The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for assessing grade-level content. Summative Interim Assessments include Beginning-of-Year, Mid-Year, and End-of-Year.

Examples of aligned assessment items include but are not limited to:

• Unit 2 Cumulative Assessment, Item 8, “In gym class students were doing the standing long jump. Lance’s jump measured 5 feet. He thinks that he jumped 50 inches. Is he correct? Explain how you know.” (4.NBT.5, 4.MD.1, 4.MD.2) 
• Unit 3 Assessment, Item 5, “a. Using your fraction circles to help you, find and name 2 fractions that are equivalent to $$\frac{1}{3}$$. b. Using your fraction circles to help you, find and name 2 fractions that are equivalent to $$\frac{2}{5}$$.” (4.NF.1)
• Unit 4 Cumulative Assessment, Item 1, “a. List the first 6 multiples of 9. b. Name two factors of 9. c. Is 9 a multiple of those numbers? Explain.” (4.OA.4)
• Unit 6 Assessment, Item 5, “For each angle, circle the type. Then use a protractor to measure each angle, and record your measurement.” (4.MD.6)

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for spending a majority of instructional time on major work of the grade. 

• There are 8 instructional units, of which 5.7 units address major work of the grade or supporting work connected to major work of the grade, approximately 71%.
• There are 112 lessons, of which 82.75 address major work of the grade or supporting work connected to the major work of the grade, approximately 74%.
• In total, there are 170 days of instruction (112 lessons, 38 flex days, and 20 days for assessment), of which 98.75 days address major work of the grade or supporting work connected to the major work of the grade, approximately 58%. 
• Within the 38 Flex days, the percentage of major work or supporting work connected to major work could not be calculated because the materials suggested list of differentiated activities do not include explicit instructions. Therefore, it cannot be determined if all students would be working on major work of the grade.

The number of lessons devoted to major work is most representative of the instructional materials. As a result, approximately 74% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Examples of supporting standards/clusters connected to the major standards/clusters of the grade include but are not limited to:

• In Lesson 1-13, Student Math Journal, students apply the area and perimeter formulas for rectangles in real-world and mathematical problems (4.MD.3) to fluently add and subtract multi-digit whole numbers using the standard algorithm (4.NBT.4) and multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations (4.NBT.5). Problem 4, “Jerry wants to build a rectangular vegetable garden with a fence around it. He wants the garden to be 8 feet long and 4 feet wide. Sketch his garden. Find the perimeter. Show your work.”
• In Lesson 2-3, Student Math Journal, students multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the properties of operations (4.NBT.5) to find factor pairs for a whole number in the range 1-100 (4.OA.4). Problem 1, “Write equations to help you find the factor pairs of each number below. 20, 16, 13, 27, and 32.” 
• In Lesson 6-7, Teacher’s Lesson Guide, students find all factor pairs for a whole number in the range 1-100 (4.OA.4) to understand finding whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (4.NBT.6). Students use partial-quotients division to divide whole numbers using factors. The teacher poses this problem, “Corey bought 162 stickers to put in gift bags. She wants each gift bag to contain 6 stickers. How many gift bags can she make?” The Student Math Journal, Problem 2, “Carpenters are installing hinges. They have 371 screws. Each hinge needs 3 screws. How many hinges can they install?” 
• In Lesson 6-2, Teacher’s lesson Guide, students apply area and perimeter formulas (4.MD.3) to find missing side lengths of rectangles (4.NBT.5, 4.NBT.6). The teacher prompt states, “A rectangular garden has an area of 450 square feet.  One side is 9 feet long. How long is the other side?” 
• In Lesson 7-13, Math Journal, students make a line plot to display a data set of measurements in fractions of a unit (4.MD.4) to understand building fractions from unit fractions (4.NF.3). Problem 3, “How many insects are longer than 7/8 inch and shorter than 1 6/8 inch? What is their combined length?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations that the amount of content designated for one grade level is viable for one year. 

