The instructional materials for Math Expressions Grade 3 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials identify Five Core Structures: Helping Community, Building Concepts, Math Talk, Quick Practice, and Student Leaders as the organizational structures of the program. “Building Concepts in the classroom experiences in which students use objects, drawings, conceptual language, and real-world situations - all of which help students build mathematical ideas that make sense to them.”

The instructional materials present opportunities for students to develop conceptual understanding. For example:

• Unit 1, Lesson 2, students write multiplication equations for equal groups from pictures and tables to find the total number. “How many bananas?” requires students to write “4 x 3 = 12” from the figure. In the following activity, students make a drawing for each problem, label the drawing with a multiplication equation, and write the answer to the problem.
• Unit 2, Lesson 9, Activity 3, students interpret products and quotients of whole numbers. “Louis put 72 marbles in 8 bags. He put the same number of marbles in each bag.” The directions state, “Write a question for the given information and solve.” Students determine which operation to use and formulate a question that requires that operation.
• Unit 4, Lesson 1, Understand Fractions, Activity 1, "Students use what they have learned about decomposing shapes as a foundation for understanding unit fractions as equal parts of a whole." Activity 2, "Students use fraction bars to visualize and represent unit fractions as the elements for building other fractions."

The instructional materials present opportunities for students to independently demonstrate conceptual understanding. For example,

• Unit 1, Lesson 16, students determine the type of multiplication or division problem represented in word problems, and solve the word problems. For example, Problem 3, “Zamir bought 21 treats to the dog park. He divided the treats equally among the 7 dogs that were there. How many treats did each dog get?” Students need to determine this is an Equal Groups Division with an Unknown Multiplier (number of groups) problem, and write and solve the equation.
• In Unit 5, Lesson 2, students find the area and perimeter of rectangles. Check Understanding problem, “Draw a rectangle with an area of 36 centimeters square and one side length of 4 centimeters. Find the unknown side length. Then find the perimeter." 
• Unit 6, Lesson 4, students engage with comparison problems. Students represent comparison problems in two ways: using a drawing and using a bar diagram and completing statements using more and fewer. For example, “Claire has 8 marbles. Sasha had 15 marbles.” Problem 17, “How many more marbles does Sasha have than Claire? Problem 18, “How many fewer marbles does Claire have than Sasha?”

The instructional materials for Math Expressions Grade 3 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials provide regular opportunities for students to attend to the standards. For example, 3.OA.7, fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division or properties of operation; and 3.NBT.2, fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

The instructional materials develop procedural skill and fluency throughout the grade-level. Each lesson includes a “Quick Practice” described as “routines [that] focus on vitally important skills and concepts that can be practiced in a whole-class activity with immediate feedback”. Quick Practice can be found at the beginning of every unit on the pages beginning with the letters QP. Student materials and instructions are also found in the Teacher Resource Book on pages beginning with Q. Examples include:

• The Introduction includes a chart “Path to Fluency: Kindergarten through Grade 6” which provides a pathway for fluency and memorization of basic facts, and operations with multi-digit operations. Also included are Reteach and Practice Sheets, Quick Practices, and Daily Routines supporting fluency.
• Unit 1, Teacher Resource Book, Multiplication Equations as Groups of 9, “The Student Leader points at the equation in the 9s column of the Multiplication Table Poster in order: 1x9=9, 2x9=18, 3x9=27, and so on. Class says: 9 is 1 group of 9 and raises 1 finger. 18 is 2 groups of 9 and raises 2 fingers. 27 is 3 groups of 9 and raises 3 fingers, and so on.”
• Unit 5, Teacher Resource Book, “Display these fractions bars. Student Leader 1 says two unit fractions with different denominators (such as 1/3 and 1/5 ) and asks which is greater and why. Class: 1/3 is greater because it has fewer unit fractions to make the same whole.” Students practice fraction fluency with students leading the activity.

