4th Grade - Gateway 3

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• Teacher's Lesson Guide, Welcome to Everyday Mathematics, explains how the program is presented. “Throughout Everyday Mathematics, emphasis is placed on problem solving in everyday situations and mathematical contexts; an instructional design that revisits topics regularly to ensure depth of knowledge and long-term learning; distributed practice through games and other daily activities; teaching that supports “productive struggle” and maintains high cognitive demand; and lessons and activities that engage all students and make mathematics fun!”

• Implementation Guide, Guiding Principles for the Design and Development of Everyday Mathematics, explains the foundational principles. “The foundational principles that guide Everyday Mathematics development address what children know when they come to school, how they learn best, what they should learn, and the role of problem-solving and assessment in the curriculum.”

• Unit 4, Multi-Digit Multiplication, Organizer, Coherence, provides an overview of content and expectations for the unit. “In Unit 1, students learn U.S. traditional addition. In Units 2-4, students practice multi-digit addition and subtraction with whole numbers in a variety of contexts. In Grade 3, students learn a variety of methods for multidigit addition and subtraction. Beginning in Unit 6, students use multidigit addition and subtraction with whole numbers as they do partial-quotients division. In Grade 5, students will learn how to multiply using U.S. traditional multiplication.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Examples include:

• Implementation Guide, Everyday Mathematics Instructional Design, “Lesson Structure and Features include; Lesson Opener, Mental Math and Fluency, Daily Routines, Math Message, Math Message Follow-Up, Assessment Check-In, Summarize, Practice, Math Boxes, and Home-Links.”

• Lesson 1-13, Finding Perimeters of Squares and Rectangles, Common Misconception teacher guidance addresses common misconceptions as students use formulas for finding the perimeter of a rectangle. “Watch for students who think they have found the answer by adding only the 2 labeled sides. Suggest that they label the lengths of the other two sides. Remind these students that even though all sides may not be labeled, all sides do need to be included in the calculation.”

• Lesson 5-8, Subtracting Mixed Numbers, Focus: Assessment Check-In, teacher guidance supports students in solving mixed number subtraction problems. “Observe students completing journal page 171. Expect most to be able to solve Problems 1 and 2 using a strategy. Encourage students who struggle to work through each step by modeling the mixed numbers, taking pieces away, then renaming them using fraction circles. For students who complete all parts successfully, suggest that they write a number story and illustrate how to solve it in two ways.”

• Lesson 8-7, More Decimal Number Stories, Focus: Solving a Perimeter Problem with Simple Decimals, Math Message Follow-Up, teacher guidance connects students' prior knowledge to new concepts. “Before students share solution strategies, guide a discussion about decimal and fraction equivalence by displaying two-name collection boxes, one for 5.6 and one for 3.8, and completing them as a class. Remind students that sometimes one form of a number is easier to work with than another when solving problems. Sometimes the “easier” form is different for different people. This is why more than one way to solve a problem is encouraged. It is important to remember that mixed numbers and decimals are simply different ways to name the same number and that numbers can be combined in different ways to find an answer.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Each Unit Organizer Coherence table provides adult-level explanations and examples of complex grade/course-level concepts so teachers can improve their content knowledge. Professional Development side notes within Lessons support teachers in building knowledge of key mathematical concepts. Examples include:

• Lesson 1-10, U.S. Customary Units of Length, Professional Development, supports teachers with concepts for work beyond the grade. “The Grade 4 standard 4.MD.1 expects students to express measurements in a larger unit in terms of a smaller unit. Thus, students may only understand this relationship as moving from larger to smaller. The Grade 5 standards expect that students will explore the relationship in both directions. It is important that students are exposed to converting measurement units from smaller to larger in Grade 4, so as to have a general understanding of the relationship. Do not expect that your students will master the concept. It will not be assessed in Grade 4.”

• Unit 2, Multiplication and Geometry, Unit 2 Organizer, 4.G.2, supports teachers with concepts for work beyond the grade. “Links to the Future: In Grade 5, students use properties of triangles to create a triangle hierarchy.”

• Lesson 4-13, Lattice Multiplication, Professional Development, explains using the lattice multiplication. “Lattice multiplication is a simple alternative to traditional algorithms that deal with whole numbers. It focuses on place value and the organization of a multiplication problem. The lattice method breaks down the numbers into place values, allowing students to work with smaller numbers while solving a multi-digit multiplication problem. Students are not required to learn this method, but they should be encouraged to try. Having choices among methods is important. Given those choices, most students will select ones that work best for them.”

