Alignment

Focus & Coherence

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  •  12 12
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Rigor & Mathematical Practices

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Meets Expectations
Partially Meets Expectations
Does not Meet Expectations
Usability

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  •  31
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  •  38 37
Meets Expectations
Partially Meets Expectations
Does not Meet Expectations

Alignment

The instructional materials reviewed for Grade 4 are aligned to the CCSSM. The assessments are focused on grade-level standards and the materials spend the majority of the time on the major work of the grade. The materials are coherent. The materials generally follow the progression of the standards and connect the mathematics within the grade level although at times off-grade level content is not identified. There is coherence within units of each grade. The Grade 4 materials include all three aspects of rigor, and there is a balance of the aspects of rigor. The MPs are used to enrich the learning, but additional teacher assistance in engaging students in constructing viable arguments and analyzing the arguments of others is needed. Overall, the materials are focused, follow a coherent plan, balance the aspects of rigor, and attend to the MPs.



GATEWAY ONE

Focus & Coherence

MEETS EXPECTATIONS

The materials reviewed for Grade 4 meet the expectations for Gateway 1. The materials do not assess above-grade-level content. Within the materials, there is enough time devoted to the major work of the grade. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are partially consistent with the mathematical progressions in the standards, and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the Grade 4 materials are focused and follow a coherent plan.

Focus

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  •  6 6

The materials for Grade 4 meet the expectations for focus. The curriculum is focused by not assessing future, grade-level content in summative assessments. The instructional time is spent on the major clusters of the grade the majority of the time; the focus on major clusters of the grade is approximately 68 percent of the instructional time.

Criterion 1a

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  •  2 2

Materials do not assess topics before the grade level in which the topic should be introduced.

The instructional materials reviewed for Grade 4 meet the expectations for assessing grade-level content. Overall, no above-grade-level content was assessed within the summative assessments provided. Summative assessments considered during the review for this indicator include unit post-assessments and Number Corner assessments that require mastery of a skill.

2/2
Indicator 1a

The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Grade 4 meet the expectations for focus within assessment. Overall, the instructional material does not assess any content from future grades within the summative assessments.

  • No above-grade-level content was assessed on summative assessments.
  • The summative assessments focus on grade-level or previous grade-level topics.

Summative assessment items reviewed did assess material in alignment with the Grade 4 content standards:

  • Unit 2 assessments include multi-digit multiplication written both vertically and horizontally; however, there is no requirement of the standard algorithm.
  • In alignment with the NF domain for Grade 4, within the fractions units (units 3 and 6), all items on the post-assessment use denominators limited to 2, 3, 4, 5, 6, 8, 10, 12 and 100. Addition and subtraction of fractions is only with like-denominators.
  • In unit 6, multi-digit multiplication and division, no assessment items require the standard algorithm; most items allow for student choice of solution strategy. For example, on the post-assessment, item 8 says, “There are 4 friends going to a concert. They are paying a group rate of $176. How much does each friend need to pay? Show your work.”

Criterion 1b

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  •  4 4

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials reviewed for Grade 4 meet the expectations for focus on the major clusters of each grade. Students and teachers using the materials as designated will devote the majority of class time to major clusters of the grade.

4/4
Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Grade 4 meet the expectations for focus. The instructional materials do allocate the majority of class time to the major work of the grade. All sessions (lessons), except summative and pre-assessment sessions, were counted and assigned 60 minutes of time. Number Corner activities were counted and assigned 20 minutes of time. When sessions or Number Corner activities focused on supporting clusters and clearly supported major clusters of the grade, they were counted. Reviewers looked individually at each session and Number Corner in order to determine alignment with major clusters and supporting clusters. Standards reported in the teacher materials for sessions and Number Corners were not always found to be accurate or representative of the actual content of the sessions and Number Corners. Reviewers determined standards alignment of the sessions and Number Corner activities based on teacher directions, student activities and work, not standards that the teacher materials claimed. Optional Daily Practice pages and Home Connection pages were not considered for this indicator because they did not appear to be a required component of the sessions.

Some calculations, specifically the number of units and the number of modules (chapters), equaled slightly less than 65 percent of the time spent on major and supporting clusters of the grade. However, looking at the sessions (lessons) and instructional time, when considering both sessions and Number Corners together, the calculations equal more than 65 percent of the time spent on major and supporting clusters of the grade.

  • Units – 5 out of 8 units spend the majority of the time on major work of the grade, which is approximately 63 percent -- Units 2, 3, 4, 6, and 7 spend all or most of the instructional time on major work of the grade. Units 1, 5 and 8 do not spend most of the instructional time on major work of the grade. Unit 1 spends most of the instructional time reviewing standards required for Grade 3 and setting up classroom expectations for Grade 4. There are multiplication review problems using arrays and number lines with one step word problems, reviews of strategies for multiplication facts, and some measurement review which is a Grade 3 supporting standard. Unit 8 is set up as project-based learning, however the content is above Grade 4 standards. In Modules 3 and 4, the students are working with concepts of scale, which is a middle school standard. Unit 5 focuses on Grade 4 content that is not major work of the grade.
  • Modules (chapters) – 20 out of 32 spend the majority of the time on major work of the grade, which is approximately 63 percent. Modules that spend the majority of the time on major work of the grade are: Unit 1, Modules 1 and 3; Unit 2, Modules 1, 2, 3 and 4; Unit 3, Modules 1, 2, 3 and 4; Unit 4, Modules 1 and 2; Unit 6, Modules 1, 2, 3 and 4; and Unit 7, Modules 1, 2, 3 and 4. Modules that spend about half the time on major or supporting clusters of the grade are: Unit 1, Modules 1 and 3; Unit 4, Module 3; and Unit 5, Module 3. There are no modules in Unit 8 that spend the majority of the time on major work of the grade.
  • Sessions (lessons) – 97 out of 147 spend the majority of the time on major work of the grade, which is approximately 66 percent. Approximately 30 percent of the 147 Bridges sessions or 44 sessions address multi-digit multiplication and division (4.NBT.5 and 4.NBT.6), a critical area and major work for Grade 4. For example, 17 out of 18 sessions in Unit 2 address concepts related to multi-digit multiplication and division. In addition, 15 out of 18 sessions in Unit 6 address multi-digit multiplication and division with a focus on strategies and application through solving problems related to perimeter and area, measurement and data, and fractions. Approximately 22 percent of the 147 Bridges sessions or 32 sessions address fractional equivalence and operations with fractions (4.NF), a critical area and major work of Grade 4. For example, all 18 sessions which make up Unit 3 address fraction equivalence, adding and subtracting decimals with like denominators, multiplication of fractions by a whole number, and an introduction to decimals and decimal notation of fractions.
  • Instructional Time, including Bridges sessions and Number Corner activities – 8,160 instructional minutes out of 12,020 instructional minutes are spent on the major work of the grade, which is approximately 68 percent. The Bridges sessions alone equate to 5,820 out of 8,820 instructional minutes devoted to major work, which is approximately 66 percent of session instructional time being spent on the major and supporting clusters of Grade 4. The Number Corner activities reflect 2,340 out of 3,200 instructional minutes devoted to major work, which is approximately 73 percent of Number Corner instructional time being spent on the major and supporting clusters of Grade 4.

Coherence

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The instructional materials reviewed for Grade 4 partially meet the expectations for coherence. The materials use supporting content as a way to continue work with the major work of the grade. The materials provide viable content for a school year, including 160 days of lessons and assessments. The materials are partially consistent with the progressions in the standards, with some above-grade-level content unidentified and interfering with grade-level work. All students are given extensive work on grade-level problems, even students who are struggling, and this work progresses in a mathematically, logical way. Knowledge from prior grades is related to grade-level standards. Connections are sometimes made between domains and clusters within the grade level with some missed opportunities to make connections. Also the materials lack visible learning objectives shaped by CCSSM cluster headings. Overall, the Grade 4 materials partially support coherence and consistency with the progressions in the standards.

Criterion 1c-1f

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Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

2/2
Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Grade 4 meet the expectations for supporting content enhancing focus and coherence simultaneously by engaging students in the major work of the grade. The materials engaged students in major clusters of the grade while focusing on supporting clusters. For Grade 4, reviewers focused on the use of factors and multiples and solving problems involving measurement and conversations in supporting operations with multi-digit whole numbers, fractions, and decimals, in both the daily Bridges Sessions and the Number Corner activities.