The instructional materials provide opportunities for student to independently demonstrate procedural skill and fluency throughout the grade-level. These include: Path to Fluency Practice, and Fluency Checks. For example:

• Unit 1, Lesson 14, Activity 2, students review strategies for multiplying and dividing. The Student Activity Book includes questions that “focus on using strategies students were introduced to in previous lessons.” “Emily knows that 4 x 10 = 40.  How can she use subtraction and multiples of 9 to find 4 x 9?” (3.OA.7)
• Unit 1, Lesson 3, Path to Fluency, students practice “count bys”, mixed up multiplication, and mixed up division facts for 5s, 2s, 10s, and 9s.
• Unit 3, Fluency Check 3, students practice single digit by single digit multiplication. For example, Problem 10, “9 x 3 = __.”  
• Student Activity Book, Unit 3, Lesson 8, students practice writing addition problems vertically by lining up the place values correctly before adding. (3.NBT.2)

In addition, Homework and Remembering activity pages found at the end of each lesson provide additional practice to build procedural skill and fluency.

The instructional materials for Math Expressions Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

Students engage with application problems in many lessons for the standards that address application in solving real-word problems. For example, in the Student Activity Book, Unit 4, Lesson 9, students solve application problems involving elapsed time. “Berto spent from 3:45 p.m. to 4:15 p.m. doing math homework and from 4:30 p.m. to 5:10 p.m. doing social studies homework. How much time did he spend on his math and social studies homework?”

Each lesson includes an Anytime Problem listed in the lesson at a glance, and Anytime Problems include both routine and non-routine application problems. For example:

• In Unit 4, Lesson 12, Anytime Problem, “Sarah spent $5 from her savings to buy a book. Then, she spent half of her remaining savings when she paid $9 for a shirt. How much money did Sarah have in savings before she bought the book and the shirt?”
• Unit 1, Lesson 2, Anytime Problem, “Dulal, Lien, and Kate have either the bird, the cat, or the fish. Dulal does not have the bird. Lien does not have the cat. Kate does not have the fish or the cat. Which pet does each person have?”

The instructional materials present opportunities for students to engage in routine application throughout the grade-level. Examples of routine applications of grade-level mathematics are found in the Student Activity Book. For example:

• Unit 1, Lesson 15, “There are 3 shelves in a bookshelf. Each shelf has 2 piles of books on it. If there are 3 books in each pile, how many books are in the bookshelf?”
• Unit 2, Lesson 2, Student Activity Book, “Ana has a ribbon that is 18 inches long. She cut the ribbon into 3 equal pieces. Then she cut each of those pieces in half. How many small pieces of ribbon are there? How long is each piece?”
• Unit 3, Lesson 3, students use place value drawings to assist in solving problems. “Scott made a batch of rolls. He gave a bag of 10 rolls to each of 7 friends. He kept 1 bag for himself. How many rolls did he bake in all?”
• Unit 6, Lesson 6, Problem 2, “Mark has 6 shirts and 5 pairs of pants. Today his aunt gave him 4 more shirts and another pair of pants. How many shirts does he have now?” 

Remembering pages at the end of each lesson are designed for Spiral Review anytime after the lesson occurs. One feature of the Remembering problems are those titled Stretch Your Thinking, which often present opportunities for students to engage with non-routine problems. For example:

• Unit 2, Lesson 15, Remembering, Stretch Your Thinking, Exercise 14, “Matt runs four days a week. On the first day he runs 30 minutes. On the second day he runs 5 minutes more than the first day. On the third day, he runs the same number of minutes as on the second day. On the fourth day, he runs 10 minutes more than the previous day. After Matt runds on the fourth day, how many minutes in all has he run?”
• Unit 3, Lesson 5, Remembering, Stretch Your Thinking, Exercise 17, “Adult and student tickets were sold for a concert. When the numbers of adult tickets and students tickets are rounded, the total number of tickets sold was about 1,200. List four different combinations of adult and students tickets that might have been sold.”
• Unit 7, Lesson 4, Remembering, Stretch Your Thinking, Exercise 7, “Jake has 12 liters of water. Name four different ways he can divide the water into buckets so each bucket has the same number of liters.”

