The instructional materials reviewed for Grade 5 are aligned to the CCSSM. The assessments are focused on grade-level standards, and the materials do 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 5 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. The vocabulary in this grade includes some instances of superficial instruction. Overall, the materials are focused, follow a coherent plan, balance the aspects of rigor, and attend to the MP.
Focus & Coherence
The materials reviewed for Grade 5 meet the expectations for Gateway 1. These materials do not assess above-grade-level content, and they spend the majority of the time on the major work of the grade level. 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 5 materials are focused and follow a coherent plan.
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The materials for Grade 5 met the criteria for focus. The curriculum does not assess future grade level content in summative assessments, and the majority of the instructional time is spent on the major work of the grade.
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Materials do not assess topics before the grade level in which the topic should be introduced.
The instructional materials reviewed for Grade 5 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.
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 5 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 5 content standards:
- In unit 3, addition and subtraction of decimals items do not require the standard algorithm (although addition and subtraction problems are sometimes written vertically). Also in unit 3, multi-digit division items do not require the standard algorithm. For example, on the post-assessment, item 10b. says, “Use numbers, words, or labeled sketches to solve the problem.”
- In alignment with standard 5.NBT.5, multi-digit multiplication is assessed using the standard algorithm.
- In Unit 5, multiplication and division of fractions, all division items on all the assessments contain only whole numbers divided by fractions or fractions divided by whole numbers.
- In Unit 6, graphing, geometry, and volume, all items are in the first quadrant of the graph.
- In Unit 7, the multiplication and division of decimals items do not require standard algorithm.
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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 5 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.
Instructional material spends the majority of class time on the major cluster of each grade.
The instructional materials reviewed for Grade 5 meet the expectations for focus by spending the majority of class time on 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 Bridges sessions or Number Corner activities focused on supporting clusters 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.
All calculations, including the number of units, the number of modules (chapters), the number of sessions (lessons), and instructional time, when considering both sessions and Number Corners together, equal more than 65 percent of the time spent on major work of the grade.
- Units: 6.5 out of 8 units spend the majority of the time on major work of the grade, which is approximately 81 percent. Units 1, 2, 3, 4, 5 and 7 spend all or most of the instructional time on major work of the grade. Unit 8 does not spend most of the instructional time on major work of the grade. Units 6 spends instructional time on major work of the grade about half of the time.
- Modules (chapters): 25.5 out of 32 Modules spend the majority of the time on major work of the grade, which is approximately 80 percent. Modules that spend about half the time on major work of the grade are: Unit 1, Modules 1, 2, and 3; and Unit 8, Modules 1 and 2. Modules that do not spend time on major work of the grade are Unit 6, Modules 1 and 2.
- Bridges Sessions (lessons): 115 out of 146 session spend the majority of the time on major work of the grade, which is approximately 79 percent. Approximately 45 percent of Bridges sessions, or 66 out of 146 Bridges Sessions, address operations of fractions, a critical area and major work in Grade 5. For example, in Unit 2 all 18 sessions address concepts related to adding and subtracting fractions and in Unit 5, 17 out of 18 sessions address concepts related to multiplying and dividing fractions. In addition, Unit 6 has 3 sessions which address dividing fractions, and Unit 7 includes an additional 9 sessions related to dividing fractions. Approximately 42 percent of Bridges Sessions, or 62 out of 146, address whole number and decimal operations which is a critical area and major work in Grade 5. For example, in Unit 3, 14 out of 18 sessions address whole number or decimal operations, including adding and subtracting decimals and whole number division using the area model. In Unit 4, 15 out of 18 sessions address multiplying and dividing whole numbers and decimals as students are introduced to strategies for multiplying and dividing whole numbers and decimals with a progression that includes using arrays to eventually introduce the standard algorithm for multi-digit multiplication.
- Instructional Time, including Bridges sessions and Number Corner activities: 9,200 out of 11,960 instructional minutes are spent on the major work of the grade, which is approximately 77 percent. While teaching just sessions alone will ensure the majority of instructional time is spent on major clusters of the grade (6,900 out of 8,760 instructional minutes or approximately 79 percent), the content in Number Corner activities is also an essential component of the curriculum and will additionally support students’ work and practice on major and supporting clusters of the grade (2,260 out of the 3,200 instructional minutes or approximately 72 percent).
The instructional materials reviewed for Grade 5 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 made between domains and clusters within the grade level, however the materials lack visible learning objectives shaped by CCSSM cluster headings. Overall, the Grade 5 materials partially support coherence and consistency with the progressions in the standards.
Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The instructional materials reviewed for Grade 5 meet the expectations for supporting content enhancing focus and coherence simultaneously by engaging students in the major work of the grade. Overall, the materials engaged students in major clusters of the grade while focusing on supporting clusters. For Grade 5, reviewers focused on the use of data and conversion of measurements as methods for supporting operations with whole numbers, fractions, and decimals, as well as understanding the place value system, in both the daily sessions and the Number Corner activities:
- In Unit 3, Module 3, session 1, students are converting megabytes to kilobytes; the teacher direction 8 on page 6 says, “Guide students to see and describe patterns in the placement of the decimal point when a decimal number is multiplied by powers of 10.” Also, the directions suggest use of place value charts to demonstrate the multiplication.
- In Unit 3, Module 3, session 2, the forum (student discussion) is centered on the work from the day before, converting megabytes to kilobytes using place value and operations involving decimals.
- In Unit 3, Module 3, session 3, when students are doing metric conversions, the visual representation shows a table with kilograms (1, 0.5, and 1.5) and grams (1,000, 500, and 1,500). It asks how many grams are equivalent to 1.5 kilograms and suggests showing multiplication by ½.
- In the December Calendar Collector during Number Corner, students are engaging in measurement conversions and working with data with fractions on a line plot, using the context of student heights and foot lengths.
- In the February Calendar Collector during Number Corner, students are working with conversions involving liters, and in the February Solving Problems, students are solving conversion problems, all involving understanding of the place value system as well as whole number and decimal operations.
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 5 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 of the sessions are 60 minutes of instruction or assessment, however some sessions, especially in Unit 8, are up to 160 minutes of instruction.
- In addition to daily sessions, daily Number Corner activities are an essential component of the curriculum.
- 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.
- 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 time 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 of days or time if necessary.
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 5 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:
- Students engage with previous, grade-level content in Unit 1: “In Unit 1, an exploration of volume serves as a bridge between fourth and fifth grade. Working with volume provides a context in which students review and extend skills and concepts from fourth grade, while introducing skills and concepts that are central to this year’s studies.” (From the unit overview, page ii.)
- Previous standards are noted in the sessions in which students engage in them. For example, in Unit 1, Module 1, Session 4, students are working with cubes to find different rectangular prism arrangements for 24 cubes, working with factors related to a base that makes 24 cubic units, and recording their arrangements with expressions that use parentheses, which connects 4.OA.B.4 with Grade 5 concepts of understanding volume and writing numerical expressions.
- In Unit 4, Module 2, Sessions 3 and 4, students engage with Grade 6 content (6.G.A) on surface area (“How Much Cardboard?” and “Cardboard Forum”). The above grade-level standard is not marked or addressed as above grade level in any way.
- 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, in Module 1, Session 5, students are “Varying the Surface Area” (6.G.A). In Module 3, Session 3, "Students use what they have learned about solar energy to design a solar house and draw it to scale.” Vocabulary listed is “Scale Factor” (page 11) and teacher direction 3 (page 12) says, “Review the terms scale and scale factor” (7.G.A.1). The above grade-level standard is not marked or addressed as above grade level in any way.
Materials give all students extensive work with grade-level problems:
- In daily sessions, 138 out 146 provide an opportunity for students to engage with grade-level problems through a Problem & Investigation or a Problem String.
- In Number Corners, 91 out of 160 provide opportunities for students to engage in grade-level problems through Calendar Grid, Solving Problems, and Problem String activities.
- Suggestions for support or challenge are noted in the Teachers Guide in the Differentiation Table at the end of the introduction pages for each unit. Students are still working with the grade-level standards, but with modifications. For example, in Unit 3, Module 2, Session 6 during Work Places, teacher direction 12 suggests for support to "Suggest specific Work Places for struggling students to work on critical skills."
- 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" (Units 1-8 last page of Unit Introduction).
- Within the sessions, many of the practice pages related to the lessons are listed as "optional daily practice" and optional "home connections." While there are additional opportunities for extensive practice of grade-level work in these components, they are always listed as optional.
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.
- Unit 2, Module 1, Session 4 clearly identifies prior, grade-level standards (4.NF.1) in using equivalent fractions to add and subtract fractions with unlike denominators (page 21).
- Unit 4, Module 1, session 2 clearly identifies prior, grade-level standards as being reviewed or extended including 4.OA.4, 4.NBT.5, and 4.NBT.6 (page 9).
- Unit 4, Module 2, sessions 1, 2 and 4 clearly identify 4.NBT.5 connecting fractions to whole numbers and playing multiplication games (pages 3, 11 and 25).
- Unit 4, Module 3, sessions 1, 2, 3, 4, 6 and 7 clearly identify 4.NBT.5 in multiplying multi-digit numbers using the algorithm with arrays and partial product strategies (pages 3, 9, 13, 19, 31 and 37).
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 5 partially meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards. Overall, the materials lack learning 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.
There are missed opportunities to attend to cluster-level headings in Bridges sessions, including:
- The "Introducing Common Denominators" session does not attend to the concept of fraction equivalence when adding and subtracting fractions.