Recommended pacing information is found on page xxii of the Teacher’s Lesson Guide and online in the Instructional Pacing Recommendations. As designed, the instructional materials can be completed in 170 days:

• There are 8 instructional units with 112 lessons. Open Response/Reengagement lessons require 2 days of instruction adding 8 additional lesson days.
• There are 38 Flex Days that can be used for lesson extension, journal fix-up, differentiation, or games; however, explicit teacher instructions are not provided.
• There are 20 days for assessment which include Progress Checks, Open Response Lessons,  Beginning-of-the-Year Assessment, Mid-Year Assessment, and End-of-Year Assessment.  

The materials note lessons are 60-75 minutes and consist of 3 components: Warm-Up: 5-10 minutes; Core Activity: Focus: 35-40 minutes; and Core Activity: Practice: 20-25 minutes.

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for being consistent with the progressions in the Standards. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades and with work in future grades, and the materials present extensive work with grade-level problems.

The instructional materials relate grade-level concepts to prior knowledge from earlier grades. Each Unit Organizer contains a Coherence section with “Links to the Past”. This section describes “how standards addressed in the Focus parts of the lessons link to the mathematics that children have done in the past.” Examples include:

• Unit 1, Teacher’s Lesson Guide, Links to the Past, “4.NBT.4: In Grade 3, students learn a variety of methods for multidigit addition and subtraction, including partial sums addition, column addition, expand-and-trade subtraction, and trade-first subtraction.” These methods have connections to the U.S. traditional algorithms that are introduced in Grade 4.”  
• Unit 5, Teacher’s Lesson Guide, Links to the Past,”4.NF.3, 4.NF.3a: In Grade 3, students use fraction strips, fraction circles, and fraction number lines to determine equivalence and to compare and order fractions.”  
• Unit 7, Teacher’s Lesson Guide, Links to the Past, “4.MD.4: In Unit 5, students review line plots and create line plots that include fractional units of length and weight. In Grade 3, children measured lengths using rulers marked $$\frac{1}{2}$$ and $$\frac{1}{4}$$ of an inch and represented the data in line plots.”  

The instructional materials relate grade-level concepts with work in future grades. Each Unit Organizer contains a Coherence section with “Links to the Future”. This section identifies what students “will do in the future.” Examples include:

• Unit 2, Teacher’s Lesson Guide, Links to the Future, “4.OA.5: In Grade 5, students use rules, tables, and graphs to extend patterns and solve real-world problems.”
• Unit 6, Teacher’s Lesson Guide, Links to the Future, “4.NBT.5: Throughout Grade 4, students solve multiplication problems involving varied contexts. In Grade 5, students learn U.S. traditional multiplication and use it to solve problems involving whole numbers.”
• Unit 8, Teacher’s Lesson Guide, LInks to the Future, “4.MD.2: In Grade 5, measurement continues to serve as a context for problem solving and for applying computational skills.”

Examples of the materials giving all students extensive work with grade-level problems include:

• In Lesson 1-4, Teacher’s Lesson Guide Volume 1, Warm Up, students identify and write the place value of an indicated digit. For example, “Display numbers using a place-value tool. Have students write their value of the indicated digit on their slates. Leveled exercises: What is the value of the 3 in 39? The 8 in 98? The 6 in 602? What is the value of the 7 in 3750? The 2 in 2,006? The 1 in 6,615? What is the value of the 4 in 13,407? The 5 in 15,247? The 1 in 104,539?” (4.NBT.2)
• In Lesson 5-5, Teacher’s Lesson Guide, Adding Tenths and Hundredths, Focus, Solving Fractions Addition Problems with Denominators of 10 and 100, “Students add unlike fractions with tenths and hundredths.” For example, Student Math Journal, “Problem 7, ‘1 $$\frac{2}{10}$$ + 6 $$\frac{35}{100}$$’.” (4.NF.6)
• In Lesson 7-3, Math Journal 2, Problem 3, “Draw a picture to represent the equations. Addition equation: $$\frac{1}{6}$$ + $$\frac{1}{6}$$ + $$\frac{1}{6}$$ = $$\frac{3}{6}$$; Multiplication equation; 3 * $$\frac{1}{6}$$ = $$\frac{3}{6}$$.” (4.NF.4)

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Focus and Supporting Clusters addressed in each section are found in the Table of Contents, the Focus portion of each Section Organizer, and in the Focus portion of each lesson. Examples include:

• The Lesson Overview for Lesson 2-3, “Students work with factor pairs, arrays, and corresponding equations,” is shaped by 4.OA.B, “Gain familiarity with factors and multiples.”
• The Lesson Overview for Lesson 2-7, “Students convert units of time to smaller units of time and solve number stories involving time,” is shaped by 4.MD.A, “Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.”
• The Lesson Overview for Lesson 3-2, “Students use an area model to recognize and generate equivalent fractions,” is shaped by cluster heading, 4.NF.A, “Extend understanding of fraction equivalence and ordering.”
• The Lesson Overview for Lesson 7-10, “Students solve multistep number stories involving fractions,” is shaped by 4.NF.A, “Extend understanding of fraction equivalence and ordering” and 4.NF.B, “Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.”

The materials include problems and activities connecting two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples include:

• Lesson 3-8 connects 4.NF.A and 4.NF.C as students explore tenths with fraction circles. In the Student Math Journal, Problems 1 and 2, students look at visual models of circles divided into tenths and “Write a fraction and a decimal to match each circle.”
• Lesson 5-5 connects 4.NF.B with 4.NF.A as students use equivalent fractions to write fractions with denominators of 10 as equivalent fractions with denominators of 100 and add fractions with like denominators. In the Student Math Journal, Problem 1, “5 tenths + 27 hundredths.” The directions state, “Use what you know about equivalent fractions to add. Write an equation to show your work.”
• Lesson 6-5 connects 4.OA.A and 4.NBT.B as students solve division story problems and interpret remainders. In the Student Math Journal, Problems 1 and 2, students solve “Elbert’s Egg Emporium: One morning, Elbert collected 151 eggs. 1. How many cartons did he need for the eggs? Show your work. Be sure to include units with your answer. 2. How many eggs did Elbert eat for breakfast? Show or explain how you know. Be sure to include units with your answer.”
• Lesson 7-7 connects 4.OA.A and 4.NBT.B as students interpret the reasonableness of remainders in multi-digit division problems. In the Student Math Journal, Problem 2, “Anna wants to put 72 baseball cards in an album. A square album fits 4 cards per page and a rectangular album fits 5 cards per page. How many more pages will she need to fit all the cards if she uses the square album rather than the rectangular album?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

All units begin with a Unit Organizer which includes Planning for Rich Math Instruction. This component indicates where conceptual understanding is emphasized within each lesson of the Unit. Lessons include Focus, “Introduction of New Content”, designed to help teachers build their students’ conceptual understanding. The instructional materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the clusters (4.NF.A, 4.NBT.A, 4.NBT.B). Examples include: 

• In Teacher’s Lesson Guide, Lesson 1-2, Place-Value Concepts, Focus, students develop conceptual understanding with place value and comparing numbers. Students compare the numbers 46,385 and 463,850. Teachers “Ask students to respond on their slates to the following questions about 46,385 using the place-value chart on Math Masters, page 2 for reference. Which digit is in the hundreds place? What is the value of the digit? Which digit is in the ones place? What is the value of the digit? Which digit is in the ten-thousands place? What is the value of the digit?” Later, students compare and order numbers. Teachers “Pose the following problem: Which number is larger, 47,899 or 48,908? Ask: How can we use expanded form to tell which is larger?” (4.NBT.2)
• In Teacher’s Lesson Guide, Lesson 3-1, Equal Sharing and Equivalence, Focus-Math Message, students model equal-sharing situations and examine equivalent names for those models. “Two brothers go to lunch and share three 8-inch pizzas equally. How much pizza does each brother get? Have students share the drawings they used to model and solve the problem.” Within the Math Journal activity, page 68, students practice using visual representations of fractions and equal sharing to include subdividing “leftover” pieces to produce fair shares. For example, “Use drawings to help you solve the problems. Solve each problem in more than one way. Show your work. 1. Three friends shared 4 chicken quesadillas equally. How many quesadillas did each friend get?” (4.NF.A)
• In Student Math Journal, Lesson 4-3, Partitioning Rectangles, students develop conceptual understanding of multiplication by partitioning rectangles. Problems 1 and 2, “Maya wants to lay tile on a floor that is 8 feet wide by 24 feet long. The tiles she wants to use are 1 square foot each. How many tiles will Maya need? Draw a picture to represent Maya’s floor. Explain how you figure out how many tiles Maya needs.” (4.NBT.5)
• In Teacher’s Lesson Guide, Lesson 5-5, Adding Tenths and Hundredths, Focus-Math Message, “Alex made a poster with 100 paper clips for the 100th day of school celebration. $$\frac{2}{10}$$ of the paper clips were gold. $$\frac{45}{100}$$ of the paper clips were silver. The rest were other colors. On your slate, write a number model with an unknown to represent the fraction of paper clips on Alex’s poster that are silver or gold.” Students develop conceptual understanding of adding unlike fractions with tenths and hundredths when writing a corresponding number model using fraction names; “2 tenths + 45 hundredths = c.” (4.NF.1)
• In Teacher’s Lesson Guide, Lesson 6-3, Strategies for Division, Focus, students use multiples to solve division problems. In the Math Message, students solve, “Mariana is in charge of seating students for an assembly. Each table seats 6. Seventy-eight students will attend the assembly. How many tables will Mariana need to seat all of the students?” Teachers “Invite students to share strategies for solving the problem, discussing the various steps they take. Emphasize the following strategies: Representing the problem concretely, subtracting groups of 6 from 78, and finding multiples of 6. Tell students that today they will use multiples to help find the answers to division problems more efficiently. Pose two more division problems for the class to try. Guide a discussion of how students make sense of the problem and think through solving the problem.” (4.NBT.6)