• Lesson 5-10, Rotations and Iterating Angles, Professional Development, explains understanding angles and protractor use. “For students to accurately use a protractor to measure angles, they must first understand what they are measuring. The attribute of angle size is a source of confusion for most students. Using a nonstandard unit like a wedge to measure an angle helps students see that measuring the size of an angle is the same as measuring any other attribute: iterating unit angles fills the spread between an angle’s rays, just as iterating unit lengths fills a given length.”

• Unit 7, Multiplication of a Fraction by a Whole Number; Measurement, Unit 7 Organizer, 4.OA.3, provides support with explanations and examples of the more complex grade/course-level concepts. “Links to the Past: In Grade 3, students solve two-step word problems involving all four operations, using a letter for the unknown quantity. They estimate to assess the reasonableness of results.”

• Lesson 8-7, More Decimal Number Stories, Professional Development, supports teachers with concepts for work beyond the grade. “Even though formal operations with decimals are not a Grade 4 mathematics topic in Indiana Academic Standards, the work in this lesson meets 4.MD.2, which specifies problems involving distances and simple decimals and provides a foundation for more formal work with decimal operations in Grade 5. The link between different representations of numbers, especially fractions and decimals, is a key concept for success with decimal computation in fifth grade.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the Correlations to the Standards for Mathematics, Unit Organizers, Pathway to Mastery, and within each lesson. Examples include:

• 4th Grade Math, Correlation to the Standards for Mathematics Chart includes a table with each lesson and aligned grade-level standards. Teachers can easily identify a lesson when each grade-level standard will be addressed. 

• 4th Grade Math, Unit 3, Fractions and Decimals, Organizer, Contents Lesson Map outlines lessons, aligned standards, and the lesson overview for each lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Lesson 6-5, (Day 2): Fruit Baskets, Core Standards identified are 4.NF.7, 4.OA.3, and 4.NBT.6. Lessons contain a consistent structure that includes an Overview, Before You Begin, Vocabulary, Warm-Up, Focus, Assessment Check-In, Practice, Minute Math, Math Boxes, and Home-Link. This provides an additional place to reference standards, and language of the standard, within each lesson.

• Mastery Expectations, 4.NBT.3, “First Quarter: Round numbers through the hundred thousands to the thousands place or larger. Second Quarter: Use place value understanding to round multi-digit whole numbers to any place. Third Quarter: Ongoing practice and application. Fourth Quarter: Ongoing practice and application.” Mastery is expected in the Second Quarter. 

Each Unit Organizer Coherence table includes an overview of content standards addressed within the unit as well as a narrative outlining relevant prior and future content connections for teachers. Examples include:

• Unit 3, Fractions and Decimals, Organizer, Coherence, includes an overview of how the content in 4th grade builds from previous grades and extends to future grades. “In Grade 3, students use visual models, such as fraction circles, drawings, and number lines to compare fractions. They learn that comparisons are possible only if the whole is the same size. In Grade 5, students apply formal strategies to add and subtract fractions and use estimates to help them assess the reasonableness of their answers.” 

• Unit 5, Fraction and Mixed-Number Computation; Measurement, Organizer, Coherence, includes an overview of how the content in 4th grade builds from previous grades and extends to future grades. “In Grade 3, students used fraction strips, number lines, and fraction circles to explore fractions that are equivalent to whole numbers and to convert whole numbers into fraction equivalents. In Grade 5, students develop formal strategies to add and subtract fractions and mixed numbers.”

• Unit 8, Fraction Operations; Applications, Organizer, Coherence, includes an overview of how the content in 4th grade builds from previous grades and extends to future grades. “In Grade 3, students use concrete and visual representations to explore fractions as equal parts of the whole. In Grade 5, students develop formal strategies, including using equivalent fractions with common denominators, to add and subtract fractions and mixed numbers.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

Instructional approaches to the program are described within the Teacher’s Lesson Guide. Examples include:

• Teacher’s Lesson Guide, Welcome to Everyday Mathematics, The University of Chicago School Mathematics Project (UCSMP) describes the five areas of the Everyday Mathematics 4 classroom. “Problem solving in everyday situations and mathematical contexts, an instructional design that revisits topics regularly to ensure depth of knowledge and long-term learning, a distributed practice through games and other activities, teaching that supports ‘productive struggle’ and maintains high cognitive demand, and lessons and activities that engage all children and make mathematics fun!” 

