5th Grade - Gateway 3

The instructional materials for Math Expressions Grade 5 meet expectations that materials distinguish between problems and exercises.

Materials provide the opportunity for students to learn new mathematics through problem solving activities. In a typical lesson, Activity 1 and Activity 2 develop the new math content of the lesson. Lessons are outlined according to an Inquiry Lesson Path based on four phases: Phase 1 Guided Introduction, Phase 2 Learning Unfolds, Phase 3 Knead Knowledge (practice stage), and Phase 4 Maintaining and Integrating Fluency. Students build mastery through practice problems/exercises. In a typical lesson, during Activity 2 and Activity 3, students complete problems in the Student Activity Book which provide practice with the math content. The purpose of each Activity within a unit is explained in the “Teaching the Lesson Section” found on the first page of each lesson.

Examples include but are not limited to:

• In Unit 3, Lesson 2, Teaching the Lesson Section, Activity 1, Extend Unit Fraction Multiplication, is stated as important because “Through the use of number lines, students will expand their understandings of the connection between multiplying by a unit fraction and multiplying by a non-unit fraction.” Activity 2, Multiplication with Fractions Practice Metric, is stated as important because “This multiplication practice with unit and non-unit fractions will use skills developed in the previous activity to help students understand the relationships in multiplicative comparison problems.”  
• In the Student Activity Book, Unit 8, Lesson 4, Activity 2, students count cubes to find the volume of a rectangular prism: “Find the number of cubes and the volume.” A picture of a prism is shown that is 5 cubes in length, 4 cubes in width, and 3 cubes in height. In Activity 3, students compare measurement of volume with standard and improvised units. Teachers prompt students to build a rectangular prism using their math books, describe the volume of the prism, and try to build another prism the same size with a different set of books.

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide tasks in an intentional sequence.

The design of the assignments follows a natural progression, leading to full understanding and mastery of new mathematics. Lessons follow a consistent pattern of two or three activities per lesson. Activity 1 usually focuses on the new learning. This learning is reinforced in Activity 2, and then students practice the new learning by completing Student Activity Book pages during Activity 3. Activity 3 either reinforces the new skill, or it reviews previously learned content.

Examples include but are not limited to:

• In Unit 2, Lesson 5, students discuss various strategies for making a new ten when adding. Next, students practice adding decimals and whole numbers and finally adding money.
• In Unit 4, Lesson 4, students multiply 2-digit numbers using the Place Value Rows Method (area models). Then students transition to multiplying using the Short Cut Method (traditional algorithm).
• In Unit 5, Lesson 4, students explore different situations and discuss what the remainder means. Then students solve two-digit division problems and interpret the remainders.

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide varied opportunities for students to present their mathematical knowledge.

Examples of how students produce answers and solutions include but are not limited to:

• Using fraction bars to solve problems 
• Using Arrays and Area Models to solve multiplication problems
• Using drawings to make sense of mathematics
• Using bar diagrams to solve division problems
• Using Place Value Charts to compare whole numbers and decimals
• Providing thinking explanations as they answer Check for Understanding questions in the Student Activity Book
• Completing fluency checks and practice in the Student Activity Book 
• Critiquing the Reasoning of others by asking “good thinker questions” and using “good justifications” 
• Practicing “good explanations” 
• Identifying the error and correcting it (Puzzled Penguin)
• Solving problems and exercises in the Student Activity Book

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide virtual and physical manipulatives that are faithful representations of the mathematical objects they represent and are connected to the written material.

Students use a variety of manipulatives including MathBoards, Secret Code Cards, base-ten blocks, Square Inch Tiles, fraction bars, number lines, coordinate grids, Math Mountains, and Math Mountain Cards. Most of the manipulatives are available virtually in the itools found in ThinkCentral. Manipulatives are often connected to written methods when appropriate.

Examples include but are not limited to:

• Unit 1, Lesson 1, students use MathBoard Fraction Bars to use unit fractions to build equivalent fractions.
• Unit 2, Lesson 2, students use Secret Code Cards to model whole numbers and decimals and write the numbers in standard, expanded, and word form.
• Unit 7, Lesson 5 students use coordinate grids to name the ordered pair for a point’s location and to place a point on the grid.