  • In Unit 1, Module 1, Session 5, students are working with area models and the relationship between multiplication and division (4.NBT.6) while applying their understanding of the area formula as a multiplication equation with an unknown factor (4.MD.3).
  • In Unit 1, Module 3, Session 4, students are working with multiplicative comparison and writing comparison statements to match multiplication equations (4.OA.1) while applying knowledge of factor pairs and multiples (4.OA.4).
  • In Unit 6, Module 3, Session 1, fractions (of elapsed time), there are solid connections made between 4.MD and 4.NF while collecting and representing data through a diagram, line plot, and number line.
  • In Unit 6, Module 3, Session 2 (Data Analysis), solid connections are made between 4.MD.4 and 4.NF.3 as students use line plots to display measurements (4.MD.4) in fractions and add mixed numbers (4.NF.3).
  • In Unit 7, Module 4, Session 3, students work with story problems involving intervals of time, liquid volume, and weight (4.MD.2) while multiplying 2-digit numbers using strategies based on place value and the properties of operations and using equations and rectangular arrays to explain their strategies (4.NBT.5).
  • In September Number Corner "Calendar Collector," students are making conversions of inches, feet, and yards involving whole numbers and fractions.
  • In October Number Corner "Computational Fluency," students are exploring the relationship between factors and multiples through the use of number lines, while focusing on rounding, estimating, place value, and multi-digit addition and subtraction.
  • In April Number Corner "Solving Problems," students are engaging in solving problems in which they add or subtract mixed fractions while working with line plots in whole and fractional amounts.
2/2
Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Grade 4 meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades. While reviewers note that there are a minimum of 80 minutes of daily instruction required for all the curriculum components to be completed, including sessions and Number Corner activities, the amount of content, specifically the number of days, is viable for one school year:

  • The materials contain 160 sessions (daily lessons) that are evenly spread across eight units of instruction, including assessments.
  • Most Bridges Session are 60 minutes in length for Grade 4, with an additional 20 minutes required for the daily Number Corner activities for a total of 80 minutes (60 minutes for Bridges Sessions and 20 minutes for Number Corner Activities).
  • There are some sessions that are explicitly designed to take two days and are indicated as such in the Teacher's Manual as seen in Unit 1, Module 2, Sessions 5-6.
  • The Number Corner Activities are an essential component of the curriculum emphasizing "opportunities to apply Common Core skills in new settings and real-world contexts."
  • For Number Corners, there are 20 days in September, October, January, February, March, April, May/June; and 15 days in November and December, which equals 170 days of Number Corner activities, including assessments.
  • Number Corner activities are daily 20-minute workouts that introduce, reinforce, and extend skills and concepts related to the critical areas of study at each grade level.
  • While a district, school or teacher would not need to make significant changes to the curriculum scope and sequence, reviewers indicated concerns for the amount of minutes necessary to complete all required components of each daily requirement, including sessions and Number Corners.
  • The materials are structured so that a teacher could make modifications to days or time if necessary.
1/2
Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials for Grade 4 partially meet the expectations for the materials being consistent with the progressions in the standards. Content from prior grades is clearly identified and related to grade-level work, however content from future grades is not clearly identified and is not always clearly related to grade-level work. Materials give all students extensive work with grade-level problems. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

Content from prior grades is clearly identified and relates to grade level work, however content from future grades is not identified and does not always relate to grade-level work:

  • The Grade 3 Operations and Algebraic Thinking Standards including 3.0A.1, 3.OA.3, 3.OA.4, 3.OA.5 and 3.OA.7 are identified for the purposes of review and extension in Unit 1, Module 1, Sessions 1-6 (Models for Multiplication and Division) alongside standards in major clusters 4.OA.A and 4.NBT.B.
  • The Grade 3 Operations and Algebraic Thinking standards including 3.OA.5, 3.OA.7, and 3.OA.9 and Grade 3 Measurement and Data standards 3.MD.7b-3 are identified for the purpose of review and extension in Unit 1, Module 2, Sessions 1 and 3-6 (Primes & Composites), alongside standards in major cluster 4.OA.B.
  • In May Number Corner “Calendar Collector,” 5.G.2 is identified alongside supporting cluster 4.MD.A, and 5.NF.4a is identified alongside standards in major cluster 4.NF.B.
  • There are references to range, which is well above grade-level and not marked or discussed as such, in Unit 8, Module 1, Session 1; Unit 8, Module 1, Session 3; and Unit 8, Module 2, Session 4 (6.SP.B.5.C).
  • There is a large amount of work integral to Unit 8 that is well above grade level and not marked or discussed as such. For example, Module 2, sessions 3, 4, and 5 has students analyzing data for mode (session 2), range (session 1 and session 3), mean and range (session 4), and range and mode (session 5). All of these are above grade-level standards and are not indicated as such. (6.SP.B.5.C).
  • References to scale and scale factor are in Unit 8, Module 3, Sessions 2, 3 and 6, which is a Grade 7 standard. (7.G.A.1). The above grade-level standard is not marked or addressed as above grade level in any way.
  • Content from future grades is not always clearly identified. For example Unit 1, Module 2, session 6 (TE page 34) asks students to "enter numbers on the ratio table" in order for students to show the Double-Double-Double strategy in another format. The above grade level standard (6.RP.A) is not clearly identified as such.

Materials give all students extensive work with grade-level problems:

  • 136 out of 147 Bridges Sessions provide an opportunity for students to engage with grade-level problems through a Problem & Investigation or a Problem String.
  • 104 out of 160 Number Corner Sessions provide an opportunity for students to engage in problem solving through Calendar Grid, Solving Problems and Problem String activities.
  • Suggestions for SUPPORT or CHALLENGE are embedded in some sessions (not consistent) and noted in the Teacher’s Guide. Students are working with grade-level content, with modifications. Examples include Unit 3, Module 2, Session 2, teacher direction 11 makes suggestions for playing "The Last Equation Wins" with modifications to support struggling learners as well as modifications to challenge others (page 14).
  • In addition to the explicit suggestions for support or challenge contained in the sessions as identified in the Differentiation Table, the Problem & Investigations and Problem Strings are "open-ended and lend themselves to differentiated instruction by the nature of the task design." (found on last page of Unit Introduction for all units)
  • Within the sessions, many of the practice pages related to the lessons are listed as "optional Daily Practice" and "optional Home Connections."

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

All grade-level standards are identified in the Skills and Concepts section at the beginning of each session, including prior, grade-level standards.

  • In Unit 1, Module 1, Sessions 1-6 (Models for Multiplication and Division), Grade 3 Operations and Algebraic Thinking Standards, including 3.OA.1, 3.OA.3, 3.OA.4, 3.OA.5 and 3.OA.7, are identified for the purposes of review and extension alongside standards in major clusters 4.OA.A and 4.NBT.B.
  • In Unit 1, Module 2, Sessions 1 and 3-6 (Primes & Composites), Grade 3 Operations and Algebraic Thinking standards, including 3.OA.5, 3.OA.7, and 3.OA.9, and Grade 3 Measurement and Data standards, 3.MD.7.B and 3.MD.7.C are identified for purpose of review and extension alongside standards in major cluster 4.OA.B.
  • In Unit 2, Module 1 (Building Multiplication Arrays), Session 3, 3.MD.7 is identified for the purpose of review and extension alongside standards in supporting cluster 4.MD.A.
  • In Unit 2, Module 1 (Multiplication Arrays), Session 4, 3.OA.4 and 3.OA.7 are identified for the purpose of review and extension alongside standards in major cluster 4.NBT.A and 4.NBT.B.
  • The Module Overview in Unit 3, Module 1, (TM, page 1) states, "Students review fraction skills and concepts from the previous grade and extend their understandings to mixed numbers, improper fractions, and more sophisticated strategies for generating equivalent fractions." The Unit 3 Module Overview is the only unit overview (of eight) to explicitly state connections to previous grade levels.
  • In Unit 3, Module 1 (Equivalent Fractions), Session 1, 3.NF.2 is identified for the purpose of review and extension alongside standards in major cluster 4.NF.B.
  • In Unit 4, Module 4 (Measurement & Data Displays), Session 1, 3.MD.4 is identified for the purpose of review and extension alongside standards in supporting cluster 4.MD.B.
1/2
Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. 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.

The instructional materials reviewed for Grade 4 partially meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards. Overall, the materials do not include objectives that are visibly shaped by CCSSM cluster headings, however the materials do include problems and activities that serve to connect two or more clusters in a domain and two or more domains in a grade.

  • Fair Shares and Fractions & Mixed Numbers, in Unit 3, Module 1, Sessions 2 and 3, could have attended more closely to the cluster heading "Extend understanding of fraction equivalence and ordering."
  • Ordering Fractions and Decimals on a Number Line, in Unit 3, Module 4, Session 3, could have attended more closely to to the cluster heading "Extend understanding of fraction equivalence and ordering."
  • Looking at Data Displays, in Unit 4, Module 4, Session 2, and Analyzing Grass Data, in Unit 8, Module 3, Session 4, could have attended more closely to the cluster heading "Represent and interpret data."
  • Introducing Angles, Benchmark Angles, and Sir Cumference and the Great Knight of Angle, in Unit 5, Module 1, Sessions 2, 3 and 5, could have attended more closely to the cluster heading "Draw and identify lines and angles, and classify shapes by properties of their lines and angles."

There are many instances of problems and activities within the materials that serve to connect two or more clusters in a domain and two or more domains in a grade:

  • Unit 2, Module 2, Sessions 1, 3 and 4 connect 4.OA.3 (solving multistep word problems with whole number using 4 operations) with 4.NBT.5 (multi-digit multiplication strategies).
  • Unit 3, Module 1, Sessions 3, 4 and 5 connect 4.NF.1 (equivalent fraction understanding) with 4.NF.3 (adding, subtracting, and decomposing fractions).
  • Unit 4, Module 2, Sessions 3, 4, and 5 connect 4.NBT.1 (Place value understanding with multi-digit numbers) with 4.NBT.4 (Fluently adding and subtracting multi-digit numbers).
  • Unit 6, Module 2, Sessions 1-5 connect 4.MD.3 (Solving area and perimeter of rectangle problems with formulas) with 4.NBT.6 (Multi-digit division).