- The "Whole Number and Decimal Place Value" session does not attend to "understanding the place value system."
- The "Multiplication and Volume" session does "relate volume to multiplication and addition" but does not attend to "understand concepts of volume."
- The "Multiply Fractions by Fractions" session does not attend to "apply and extend previous understandings of multiplication and division to multiply and divide fractions."
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 1, Module 2, Session 2: 5.OA.2, 5.NF.5.A, 5.MD.3.A, 5.MD.B and 5MD.5.A (making connections between volume and multiplication).
- Unit 2, Module 2, Session 4: 5.NF.3 and 5.NBT.7 (word problems involving division, fractions, and decimals).
- Unit 4, Module 3, Session 7: 5.NBT.6, 5.NBT.7 and 5.MD.5.B (problem solving including division, decimals, and volume).
Rigor And Mathematical Practices
The materials reviewed for Grade 5 partially 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 three 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. While the materials attend to the specialized language of mathematics, there are places where above grade-level vocabulary is superficially taught. Overall, the materials for Grade 5 partially meet the expectations for Rigor and Mathematical Practices.
Rigor and Balance
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The materials reviewed for Grade 5 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 5, fluency and procedural work includes 5.NBT.B.5 ("Multi-digit Multiplication"). Application problems occur regularly throughout both the Bridges sessions and the Number Corner activities.
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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.
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 5 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 throughout 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, equations, and ratio tables. Problem Strings address conceptual understanding in Units 1, Unit 2, Unit 3, Unit 4, Unit 5 and Unit 7 to address 5.NBT.5, 5.NBT.6, 5.NBT.7 and cluster 5.NF.B.
- A Math Forum structure is used throughout the units, which allows students to share their thinking, ask questions, and explore key concepts. For example, in Unit 5, Module 3, Session 2, Multiplying Fractions Forum, teacher direction 2, (page 8) suggests the teacher partner students who worked on the same problems, have them compare their models and answers, figure out correct answers, and compare and contrast their models “How are they similar and how are they different?” Direction 4 says just a few students will share their work during the forum while the rest of the class listens, makes observations, and ask questions. Direction 5, says “Help students be explicit about where they showed the fractions in their models.”
- 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 5, Module 4, Sessions 1, 2 and 4, begin with "Reviewing Sharing and Grouping Division" (Session 1), showing pictorial models for the types of division, then moves into "Grouping Stories" (Session 2) modeling division of whole numbers by fractions with number lines, tape diagrams, and equations. Then "Sharing Stories" (Session 4) models fractions divided by whole numbers with area models, number lines, and equations.
- In Unit 7, the Teaching Tips (page iv) include a heading titled "Models and Strategies Rather than Formal Procedures" which encourages teachers to "Refrain from making fraction division, as well as decimal multiplication and division, procedural (as in "do this and then do that"). Allow students the time to experience and process the models and the strategies so that they become natural extensions of using the relationships to think about the problems."
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 5 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 designed to develop and maintain fluency." (page v, Teacher Manual)
- Each month's Number Corners contains a Computational Fluency component, for example: "Multiple Game" in August/September, "Quotient Bingo" in March, and "Fraction Splat" in May/June. Each month also contains a Problem String component that focuses on computation with a strategy, for example: "Multiplication and Division" in December, "Multiplying Whole Numbers by Fractions" in February, and "Fraction Addition & Subtraction" in March.
- Problem strings are used in sessions throughout the school year. "The goal is to help students develop more efficient ways of solving a particular kind of problem." (Teachers Manual Unit 1, Introducing Bridges Mathematics). For example, in Unit 4, Module 1, Session 2, "The Product Game, Version 2," students begin the session with a problem string called “half-ten facts” in which they apply the strategy of multiplying double- and triple-digit numbers by five by first multiplying by 10 and then halving the product.
- Attention to the Grade 5 fluency requirement of multi-digit multiplication using the standard algorithm (5.NBT.B.5) includes Unit 4, Module 3, Sessions 3 and 4, "Array to Algorithm, Part 1 and Part 2," when students translate from the array model of multi-digit multiplication to the standard algorithm. In Session 5, students are "Practicing the Standard Algorithm."
- There is a multiplication algorithm check point assessment in Unit 4, Module 4, Session 1.
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 5 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 vi, Number Corner Volume 1).
- 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 multiple sessions 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 2, Module 2, Sessions 4 and 5, Problems & Investigations and Math Forum, Better Buy 9 (5.NF.3 and 5.NBT.7) and Unit 4, Module 1, Sessions 3 and 4, Problems and Investigations, Callie’s Cake Pops and Math Forum (5.NBT.7).
- Unit 8 is a complete project-based application unit where students are working on designing solar homes. Sessions mostly combine measurement and data standards, with geometry and fractions standards, in real-world, problem-solving tasks and situations.