• Teacher’s Lesson Guide, About Everyday Mathematics, An Investment in How Your Children Learn, The Everyday Mathematics Difference, includes the mission of the program as well as a description of the core beliefs. “Decades of research show that students who use Everyday Mathematics develop deeper conceptual understanding and greater depth of knowledge than students using other programs. They develop powerful, life-long habits of mind such as perseverance, creative thinking, and the ability to express and defend their reasoning.”

• Teacher’s Lesson Guide, About Everyday Mathematics, A Commitment to Educational Equality, outlines the student learning experience. “Everyday Mathematics was founded on the principle that every student can and should learn challenging, interesting, and useful mathematics. The program is designed to ensure that each of your students develops positive attitudes about math and powerful habits of mind that will carry them through college, career, and beyond. Provide Multiple Pathways to Learning, Create a System for Differentiation in Your Classroom, Access Quality Materials, Use Data to Drive Your Instruction, and Build and Maintain Strong Home-School Connections.”

• Teacher’s Lesson Guide, About Everyday Mathematics, Problem-based Instruction, approach to teaching skills helps to outline how to teach a lesson. “Everyday Mathematics builds problem solving into every lesson. Problem solving is in everything they do. Warm-up Activity: Lessons begin with a quick, scaffolded Mental Math and Fluency exercise. Daily Routines: Reinforce and apply concepts and skills with daily activities. Math Message: Engage in high cognitive demand problem-solving activities that encourage productive struggle. Focus Activities: Introduce new content with group problem solving activities and classroom discussion. Summarize: Discuss and make connections to the themes of the focus activity. Practice Activities: Lessons end with a spiraled review of content from past lessons.” 

• Teacher’s Lesson Guide, Everyday Mathematics in Your Classroom, The Everyday Mathematics Lesson, outlines the design of lessons. “Lessons are designed to help teachers facilitate instruction and engineered to accommodate flexible group models. The three-part, activity-driven lesson structure helps you easily incorporate research-based instructional methods into your daily instruction. Embedded Rigor and Spiraled Instruction: Each lesson weaves new content with the practice of content introduced in earlier lessons. The structure of the lessons ensures that your instruction includes all elements of rigor in equal measure with problem solving at the heart of everything you do.”

Preparing for the Module provides a Research into Practice section citing and describing research-based strategies in each unit. Examples include:

• Implementation Guide, Everyday Mathematics & the Common Core State Standards, 1.1.1 Rigor, “The Publishers’ Criteria, a companion document to the Common Core State Standards, defines rigor as the pursuit, with equal intensity, of conceptual understanding, procedural skill and fluency, and applications (National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CCSSO], 2013, p. 3).

• Implementation Guide, Differentiating Instruction with Everyday Mathematics, Differentiation Strategies in Everyday Mathematics, 10.3.3, Effective Differentiation Maintains the Cognitive Demand of the Mathematics, “Researchers broadly categorize mathematical tasks into two categories; low cognitive demand tasks, and high cognitive demand tasks. While the discussion of cognitive demand in mathematics lessons is discussed widely, see Sten, M.K., Grover, B.W. & Henningsen, M. (1996) for an introduction to the concept of high and low cognitive demand tasks.”

• Implementation Guide, Open Response and Re-Engagement, 6.1 Overview, “Research conducted by the Mathematics Assessment Collaborative has demonstrated that the use of complex open response problems “significantly enhances student achievement both on standardized multiple-choice achievement tests and on more complex performance-based assessments” (Paek & Foster, 2012, p. 11).”

• The University of Chicago School Mathematics Project provides Efficient Research on third party studies. For example:

  • A Study to Explore How Gardner’s Multiple Intelligences Are Represented in Fourth Grade Everyday Mathematics Curriculum in the State of Texas.

  • An Action-Based Research Study on How Using Manipulatives Will Increase Student’s Achievement in Mathematics.

  • Differentiating Instruction to Close the Achievement Gap for Special Education Students Using Everyday Math.

  • Implementing a Curriculum Innovation with Sustainability: A Case Study from Upstate New York.

  • Achievement Results for Second and Third Graders Using the Standards-Based Curriculum Everyday Mathematics.

  • The Relationship between Third and Fourth Grade Everyday Mathematics Assessment and Performance on the New Jersey Assessment of Skills and Knowledge in Fourth Grade (NJASK/4).

  • The Impact of a Reform-Based Elementary Mathematics Textbook on Students’ Fractional Number Sense.