GATEWAY TWO

Rigor And Mathematical Practices

MEETS EXPECTATIONS

The materials reviewed for Grade 4 meet the expectations for Gateway 2, Rigor and Mathematical Practices. All three of the aspects of rigor are present and focused on in the materials. There is a balance of the 3 aspects of rigor within the grade, specifically where the Standards set explicit expectations for conceptual understanding, procedural skill and fluency, and application. All eight MPs are included in a way that connects logically to the mathematical content. However, the MPs are not always identified correctly and/or the full meaning of the MPs is sometimes missed. The materials set up opportunities for students to engage in mathematical reasoning and somewhat support teachers in assisting students in reasoning, however there are missed opportunities to assist teachers in supporting students to critique the arguments of others. The materials attend to the specialized language of mathematics and provide explicit instruction in how to communicate mathematical reasoning using words, diagrams and symbols. Overall, the materials for Grade 4 meet the expectations for Rigor and Mathematical Practices.

Rigor and Balance

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  •  0
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  •  5
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  •  8 8

The materials reviewed for Grade 4 meet the expectation for this criterion by providing a balance of all three aspects of rigor throughout the materials. Within the Bridges sessions and Number Corners, key concepts related to the work of the grade are developed with a variety of conceptual questions, different concrete and pictorial representations and student explanations. In Grade 4, fluency and procedural work includes 4.NBT.B.4 (addition and subtraction using the standard algorithm). Application problems occur regularly throughout both the Bridges sessions and the Number Corner activities.

Criterion 2a-2d

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  •  0
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  •  5
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  •  8 8

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

2/2
Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials reviewed for Grade 4 meet the expectations by attending to conceptual understanding. Overall, the instructional materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

  • Problem strings are used in seven sessions during the year to provide a conceptual understanding approach to teaching procedural skills and computational fluency with an emphasis on making connections across representations including number lines, arrays, and equations. Problem Strings address conceptual understanding in Unit 2 (Module 3, Session 3), Unit 4 (Module 1, Session 4), and Unit 6 (Module 1, Sessions 3, 4, 7; Module 3, Session 4; Module 4, Session 1) to address 4.NBT.3, 4.NBT.5, 4.NBT.6.
  • A Math Forum structure is used in nine sessions during the year, which provides "a formal and structured time for students to share and discuss their work." For example, in Unit 6, Module 2, Session 2, Area Challenges Forum, teacher direction 2, (page 10), says, "students share and compare solutions and strategies with classmates other than their partners from the previous session. Did they get the same answers? Can they understand each other's strategies?" Direction 3, says "If a student shares something similar that elevates the level of discussion, model what the student did with sketches, numbers, and word," also, "Invite the rest of the class to ask questions, and have the presenters respond to those questions."
  • The Bridges Introduction pages of the Teacher Manual outline a variety of models that students access throughout the school year in order to demonstrate their understanding.
  • Many representations are used throughout the sessions. For example, Unit 3, Module 1, Sessions 3, 4 and 5 (4.NF.1, 4.NF.2, 4.NF.3), "Fractions and Mixed Numbers" (Session 3) begins with making construction paper fraction strips and using them to investigate equivalent fractions, mixed numbers, and improper fractions. Session 4 moves into "If This Is One Third..." where students use a strip of paper to represent 1/3 of a piece of licorice, and are asked to create a paper strip to represent the whole. In addition, students create thirds, sixths, and twelfths to place on the class number line. In Session 5, "Egg Carton Fractions," students take what they have learned in the previous sessions and apply it to the context of an egg carton model.
  • In Unit 2, Introduction, page vi, includes a heading titled "Using Various Models & Strategies" which explains "Some students will use all of the models to solve problems, and some won't, and that's OK. Keep modeling student strategies, and as students get used to the models and solidify numerical relationships, they will begin to use the models as tools to solve problems."
2/2
Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials reviewed for Grade 4 meet the expectations by attending to procedural skill and fluency. Overall, the instructional materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

  • The Number Corner component of the Bridges curriculum "engages students and contributes to a math-rich, classroom environment that promotes both procedural fluency and conceptual understanding." (Bridges Introduction pages).
    • The Computational Fluency component of Number Corner focuses on "activities, games, and practice pages to help students work efficiently, flexibly, and accurately with numbers." (page v, Number Corner Teacher Manual)
    • Each month's Number Corners contains a Computational Fluency component, for example: "Playing Splat! with a Partner" in September, "Playing Division Capture" in January, and "Decimal Draw" in May. Each month also contains a Problem String component that focuses on computation with a strategy, for example: "Multi-Digit Addition Strategies" in November, "Generating Equivalent Fractions" in March, and "Multiplying Fractions & Whole Numbers" in May.
  • Problem strings are used in Bridges Units 2, 4 and 6 (seven total sessions) during the school year. "The goal is to help students develop more efficient ways of solving a particular kind of problem, based upon the connections they see among the problems in the string." (Teachers Manual Unit 1, Introducing Bridges Mathematics). For example, Unit 2, Module 3, Session 3, "Doubling & Halving," students begin the session with problems that double or half and then move to problems that double and half, resulting in the same product.
  • Attention to the Grade 4 fluency requirement of addition and subtraction of multi-digit whole numbers using the standard algorithm (4.NBT.B.4) includes Unit 4, Module 1, Sessions 5 and 6, "The Standard Algorithm for Multi-Digit Addition (Session 5)," when students work in pairs to solve a 3-digit addition story problem, followed by class work solving a variety of addition problems. In Session 6, "Think Before You Add," students explore which number combinations in multi-digit addition problems lend themselves to particular strategies, including the standard algorithm. Also, in Unit 4, Module 2, Session 3, Problems and Investigation, the activity titled “The Standard Algorithm for Multi-Digit Subtraction” (4.NBT.A.1, 4.NBT.B.4) provides students with an opportunity to solve a 4-digit subtraction story problem after which students share their strategies, and the teacher records each method on a poster. The teacher then provides direct instruction on the standard regrouping algorithm and provides the class with opportunities to practice by solving a variety of subtraction problems.
  • There is a Place Value & Addition Checkpoint assessment in Unit 4, Module 1, Session 7, which assesses the standard algorithm for multi-digit addition.
  • There is a Subtraction Checkpoint in Unit 4, Module 3, Session 1, which assess the standard algorithm for multi-digit subtraction.
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Indicator 2c

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

The instructional materials reviewed for Grade 4 meet the expectations by attending to application. Overall, the instructional materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

  • In the Solving Problems component of the Bridges Number Corner, students spend time working on application problems. "Often the problems connect to another workout in the same month, which enables students to apply skills they learned elsewhere to a problem-solving context." (page vii, Number Corner Volume 1 Introduction). For example, the March Solving Problem activities, “Fractions in the Garden,” “More Garden Fractions,” and “Sharing and Matching,” are a series of problems addressing 4.NF.4.C in the context of planting flowers in one part of a garden.
  • Teachers pose contextual Problem Strings and Problems & Investigations throughout the Bridges curriculum that are grounded in real-world application in which students model, discuss, reason, and defend their thinking.
    • Within the materials, there are opportunities where a Problem & Investigation of a real-world scenario is paired with a Forum for discussion and exploration of the problem-solving strategies, for example, Unit 6, Module 3, Sessions 4 and 5, Problems & Investigations and Math Forum, Present Purchase (4.OA.3 and 4.NBT.6, 4.NF.6). In Session 4, students explore the connections between division, decimals, and fractions by solving problems related to sharing money. In Session 5, students share and discuss the various strategies that they used to solve the Session 4 problem string. Time is provided for students to look for relationships across the various division problems.
    • In Unit 3, Module 2, Sessions 5 and Session 6, Problems and Investigations titled “How Many Candy Bars” (4.NF.B.3.A, 4.NF.B.3.C, 4.NF.B.3.D) provides students an opportunity to engage with a word problem which requires them to add fractions on an open number line and solve the problem to find out how many candy bars independently or in pairs. The word problem is a multi-step and non-routine problem. This is followed by a Math Forum in Session 6 in which specific students chosen by the teacher discuss their strategies and solutions during a class-wide discussion orchestrated by the teacher.
  • Unit 8 is a complete project-based application unit where students are working on designing and building a model playground. Sessions combine measurement data standards with geometry standards, in real-world problem solving tasks and situations.
  • Many student pages and sessions throughout the materials contain application problems aligned to major work of the grade, however nearly all of the multi-step, non-routine problems are at the end of the pages and labeled "Challenge.
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Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials reviewed for Grade 4 meet the expectations for balancing the three aspects of rigor. Overall, within the instructional materials, the three aspects of rigor are not always treated together and are not always treated separately.

  • The Problems & Investigations within the sessions call for students to apply procedural skill & fluency and conceptual understanding to solve application problems. In Unit 2, Module 1, Session 4, students begin by using their multiplication/division skills to play "What's Missing? Bingo" with a partner. Next, the class is posed with Flora's Problem, in which they use base-ten area pieces to model multiplication arrays relating to the given story. Then students further practice their skills through the "More Multiplying by Ten" Home Connection (pages 25-26).
  • Application problems often call for students to model their thinking through the use of open number lines, array or area models, base-ten pieces, ratio tables, geoboards, etc.
  • Procedural skill and fluency is often noted side-by-side as students are working in conceptual models. For example, in the Unit 4, Module 1, Session 4 Problem String, students write equations as well as an open number line to represent their thinking related to the compensation strategy for addition.
  • Problem strings (found in Unit 2, 4 and 6) target procedural skill and fluency by targeting opportunistic strategies. Teachers represent student thinking with a variety of conceptual models. "Each time, students solve the problem independently using any strategy they like, and then the teacher uses a specific model (a number line, for example) to represent students' strategies." (Teachers Manual Unit 1, Introducing Bridges in Mathematics).
  • Application is the focus in Unit 8, when students are designing a model playground.
  • Procedural skill and fluency is attended to separately in the Number Corner component "Computational Fluency."