The instructional materials for Math Expressions Grade 3 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All three aspects of rigor are represented in the materials, for example:

• Each lesson has a 5-minute Quick Practice providing practice with skills that should be mastered throughout the year.
• There are Performance Tasks throughout the series, where students use conceptual understanding to perform a mathematical task. For example, Unit 3, Problem 4, “June, Ella, and Joshua are collecting pennies for the service project. June collected 324 pennies. Ella collected 442 pennies, and Joshua collected 248 pennies… Part C: How many more pennies do the students need to collect to buy a second flat of flowers? Show your work.”
• Fluency Checks are included throughout the series, where students practice procedural skills and fluency. For example, Student Activity Book, Lesson 19, Fluency Check, Problem 3, “Add: 8+5=_.”
• Application problems are embedded into practice in the Student Activity Book. For example, Unit 5, Lesson 5, Solve Perimeter and Area Problems, Problem 1, “The dimensions of a rectangular picture frame are 9 inches and 6 inches. What is the greatest size picture that would fit in the frame?”

Examples where student engage in multiple aspects of rigor:

• Unit 3, Lesson 7, students are introduced to proof drawing addition. This lesson builds students conceptual understanding of addition by asking students to visualize and model regrouping by making proof drawings to illustrate adding two 3-digit numbers. Students are introduced to four methods for solving addition problems: Show All Totals Method, New Groups Below Method, New Groups Above Method, and Proof Drawing. Students use each method to solve: “Tonya and Mark collect seashells. Tonya has 249 shells and Mark has 386 shells. How many do they have in all?”
• Unit 3, Lesson 16, students practice and discuss addition and subtraction methods. In Activity 1, students create an addition word problem using the given numbers 672 and 228. Students then create a subtraction problem using the given numbers 814 and 439. Students create a Math Mountain for those numbers, and then find addition problems that match the mountain. Students apply learned skills by creating word problems. They use procedures to solve the equations in the problems.

The instructional materials for Math Expressions Grade 3 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 4, Teaching the Lesson Section, Activity 1, Quick Practice Routines: Represent Hundreds, Tens, and Ones, is stated as important because “Students create place value drawings to help conceptualize 2-digit and 3-digit numbers.” Activity 2, Place Value Drawings for Thousands, Hundreds, Tens, and Ones, is stated as important because “Students represent numbers through thousands with place value drawings.”  Activity 3, Use Place Value to Compare Numbers, is stated as important because “Students use place value drawings to visualize and compare numbers of increasing value.”
• In the Student Activity Book, Unit 6, Lesson 2, Activity 2, students solve real world problems with unknown addends: “There were 90 girls and some boys in an after-school program. 160 children were in the program in all. How many boys were in the after-school program?” In Activity 3, students are introduced to word problems with unknown factors: “A toymaker has 36 boxes of toy trains to ship to 4 toy shops. Each shop will get the same number of boxes. How many boxes of toy trains will each shop get?”

The instructional materials for Math Expressions Grade 3 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 11, students review multiplication facts for 2, 3, 5, 9, and 10 using strategy cards (flashcards). Next students write multiplication equations to represent areas of rectangles and draw rectangles whose areas represent multiplication equations. Finally, students use the area model and distributive property to develop a multiplication strategy.
• In Unit 4, Lesson 4, students compare fractions using fraction bars looking for patterns, then students use number lines to compare fractions.
• In Unit 6, Lesson 1, students use Math Mountains to break-apart numbers and turn the Math Mountains into addition equations with total and 2 addends. Then students are able to solve story problems in the Student Activity Book.

The instructional materials for Math Expressions Grade 3 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
• Representing 3-digit numbers with Secret Code Cards
• 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 3 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 3, students use arrays to solve and write multiplication problems. 
• Unit 1, Lesson 11, students use Square Inch Tiles to make area models to write and solve multiplication problems. 
• Unit 4, Lesson 2 students use fraction bars to break apart 1 whole into equal fractional parts and use number lines to represent fractions less than 1 whole.