  • A Study of the Effects of Everyday Mathematics on Student Achievement of Third, Fourth, and Fifth-grade students in a Large North Texas Urban School District.

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

A year-long list of materials needed is provided in the Teacher’s Lesson Guide, Getting to Know Your Classroom Resource Package, Manipulative Kits, and eToolkit. “The table below lists the materials that are used on a regular basis throughout Fourth Grade Everyday Mathematics.” Each unit includes a Materials Overview section outlining supplies needed for each lesson within the unit. Additionally, specific lessons include notes about supplies needed to support instructional activities, found in the overview of the lesson under Materials. Examples include:

• Unit 1, Place Value; Multidigit Addition and Subtraction, Unit 1 Organizer, Unit 1 Materials, teachers need, “number cards 0-9 (4 of each); tape measure; yardstick; 12-inch ruler; place-value tool; calculator in lesson 10.” 

• Lesson 1-10, U.S. Customary Units of Length, Math Message, “Display the measurement scales on journal page 24, along with the yardstick and 12-inch ruler.” Focus: Measuring Lengths in U.S. Customary Units, “Show students the yardstick and 12-inch ruler to remind them of the relative sizes of yards, feet, and inches.”

• Unit 4, Multi-Digit Multiplication, Unit 4 Organizer, Unit 4 Materials, teachers need, “graduated cylinder; beakers (optional); fraction circles or number line; eyedropper; containers; 1-liter pitcher or beaker; water; calculator; paper in lesson 4.” 

• Lesson 7-3, A Fraction as a Multiple of a Unit Fraction, Overview, Materials, “slate; fraction circles; Math Journal 2, pp. 234-235; Student Reference Book p. 267; Math Masters p. G35; number cards 0-9 (4 of each) or one 10-sided die labeled 0-9; Math Journal 2, p. 236; half-circle protractor; Math Masters, p. 266.” Math Message, “Use centimeter cubes and grid paper to show your thinking.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

Beginning-of-Year Assessment, Unit Assessments, Open Response Assessments, Cumulative Assessments, Mid-Year Assessment and End-of-Year Assessment consistently and accurately identify grade-level content standards along with the mathematical practices within each Unit. Examples from formal assessments include:

• Unit 1, Place Value; Multidigit Addition and Subtraction, Open Response Assessment, denotes standards addressed for the open response. “Emma and Cody solved this subtraction problem in different ways. 904-795= ___. 1. Explain why Emma wrote 5+100+4=109. 2. When Cody saw 4-5 in the ones, he realized he needed to make a trade. Why did he write 14 above the ones place?” (4.NBT.4) 

• Unit 4, Multi-Digit Multiplication, Cumulative Assessment, denotes standards addressed for each problem. Problem 15, “Draw a shape that has 2 pairs of parallel sides and 4 right angles. Give two names for this shape.” (4.G.2)

• Mid-Year Assessment, denotes standards addressed for each problem. Problem 11, “a. Use fraction circles to help you find and write an equivalent fraction for \frac{4}{12}. b. Draw a picture to show that \frac{4}{12} is equivalent to the fraction you wrote above.” (4.NF.1)

• Unit 7, Multiplication of a Fraction by a Whole Number; Measurement, Unit Assessment, denotes mathematical practices for each problem. Problem 1, “Solve the number stories using pictures or equations. a. We have 8 cans of pineapple chunks in our pantry. Each can weighs \frac{5}{8} pound. How much do the cans weigh together? b. Lori runs \frac{6}{10} mile every day. How many miles does she run in a week? c. Patrick’s pancake recipe calls for 1\frac{1}{2} cups of blueberries. If he wants to triple the recipe, how many cups of blueberries will he need?” (SMP4)

• End-of-Year Assessment, denotes mathematical practices for each problem. Problem 12, “Use a formula to find the perimeter of the rectangle. Show your work in the space provided.” A rectangle is shown with 7 m as the length and 4 m as the width. (SMP7)

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative Assessments include Beginning-of-Year Assessment and Preview Math-Boxes. Summative Assessments include Mid-Year Assessment, End-of-Year Assessment, Unit Assessments, Open Response Assessment/Cumulative Assessments. All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types: multiple choice, short answer, and constructed response. Examples include:

• Unit 4, Multi-Digit Multiplication, Cumulative Assessment, develops the full intent of standard 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm. Problem 9, “Solve using U.S. traditional addition. a. 45,187+12,931. b. 53,214+98,926.” Problem 10, “Solve using U.S. traditional subtraction. A. 38,000-23,177. b. 17,142-9,663.”

