Lesson Video reference

From Grade 1 Teacher’s Edition Lesson 1-1, p. 5A





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Indicator 5

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

Evidence

Manipulatives, also called “math tools” in enVision® Mathematics, are provided to support student learning and to correspond to the math tools that are commonly described and depicted in lesson activities. Manipulatives are available as individual manipulative kits, basic classroom manipulatives kits, and as Teaching Tools, which are paper versions of the manipulatives. Teaching Tools can be printed from blackline masters available in Volume 2 of each grade’s Teacher’s Resource Masters component. In addition, there are online digital manipulatives called Math Tools available on www.pearsonrealize.com. (See an example of a digital fraction manipulative below.)

Screenshots of the Math Tools that are available online at PearsonRealize.com





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The connections between mathematical objects and written methods appear throughout enVision® Mathematics, especially in the Solve & Share and Visual Learning Bridge parts of each lesson. Here are some examples from Grade 1 and Grade 4:







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Indicator 6

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

Evidence

enVision® Mathematics provides students with multiple ways to show what they have learned.

DIAGNOSTIC ASSESSMENT

• At the start of the YEAR – Readiness Test online at PearsonRealize.com – Readiness Test Masters in Assessment Sourcebook

• At the start of a TOPIC – Diagnostic Test Masters in Math Diagnosis and Intervention System – Review What You Know in Student Edition or eText

FORMATIVE ASSESSMENT

• During a LESSON – Solve & Share Observation Tool in Teaching Tools section of the Teacher’s Resource Masters

– Realize Scout Observational Assessment – Questions in the Visual Learning Animation Plus online at PearsonRealize.com – Questions to use with the Visual Learning Bridge in Teacher’s Edition or eText – Convince Me! (Grade K) in Student Edition or eText – Convince Me! (Grades 1–5) in Teacher’s Edition or eText – Guided Practice in Student Edition or eText

• At the end of a LESSON – Quick Check in Teacher’s Edition or eText – Quick Check online at PearsonRealize.com

– Lesson Self-Assessment (Grades 3–5) online at PearsonRealize.com – Self-Assessment Tool in Teaching Tools section of the Teacher’s Resource Masters





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SUMMATIVE ASSESSMENT

• At the end of a TOPIC – Topic Assessment Practice in Student Edition or eText – Topic Assessment Masters in Assessment Sourcebook – Topic Assessment online at PearsonRealize.com – Topic Assessment by ExamView® Test Generator – Topic Performance Task in Student Edition or eText – Topic Performance Task Masters in Assessment Sourcebook – Fluency Practice/Assessment Masters in Teacher’s Resource Masters – Fluency Assessment by ExamView® – Fluency Assessment by Practice Buddy Online (Grades 3–5) – Basic-Facts Timed Test Masters (Grades 1–5) in Assessment Sourcebook • After a group of TOPICS – Cumulative/Benchmark Assessment online at PearsonRealize.com – Cumulative/Benchmark Assessment Masters in Assessment Sourcebook • At the end of the YEAR – End-of-Year Assessment online at PearsonRealize.com – End-of-Year Assessment Masters in Assessment Sourcebook – Progress Monitoring Assessment Forms A, B, and C at PearsonRealize.com

– Progress Monitoring Assessment Forms A, B, and C in Assessment Sourcebook

SELF-ASSESSMENT

– My Fluency Progress online at PearsonRealize.com – My Fluency Progress in Teaching Tools section of the Teacher’s Resource Masters – Self-Assessment Tool online at PearsonRealize.com – Self-Assessment Tool in Teaching Tools section of the Teacher’s Resource Masters





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Indicator 7

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

Evidence

enVision® Mathematics provides comprehensive assessment resources for interpreting student performance.

• Online assessments are auto-scored and the 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 Rubrics are provided in the Teacher’s Edition section for the Math Practices and Problem Solving Handbook. These can be used at any time.

• Scoring guides/rubrics/answer keys (including some sample student responses) for assessments are given in the Teacher’s Edition. Answer keys and some sample student responses are provided in the Assessment Sourcebook.

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

From Grade 4 Teacher’s Edition Topic 9, p. 379-380

• Rubric to interpret student-written responses on assessments, available online, can be used for any assessment that includes student-written responses.





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Indicator 8

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

Evidence

Assessments in enVision® Mathematics use appropriate grade-level language and are presented in ways that are consistent with the work students did in the lessons. Note that at any time, teachers can customize the online assessments to address the needs of specific students or the entire class.

The formats of the assessment items, which prepare students for high-stakes 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

• Matching or table completion

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

• Performance tasks

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 instruction and differentiation.