Mathematical Practice-Content Connections

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The materials reviewed for Grade 4 partially meet the criteria for practice-content connections. The MPs are identified and used to enrich mathematics content. The materials often attend to the full meaning of each practice; however, there are instances where the students are not using the practices as written. For example, when speed/fluency games are labeled MP1 throughout the materials, and when some lessons identify MP4 incorrectly because students are not modeling contextual problems. The materials reviewed for Grade 4 partially attend to the standards' emphasis on mathematical reasoning. Overall students are prompted to construct viable arguments and analyze the arguments of others. However, there are missed opportunities to assist teachers in helping students to critique the arguments of others. The materials attend to the specialized language of mathematics and provide explicit instruction in how to communicate mathematical reasoning using words, diagrams and symbols. Overall, the materials partially meet the criteria for practice-content connections.

Criterion 2e-2g

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Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

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Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Grade 4 meet the expectations for the MPs being identified and used to enrich the mathematics content within and throughout the grade. The instructional materials identify the MPs, with two to four identified for each Bridges session and Number Corner activities. Students using the materials as intended will engage in the MPs along with the content Standards for the Grade.

  • The Introduction to Bridges Grade 4 includes a table describing what the MPs look like for students in Grade 4 for each practice.
  • All eight MPs are identified throughout the curricular materials.
  • There are no Bridges Sessions or Number Corner Activities in which the MPs are treated in isolation from grade-level, content standards.
  • The Bridges overall Scope & Sequence for Units does not note the practice standards, however, between two and four MPs are identified in the “Skills and Concepts” section at the beginning of every Bridges Session.
  • The MPs are identified in the “Target Skills” section at the beginning of every Number Corner month and within the "Skills and Concepts" section at the beginning of the Number Corner activity types. There is a "Math Practices & the Number Corner Learning Community" section at the beginning of the first Number Corner binder, which describes how students engage with the MPs during Number Corners.
  • "Math Practices in Action" are located in the margin of the teacher notes within the Bridges sessions. They identify how the students engage with the MPs along with the content standards. For example:
    • In Unit 2, Module 3, Session 5, "Math Practices in Action - MP3" calls teachers' attention to the fact that "while students are sharing and reflecting upon a variety of strategies for solving the problem, students construct viable arguments and critique the reasoning of others. In doing so, they are comparing the strategies, to each other and to their own thinking. This dramatically deepens their understanding of multiplication and contributes to computational fluency. (4.NBT.5)
    • In Unit 3, Module 2, Session 3, "Math Practices in Action" says, "Asking students to describe patterns they notice on the chart invites them to look for and express regularity in repeated reasoning. In their search for regularity, students will make observations related to equivalent fractions and adding fractions with like denominators," thus engaging students in MP8. (4.NF.1, 4.NF.2, 4.NF.3)
    • In Unit 6, Module 1, session 7, MP8 is identified in "Math Practices in Action - MP8." It discusses how problem strings involve repeated reasoning, which allows students to begin to recognize patterns that help them generalize methods (4.NBT.5).
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Indicator 2f

Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Grade 4 partially meet the expectations for carefully attending to the full meaning of each practice standard. Overall, the materials often attend to the full meaning of each standard. However there are instances where the students are not using the MPs as written.

  • "Math Practices in Action" are located in the margin of the teacher notes within the Bridges sessions. They call the teachers' attention to how the activities within the Bridges sessions engage students with a particular MP. For example, in Unit 2, Module 2, Session 3, one calls the teacher’s attention to the fact that having students work together first provides scaffolding to help students make sense of the problem and persevere in solving it. Students can continue to use the number line model if they like when they begin working independently. They can choose to use models and strategies of their own. This is one way of engaging them with MP1.
  • In many cases, the materials attend to the full meaning of the MPs:
    • In Unit 4, Module 3, Session 3, "It's About Time," students are engaging in MP4 while using a number line to model elapsed time in real world contexts.
    • In Unit 6, Module 2, session 2, "Area Challenge Forum," students are engaging in MP3 through a perimeter investigation. Teacher direction 3 invites students to present their work and prompts the teacher to "invite the rest of the class to ask questions, and have the presenters respond to those questions."
    • In Unit 7, Module 4, Session 1, Problems & Investigations, "Working with a Two-Part Area Model," students are engaging in MP7 while drawing connections between the partial products in an array model and a standard algorithm and using their understandings of the distributive property and of place value to inform their work with the algorithm.
    • In Unit 3, Module 4, Session 3, MP6 is identified, and the "Math Practice in Action - MP6" explains that having students place decimal and fraction cards on number line between 0 and 1 whole and 1 whole and 2 wholes allows students to pay attention to precision using a flexible model that engages them in carefully considering the relationship between the different decimals and fractions.
    • In Unit 4, Module 2, Session 1, "Problem String: How Old? Part 1 and 2," MP8 is identified, and students are given an opportunity in solving several subtraction problems over two problem strings to notice when it is best to apply either the "Finding the Difference" strategy or the "Removing (Taking Away)" strategy with the use of open number line models as a representation.
  • However there were instances of not attending to the full meaning of MP4:
    • In Unit 1 Module 2 Session 5, MP 4 is identified. In this Session students are working on strategies for multiplying by 4 and 6, and students are working with arrays. Students are not applying mathematics to real-world situations, thus not attending to the full meaning of MP4.
    • In Unit 2, Module 1, session 1, "The Great Wall of Base Ten", MP4 is identified as the MP in which students are engaging. Students are using base ten area pieces to build models through 10,000. Students are not applying mathematics to real-world situations, thus not attending to the full meaning of the standard.
  • There were instances of MP5 where the full meaning was not attended to:
    • In Unit 3, Module 3, Session 1, "Introducing Decimal Numbers", MP5 is identified, and the teacher directs students to use bas-ten pieces on mats to represent 1, strips to represent tenths, and units to represent hundredths, all of which the teacher directs students to use to represent various decimals. Since students are not given a choice of which tool to solve problems, this session does not attend to the full meaning of MP5.
    • In Unit 4, Module 4, Session 4, "Measuring Hand Spans and Arm Spans," MP5 is identified, and in the teacher directions Step 6, the teacher is directed to have students get out their rulers and measure their hand spans as well as those of their classmates and record the data in a table of their design. Since students are not given a choice of which tool to solve problems, this session does not attend to the full meaning of MP5.
Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

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Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Grade 4 meet the expectations for prompting students to reason by constructing viable arguments and analyzing the arguments of others. The student materials in both Bridges and Number Corner provide opportunities throughout the year for students to reason by both constructing viable arguments and analyzing the reasoning of others, however more opportunities could be provided in the student materials, other than formative and summative assessments, to engage in analyzing the arguments of others.

  • In Unit 2, Module 3, Session 1, Eggs & Apples, problem 1c and problem 2c ask “What strategy did you use? Why did you choose this strategy?”
  • In Unit 3, Module 2, Session 1, Equivalent Fractions Checkpoint, problems 2a and 2b ask, “LaTonya says that ½, 2/4, and 3/6 can be all worth the same amount. Do you agree with her? Use labeled sketches to explain your thinking.”
  • Unit 4, Module 1, Session 5, Addition Algorithm Practice problem 4 states and asks, “Alexis finished her homework in a hurry before the performance. Did she do the problem below correctly? Why or Why Not?“
  • Unit 6, Module 1, Session 4, Multiplication Strategies, states and asks, “Edie says she can solve 27 x 99 by solving 27 x 100 and then taking away 1 x 27. Do you agree or disagree? Explain.”
  • There are many prompts for students to explain how they got their answers or show their work. In assessments and on assignments the directions say, "Show your work" or "Use numbers, labels, models or words to show your thinking." For example, in Unit 3, Module 2, session 3, in the student workbook, questions 1 - 5 require students to "Use numbers, labeled sketches, or words to show your thinking."
  • In the October Number Corner, Solving Problems, Activity 3 "More Stamps & Beads," Christopher's work and solution is demonstrated relating to a multiplication story problem. Question 1 says "Do you agree with Christopher? If you agree with Christopher, explain why. If you disagree with Christopher, explain why and show how you would solve the problem."
  • In the December Solving Problems, Activity 40, it says “Max used a Venn Diagram to organize the shapes below. Do you agree with how Max organized the shapes? Why or Why Not?
  • January Checkup 2, items 12 and 13 say, “Solve this problem with the standard algorithm for addition/subtraction. Do you think the standard algorithm is the most efficient way to solve this problem? Why or Why not?”
  • March Checkup 3, Item 5 says, “Mara says that 0.45 is greater than 0.5 because 45 is greater than 5. Do you agree or disagree with Mara? Explain. Show you thinking with numbers, sketches, or words.”
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Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Grade 4 partially meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, there is assistance for teachers in engaging students in constructing viable arguments, however there is minimal assistance for teachers in supporting their students in analyzing the arguments of others.