- Many student pages and sessions throughout the materials contain application problems aligned to the work of the grade, however most of the multi-step, non-routine problems are at the end of the pages and labeled "Challenge."
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 5 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 and fluency and conceptual understanding to solve application problems. In Unit 2, Module 2, Session 1, students begin by modeling with arrays of tiles to multiply whole numbers by fractions. Next, they are posed with the River Trail investigation in which students create a course and label landmarks according to distance traveled. Then students further practice their skills through the Cafeteria Problems Home Connection (page 9).
- Application problems often call for students to model their thinking through the use of area models, number lines, ratio tables, etc.
- Procedural skill and fluency is often noted side-by-side as students are working in conceptual models. For example, in the Unit 7, Module 2, Session 3 Problem String, students write equations and use clock models while working with fraction division.
- Problem strings 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 or ratio table, for example) to represent students' strategies." (Teachers Manual Unit 1, Introducing Bridges Mathematics).
- Application is the focus in Unit 8, when students are designing solar homes.
- Procedural skill and fluency is attended to separately in the Number Corner component "Computational Fluency."
Mathematical Practice-Content Connections
The materials reviewed for Grade 5 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 5 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. While the materials attend to the specialized language of mathematics, there are places where above grade-level vocabulary is superficially taught. Overall, the materials partially meet the criteria for practice-content connections.
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
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 5 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 MPs identified for each Bridges session and Number Corner activity. Students using the materials as intended will engage in the MPs along with the content standards for the grade.
- The Introduction to Bridges Grade 5 includes a table describing what the MPs look like for Grade 5 for each practice.
- All eight MPs are identified throughout the curricular materials.
- The Bridges overall scope and 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. The MPs are also noted in the Solving Problems component of the Number Corner scope and sequence for the months of: November, December, and May/June, and they are identified in the “Skills and Concepts” section at the beginning of every Number Corner activity type.
- "Math Practices in Action" is 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 1, Module 4, Session 3, the "Math Practices in Action" calls teachers' attention to the fact that having students generate their own division story problem is one way of engaging them with MP2 (5.NBT.6).
- In Unit 3, Module 3, Session 4, "Place Value Patterns," MP7 and MP8 are identified and in the "Math Practices in Action - MP8," it discusses how problem strings are a wonderful opportunity to engage students in looking for and expressing regularity in the context of the Problem String (5.NBT.1, 5.NBT.2, and 5.NBT.7).
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 5 partially meet the expectations for carefully attending to the full meaning of each MP. Overall, the materials often attend to the full meaning of each MP, but there are instances where the students are not using the MPs as written.
- "Math Practices in Action" (MPiA) is located in the margin of the teacher notes within the Bridges sessions. They call the teacher’s attention to how the activities within the Bridges sessions engage students with a particular MP. For example, in Unit 1, Module 4, Session 3, the MPiA calls the teacher’s attention to the fact that having students generate their own division story problem is one way of engaging them with MP2.
- In many cases, the materials attend to the full meaning of the MPs:
- In Unit 2, Module 2, session 5, MP2 is identified, and in the "Math Practice in Action - MP2," it explains that the Problem String allows students to embed the subtraction problems in a money or time context and then write an equation.
- In Unit 3, Module 2, session 3, MP3 is identified, and the "Math Practices in Action - MP3" provides an explanation for how students will be engaged this practice by looking at a hypothetical student's work, which presents common misconceptions.
- In Unit 3, Module 2, session 6, MP8 is identified and the "Math Practices in Action - MP8" discusses how a teacher can guide an inquiry into filling out a fraction and decimal chart provides students with the opportunity to look for and express regularity in reasoning by identifying fraction and decimal equivalents.
- There is ambiguity over whether "model" means to draw a picture representing the problem or whether "model" means to create a mathematical representation. For example, Unit 2, Module 1, Session 3, “Clock Fractions,” lists MP4, Model with Mathematics. In the lesson, students are finding equivalent fractions using a clock model. There are no real-world problem situations presented in the lesson.
- In some cases, when MP1 is identified, students are not engaging in the full meaning of the MP. For example, Unit 3, Module 1, Session 2 “Beat the Calculator: Fractions” lists MP1. Within the session, students learn and play a game called Beat the Calculator where, “one player uses a calculator and the other uses learned strategies as they race to find the sum or difference of fractions shown on a card.” This does not attend to the full meaning of MP1.
- The use of the "Math Forum" structure throughout the Bridges program is a missed opportunity to fully attend to MP3. For example, in Unit 2, Module 2, Session 2 - "River Forum," students meet in small groups to compare answers and share strategies for solving the problems. MP3 is identified. Although students are sharing their strategies for how they got their answers, there is no mention of students justifying their thinking or questioning their peers. All of the questions are posed by the teacher. An opportunity is missed in that the teacher notes do not provide information on how to facilitate a discussion in which students justify their thinking and critique the reasoning of each others.