The instructional materials for Math Expressions Grade 5 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Examples of teacher support include but are not limited to:

• Questions for teachers to pose are consistently included in the lesson narrative. They are italicized, making them easily visible. 
• MathTalk in Action boxes include questions for the teacher to ask and potential student responses. For example, in Unit 1, Lesson 5, the teacher is guided to ask the questions: “Rashida, will you explain your work? How did you change 8/3 to a mixed number? What did you do next?”
• Teacher Notes are also provided at the bottom of the lesson pages and include questions to deepen students understanding of the mathematics. For example, in Unit 4, Lesson 1, Inquiry Notes state, “As you discuss the generalizations outlined in the three bulleted points use questions such as ‘How do you determine what number to use as the exponent when you write a number as a power of ten? When multiplying by a whole number by a power of ten, how is the exponent related to the number of zeros in the product? When you multiply a whole number by a power of ten, what does the exponent of the power of ten tell you?’”  

The instructional materials for Math Expressions Grade 5 meet expectations that 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. Materials also, when necessary provide teacher guidance for the use of embedded technology to support and enhance student learning.

Ample guidance is provided in the Teacher Guide for planning. The Pacing Guide provides guidance for each unit. Charts show the Learning Progression for the Content Standards Across Grades for the standards addressed in the Unit. A Planning Chart for each Unit that includes Math Activity Center Resources, Big Idea Resources, and Lesson Resources is provided. The Planning Chart also includes the standards addressed in each lesson, the digital and print resources for each lesson, and the assessments for the Unit. A table of the Standards for Mathematical Practice and the lessons where each is embedded is included. Also, a Table of the Math Content Standards and the lessons where they are taught is provided. Finally, a list of Assessment, Review, and Intervention Resources for the Unit is provided.

Examples include but are not limited to:

• Each lesson includes guidance on the focus of each Activity and why it is important. For example, in Unit 2, Lesson 6, Activity 1, Ungroup With Zeros, is stated as important because “By providing several methods to subtract when ungrouping with zeros, students can choose a method that makes the most sense for them and use it to solve subtraction problems with decimals.”
• Each Activity includes an explanation of what the teacher should do or say and includes possible correct responses to questions posed by the teacher. 
• Formative Assessment and Check for Understanding questions are highlighted in the Teacher Guide.
• Math Practices are highlighted in the lesson narratives.
• A list of questions that can be used to build a Math Talk community is included at the beginning of each Unit. 
• Notes at the bottom of each page of the lesson narrative give useful suggestions for implementing the lesson, asking questions, acquiring vocabulary, and building concepts. For example, in Unit 3, Lesson 1, the Teaching Notes for Language and Vocabulary states, “Some students may not equate the word ‘of’ with multiplication. To help them, focus on the ‘groups of’ language below the pictures for Exercises 3 and 4. For example, say and write: 3 groups of 6 = 3 * 6 = 18. Give students a chance to say and write other ‘groups of’ phrases in the same way.”
• Digital Resources for each lesson are highlighted on the first page of the lesson, and itools, which include virtual manipulatives, are shown in the lesson narrative when it may be beneficial to use them. For example, in Unit 6, Lesson 4, a picture of itools Number Lines are shown because they may be used in the lesson.

The instructional materials for Math Expressions Grade 5 meet expectations that materials contain a teacher’s edition that contains full, adult-level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.

Notes are provided at the bottom of each lesson narrative in the Teacher Edition to deepen teacher understanding of the mathematics and to improve instruction. Math Background Notes provide information about the math topic to deepen teacher’s understanding. Watch For! Notes provide information about potential misconceptions and things to watch for as students complete the lesson. What to Expect from Students notes provide information about how students might engage with the math and why the math is important. Building Concepts notes provide explanations of the math and how students learn.