Every lesson includes Common Core practice items in formats that help prepare students for high-stakes tests. “Problem Solving” lessons include opportunities for students to engage specific math practices as they solve problems in performance-based assessments.





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Indicator 9

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, many of whom were involved in the writing of the CCSSM and Progressions, were directly responsible for the creation of enVision® Mathematics. They are experienced elementary math educators, nationally recognized math education leaders, and experts in key math education issues. The program authors are shown below and also listed at the beginning of the core program components (e.g., Student Edition, Teacher’s Edition, and Teacher’s Edition Program Overview). The authors provide their perspective on the program goals and organization, as well as the instructional model, on pages 24–27 of the Teacher’s Edition Program Overview.

AUTHORS

Randall I. Charles Professor Emeritus Department of Mathematics San Jose State University San Jose, California Jennifer Bay-Williams Professor of Mathematics Education College of Education and Human Development University of Louisville Louisville, Kentucky Robert Q. Berry, III Professor of Mathematics Education Department of Curriculum, Instruction and Special Education University of Virginia Charlottesville, Virginia Janet H. Caldwell Professor Emerita Department of Mathematics Rowan University Glassboro, New Jersey

Zachary Champagne Assistant in Research Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) Jacksonville, Florida Juanita Copley Professor Emerita College of Education University of Houston Houston, Texas Warren Crown Professor Emeritus of Mathematics Education Graduate School of Education Rutgers University New Brunswick, New Jersey Francis (Skip) Fennell Professor Emeritus of Education and Graduate and Professional Studies McDaniel College Westminster, Maryland

Karen Karp Professor of Mathematics Education School of Education Johns Hopkins University Baltimore, Maryland Stuart J. Murphy Visual Learning Specialist Boston, Massachusetts Jane F. Schielack Professor Emerita Department of Mathematics Texas A&M University College Station, Texas Jennifer M. Suh Associate Professor for Mathematics Education George Mason University Fairfax, Virginia Jonathan A. Wray Mathematics Supervisor Howard County Public Schools Ellicott City, Maryland





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Reviewers for enVision® Mathematics:

Katina Arnold Teacher Liberty Public School District Kansas City, Missouri

Andrea Hamilton K–5 Math Specialist Lake Forest School District Felton, Delaware

Jamie Pickett Teacher Platte-County School District Kansas City, Missouri

Christy Bennett Elementary Math and Science Specialist DeSoto County Schools Hernando, Mississippi

Susan Hankins Instructional Coach Tupelo Public School District Tupelo, Mississippi

Mandy Schall Math Coach Allegany County Public Schools Cumberland, Maryland

Shauna Bostick Elementary Math Specialist Lee County School District Tupelo, Mississippi

Barb Jamison Teacher Excelsior Springs School District Excelsior Springs, Missouri

Marjorie Stevens Math Consultant Utica Community Schools Shelby Township, Michigan

Samantha Brant Teacher Platte County School District Platte City, Missouri

Pam Jones Elementary Math Coach Lake Region School District Bridgton, Maine

Shyree Stevenson ELL Coach Penns Grove-Carneys Point Regional School District Penns Grove, New Jersey

Jamie Clark Elementary Math Coach Allegany County Public Schools Cumberland, Maryland

Sherri Kane Secondary Mathematics Curriculum Specialist Lee’s Summit R7 School District Lee’s Summit, Missouri

Kayla Stone Teacher Excelsior Springs School District Excelsior Springs, Missouri

Shauna Gardner Math and Science Instructional Coach DeSoto County Schools Hernando, Mississippi

Jessica Leonard ESOL Teacher Volusia County Schools DeLand, Florida

Sara Sultan PD Academic Trainer, Math Tucson Unified School District Tucson, Arizona

Kathy Graham Educational Consultant Twin Falls, Idaho

Jill K. Milton Elementary Math Coordinator Norwood Public Schools Norwood, Massachusetts

Angela Waltrup Elementary Math Content Specialist Washington County Public Schools Hagerstown, Maryland





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Mathematician Reviewers for enVision® Mathematics: Roger Howe Professor of Mathematics Yale University New Haven, Connecticut

Gary Lippman Professor of Mathematics and Computer Science California State University, East Bay Hayward, California

ELL Consultants for enVision® Mathematics:

Janice R. Corona Independent Education Consultant Dallas, Texas

Jim Cummins Professor The University of Toronto Toronto, Canada





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Indicator 10

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

Evidence

The visual design in enVision® Mathematics engages students and supports their learning using a wide variety of clear, well-organized elements and features that are easy for students to interpret and understand. The following are descriptions of features found in the Student Edition and the Interactive Student Edition.