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
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 5 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 provided opportunities throughout the year for students to reason by both constructing viable arguments and analyzing the reasoning of others, however, while the student materials often prompt students to reason by constructing viable arguments, there is less consistency and opportunities for students to analyze the arguments of others, which leads to a lack of balance. More opportunities could be provided in the student materials, other than formative and summative assessments, to engage in analyzing the arguments of others.
- In the Unit 2 "Working with Fractions Checkpoint," question 4c says, “When Erik solved this problem, he got 5/4 of a mile for his answer. Is this a reasonable answer? Why or why not?"
- In the Unit 3 Post-Assessment, problem 7c says, “Sara was looking at the table above. She noticed that when you multiply by a power of 10 (10, 100, 1000, and so on), the decimal point moves over to the right by one place every time. Explain to Sara why it works this way.”
- In the Unit 3, Module 2, Session 3 Decimal Equivalencies Work Sample, problem #1 says, “Jacob says that 0.400 and 0.004 are equal. Do you agree with him? Use numbers, words, or labeled sketches to explain your answer.” and problem 2 says, “Ivy says that 0.6 and 0.600 are equal. Do you agree with her? Use numbers, words, or labeled sketches to explain your answer."
- In Number Corners, the October Calendar Collector Activity 4, Question 8 asks the student to "Write at least four statements about how this carrot experiment was similar to and different from your own class's carrot experiment." The November Calendar Collector Activity 3, Question 5 states "Cameron said, 'Since there is 1,000 meters in a kilometer, meters must be bigger than kilometers.' Respond to Cameron."
- “Math Forums, which occur a few times in most units, are a more formal and structured time for students to share and discuss their work. Students who are not sharing their own work are expected to listen carefully, compare their classmates’ work to their own, and ask questions to better understand each students’ ideas.” For example, Unit 6, Module 3, Session 3, “Matt’s Marbles Math Forum,” he teacher directions say, “Invite students to present their work. After they have finished, ask the other students if they understood what the students did and whether anyone else used the same or similar approach. Invite the rest of the class to ask questions, and have the presenters respond to the questions.”
- There are many prompts for students to explain how they got their answers or show their work, but there are also missed opportunities to evaluate the thinking of others. In most of the problems on assignments, the directions say, "Show your work" or "Use numbers, labels, models or words to show your thinking." For example, Unit 6, Module 1, Session 5, “Tile Pool Challenge,” problems 1 and 2 say, “Use numbers, words, or labeled sketches to explain how you got your answer.” And, in the Unit 1 Multiplication & Volume Checkpoint, question 5 says, “Show your thinking using words, numbers, or labeled sketches.”
- There are numerous examples in Number Corner in which students are asked to "Use the space to solve the problem and record your thinking with numbers, words, equations, or models." For example, in the September Number Corner Calendar Collector Activity 3, question 3 states "Raj and his partner disagree about the following equations. Tell whether each is true or false." The students need to label each of the three equations as true or false, but are not prompted to explain the reasoning for their decision.
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 5 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, sample dialogue is provided to assist the teacher in engaging students in constructing viable arguments. For example:
- November Number Corner Solving Problems Activity 2 says, “And how did you find ¼ of 36?” and “It’s interesting that you cut yours a different way than the first pair did. Can you tell us why?”
- In Number Corner February Solving Problems Activity 4, the teacher asks, “How did you know that?” and “Can you say more about how you knew that 0.1 liters was the same as 100 milliliters? I didn’t hear how you knew that.” The teacher in the sample dialog asks “Serafina, tell us about your thinking please. I noticed some similarities in how you and Willie got started.” Later the teacher states, “I am curious about how you knew that 600 ml was 0.6 liters. Is anyone else wondering about that?”
- In Bridges session 2, Unit 3, Module 4, the teachers says, "I'm really stuck on this problem... how could you help me figure out what to do without just telling me? How could you help me help myself through this problem?" and, "What do you mean?"
- In the Bridges Number Corner, the October Teaching Tips prompts teachers to build a sense of community this month. "In particular, help students develop their discussion skills. Encourage them to respond to each other's ideas, whether they agree and add on or offer a different idea. Help students understand that they are accountable for participating in Number Corner conversations."
- In February Number Corner Solving Problems (Day 1), direction 7 says "As students finish the problems, have them visit with another student about their thinking. Encourage them to look for similarities and differences in how they solved the problems.
- MP3 is mentioned specifically 8 times throughout the Bridges sessions in the "Math Practices in Action," which support teachers in understanding how the MP is applied in the sessions.