  • Throughout the Bridges curriculum, many sessions include sample dialogue that is provided to assist the teacher in engaging students in reasoning and constructing viable arguments. Although activities are evident that allow students to analyze the arguments of classmates, the teacher is not provided enough support to help students with this analysis. For example:
    • In "The Giant's Castle," in Unit 1, Module 3, Session 3, Problems & Investigations, while engaging in an activity related to multiplicative comparisons, students share and compare their work. Teacher direction 3 says, "Record any and all responses without comment, setting the stage for student discussion and debate." Direction 4 prompts the teacher to invite students to present and defend each response. Direction 7 requires students to agree or disagree with a fictional student's thinking. Volunteers are called upon to explain and defend both positions.
    • In Unit 2, Module 4, session 4, "Spilled Ketchup," step 5 sample dialogue says, "Teacher: I’m thinking about what we talked about yesterday, that the story affects how we show remainder. Does anyone have any thoughts about the remainder that is written? Teacher: Does anyone agree or disagree with that idea?..."
    • In Unit 6, Module 2, Session 2, students are participating in a Math Forum. Teacher direction 2 says "Give them a minute and then have them share and compare solutions and strategies with classmates other than their partners from the previous session. Did they get the same answers? Can they understand each other's strategies?" Direction 2 invites students to share their work. The class is questioned whether they used the same or a similar approach. As approaches are shared, the teacher documents it through sketches, numbers, and words. The class is prompted to ask questions, and have the presenters respond to those questions. This activity allows students to analyze the arguments of classmates, however the teacher is not provided enough support to guide students with this analysis.
  • In the Bridges Number Corner curriculum, teachers are provided support in orchestrating discussions that engage students in constructing viable arguments, and minimal support that engages students in analyzing the arguments of others, through key question side notes, teacher directions, and sample dialog, especially during the Solving Problems and Problem String activities.
    • In the February Number Corner Teaching Tips, the teacher is prompted to: "Encourage more class participation by inviting students to comment on and ask questions about other students' contributions, and having them restate important ideas and directions. Provide positive feedback for participation, especially for students who have a harder time speaking up."
    • The Number Corner October Teaching Tips prompts teachers to build a sense of community this month. "In particular, help students develop their discussion skills. Encourage students to respond to each other's ideas: if they agree with a classmate, they might still have some ideas to add or some ways to extend their thinking; if they disagree, they can suggest a different idea and explain why it makes sense to them. Help students understand they are accountable for participating in conversations during Number Corner."
    • "Key Questions" side notes in margins support teachers engaging students in constructing viable arguments, For example: in September they say, “Is your answer reasonable?”, “How can you check your work after you have solved the problem?” In December they say, “What is the most efficient way to solve this problem?” and “Do you see a relationship between multiplication and division? Explain.” In February, they say, "How can you show your thinking? What model could you use to show your thinking? Why is this a useful model?"
  • MP3 is mentioned specifically eight times throughout the Bridges sessions in the "Math Practices in Action," which supports teachers in understanding how the MP is applied in the sessions. In Unit 2, Module 4, Session 3, "Math Practices in Action" says, "When they are able to pinpoint and clarify things they don't understand about others' work, it deepens their conceptual understanding, permits them to think carefully about others' reasoning, and improves their ability to construct clear and viable mathematical arguments of their own." This is vague guidance about how the teacher is to facilitate student reasoning.
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Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Grade 4 meet the expectations for the materials explicitly attending to the specialized language of mathematics. Overall, the materials provide explicit instruction in how to communicate mathematical reasoning using words, diagrams and symbols; however more explicit instruction related to precise communication is needed.

  • In the introduction to the series, “The curriculum includes a set of Word Resource Cards for every classroom. Each card features a mathematical term accompanied by illustrations, with a definition on the back. The cards are integrated into lessons and displayed in the classroom to support students’ acquisition and use of precise mathematical language.”
  • In Section 3 of the Assessment Guide, Assessing Math Practices, an app called Math Vocabulary Cards, which included the same illustrated terms and illustrations as the Word Resource Cards, is also available to serve as a compact and convenient math dictionary.
  • At the beginning of the sessions, a sidebar lists the vocabulary for the lesson, with an asterisk that identifies "those terms for which Word Resource Cards are available." Sessions contain directions for use of the cards. For example, in Unit 2, Module 1, session 3, teacher direction 8 says, "Post the Word Resource card for dimension and review the term. Then have students add the word to their handbooks.”
  • “Bridges Grade 4 incorporates manipulatives and visual models that provide a variety of ways for students to make sense of mathematical concepts, represent and solve problems, attend to precision in their efforts, and communicate about their thinking” (Bridges Introduction). Below are some examples:
    • In Unit 2, Module 1, Session 3, Metric Units of Linear and Area Measurement, the focus for specialized mathematical language is on the base-ten model and the terms dimension, area, centimeter, and meter.
    • In Unit 3, Module 4, Session 3, Ordering Fractions and Decimals on the Number Line, the focus on specialized math language is on attending to precision in the placement of fractions and decimals on a number line as described in “Math Practices in Action: MP6.”
    • In Unit 4, Module 1, Session 6, Think Before You Add, the focus on specialized math language is on specific multi-digit addition strategies including the “Standard Algorithm,” “Give and Take,” and “Split Them All Up.” The “Give and Take” strategy is accompanied by a number line.
  • Problem Strings, in both Bridges (Unit 2, 4 and 6) and Number Corner, provide students with opportunities to make mathematical arguments, explanations, and generalizations using an intentionally designed string of whole number and fractions problems. Examples below:
    • In Unit 4, Module 2, Session 1, "Removal vs. Finding the Difference," students use the “Find the Difference” and “Removing” strategies to subtract after they are modeled by the teacher using an open number line. The teacher poses specific questions to support students in generalizing about when one strategy might be used over the other. A sidebar provides teachers with the background needed to facilitate this discussion.
    • In Unit 6, Module 1, Session 3, "Packs of Pens," students solve a string of multi-digit multiplication problems using various strategies including the distributive property which is modeled using open arrays. The teacher also models students’ strategies with a ratio table for the whole string. Students make generalizations at the end of the string.
    • In the May Number Corner Problem String 25 and 26, students use the strategy of scaling up the product of a unit fraction and whole number to multiply non-unit fractions and whole numbers. The teacher models student thinking with a tile array or ratio table afterward which students may use as a tool of thinking in their own mathematical arguments or explanations. Teacher direction 6 says: “Encourage them (students) to refer to arrays in their explanations and describe how they used them to get or confirm their answer.”
  • Vocabulary Side Notes appear in the margins of the teacher directions in some Number Corner Activities to suggest ways in which mathematics vocabulary can be taught or explained to students.
    • December Solving Problems Activity 2, An Etymology Side Notes for the teacher on the greek roots “Sym” and “met” was provided.
  • The January Number Corner, Calendar Grid, "Similar Figures," provides examples in which students are explicitly taught the specialized language of mathematics, for example:
    • Mathematical Background - References similar figures, equilateral, scalene, isosceles, isosceles right triangle, scalene right triangle.
    • Key Questions - What attributes can you use to describe these shapes? What does similar mean? How can you tell if two figures are similar?
    • In Activity 1 - Teacher direction 2 says "If students don't use the word similar let them know that the shapes on marker 1 and 2, and on 3 and 4, are similar, which means that they are exactly the same shape (although they are not the same size).
  • Math Practices in Action that identify MP6 nearly all give direction to the teacher about accuracy with calculation or measurement, not precise communication. For example, in Unit 4, Module 4, session 5, the "Math Practices in Action - MP 6" says, "When applying strategies that involve decomposing numbers and adding the same amount to both numbers, students must attend to precision to ensure that they arrive at the correct answer..."

GATEWAY THREE

Usability

MEETS EXPECTATIONS

The materials reviewed meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially and each activity has a mathematical purpose. Students are asked to produce a variety of assignments. Manipulatives and models are used to enhance learning and the purpose of each is explained well. The visual design is not distracting or chaotic and supports learning. The materials support teachers in learning and understanding the standards. All materials include support for teachers in using questions to guide mathematical development. Teacher editions have many annotations and examples on how to present the content and an explanation of the math of each unit and the program as a whole.

A baseline assessment allows teachers to gather information on student's prior knowledge, and teachers are offered support in identifying and addressing common student errors and misconceptions. Materials include opportunities for ongoing review and practice. All assessments include information on standards alignment and scoring rubrics. There are limited systems or suggestions for students to monitor their own progress. Activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.

All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide technology based assessment opportunities, and the digital resources are not easily customized for individual learners.

Overall, the materials meet the expectations for usability.

Criterion 3a-3e

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Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

Materials are well-designed, and lessons are intentionally sequenced. Typically students learn new mathematics in the Problems & Investigations portion of Sessions while they apply the mathematics and work towards mastery during the Work Station portion of Sessions and during Number Corner. Students produce a variety of types of answers including both verbal and written answers. Manipulatives such as fraction strips, number lines and geoboards are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

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Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that 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.

The Sessions within the Units distinguishes the problems and exercises clearly. In general, students are learning new mathematics in the Problems & Investigations portion of each session. Students are provided the opportunity to apply the math and work toward mastery during the Work Station portion of the Session as well as in daily Number Corners.

For example, in Unit 6, Module 1, Sessions 2 and 3 the students are “Discussing Larger Division with Money.” During the Problems & Investigation section of the lesson students are first working in groups and pairs to solve money story problems involving multiplication and division. As students work and record their answers, they pause for discussion to share their thinking with the class, using paper bills and base ten area pieces. In Session 3, students share and compare their solutions and strategies from some of the problems from Session 2. Students complete a Problem String to review, model, and solve larger multiplication problems. During the Work Place activity, the students are working independently to solve larger division problems.

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Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.

The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is continued to be reviewed, practiced and extended throughout the year.

