5th Grade - Gateway 2

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."

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.

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."

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."

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).

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.

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.