- MP3 is listed in Unit 1, Module 3, Session 2, where students work on a problem string. The "Math Practices in Action" explains, "Discussing how different strategies work and when they might be most useful is part of constructing viable arguments and critiquing the reasoning of others." Within the lesson, the teacher directions say, "Use open arrays, expressions, and equations to represent students' thinking." and "Write the term partial products and summarize the key idea of using partial products efficiently to solve multiplication problems." The teacher directions lead the teacher only to prompt the students to explain how they solved the problems in the Problem String, and then direct the students to the partial products method. The final direction says, "Have the students turn to a partner and summarize how to use partial products to solve a multiplication problem." There is a missed opportunity to assist the teacher in having students construct an argument or critique the arguments of others.
- In the February Number Corner Solving Problems (Day 5) direction 4 prompts the teacher to invite students to put their work under a document camera to show how they solved a problem. The teacher should look for students to describe strategies that demonstrate multiplicative thinking in their descriptions. There are no directions prompting students to ask questions that would require critiquing, justifying, or responding to feedback in any way.
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Grade 5 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. There are also some instances in the materials where vocabulary that is not grade appropriate is introduced in basic, incomplete ways.
- The introduction to the series explains, “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.
- Students using these materials keep a math journal with a “handbook” section where they record mathematics vocabulary. For example, Unit 2, Module, 1, Session 1 gives the teacher direction, “Wrap up the string by having students add the terms numerator and denominator to the handbook section of their math journals. Record each term, along with a class definition of each and at least one example generated by the group on the board or at the projector as students do so in their handbooks.”
- Students are often supported to show their mathematical reasoning using words, diagrams and symbols. For example:
- Unit 2, Module 3, Session 4, the teacher guides the students in recording and discussing different strategies for adding and subtracting fractions. The teacher directions say, "Ask a few students to share how they solved the problems while you record their strategies on the chart paper." The teacher is prompted to lead a discussion about which strategies work work well for different problems based on the numbers, including sample questions such as: "Which problems lend themselves to using money to think about finding common denominators?" and "Which problems lend themselves to using a double number line to think about finding a common denominators?"
- Unit 6, Module 1, Session 5, “Tile Pool Challenge” on the student page, problems 1 and 2 say, “Use numbers, words, or labeled sketches to explain how you got your answer.” Problem 3 says, “What do you have to do to figure out how many tiles it takes to build the water for any arrangement in this sequence? Include a labeled sketch in your explanation.”
- In the September Number Corner Problem Strings the teacher directions say, "Provide guidance for the class in thinking about which strategies make more sense in solving certain kinds of problems." In Problem String 2 the suggestion states, “Work with the class to think about combinations for which removal or take away—which can be represented by backward hops on the open number line—makes more sense than finding the difference and vice versa.”
- The Number Corner November Teaching Tip prompts the teacher to emphasize the mathematical practices of persevering to solve problems and using precise mathematical language. "Encourage students by asking them to justify their responses and echoing back to them mathematical language in the discussions. For example, as students describe a triangle as having all equal sides, respond that the sides are congruent or that the measure of the sides are equal." Another example is, "As students talk about sliding or shifting triangles, use the word translation as you discuss their ideas." While teachers are prompted to use accurate vocabulary, the concepts of translation and congruence are above grade level and not fully developed in these lessons.
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.
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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 unit cubes, geoboards and protractors are used throughout the instructional materials as mathematical representation and to build conceptual understanding.
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 3, Module 1, Session 3 “Multiplying and Dividing by Ten,” students are solving problems using money notation and considering the place value patterns that emerge when multiplying and dividing decimal numbers by 10. During the “Problems and Investigations” portion of this lesson students work with a scenario in which ten friends are planning an outing to the art museum and have some questions about the money they will spend. During this work time, students are asked clarifying questions to check understanding, then they share answers and sketches as a group. They also discuss their finding of patterns they noticed when using the number ten and the place value patterns that emerge within these findings. On the Student Book page, the students use a calculator to predict and record the results of multiplying and dividing by ten. They look for the patterns and record those observations.
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, standard 5.NF.1 (add and subtract fractions with unlike denominators, include mixed numbers) is introduced in Grade 4, developed and mastered in Unit 2, and is reviewed/practiced/extended in Units 3 and 5. This standard continues to be developed in Number Corners from October through January, and in March through May. Another example is standard 5.NBT.2 (explain patterns in the number of zeros in the product when multiplying by powers of 10). This standard is developed in Unit 3, and mastered in Unit 7. It also is practiced during Number Corners from November through February.
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 5, Module 1, Session 2 students are considering ways to multiply a whole number by a unit fraction. They begin with a partner discussion as to how they might find the answer to the given combination. To further their understanding, the students are asked to work with a problem using a common rather than a unit fraction. Students pair up to share ideas before solving. Once they have their strategies in place, they play a game practicing and reinforcing the learning.
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 unit cubes to fill a rectangular prism, rulers to measure, geoboards, protractors, and graph paper to accurately represent problems.