The "Skills Across Grade Level" table is present at the beginning of each Unit. This table shows the major skills and concepts addressed in the Unit. The table also provides information about how these skills are addressed elsewhere in the Grade, including Number Corner, and in the grade that follows. Finally, the table indicates if the skill is introduced (I), developed (D), expected to be mastered (M), or reviewed, practiced or extended to higher levels (R/E).

Concepts are developed and investigated in daily lessons and are reinforced through independent and guided activities in work places. Number Corner, which incorporates the same daily routines each month (not all on the same day) has a spiraling component that reinforces and builds on previous learning. Assignments, both in class and for homework, directly correlate to the lesson being investigated within the unit.

The sequence of the assignments is placed in an intentional manner. First, students complete tasks whole group in a teacher directed setting. Then students are given opportunities to share their strategies used in the tasks completed in the Problems & Investigations. The Work Places activities are done in small groups or partners to complete tasks that are based on the problems done whole group in the Problems & Investigations. The students then are given tasks that build on the session skills learned for the home connections.

For example, 4.NBT.5 (multiply a 2 or 3 digit whole number by a 1 digit whole number using strategies based on place value and the properties of operations) is introduced in Grade 3, developed in Unit 2, mastered in Unit 6, and is reviewed/practiced/extended in Unit 7. The standard continues to be developed during Number Corners in September, October and January, as well as in Grade 5.

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Indicator 3c

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

There is variety in what students are asked to produce. Throughout the grade, students are asked to respond and produce in various manners. Often, working with concrete and moving to more abstract models as well as verbally explaining their strategies. Students are asked to produce written evidence using drawings, representations of tools or equations along with a verbal explanation to defend and make their thinking visible.

For example, in Unit 3, Module 2, Session 3 students are working together as a class to create a chart of equivalent fractions for 1/4, 1/2, and 3/4. The chart contains visual representations, words, and symbols all standing for each fraction. The students make observations about the chart and the fractions, then use those fractions as benchmarks when making comparisons among fractions with like and unlike denominators. After a class discussion, they practice adding and subtracting fractions with like denominators during their work time.

Also, in the December Number Corner during the Calendar Grid lesson, the students use a variety of ways to explore congruence, line symmetry, parallel and perpendicular lines. They use pentominos to explore and look for patterns and have a discussion about parallel, intersecting, and perpendicular lines and sides using the first four pentominos. As they work through the activities, they come up with definitions, create pentominoes, and discuss their findings with each other.

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Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods. Manipulatives are used and provided to represent mathematical representations and provide opportunities to build conceptual understanding. Some examples are the fraction strips, number lines, and geoboards. When appropriate, they are connected to written representations.

For example, in Unit 3, Module 1, Session 3, students are working on fractions and mixed numbers. They are asked to make a set of construction paper fractions strips and then use those strips to investigate equivalent fractions, mixed numbers, and improper fractions. They use these fraction strips throughout other lessons to strengthen their understanding of fractions. For example, in Unit 3, Module 2, Session 1, the students are naming fractional parts of the geoboard and describing the parts’ relationships to one another by using their strips to check for understanding.

Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The material is not distracting and does support the students in engaging thoughtfully with the mathematical concepts presented. The visual design of the materials is organized and enables students to make sense of the task at hand. The font, size of print, amount of written directions and language used on student pages is appropriate for Grade 4. The visual design is used to enhance the math problems and skills demonstrated on each page. The pictures match the concepts addressed such as having the characters that are in the story problems placed in picture format on the page as well. Some problems may even require students to use the pictures to solve the story problems.

For example, in Unit 7, Module 1, Session 4, the design of the students’ Pattern Chart sheet supports the students in engaging thoughtful work with tile sequences. The students examine the relationship between each arrangement number in the sequence and the number of tiles in that arrangement, then they test conjectures about the relationship on their worksheets. They fill in a chart which is clearly marked with spaces for the work. There is an example with a picture to provide scaffolding if needed.

Criterion 3f-3l

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  •  8 7

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials support teacher learning and understanding of the standards. The instructional materials provide questions and discourse that support teachers in providing quality instruction. The teacher's edition is easy to use and consistently organized and annotated. The teacher's edition explains the math in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.

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Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Lessons provide teachers with guiding questions to elicit student understanding and discourse to allow student thinking to be visible. Discussion questions provide a context for students to communicate generalizations, find patterns, and draw conclusions.

Each unit has a Sessions page, which is the Daily Lesson Plan. The materials have quality questions throughout most lessons. Most questions are open-ended and prompt students to higher level thinking.

In Unit 2, Module 2, Session 2 as students are learning to represent arrays pictorially, after working with concrete models, teachers are prompted to ask the following questions:

  • "Why is the first rectangle labeled with the number 40? Where can you see that part of the array in your own base ten area pieces?"
  • "Why is the second, smaller rectangle labeled with the number 12? Where can you see that 12 in your pieces?"

In Unit 3, Module 1, Session 2 - As students are exploring fractions on the number line, teachers are prompted to ask the following questions:

  • "How might the number line be used to represent the fruit strip problem? The money problem? The practice time problem?"
  • "What does the 1 (whole) represent for each problem?"
  • "What does the 3/4 represent for each problem?"

In Unit 6, Module 3, Session 3, - When discussing a fraction game, the teacher is prompted to ask the following questions:

  • "How can you represent each of your spins individually?"
  • "What fractions are equivalent to the ones you spun?"
  • "What trades can you make so you have as few pieces as possible on your record sheet?"

In the November Number Corner Computational Fluency activity, "Roll & Compare" game, the following questions are provided in the "Key Questions" section in the margin:

  • "If the number in the ten thousand place of your number is larger than the number in the ten thousand place of your partner's number, do the other digits in the other places matter, if you are trying to make the biggest number?"
  • "If you roll a zero, and you want to make the smaller number, where should you put the zero? What about the larger number?"
  • "Do bigger numbers have more or fewer multiples within a certain range of numbers? Why? (For example, are there more multiples of 5 or 10 between 0 and 100? Why?)"
  • "How can knowing your 5 and 10 facts help you with other multiplication problems?"
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Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials; however, additional teacher guidance for the use of embedded technology to support and enhance student learning is needed.

There is ample support within the Bridges material to assist teachers in presenting the materials. Teacher editions provide directions and sample scripts to guide conversations. Annotations in the margins offer connections to the math practices and additional information to build teacher understanding of the mathematical relevance of the lesson.

Each of the 8 Units also have an Introductory section that describes the mathematical content of the unit and includes charts for teacher planning. Teachers are given an overview of mathematical background, instructional sequence, and the ways that the materials relate to what the students have already learned and what they will learn in the future units and grade levels. There is a Unit Planner, Skills Across the Grade Levels Chart, Assessment Chart, Differentiation Chart, Module Planner, Materials Preparation Chart. Each unit has a Sessions page, which is the Daily Lesson Plan.

The Sessions contain:

  • sample Teacher/Student dialogue;
  • Math Practices In Action icons as a sidebar within the sessions - These sidebars provide information on what MP is connected to the activity;
  • a Literature Connection sidebar - These sidebars list suggested read-alouds that go with each session;
  • ELL/Challenge/Support notations where applicable throughout the sessions
  • Vocabulary section within each session - This section contains vocabulary that is pertinent to the lesson and indicators showing which words have available vocabulary cards online

Technology is referenced in the margin notes within lessons and suggests teachers go to the online resource. Although there are no embedded technology links within the lessons, there are technology resources available on the Bridges Online Resource page such as videos, whiteboard files, apps, blogs, and online resource links (virtual manipulatives, images, teacher tip articles, games, references). However, teacher guidance on how to incorporate these resource is lacking within the materials. It would be very beneficial if the technology links were embedded within each session, where applicable, instead of only in the online teacher resource. For instance, the teacher materials would be enhanced if a teacher could click on the embedded link, (if using the online teacher manual) and get to the Whiteboard flipchart and/or the virtual manipulatives.

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Indicator 3h

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Materials contain adult-level explanations of the math concepts contained in each unit. The introduction to each unit provides the mathematical background for the unit concepts, the relevance of the models and representations within the unit, and teaching tips. When applicable to the unit content, the introduction will describe the algebra connection within the unit.

At the beginning of each Unit, the teacher's edition contains a "Mathematical Background" section. This includes the math concepts addressed in the unit. For example, Unit 3 states, "Unit 3 takes an applied visual approach to fractions and decimals. Over the course of 20 sessions, students make extensive use of concrete manipulatives and visual models to explore unit fractions, common fractions, mixed numbers, improper fractions, equivalent fractions, and decimals. They come to understand that two fractions with unlike numerators and denominators such as 4/6 and 8/12, can be equal..."

The Mathematical Background also includes sample models with diagrams and explanations, strategies, and algebra connections. There is also a Teaching Tips section following the Mathematical Background that give explanations of strategies, tools and representations within the sessions such as geoboards, protractors, mathematical language, and modeling. There are also explanations and samples of the various representations used within the unit such as area model, base ten pieces, ratio tables, and number line.

In the Implementation section of the Online Resources, there is a "Math Coach" tab that provides the Implementation Guide, Scope & Sequence, Unpacked Content, and CCSS Focus for Grade 4 Mathematics.

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Indicator 3i

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Materials contain a teacher’s edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

In the Unit 1 binder there is a section called "Introducing Bridges in Mathematics." In this section there is an overview of the components in a day (Problems & Investigations, Work Places, Assessments, Number Corner). Then there is an explanation of the Mathematical Emphasis in the program. Content, Practices, and Models are explained with pictures, examples and explanations. There is a chart that breaks down the mathematical practices and the characteristics of children in that grade level for each of the math practices. There is an explanation of the skills across the grade levels chart, the assessments chart, and the differentiation chart to assist teachers with the use of these resources. The same explanations are available on the website. There are explanations in the Assessment Guide that goes into they Types of Assessments in Bridges sessions and Number corner.