A "community" of relatable, diverse, math-loving avatars recur throughout the program to present problems, share tips, and provide explanations. The avatars remind students about the math practices and “habits of mind.” The avatars support students, promoting success, by demonstrating that thinking mathematically is fun and engaging.

From Grade 1 Student Edition Lesson 4-9, p. 195

All art is visually pleasing, engaging, and functional. A clean, uncluttered page reduces students’ cognitive load and enables them to focus on the mathematics. See the example at the right from Grade 1.






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There is a clear, consistent layout for all lessons. For example, for all grades, each lesson begins with the Solve & Share, which is designed for a consistent visual experience. In Grades 1 and 2, for example, the avatar always presents the problem in a speech bubble and students always show their solutions in a designated work area that is typically an open workspace. This predictable presentation enables students to immediately begin working on the Solve & Share rather than trying to figure out the page. See the example from Grade 1 shown below.


The Visual Learning Bridge presents new information in sequential frames. This enables teachers and students to break down learning into manageable chunks so that students can process and incorporate new information. Every Visual Learning Bridge in enVision®

Mathematics is available as an online Visual Learning Animation Plus, thus engaging students in the motion of the learning experience and facilitating teacher-led classroom conversations. Also, every frame within each Visual Learning Bridge is accompanied in the teacher materials by high-cognitive-level questions for teachers to use with students. See the example from Grade 1 shown on the next page





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From Grade 1, Student Edition Lesson 1-2, p. 10

Bar diagrams, place-value charts, and other visual tools support conceptual development from concrete to pictorial to symbolic thinking. These visual tools appear throughout the grades so that using them becomes second nature to students. See the examples below.



Text is interspersed with illustrations and diagrams that provide visual cues to support understanding. Avatars coach students and model using the manipulatives that students use in the lesson. Key vocabulary is highlighted in yellow. Color is used to support student understanding, often connecting real-world objects to the manipulatives shown, or to draw attention to certain numbers in equations.





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The online Interactive Student Edition has an intuitive user interface. Students find it easy to point and click to navigate from lesson to lesson and from one lesson activity to another. See the sample menu, shown at the right. The online digital tools are "universal" and easy to use, especially for digital natives. Students have access to a drawing tool, manipulatives, a camera and a typing tool. The tools can motivate students who typically find it challenging to write or draw to explain their thinking. Other components in enVision® Mathematics that demonstrate how the visual design of the program supports student engagement and the full range of learners and learning styles: Interactive Math Stories invite students to write and color to complete their own books (1 per Topic, at Grades K–2). See the Grade 1 Topic 5 example shown below, at the far left. My Word Cards are illustrated vocabulary cards that give students something visual and tangible to study. For example, see the front and back of a card (purple frame) from Grade 1 Topic 1 below, center. Pick a Project engages students with interesting photos and provocative questions that inspire students to research, create, design, or plan something related to a real-world context. 3-ACT Math sparks students’ interest with a compelling real-world problem situation presented with a video. Students step through a consistent activity format, and they are provided with a recording sheet that helps them organize their thinking process. Problem-Solving Reading Mats are visually stunning presentations of data related to a single Topic or theme, which students use to solve data-related problems. See the Grade 1 Topic 5 example (blue frame) below, at the far right.





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Another Look! Videos, available at PearsonRealize.com, also engage students and support the visual design. Indicator 11

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 and the Visual Learning Animation Plus provide useful resources for parents to understand the concepts as they were taught in the lesson. Student assignments are available in the Student Edition as well as online for parents to review. The assignments consist of routine problems as well as rich problems that help students develop their critical thinking skills. Fluency-building activities are included in the materials that get sent home to parents.

The Additional Practice Workbook includes Another Look!, which has a worked-out example to remind students and parents about important ideas in the lesson. The Another Look! Video provides online help with an engaging video that presents an example similar to the one in Another Look! in the Additional Practice Workbook.

In addition, every Topic in every grade also has a Home Connection letter (in English and Spanish) with an activity for parents and families to do with their children to reinforce the concepts and skills being learned in the classroom.

Teaching Tool 2 in the Teacher’s Resource Masters is a “Dear Family” letter that serves as a template for communicating student progress at the end of each Topic.





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APPENDIX: Scope and Sequence for enVision® Mathematics

From the Grade 4 Teacher’s Edition Program Overview, pp. 102–103





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From the Grade 4 Teacher’s Edition Program Overview pp. 112–113





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instructional materials evaluation toolkit (imet)

Documents