For example, in Unit 6, Module 3, students are moving from using discrete counting to the continuous measurement definition of volume. Students are working to pack boxes of 1-inch marbles using paper cut-outs as boxes. They investigate how many marble boxes they can pack in each of the larger boxes for shipping. They predict, measure and review dimensions to solve the task. Once complete, they have a class forum (discussion) to share the results and check for accuracy.
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 5. 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 6, Module 4, Session 2, the students are reading a passage about flags in which they learn that flag-makers use special terms for the width and length of a flag. The font size and print are correct for a typical 5th grade student. The students are determining the dimensions and areas of various flags. The student pages are clearly marked and enough work space is given to complete the problem.
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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.
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 1, Session 3 during a number string on fraction addition and subtraction, teachers are prompted to ask the following questions:
- "Which representation are you most likely to use?"
- "Is it easier to visualize 1/5 or 4/20?"
- "Which would you rather have in your pocket, 2 dimes or 20 pennies? Why?"
In Unit 3, Module 2, Session 2 - When discussing a fraction game, the teacher is prompted to ask the following questions:
- "How can you be sure you made the smallest or largest decimal possible?"
- "Is there ever a reason to not make the smallest or largest decimal possible?"
- "How do the wild cards change the game?"
In Unit 5, Module 2, Session 4, - As students are exploring multiplication with fractions, the teacher is prompted to ask the following questions:
- "How is multiplying with fractions different from multiplying with whole numbers?"
- "How is multiplying with fractions similar to multiplying with whole numbers?"
- "If you multiply a fraction less than 1 by another fraction less than 1, what kind of product will you get?"
In Unit 7, Module 4, Session 3, - students are working with decimal operations. The teacher is prompted to ask the following questions:
- "Given a situation that involves dividing $94.00 by 8, would an answer of $1.175 make sense? Why or why not? What about $11.75 or $117.50?"
In the December Number Corner Calendar Collector activity, the following questions are provided in the "Key Questions" section in the margin:
- "What does the ordered pair (x, y) represent in this situation?"
- "What does this line plot show about the heights of students in our class?"
- What is similar about these line plots? What is similar about them?"
- Imagine we added data about 10 more 5th grade classrooms to each graph. How do you think that would change the graphs? In what ways would the graphs remain similar?"
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 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
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, "Understanding equivalence is critical to adding and subtracting fractions. We want students to have many meanings come to mind when they see a fraction. For 1/4, for example, a student might think of 1 quarter, 25 cents, $0.25, half of 1/2, double 1/8, 25%, dividing something by 4, 1/4 of an hour, 15 minutes out of 60 minutes, a distance 1/4 of a unit from 0, and so on. Then when students see 1/4 added to another fraction, they can use the meaning that is most helpful, given the denominator of the other fraction."
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 5 Mathematics.
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 the Types of Assessments in Bridges sessions and Number corner.
The CCSS Where to Focus Grade 5 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.
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.
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 5 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 5 Unit Overviews for Units 1-8, in English and Spanish.
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."
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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.
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 fourth grade – fluency with basic multiplication and division facts, solve story problems involving all four operations, compare and order fractions, add and subtract fractions and mixed numbers with like denominators, multiply fractions by whole numbers, and write and compare decimals to hundredths. The Comprehensive Growth Assessment contains 41 written items, addressing every Common Core standard for Grade 5. 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.
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 4 Session 2 “Support” section gives suggestions for struggling students. The materials suggest that the teacher should encourage students who want or need to build with the base ten area pieces before sketching to do so. An additional suggestion is that pairs have one partner model the problem with base ten area pieces while the other makes the sketch on the problem sheet.
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 4, Module 3, Session 4, the teacher leads students to apply what they already know about area models to using the standard algorithm to multiply multi-digit whole numbers (5.NBT.5). Ongoing review and practice is often provided through Number Corner routines. Each routine builds upon the previous month’s skills and concepts. For example, in the Number Corner November Problem Strings students revisit the previous month’s work with money and clock models in order to subtract unit and non-unit fractions with unlike denominators (5.NF.1).
Materials offer ongoing formative and summative assessments:
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 2, Module 2, Session 6, Fraction Addition & Subtraction Checkpoint includes a Checkpoint Scoring Guide that lists each prompt, the correct answer, the standard, and the points possible. The Unit 2, Module 3, Session 6 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 4 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 January Number Corner Checkup 2 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.
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.
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.
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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.
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 4, Session 1, students are working on the activity "Writing Division Story Problems." Support is offered: "...give students who are unable to finish during the math period extra time to complete the assignment before the end of the day." Challenge is offered: "Ask early finishers to prepare a second and even a third problems for their classmates."
In the Unit 2, Module 2, Session 1 "River Trail" activity, the following suggestions are provided:
- ELL: "Gather students and have them restate the task. Ensure that students understand the information they need to place at each landmark location.”