• Mid-Year Assessment, supports the full intent of MP6, attend to precision, as students use clear and precise language to explain how they found the perimeter of the rectangle. Problem 4, “a. Find the perimeter of the rectangle with a length of 8 ft and a width of 4 ft. b. Find the area of the rectangle. C. Explain how you found the perimeter and the area.” 

• Unit 6, Division; Angles, Unit Assessment, supports the full intent of MP4, model with mathematics, as students model real-world situations using graphs, drawings, tables, symbols, numbers, diagrams, or other representations. Problem 3, “There are 38 crackers in a box. Tina and her two sixers decided to share them equally. How many crackers will each girl get?” 

• End-of-Year Assessment, develops the full intent of 4.NF.6, use decimal notation for fractions with denominators 10 or 100. Problem 11, “a. Write seven-tenths as a decimal and as a fraction. b. Write \frac{7}{100} as a decimal and in words. c. Write a number sentence using <, =, or > to compare the decimals in Problems 11a and 11b. d. Explain how you know your number sentence for 11c is correct. e. Add the fractions from Problems 11a and 11b. Show your work. f. Color part of the grid to show your answer to Problem 11e.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. Implementation Guide, Differentiating Instruction with Everyday Mathematics, 10.1 Differentiating Instruction in Everyday Mathematics: For Whom?, “Everyday Mathematics lessons offer specific differentiation advice for four groups of learners. Students Who Need More Scaffolding, Advance Learners, Beginning English Language Learners, and Intermediate and Advanced English Language Learners.” Differentiation Lesson Activities notes in each lesson provide extended suggestions for working with diverse learners. Supplementary Activities in each lesson include Readiness, Enrichment, Extra Practice, and English Language Learner. 

For example, the supplementary activities of Unit 5, Fraction and Mixed-Number Computation; Measurement, Lesson 8, include:

• Readiness, “To explore mixed-number subtraction concepts, students decompose mixed numbers in multiple ways. Display the mixed number 2\frac{2}{4} and work with students to decompose it using fraction circles. For example, show 1 whole and six-fourths using fraction circles (1 red and 6 yellows). Display the equation 1+\frac{6}{4}=2\frac{2}{4}. Have students find another solution and record the appropriate equation. Repeat with other mixed numbers. Suggestions: 2\frac{3}{5}, 3\frac{1}{2}.”

• Enrichment, “To further explore mixed-number subtraction, students solve number stories involving unlike denominators. Note: As students have not been introduced to computing with unlike denominators beyond 10 and 100, consider doing a few examples as a group to help students understand that they can use the Equivalent Fractions Rule to find equivalent fractions in order to subtract.”

• Extra Practice, “To practice mixed-number subtraction, students complete Frames-and-Arrows diagrams. Create problems to meet the needs of individual students or have them create and solve their own problems.”

• English Language Learner, Beginning ELL, “Building understanding of like using the terms same and alike. Show pairs of objects and model comparing them by pointing to like attributes and using think-alouds: ___ is like ___. They have the same ___, ___, and ___ are alike because ___. Give students various objects and direct them to find the two objects that are alike, like each other, or the same. Encourage students to repeat statements in which the terms are used.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Materials provide multiple opportunities for advanced students to investigate the grade-level content at a higher level of complexity rather than doing more assignments. The Implementation Guide, Differentiation Instructions with Everyday Mathematics, 10.4 Working with Advanced Learners, “Nearly all Everyday Mathematics lessons include a set of high cognitive demand tasks with mathematical challenges that can be extended. Every regular lesson includes recommended enrichment activities related to the lesson content on the Differentiation Options page opposite the Lesson Opener Everyday Mathematics lessons incorporate varied grouping configurations which enables the kind of flexibility that is helpful when advanced learners in heterogeneous classrooms. Progress Check lessons include suggestions for extending assessment items for advanced learners and additional Challenge problems.” The 2-day Open Response and Re-Engagement lesson rubrics provide guidance for students in Exceeding Expectations. Examples include:

• Unit 3, Fractions and Decimals, Challenge, “Use three different strategies to show how you know that \frac{3}{5} is greater than \frac{4}{10}.”

• Lesson 6-13, Extending Understandings of Whole Number Multiplication, Enrichment, “To explore finding an unknown in a number sentence that involves multiplying a fraction by a whole number, students use a variety of structures. They fill in multiplication/division diagrams, draw representations, and write multiplication equations to solve the problems.” Teachers are provided guidance to help advanced learners solve missing groups number stories.”

