4th Grade - Gateway 3

The instructional materials for Math Expressions Grade 4 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 5, Lesson 2, Teaching the Lesson Section, Activity 1, Metric Units of Liquid Volume, is stated as important because “Students develop a sense of the relative sizes of metric units of liquid volume.” Activity 2, Metric Units of Mass, is stated as important because “Students develop a sense of the relative sizes of metric units of mass.”  Activity 3, Real World Problems with Metric Measurements, is stated as imortant because “Students apply their understanding of metric units of liquid volume and mass to solve real world problems.”
• In the Student Activity Book, Unit 3, Lesson 5, Activity 1, students discuss word problems from Lesson 4 homework: “A fruit stand sells packages containing 1 peach, 1 pear, 1 apple, 1 banana, and 1 mango each. One week they sold a total of 395 pieces of fruit. How many packages did they sell?” In Activity 2, students explore division with 4-digit dividends: “Use any method to solve. 3,248 ÷ 5 =.”

The instructional materials for Math Expressions Grade 4 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 1, Lesson 6, students share addition methods for adding 3-digit whole numbers. Then students practice using various methods of addition in their Student Activity Book: New Group Above, New Groups Below, Subtotal Left to Right, and Subtotal Right to Left.
• In Unit 4, Lesson 1, students use expressions to simplify and solve equations. Then in the Student Activity Book, students use the identity property to simplify expressions and solve equations using the associative property.
• In Unit 6, Lesson 4, students practice adding and ordering unit fractions. They discuss uses for mixed numbers and then convert between mixed fractions and fractions greater than one.

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 Math Mountains to put together and take apart numbers
• Using Arrays and Area Models to solve multiplication problems
• Using drawings to make sense of mathematics
• Using Place Value Charts to compare numbers
• 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 4 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, 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 4, students use Secret Code Cards to build numbers to the millions place then write in standard, expanded, and word form. 
• Unit 2, Lesson 1, students use MathBoards to make area models to write and solve multiplication problems. 
• Unit 5, Lesson 1 students make and use meter strips to help them understand and convert metric units of length.

The instructional materials for Math Expressions Grade 4 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 4, Lesson 3, the teacher is guided to ask the questions: “What do we know about the problem? What do we need to find?”
• 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 6, Lesson 8, Inquiry Notes state, “As students discuss their work, ask questions like ‘How did you decide what model you would use to represent the problem? Why is an equation an efficient way to represent a problem? Do you see another way to solve the problem? How can you check the answer to the problem?’”
• Teacher Notes at the bottom of lesson pages also include information about best practices to strengthen teachers questioning techniques. For example, in Unit 1, Lesson 3, the Inquiry note provides the following directions concerning using questions: “The Math Talk sample discussions are intended to provide ideas that enhance an inquiry approach to learning in your classroom. Use questions like those in the sample discourse to guide discussions of mathematical concepts and skills.”

The instructional materials for Math Expressions Grade 4 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 1, Lesson 4, Activity 2, Understand Greater Numbers, is stated as important because “Students need to read and write numbers to one million in standard form, word form, and expanded form.” 
• 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 5, Lesson 5, the Teaching Notes for Watch For! states, “Students may confuse an ounce, a unit of weight, with a fluid ounce, a unit of liquid volume. A fluid ounce is often shortened to ounce when the context is known.”
• 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 4, Lesson 10, a picture of itools Counters are shown because they may be used in the lesson.

The instructional materials for Math Expressions Grade 4 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 4 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 4 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.
• The Math Background section for each unit alerts teachers to prior knowledge opportunities. For example, in Unit 5, the Math Background for Time states, “Working with time provides opportunities to review what students have learned about fractions. It is common to relate fractions of an hour to minutes in order to determine that 1/4 hour is 15 minutes and 3/4 hour is 45 minutes. Students apply this understanding as they interpret line plots that include fractions of an hour.” 
• 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 student addition of multi-digit numbers (4.NBT.4).
• 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 4 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 3 Review/Test, if a student misses Questions 1, 11, or 18, the common error identified states, “Student does not understand how to use division methods.” Teachers are directed to “Review with students the three methods suggested in this Unit. Have students work in pairs to solve the division problems using each of the three methods.”
• 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 6, Lesson 2, the Watch For! note states, “Students often misunderstand fraction drawings that have parts colored or shaded. Instead of seeing the relationship of the shaded parts to the whole, they see shaded versus unshaded parts. Classroom research has shown that having students write chains of unit fractions and identify fraction partners that make 1 whole can help them to overcome this misunderstanding.”

The instructional materials for Math Expressions Grade 4 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 4 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 4 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 4 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 2, Lesson 1, students use arrays and area models. 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, “Draw a 4 x 2 rectangle showing 4 unit squares down and 2 unit squares across. Write array and area on the board.” Emerging: “The array of unit squares is 4 down by 2 across. The area of the rectangle is 4 x 2. What is 4 x 2?” Expanding: “What is the array of unit squares? How do you find the area of the rectangle? What is the area?” Bridging: “Students can work in pairs. One partner draws an array of unit squares and the other names the area.”
• In Unit 5, Lesson 1, the Universal Access/Extra Help teaching note instructs teachers, “If students have difficulty overlapping the ends to make their meter strips, have them cut off the ends and butt each part to the next.”
• In Unit 6, Lesson 2, students are adding and subtracting fractions with like denominators. The Universal Access/Special Needs teacher note states, “For students who have difficulty reading a problem line by line, give them an index card they can use to uncover one line of a problem at a time.”

The instructional materials for Math Expressions Grade 4 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 4 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, Unit 2, Lesson 4, MathTalk, Best Practices states, “Review with the class what makes a good explanation. 1. Write your work so everyone can see it. 2. Talk loud enough for other students to hear. 3. Use a pointer to point to your work. 4. Say how you arrived at the answer, not just your answer. 5. Stand to the side of your work when you talk.”

The instructional materials for Math Expressions Grade 4 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 1, the word simplify is explained for EL students. Teachers are instructed to “Write ‘5n + 9 - 2 = 5n + 7’. Write ‘simplify’. Simplify means ‘to make easier’. Point to the 9 and then the 2. These are called like terms. To simplify combine like terms. Have students repeat.”
• Extra support is provided for EL students. For example, in Unit 2, Lesson 5, students use rounding to find estimates. The teacher prompt states, “If students have difficulty recognizing the rounding frames, suggest they use color pencils or markers to highlight the 4 x 70 and 4 x 60 rounding frames,”
• 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 4 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 4 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.