The instructional materials for Math Expressions Grade 3 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 2, Lesson 6, the teacher is guided to ask the questions: “What patterns do you see in the square numbers?, Why do you think those numbers are the same? Who can give an example?, Why is there no matching product for a square number?”
• 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 4, the Language and Vocabulary notes provide several probes to promote student thinking and discussion: “How many more does B have than A? How many fewer does A have than B?”

The instructional materials for Math Expressions Grade 3 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 1, Activity 1, 6s Multiplications and Divisions, is stated as important because “Students use patterns as a strategy to multiply and divide by 6.”
• 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 Faster Ten Sticks states, “Point out the small circles along two edges of the dot array. Explain that there are 5 dots between each pair of circles. Students can use the circles as a guide to help them draw ten sticks quickly.”
• 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 2, a picture of itools Fraction Bars and Number Lines are shown because they may be used in the lesson.

The instructional materials for Math Expressions Grade 3 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 3 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 3 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 4, the Math Background for Time states, “In the previous grade, students found elapsed time in hours and half hours from a start time and an end time. In this unit students find elapsed time in hours and minutes and use these skills to solve real world problems.” It is noted that number lines and clock models may be needed to help students calculate elapsed time.
• 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 9 assesses student multiplication and division facts (3.OA.7) and addition and subtraction of 3-digit numbers (3.NBT.2).
• 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 3 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 4 Review/Test, if a student misses Question 6 or 9, the common error identified states, “Students can’t use information from graph to solve.” Teachers are directed to “Review how to read a bar graph, pictograph, or line plot. Have students look at headings, titles, and labels before reading the numbers presented in the data.”
• 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 1, the Watch For! note states, “Be sure students understand that a thousand bar doesn’t have to be the size of 10 hunderd boxes. Whenever they see or draw a long, skinny stip in a place value drawing, they will know that it is a thousand bar and it is equivalent to 10 hundreds.” 

The instructional materials for Math Expressions Grade 3 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 3 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 3 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 3 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 6, students build fluency with 2s and 5s. 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 3 x 4 array. Write the words row and column.” Emerging: “Point to the rows. Rows go across. There are 3 rows. Point to the columns. Columns go down. There are 4 columns. Have students repeat.” Expanding: “Rows go across. How many rows are there? Columns go down. How many columns are there?” Bridging: “Have students tell the difference between rows and columns.”
• In Unit 3, Lesson 1, the Universal Access/Extra Help teaching note instructs teachers, “If students are not convinced the 5 x 20 boxes represent 100, have them draw horizontal ten sticks.”
• In Unit 5, Lesson 6, students use tangrams to find area. The Universal Access/Special Needs teacher notes state, “If students have difficulty manipulating paper manipulatives, laminate the page before you cut out the materials. Pair the students with a Helping Partner so they can work through the activity together.”

The instructional materials for Math Expressions Grade 3 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 3 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 1, Lesson 2, the MathTalk states, “The Solve and Discuss structure of conversation is used throughout the Math Expressions program. The teacher selects four or five students to go to the board and solve a problem using any method they choose. The other students work on the same problem at their desks. Then the teacher asks the students at the board to explain their methods. Students at their desks are encouraged to ask questions and to assist each other in understanding the problem. Thus students solve, explain, question, and justify. It is best to ask only two or three students at the board to explain. This avoids having the seated students become restless. Time is better spent on the next issue. Be sure to have students stand beside their work and point to parts of it as they explain it. Using a pointer that does not obscure any work enables watchers to see the part of the drawing or math symbols that are being discussed at that moment.”

The instructional materials for Math Expressions Grade 3 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 3, Lesson 1, the words circles, ten sticks, and boxes are explained for EL students. Teachers are instructed to say, “‘This is a circle. It equals 1. Its value is 1.’ Have students repeat. Continue with ten sticks and boxes.”
• Extra support is provided for EL students. For example, in Unit 5, Lesson 7, students are introduced to equivalency with fractions. Teachers are instructed to “Point out that the word equivalent has equal in it. Cross out the ‘i’, ‘v’, and ‘ent’ in equivalent to see the word equal. We show that fractions are equivalent using an equal sign.” 
• 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 3 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 3 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.