Examples include but are not limited to:

• Path to Fluency Charts are provided.
• Chart of the Addition/Subtraction and Multiplication/Division problem types is provided.
• Table of the Major Work and Major Clusters of the Grade is provided.
• Table of the Common Core State Standards for Mathematical Content is provided.
• Table of the Common Core State Standards for Mathematical Practice with an explanation for each Mathematical Practice is provided.
• The Putting Research into Practice section at the beginning of each unit provides research about best practices in teaching children mathematics.
• The Math Background section, prior to each unit, includes sections that deepen teacher knowledge of the math in the unit. Examples include Learning Path in the Common Core Standards, Help Students Avoid Common Errors, Effective Practice Routines, Relate Mathematics to the Real World, and Focus on Mathematical Practices. 
• The Math Background section, prior to each unit, provides excerpts from the Progressions for the Common Core State Standards.
• The Mathematical Practices section, prior to each unit, provides information on how students will engage with the Practice Standards throughout the unit.
• A Teacher Glossary is provided.

The instructional materials for Math Expressions Grade 5 meet expectations that materials contain a teacher’s edition that explains the role of the specific mathematics standards in the context of the overall series.

A Path to Fluency: Kindergarten through Grade 6 Chart is provided and highlights the fluency requirements of each grade level, activities that target fluency, and interventions for Grades 3, 4, 5, and 6. Also, a Major Work and Major Clusters of the Grade Chart for Grades K-6 is provided. Finally, for each unit, a Learning Progressions for the Common Core State Standards Chart for the domains addressed in the unit, which includes the current, prior, and next grade level standards is provided.

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Examples include but are not limited to:

• The Assessment Guide contains a Prerequisite Skills Inventory Test, organized by Domains, and a corresponding Prerequisite Skills Inventory Test Correlation document. The correlation aligns each question with a description of the prerequisite skill addressed, as well as the DoK level of the question. This correlation document is formatted as a table so each student’s performance by question/skill can be recorded. The Prerequisite Skills Inventory Test is designed to be administered at the beginning of the school year.
• When a student completes practice opportunities and tests in the Personal Math Trainer, all of the performance data and adaptive learning information follows each student to the next grade. 
• Quick Practice activities at the beginning of each lesson are designed to “provide opportunities for students to call to mind their prior understanding of a topic that has already been discussed in class or to begin to build a prerequisite skill for a topic that is to come later” (Teacher Edition page I4). 
• Quick Quizzes and Fluency Checks are embedded within the units to check understanding of Big Ideas prior to moving on to the next Big Idea instruction, and to monitor progress toward computational fluency. For example, Fluency Check 1 assesses multi-digit multiplication (5.NBT.5).
• Students take three progress monitoring assessments to assess grade level skills and concepts students have learned. The Beginning of Year test assesses concepts they will learn throughout the year, the Middle of Year Test shows progress made in the first half of the year, and the End of Year Test measures growth throughout the school year.

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide support for teachers to identify and address common student errors and misconceptions.

Examples include but are not limited to:

• Common student errors are identified for each Unit Review/Test question along with a direction on how to help students. For example, on the Unit 2 Review/Test, if a student misses Questions 5, 8, or 9, the common error identified states, “Student does not read or write the expanded form correctly.” Teachers are directed to “First, decompose the number using the total value of each digit. Then, ask students to factor each value such as 90 = 9 x 10. Finally, show them how to use addition to group the factors.”
• The Math Background section of each Unit provides a narrative called “Help Students Avoid Common Errors”. 
• Puzzled Penguin activities highlight typical student mistakes and misconceptions by challenging students to find the Puzzled Penguin’s mistake and correct it. Teachers are provided questions in order to lead classroom conversations through a MathTalk format that revolve around the mistake and its correction, helping students understand the mathematics.
• Watch For! are teaching notes periodically found in each unit. These notes alert teachers to common misconceptions they should be on the lookout for. For example, in Unit 3, Lesson 3, the Watch For! note states, “Some students may benefit from a brief review of how to multiply a unit fraction by a whole number.”