- Support: "Model how to create a 30-inch line with a measuring tape to simulate the trail and help students understand that it represents the full river length. Suggest that students attach their measuring tape to the butcher paper with tape.”
- Challenge: “Pair similar ability students together. Ask students to look at the list of landmarks, predict markers that will be placed at the same location, and then justify their thinking mathematically.”
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 is 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.
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 4, Module 1, Session 4, students are sharing their strategies for the problems from the previous day’s activity about a fundraiser "Callie’s Cake Pops" in a math forum. Students worked in pairs to find the cost and profit for the fundraiser using any strategy and representation they wanted in multiplying decimals and whole numbers and then created a poster to present their solutions and strategies. In this session the teacher has pairs of students meet and compare their strategies with a particular focus on how they dealt with the decimals. Several pairs of students are asked to share their posters to highlight specific strategies they used to find the cost and profit for the fundraiser including: 1) splitting the dollars and cents, 2) changing the money to equivalent fractions, 3) using a ratio table to keep track of the money and cake pops, and finally 4) the doubling and halving strategy.
Another example is found in the Number Corner February Problem Strings. Students are solving a several strings of problems involving multiplication of whole number by fractions posed by the teacher using any strategy they want. Students’ strategies are modeled using various arrangements of counters and also various equations, which highlight the associative and distributive properties. The teacher solicits students’ strategies by prompting students to “Tell me how you thought about this one.” The teacher follows this question with, “ First, is the answer going to be greater than 16 or less than 16?”, in order to support students generalizations about multiplying fractions by whole numbers.
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 2, Session 3 provides a ELL suggestion. The suggestion reads as follows: "Make sure students understand the problem. Write the problem in numbers as you say it in words. Draw the double number lines and the jumps as you talk about what is happening. Point to the jumps and the landing places as you write the corresponding numbers and fractions.”
- In Unit 5, Module 4, Session 3, students are working on solving problems where they divide a whole number by a fraction. The following Support suggestion is provided: "Encourage students who may be getting frustrated at this point to see if they can draw a picture that might help them understand the situation."
- In Unit 6, Module 2, Session 2, students are classifying quadrilaterals by their characteristics. The following ELL suggestions is provided as follows: “This is a particularly language-intensive assignment. You will almost certainly want to pair ELL students with partners who speak both languages. You might also have students record the name of each shape on the hierarchy in their own language as well as, or instead of, English.
- In the Number Corner April Computational Fluency, students are placing decimal on a number line using various strategies. The ELL suggestion is: "Make sure students understand the problem. Emphasize or review key vocabulary. You may want to have students work in pairs to discuss and solve the problems."
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 1, Session 1, students are solving problems that require addition of fractions with unlike denominators. The “Challenge” suggestions is as follows: “Encourage students to use the most efficient or sophisticated strategy they can think of. Then encourage them to look back at their work and see if they can think of an even more efficient strategy.”
- In Unit 4, Module 2, Session 4, students are solving equations that require multiplying whole numbers by fractions as part of the activity “Over & Under”. The "Challenge" suggestion is as follows: "Ask students to partner with another student and discuss the order in which they solved the problems. Ask them to verbalize a “path” they could take that minimizes the amount of work to solve one or both sets of problems.
- In the Number Corner January Calendar Grid, students are graphing patterns. The Challenge suggestions are as follows: "Invite students to make up their own number pattern and graph it instead of doing a pattern from the Calendar Grid. They can work independently, trade graphs with another student, and figure out what the rule was for generating the ordered pairs."
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, such as baseballs, marbles, skateboards, money, apples, pet supplies, art supplies, and turtles, ducks. When people are shown, they are cartoons that appear to show a balance of demographic and personal characteristics.
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 independently solve problems involving addition of fractions and then share their strategies during the Problem String “Clock Fractions” while the teacher models their thinking with analog clock faces and fraction equations. Next the whole class plays a round of “Clock Fractions” against the teacher to familiarize students with the game and review strategies for adding fractions. Students then summarize the directions to the game and play “Clock Fractions” in pairs. Finally the teacher brings the class together to make observations about the game.
In Unit 3, Module 1, Session 2, students independently solve problems involving addition of decimals and then share their strategies while the teacher models the Give and Take strategy with an open number line. Next the whole class plays a few rounds of “Beat the Calculator: Fractions” against the teachers and share summarize the directions to the game with a partner. Next students work independently or with a pair one from a menu of activities including “Beat the Calculator: Fractions” introduced today.
In the Number Corner March Computational Fluency, the class plays "Quotient Bingo." The students against the teacher, then in pairs, and then discuss their strategies for playing the game and any insights or discoveries they made about the game as a whole class.
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.
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.
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 Angle Shoot, Battleship Numberline: Decimals, Interactive math dictionary, Clara Fraction Ice Cream Shop, and Dirt Bike Comparing Fractions
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.
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 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.
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.
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.
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.