• Lesson 8-2, Real-Life Angle Measures as Additive, Enrichment, “To extend their understanding of the additive nature of angle measures, students determine the measure of the angles of fraction circle pieces. They explore sums of angle measures of various combinations of fraction circle pieces.”

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The Teacher’s Lesson Guide and ConnectED Teacher Center include guidance for the teacher in meeting the needs of English Language Learners. There are specific suggestions for making anchor charts or explaining new vocabulary. The Implementation Guide, English Language Learners, Everyday Mathematics addresses the needs of three groups of ELL based on their English language proficiency (beginning, emerging, and advanced), “Beginning English language learners fall into Entering (level 1) and Emerging (level 2) proficiencies. This group is typically within the first year of learning English; students' basic communication skills with everyday language are in their early development. These students require the most intensive language-related accommodations in order to access the mathematics in most lessons. Intermediate and Advanced English learners represent Levels 3, 4, and 5 (Developing, Expanding, and Bridging) in the English language proficiencies identified above. Students in this category are typically in their second to fourth year of learning English. They may be proficient with basic communications skills in English and able to carry on everyday conversations, but they are still developing proficiency with more cognitively demanding academic language of the mathematics class.” The ConnectED Teacher Center offers extended suggestions for working with diverse learners including English Language Learners. The Teacher’s Lesson Guide provides supplementary activities for beginning English Language Learners, Intermediate, and Advanced English Language Learners. In every lesson, there are Differentiation Support suggestions, English Language Learner for Beginning ELL located on the Differentiation Options Page and Focus section. Examples include:

• Lesson 1-9, U.S. Traditional Subtraction, English Language Learner Beginning ELL, “Demonstrate the meaning of columns on a grid, using up and down gestures as you say the term. Have students trace columns from top to button with their fingers. Show examples of columns and non-examples (rows) on the grid, asking yes/no questions like: Is this a column? Choose a number from the top of the grid and tell students to point to that column and trace it up and down.”

• Lesson 5-12, Creating Symmetric Figures, Differentiation Options, English Language Learner Beginning ELL, “To scaffold students’ understanding of terms used in the lesson, including line, fold, horizontal, vertical, and mirror image, provide vocabulary cards for each term with the corresponding illustration. Use Total Physical Response prompts to model each term, directing students to find classroom examples that help illustrate each of these terms. For example, point to a vertical line as you say the term, and then ask students to point to another vertical line.” 

• Lesson 6-7, Partial-Quotients Division, Part 2, Differentiating Lesson Activities, Using Partial-Quotients Division, “Encourage students to use academic and content terms as they discuss their strategies. Provide a word bank that includes the terms at least, multiply, divisor, dividend, quotient, partial, estimate, and reasonable. Post sentence frames that encourage the use of the terms to describe the order of the steps they followed and that incorporate terms such as first, then, also, in addition, and finally. For example: First I ___ Then I ___ I also ___ In addition ___ Finally I ___. Encourage them to use Easy Multiples + and its guide in conjunction with the word bank and sentence frames.”

• The online Student Center and Student Reference Book use sound to reduce language barriers to support English language learners. Students click on the audio icon, and the sound is provided. Questions are read aloud, visual models are provided, and examples and sound definitions of mathematical terms are provided. 

• The Differentiation Support ebook available online contains Meeting Language Demands providing suggestions addressing student language demands for each lesson. Vocabulary for the lesson and suggested strategies for assessing English language learners’ understanding of particularly important words needed for accessing the lesson are provided.

The materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade level math concepts. Examples include: 

• Lesson 2-6, (Day 2): Little and Big, Focus: Solving the Open Response Problem, materials reference the use of paper clips. “How many paper clips tall is Little? Is Big’s height greater than or less than Littles’ height in dog treats? In paper clips? How do the paper clips line up with the dog treats?”

• Lesson 3-2, Fraction Circles and Equivalence, Focus: Starting a Collection of Fraction Names, materials reference use of fraction circles. “Ask students to cover the region in Problem 2 on Math Masters, page 100 with fraction circle pieces. They may use only one color at a time and should try all of the colors.” 

• Lesson 6-9, Measuring Angles, Focus: Making an Angle Measurer, materials reference using an angle measurer and protractor. “Tell students that today they will make a tool for measuring angles. Their tool will be similar to a full-circle protractor, but will only have larger intervals marked.”