4th Grade - Gateway 2

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

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

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

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

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

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