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Examples include but are not limited to:

• Homework and Remembering pages provide a review of recently taught topics as well as a spiral review throughout the year. The Personal Math Trainer online platform allows students to complete homework tasks for each lesson, receive instant feedback, and  step-by-step guidance if needed.
• Unit Review/Test and Performance Tasks for each unit are found in the Student Activity Book. The author states, “You can use this Unit Review/Test as an end-of-unit review to determine if children have mastered the content of the unit.  You can assess children’s knowledge with one of the forms of the Unit 1 Test in the Assessment Guide.” Teachers are provided with a Data-Driven Decision Making Table which suggests specific reteaching activities for students who incorrectly answer the correlated questions, as well as suggestions for which Standards Quiz to assign in the Personal Math Trainer which provides a personalized intervention for the student. The Performance Task includes a detailed scoring rubric which can be used to provide feedback to students.
• The Personal Math Trainer can be used for homework practice, fluency practice, standards practice, unit pre-tests with instant feedback, and step-by-step guidance when needed. Everything a student completes in the platform helps to improve the adaptive workflow (powered by Knewton Adaptivity) for the student throughout the year.  
• The Knewton Adaptivity, Homework with Daily Intervention and Enrichment can be used in multiple ways in the classroom. A 5-minute Warm-Up provides students with personalized review prior to the assignment. On-level and advanced students may receive less or no warm-up, as determined by Knewton. After the warm-up, the HMH pre-built assignment is given to students. A 10-minute personalized enrichment is provided for students who demonstrate mastery (95% or higher) on the assignment.  Enrichment shows students proximate, forward-looking concepts based on the assignment content.
• Other Formative Assessment opportunities include: daily Check Understanding tasks on select Student Activity Book pages, daily observation with anecdotal notes, observations during Math Talk conversations, and analyzing student work samples and student responses in the Student Activity Book. Portfolio suggestions are also provided at the end of each unit.

The materials reviewed for Math Expressions Grade 5 meet the expectation for offering ongoing assessments that clearly denote which standards are being emphasized.

Examples include but are not limited to:

• Every unit includes two versions of a Unit Assessment, Form A and Form B, found in the Assessment Guide. Both assessments provide PARCC and Smarter Balance question formats and a Standards Correlation Document which can be used to collect student performance data. This document also aligns each question to a DoK Level and Standard(s). 
• Each unit contains a Performance Assessment which can be found in the Assessment Guide. The standards are clearly noted for the assessment as a whole, and not by specific question. 
• There are three Benchmark Assessments (Beginning of the Year Inventory, Middle of the Year Inventory and End of Year Assessment) found in the Assessment Guide. Standards for these assessments are clearly noted on the Correlation Document and DoK Levels are noted.

The instructional materials for Math Expressions Grade 5 meet expectations that assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Examples include but are not limited to:

• Scoring Guides are provided for each Unit Performance Assessment found in the Assessment Guide. Each question is assigned a point value and a rubric is provided to determine Performance Levels 0-3 based on the number of points earned. Additionally, each Performance Level is further defined on a task-specific basis and indicates specifics about student understanding to assist teachers in interpreting student work. Sample student work for each Performance Level is also provided in the Assessment Guide. 
• Answer keys for the Unit Assessments, Form A and Form B, are located in the back of the Assessment Guide. However, no guidance or suggestions for follow-up instruction are included in the Assessment Guide. 
• The online Personal Math Trainer can be utilized to administer Beginning, Middle and End of Year Tests, Unit Assessments, and Fluency Checks. The data from these assessments is collected and analyzed, and a Personal Study Plan is prescribed through Adaptive Workflow settings (through Knewton Adaptivity) based on the data and the mastery threshold percentage established for the assessment. The primary use is for end of the unit assessments, or to provide targeted students with occasional review, intervention, and re-assessment opportunities. Students must complete an initial assignment (test). Students who do not demonstrate mastery receive a Personal Study Plan, consisting of a personalized review and intervention assignment lasting 15 minutes. After completing the Personal Study Plan, the initial assignment is given again, but numbers in the assessment are changed.

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Teachers guide students through an inquiry path to become mathematically proficient. The four stages of the path to learning are guided introduction, learning unfolds, knead knowledge through practice, and maintain fluency. As stated by the publisher, “Within the curriculum, a series of learning progressions reflect research on students’ natural learning stages when mastering concepts such as computation and problem-solving strategies. These learning stages informed the order of concepts, the sequence of units, and the positioning of topics in Math Expressions.”