The CCSS Where to Focus Grade 4 Mathematics document is provided in the Implementation section of the Online Resources. This document lists the progression of the major work in grades K-8.

Each unit introduction outlines the standards within the unit. A “Skills Across the Grade Level” table provides information about the coherence of the math standards that are addressed in the previous grade as well as in the following grade. The "Skills Across the Grade Level" document at the beginning of each Unit is a table that shows the major skills and concepts addressed in the Unit and where that skill and concept is addressed in the curriculum in the previous grade as well as in the following grade.

Indicator 3j

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The materials provide a list of lessons in the teacher's edition cross-referencing the standards covered and providing an estimated instructional time for each lesson and unit. The "Scope and Sequence" chart lists all modules and units, the CCSS standards covered in each unit, and a time frame for each unit. There is a separate "Scope and Sequence" chart for Number Corners.

Indicator 3k

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Home connection materials and games sometimes include a “Note to Families” to inform them of the mathematics being learned within the unit of study.

Additional Family Resources are found at the Bridges Educator's Site.

  • Grade 4 Family Welcome letter in English and Spanish- This letter introduces families to Bridges in Mathematics, welcomes them back to school, and contains a broad overview of the year's mathematical study.
  • Grade 4 Unit Overviews for Units 1-8, in English and Spanish.
Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials contain explanations of the instructional approaches of the program. In the beginning of the Unit 1 binder, there is an overview of the philosophy of this curriculum and the components included in the curriculum. There is a correlation of the CCSS and Math Practices as the core of the curriculum in the Mathematical Emphasis section. The assessment philosophy is given in the beginning of the Assessment binder. The types of assessments and their purpose is laid out for teachers. For example, informal observation is explained as "one of the best but perhaps undervalued methods of assessing students...Teachers develop intuitive understandings of students through careful observation, but not the sort where they carry a clipboard and sticky notes. These understandings develop over a period of months and involve many layers of relaxed attention and interaction."

Criterion 3m-3q

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  •  10 10

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials offer teachers resources and tools to collect ongoing data about student progress. The September Number Corner Baseline Assessment allows teachers to gather information on student's prior knowledge, and the Comprehensive Growth Assessment can be used as a baseline, quarterly, and summative assessment. Checkpoints and informal observation are included throughout the instructional materials. Throughout the materials Support sections provide common misconceptions and strategies for addressing common errors and misconceptions. Opportunities to review and practice are provided in both the Sessions and Number Corner routines. Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments clearly indicate the standards being assessed and include rubrics and scoring guidelines. There are, however, limited opportunities for students to monitor their own progress on a daily basis.

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Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The September Number Corner Baseline Assessment is a 6 page, written baseline assessment that is designed to ascertain students' current levels of key number skills and concepts targeted for mastery in third grade – multiplication and division concepts, multiplication fact fluency; 3-digit addition and subtraction; comparing and ordering fractions; use of the area model for multiplication; and line plots (found in Assessment Guide, Number Corner Assessments pg. 1). The Comprehensive Growth Assessment contains 49 written items, addressing every Common Core standard for Grade 4. This can be administered as a baseline assessment as well as an end of the year summative or quarterly to monitor students' progress.

Informal observation is used to gather information. Many of the sessions and Number Corner workouts open with a question prompt: a chart, visual display, a problem, or even a new game board. Students are asked to share comments and observations, first in pairs and then as a whole class. This gives the teacher an opportunity to check for prior knowledge, address misconceptions, as well as review and practice with teacher feedback. There are daily opportunities for observation of students during whole group and small group work as well as independent work as they work in Work Places.

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Indicator 3n

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Most Sessions have a Support section and ELL section that suggests common misconceptions and strategies for remediating the misconceptions that students may have with the skill being taught.

Materials provide sample dialogues to identify and address misconceptions. For example, the Unit 3 Module 1 Session 2 “Support” section gives suggestions for struggling students. The materials suggest that the teacher should help students understand what the problems are asking. The materials say to encourage students to use the materials available or make sketches to enact and visualize each situation.

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Indicator 3o

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review and practice, with feedback for students in learning both concepts and skills.

The scope and sequence document identifies the CCSSM that will be addressed in the sessions and in the Number Corner activities. Sessions build toward practicing the concepts and skills within independent Work Places. Opportunities to review and practice are provided throughout the materials. For example, in Unit 3, Module 1, Session 2, the teacher leads students in activities that serve to bridge the division work students did at the end of Unit 2 and the fraction work they’re about to undertake in Unit 3 (4.NF.3).

Ongoing review and practice is often provided through Number Corner routines. Each routine builds upon the previous month’s skills and concepts. For example, the Number Corner March Problem Strings build on fraction models that were introduced in February, helping students to deepen and refine their understandings of equivalent fractions (4.NF.5).

Indicator 3p

Materials offer ongoing formative and summative assessments:

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Indicator 3p.i

Assessments clearly denote which standards are being emphasized.

All assessments, both formative and summative, clearly outline the standards that are being assessed. In the assessment guide binder, the assessment map denotes the standards that are emphasized in each assessment throughout the year. Each assessment chart details the CCSS that is addressed.

For example, the Unit 1, Module 3, Session 2, Multiplication & Division Checkpoint includes a Checkpoint Scoring Guide that lists each prompt, the correct answer, the standard, and the points possible. The Unit 3, Module 4, Session 4 Post-Assessment includes a Scoring Guide that lists all items, correct answers, standards, and the possible points, as well as a Student Reflection Sheet. The Unit 6, Module 4, Session 3 Post-Assessment includes a Post-Assessment Scoring Guide that lists all items, correct answers, standards and the possible points, as well as a Student Reflection Sheet. The October Number Corner Checkup 1 includes a Scoring Guide that contains the item, the CCSSM, and the possible points. The May Number Corner Checkup 4 includes a Scoring Guide that contains the item, the CCSSM, and the possible points.

Also, each item on the Comprehensive Growth Assessment lists the standard emphasized in the Skills & Concepts Addressed chart as well as on the Comprehensive Growth Assessment Scoring Guide.

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Indicator 3p.ii

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting students' performance and suggestions for follow-up.

All Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments are accompanied by a detailed rubric and scoring guideline that provide sufficient guidance to teachers for interpreting student performance. There is a percentage breakdown to indicate Meeting, Approaching, Strategic, and Intensive scores. Section 5 of the Assessments Guide is titled "Using the Results of Assessments to Inform Differentiation and Intervention.” This section provides detailed information on how Bridges supports RTI through teachers' continual use of assessments throughout the school year to guide their decisions about the level of intervention required to ensure success of each student. There are cut scores and designations assigned to each range to help teacher identify students in need of Tier 2 and Tier 3 instruction. There is also a breakdown of Tier 1, 2, and 3 instruction suggestions.

Indicator 3q

Materials encourage students to monitor their own progress.

There is limited evidence in the instructional materials that students are self-monitoring their own progress on a frequent basis. The materials do not provide daily exit tickets (other then three generalized writing prompts) or daily formative checks. Bridges Units 1-7 do each include a Student Reflection sheet for both pre and post unit assessments. Each skill is listed in a graphic organizer, along with the problem number. Students reflect upon their work utilizing the following criteria: I can do this well already, I can do this sometimes, and I need to learn to do this. After the Pre-Assessment, students are asked to make a star next to the two skills that he/she needs to work on the most during the unit. After the Post-Assessment, students are asked to reflect upon 1-2 things that they improved upon, as well as areas still in need of work.

Section 4 of the Assessment Guide is titled, "Assessment as a Learning Opportunity." This section provides information to teachers guiding them in setting learning targets, communicating learning targets, bringing sessions to closure, and facilitating student reflection based upon the results of pre and post assessments.

Criterion 3r-3y

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  •  12 12

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

Session and Number Corner activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The Bridges and Number Corner materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.

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Indicator 3r

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Units and modules are sequenced to support student understanding. Sessions build conceptual understanding with multiple representations that are connected. Procedural skills and fluency are grounded in reasoning that was introduced conceptually, when appropriate. An overview of each unit defines the progression of the four modules within each unit and how they are scaffolded and connected to a big idea.

In the Sessions and Number Corner activities, there are ELL strategies, support strategies, and challenge strategies to assist with scaffolding lessons and making content accessible to all learners.

For example, in Unit 3, Module 1, Session 2, students are working on the activity "Sharing Situations." Support is offered: “Help struggling students understand what the problem is asking. Encourage these students to use the materials available, or make sketches to enact and visualize each situation." Challenge is offered: "Invite pairs who complete all three problems to start the challenge problem on the second sheet."

In the Unit 6, Module 2, Session 2 "Investigating Perimeter" activity, the following suggestions are provided:

  • ELL: "Use the Word Resource Cards to review the definitions of perimeter and dimension.
  • Support: "Some students might benefit from using their 60” classroom measuring tapes in addition to, or even instead of, the string so they can see the numbers more easily. If students are really struggling, it might help to sketch a rectangle on the board with one dimension labeled 12” and question marks for other dimensions.”
  • Challenge: “Some students might see right away that they can double the known dimension, subtract that from the perimeter (60”), and then divide the difference by 2 to find the unknown dimension. If students solve the problem this way, challenge them to see if their strategy works every time. Ask them to find the unknown dimension for other rectangles with 60” perimeters. Use 4”, 23”, and 16” for the known dimensions. Also, encourage students to consider the semi-perimeter (half of the perimeter) and its relationship to the dimensions.”
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Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials provide teachers with strategies for meeting the needs of a range of learners.

