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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Challenge activities found throughout the instructional materials include the following:

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

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