 Examples include but are not limited to:

• Unit 1, Lesson 4, students find strategies for comparing fractions. Prompts are given for EL students at three different levels: emerging, expanding, bridging. A teaching note is included to help teachers diagnose if students are at an Emerging, Expanding, or Bridging level. Teachers are instructed, “Write common denominator on the board. Write 7/9 and 5/6 on the board. A common denominator for a pair of fractions is a number that each denominator divides into evenly.” Emerging: “18 is divisible by both denominators 9 and 6. Write 7/9 = 14/18 and 5/6 = 15/18 on the board. 18 is a common denominator for 7/9 and 5/6. Have students repeat.” Expanding: “ Write 7/9 = 14/18 and 5/6 = 15/18 on the board. Why is 18 the common denominator?” Bridging: “What is a common denominator?”
• In Unit 5, Lesson 6, the Universal Access/Extra Help teaching note instructs teachers, “Some students may need to use play money to act out these problems. Pair them with a Helping Partner so they can talk through the problem together.”
• In Unit 3, Lesson 5, students are using multiplication strategies. The Universal Access/Special Needs teacher note states, “If a student has difficulty maintaining focus, stand near the student as you speak to the class. Give the student the opportunity to get involved with the activity by asking the student to write examples on the board or repeat a description in his or her own words.”

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Examples include but are not limited to:

• An explanation of differentiated instruction is provided in the Teacher Edition.
• A list of intervention resources is provided for each unit in the Unit Overview Assessment.
• Math Activity Centers resources for on-level, challenge, and intervention are provided for each unit’s lessons. 
• Teaching notes for English Learners are provided for emerging, expanding, and bridging students and are provided for each unit’s lessons.
• Some lessons have Differentiated Instruction notes provided for universal access/extra help.

The instructional materials for Math Expressions Grade 5 meet expectations that materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations.

MathTalks provide “an inquiry environment that encourages constructive discussion of problem-solving methods through well-defined classroom activity structures. . . comprises four components: questioning, explaining math thinking, contributing math ideas, and taking responsibility for learning” (Teacher Edition page I3). Initially, teachers model MathTalks and then students run the MathTalk. For example, in Unit 4, Lesson 4, MathTalk, Best Practices states, “You must direct student math talk for it to be productive. Over time, as students become more skilled at discussing their thinking and talking directly with one another, you will fade into the background more. But you will always monitor, clarify, extend, and ultimately make the decisions about how to direct math conversation so that it is productive for your students.”

The instructional materials for Math Expressions Grade 5 meet expectations that 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.

Examples include but are not limited to:

• Scaffolding of vocabulary is provided. For example, in Unit 4, Lesson 6, the word twice is explained for EL students. Teachers are instructed to “Write ‘twice’ on the board. Twice means 2 times as many. Write x 2 on the board. Have one volunteer draw between 1 and 5 circles. Ask another volunteer draw twice as many.”
• Extra support is provided for EL students. For example, in Unit 6, Lesson 4, students use rounding to find estimates. The teacher prompt states, “Write ‘benchmark’ on the board. A benchmark is a number we use to compare. On the board, draw and label a number line from 0 to 1 by eighths. Draw a dot at 3/8, and then highlight 0, 1/2, and 1. Explain that the highlights are benchmarks - numbers we use to compare.”
• Each unit lesson contains a Math Activity Center with activities and resources for students who are on-level and those needing challenge and intervention.
• Teaching notes included in some lessons provide specific guidance for teachers to support students who are emerging, expanding, and bridging language acquisition.

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Examples include but are not limited to:

• Math Lessons contain Differentiated Instruction Math Activity Centers. Challenge Resources specify which Activity Card will challenge advanced students.
• The online Personal Math Trainer provides personalized enrichment with learning supports.
• Challenge worksheets for each lesson are available in print and digitally and are noted on the Differentiated Instruction page for each lesson.
• Math Readers, books in the Math Activity Center, place math content in the context of stories and support higher levels of critical thinking.

The instructional materials for Math Expressions Grade 5 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics. 

Examples include but are not limited to:

• Puzzled Penguin appears throughout the unit to provide opportunities to help students avoid common errors. These errors are presented as letters to students. Students teach Puzzled Penguin the correct way and explain why the penguin is wrong. 
• Math Readers contain a variety of animals, children, and adults.