A chart at the beginning of each unit indicates which sessions contain explicit suggestions for differentiating instruction to support or challenge students. Suggestions to make instruction accessible to ELL students are also included in the chart. The same information is included within each session as it occurs within the teacher guided part of the lesson. Each Work Place Guide offers suggestions for differentiating the game or activity. The majority of activities are open-ended to allow opportunities for differentiation. Support and intervention materials are provided online and include practice pages, small-group activities and partner games.

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Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations. Tasks are typically open-ended and allow for multiple entry-points in which students are representing their thinking with various strategies and representations (concrete tools as well as equations).

In the Problems and Investigations section, students often are given the opportunities to share strategies they used in solving problems that were presented by the teacher. Students are given multiple strategies for solving problems throughout a module. They are then given opportunities to use the strategies they are successful with to solve problems in Work Places, Number Corner, and homework.

For example, in Unit 6, Module 2, Session 3, students are sharing their strategies for the “PJ’s String Perimeter Problem.” Students share in pairs and then with the whole class about how they would solve the problem. In this session the teacher then chooses several students to share their strategy for solving the problem short of providing the answer before they are tasked with solving the problem using a strategy shared by a classmate. In a sample dialog three students shared their strategies while the teacher recorded on the teacher master projected for students. Afterwards students solved the perimeter problem individually, compared their solutions, and strategies with a partner, and the teacher chose several student to explain their final calculations to find the unknown dimension.

Another example is found in the Number Corner February Problem Strings. Students are solving several strings of problems involving adding and subtracting fractions with like and unlike denominators posed by the teacher using any strategy they want. Students’ strategies are modeled using fractions bars, cents, coins, common denominator of 100 , common denominator of 4, and clock faces along with various equations, which highlight the use of equivalent fractions and decomposing fractions. The teacher solicits students’ strategies by prompting students with the questions provided in a side note on the margin of the problem string introduction including: 1) “Which strategy could you use?”, 2) “How can you show your thinking?”, and 3) “What model could you use to show your thinking?” The teacher ends the second of part of Problem String 16 by asking them to discuss
“Why it is useful to decompose fraction?” to support them with making generalizations about the big ideas of the string.

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Indicator 3u

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials suggest supports, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Online materials support students whose primary language is Spanish. The student book, home connections and component masters are all available online in Spanish. Materials have built in support in some of the lessons in which suggestions are given to make the content accessible to ELL students of any language.

There are ELL, Support, and Challenge accommodations throughout the Sessions and Number Corner activities to assist teachers with scaffolding instructions. Examples of these supports, accommodations, and modifications include the following:

  • Unit 2, Module 3, Session 1 provides a ELL suggestion. The suggestion reads as follows: "Provide opportunities for students to request clarification or rephrasing. Ask all students to justify their thinking to provide an atmosphere where students are comfortable asking questions and sharing.”
  • In Unit 7, Module 3, Session 2, students are working on solving multi-digit multiplication problems using the standard algorithm. The following Support suggestion is provided: "Depending on the needs and strengths of your class you may want to have some students solve additional problems with you, while others work independently in their books." The following ELL suggestions is made: “Review the directions with students and do one of the problems together. Describe students’ actions aloud as you work together.”
  • In Unit 4, Module 2, Session 3, students are solving multi-digit subtraction problems with the standard algorithm. The following ELL/Support suggestions is provided as follows: “The modeling you will do with the base ten area pieces will help ELL students. Make sure they are seated near the display so they can see what you are doing. Try to engage them as much as possible. For example, invite them to help you move strips and record answers. Ask them if they have seen this method before.”
  • In the Number Corner March Solving Problems, students are solving problems involving multiplying fractions by whole numbers using various strategies. The ELL/Support suggestion is: "Quickly sketch an open number line on the board to help clarify what it means to identify the two whole numbers between which the answer will lie. Mark and label the numbers from 0 through 5. Then ask students to estimate how much garden space would be planted in flowers if Gloria had 7 garden beds instead of 5, and give them some parameter to consider. Would it be as much as 2 whole beds? Why or why not? Would it be as much as 3 whole beds? Why or why not?"
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Indicator 3v

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The instructional materials provide opportunities for advanced students to investigate mathematics content at greater depth. The Sessions, Work Places, and Number Corners include "Challenge" activities for students who are ready to engage deeper in the content.

Challenge activities found throughout the instructional materials include the following:

  • In Unit 2, Module 2, Session 2, students are making generalizations about what happens when the multiplier is 10, then 100, then 100. The “Challenge” suggestions is as follows: “Ask students who seem to have generalized their understanding of multiplying by 10, 100, and 1,000 to create story problems about the expressions they are working with during the session. Challenge these students to create one problem that includes both x 10 and x 100.”
  • In Unit 4, Module 1, Session 6, students are being assessed formatively through a work sample which require them to use the standard algorithm for multi-digit addition. The "Challenge" suggestion is as follows: "Encourage students to generalize what numbers work best for which strategy.”
  • In the Number Corner November Calendar Grid, students predicting patterns based on previous tracking of elapsed time on analog clocks. The Challenge suggestions are as follows: “Challenge students by encouraging them to explore 24 hour clocks. Ask them to figure sample times on a 24 hour clock and vice versa. For example, ask them what time it is on a 24-hour clock when it is 3:00 pm. What time is on a 12 hour clock when it is 18:00?”
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Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.

The materials provide a balanced portrayal of demographic and personal characteristics. Many of the contexts of problem solving involve objects and animals familiar to students, such as marbles, school supplies, gardens, chickens and pigs. When people are shown, they are cartoons that appear to show a balance of demographic and personal characteristics.

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials offer flexible grouping and pairing options. Throughout the Units, Work Places, and Number Corners, there are opportunities to group students in various ways such as whole group on the carpet, partners during pair-share, and small groups during Problem & Investigations and Work Places.

  • In Unit 2, Module 1, Session 4, students are introduced to the game “Remainders Win” which provides opportunities for students to observe the relationship between multiplication and division and work with division with remainders. The teacher summarizes the game and plays a game of “2D Remainders Win” against the class to familiarize students with the game and review strategies for multiplication and division, while also predicting if there will be a remainder. Students then play the game in pairs. Finally students turn and talk with a partner about one division strategy they used and what strategies might be more efficient.
  • In the Number Corner April Computational Fluency, the class plays "Color Ten" a game that requires players to generate fractions and mixed numbers and then add or subtract them to get as close to 10 as possible. Students play against the teacher and then independently complete exercises involving the conversion of mixed numbers to fractions on day 1. Then students play the game in pairs on subsequent days while the teacher supports students proficiency with adding and subtracting fractions with like denominators through playing against students and guiding discussion of their mathematical thinking with questions while they are playing in pairs or as a class against the teacher.
Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.

There is limited evidence of the instructional materials encouraging teachers to draw upon home language and culture to facilitate learning. The materials provide parent welcome letters and unit overview letters that are available in English and Spanish.

Criterion 3z-3ad

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

All of the instructional materials available in print are also available online. Additionally, the Bridges website offers resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide technology based assessment opportunities, and the digital resources are not easily customized for individual learners.

Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Each session within a module offers online resources that are in alignment with the session learning goals. Online materials offer an interactive whiteboard file as a tool for group discussion to facilitate discourse in the Mathematical Practices. Resources online also include virtual manipulatives and games to reinforce skills that can be used at school and home.

In the Bridges Online Resources there are links to the following:

  • Virtual Manipulatives - a link to virtual manipulatives such as number lines, geoboard, arrays, number pieces, number racks, number frames, and math vocabulary
  • Interactive Whiteboard Files - Whiteboard files that go with each Bridges Session and Number Corner
  • Online Games – online games such as Alien Angles, Amoeba Addition, Bamzooki Multiplication & Division, Prime Time, Roboid Symmetry

Within the Teacher's Edition, there is no direct reference to online resources. If embedded within the Teacher's Edition, the resources would be more explicit and readily available to the teacher.

Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The digital materials are web-based and compatible with multiple internet browsers. They appear to be platform neutral and can be accessed on tablets and mobile devices.

All grade level Teacher Editions are available online at bridges.mathlearningcenter.org. Within the Resources link (bridges.mathlearningcenter.org/resources) there is a sidebar that links teachers to the MLC, Math Learning Center Virtual Manipulatives. These include games, Geoboards, Number Line, Number Pieces, Number Rack, Number Frames and Math Vocabulary. The resources are all free and available in platform neutral formats: Apple iOS, Microsoft and Apps from Apple App Store, Window Store, and Chrome Store. The apps can be used on iPhones and iPads. The Interactive Whiteboard files come in two different formats: SMART Notebook Files and IWB-Common Format. From the Resource page there are also many links to external sites such as ABCYA, Sheppard Software, Illuminations, Topmarks, and Youtube.

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Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials do not include opportunities to assess students’ mathematical understanding and knowledge of procedural skills using technology.

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Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials are not easily customizable for individual learners or users. Suggestions and methods of customization are not provided.

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Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The instructional materials provide opportunities for teachers to collaborate with other teachers and with students, but opportunities for students to collaborate with each other are not provided. For example, a Bridges Blog offers teacher resources and tools to develop and facilitate classroom implementation.