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. These include problems from Math Boxes, Home Link, and Practice. Examples include:

• In Math Masters, Lesson 3-1, Equal Sharing and Equivalence, Home Link 3-1, students independently demonstrate conceptual understanding of fraction equivalence through sharing. Students solve equal-sharing number stories and generate equivalent-fraction answers. Problem 2, “Five kittens are sharing 6 cups of milk equally. How much milk does each kitten get?” Students are asked to show two ways to solve the problem. (4.NF.1)
• In Math Masters, Lesson 3-6, Comparing Fractions, Home Link 3-6, students compare fractions to solve number stories. Problem 1, “Tenisha and Christa were each reading the same book. Tenisha said she was $$\frac{3}{4}$$ of the way done with it, and Christa said she was $$\frac{6}{8}$$ of the way finished. Who has read more, or have they read the same amount? How do you know?” (4.NF.A)
• In Math Masters, Lesson 4-3, Partitioning Rectangles, Home Link 4-3, students solve 2-digit by 1-digit multiplication problems by partitioning rectangles. “Solve the multiplication problems by partitioning a rectangle. Then add each part of the rectangle to get the product. Problem 2. 6 * 83 =?” (4.NBT.B)
• In Student Math Journal, Lesson 4-6, Introducing Partial-Products Multiplication, students use the partial-products multiplication strategy to extend their conceptual understanding of multiplication and place value. “Helen wants to paint the sidewalk for her block party. She needs to know the area of the sidewalk so she’ll know how much paint to buy. The sidewalk is 5 feet wide and 660 feet long. What is the area of Helen’s sidewalk? ___square feet. 1. Draw a picture to represent Helen’s sidewalk. 2. Show how you figured out the area of the sidewalk.” (4.NBT.2,4,5)
• In Math Masters, Lesson 7-6, Three-Fruit Salad, students demonstrate conceptual understanding of fractions. In the Open Response, students create recipes for a fruit salad. The directions state, “The school cook asks you to create recipes for Three-Fruit Salad. Follow these rules: Each recipe must use exactly 3 different fruits. The combined weight of the fruit for one recipe must be exactly 5 pounds. Make up two recipes that follow the rules. Show that each recipe weighs 5 pounds by using tools such as fraction circles, fraction number lines, drawings, or number models. Use multiplication when possible.” (4.NF.B)

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. Each Unit begins with Planning for Rich Math Instruction where procedural skills and development activities are identified throughout the unit. Each lesson includes Warm-Up problem(s) called Mental Math Fluency. These provide students with a variety of leveled problems to practice procedural skills. The Practice portion of each lesson provides students with a variety of spiral review problems to practice procedural skills from earlier lessons. Additional procedural skill and fluency practice is found in the Math Journal, Home Links, Math Boxes, and various games. Examples include:

• In Student Math Journal, Lesson 3-4, An Equivalent Fraction Rule, Practice, students use the four operations to solve number stories. Problem 1, “Each day a company delivers newspapers to the town of Wayland. It has 158 customers on the north side of town, and 237 customers on the south side. The company receives 900 newspapers to deliver. How many will be left over?” This activity provides an opportunity for students to develop 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”
• In Teacher’s Lesson Guide, Lesson 3-8, Modeling Tenths with Fraction Circles, Warm-Up, teachers state numbers in expanded form and students write the numbers in standard form on their slates. This activity provides an opportunity for students to develop fluency of 4.NBT.2, “Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.”
• In Teacher’s Lesson Guide, Lesson 4-3, Partitioning Rectangles, Focus, students are introduced to a multiplication strategy based on partitioning a rectangle. “Remind students that there are many ways to partition the rectangle. Point out that the numbers can be decomposed into tens and ones or even smaller, friendlier chunks.” This activity provides an opportunity for students to develop procedural skill of 4.NBT.5, “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.” 
• In Teacher’s Lesson Guide, Lesson 6-13, Extending Understandings of Whole-Number Multiplication, Warm Up, students add and subtract fractions with like denominators. Examples of problems include, “$$\frac{1}{4}$$ + $$\frac{2}{4}$$ = , $$\frac{5}{8}$$ + $$\frac{2}{8}$$ = , $$\frac{2}{3}$$ + $$\frac{2}{3}$$ =, 1 $$\frac{3}{5}$$ + 2 $$\frac{1}{5}$$ = .” This activity provides an opportunity for students to develop fluency of 4.NF.3d, “Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.”
• In Teacher’s Lesson Guide, Lesson 8-10, Fractions and Liquid Measures, students divide larger numbers and interpret remainders. In the Student Math Journal, Problem 7, “The school purchased 1,245 new fiction books for the third, fourth, and fifth grade classrooms at Portland South School. There are 3 classrooms at each grade level. a. Can the school divide the books evenly among the classrooms? Why or why not? b. What would be a fair way to divide the books among the classrooms?” This activity provides an opportunity for students to develop procedural skill of 4.OA.3, “Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level as identified in 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.” Examples include:

• In Math Masters, Lesson 5-6, Queen Arlene’s Dilemma, Home Link, students solve 4 addition or subtraction problems using the standard algorithm. Problem 3, “8,936 + 6,796.” The numbers are listed horizontally and students are instructed to use the standard algorithm to solve. (4.NBT.4)
• In Math Masters, Lesson 6-10, Using a Half-Circle Protractor, Home Link, students practice adding or subtracting multi-digit numbers vertically using the standard algorithm. Problem 9, “87,942 - 23,851” (4.NBT.4)
• In Student Math Journal, Lesson 8-12, Applying Understandings of Place Value and Operations, students use place value strategies to solve addition and subtraction puzzles. “Solve the puzzles below. Remember to rewrite the puzzle each time you figure out a digit.” (4.NBT.4)
• The Student Reference Book, includes a game, Subtraction Target Practice, that students can play independently or with a partner. This game provides an opportunity for students to demonstrate subtraction skills (4.NBT.4). Directions: “1. Shuffle the cards and place the deck number-side down on the table. Each player starts at 250. 2. Players take turns. Each player has 5 turns in a game. When it is your turn, do the following: Turn 1: Turn over the top 2 cards and make a 2-digit number (You may place the cards in either order). Subtract this number from 250 on scratch paper. Check the answer on a calculator. Turns 2-5: Take 2 cards and make a 2-digit number. Subtract this number from the result obtained in your previous subtraction problem. Check the answer on a calculator. 3. The player whose final result is closest to 0, without going below 0, is the winner. If there is only 1 player, the object of the game is to get as close to 0 as possible, without going below 0.” 
• There is an online game, Multiplication Wrestling, that provides students with the opportunity to demonstrate procedural skills and fluencies with 4.NBT.1 and 4.NBT.5. In this game, students multiply 2-digit numbers. Directions: “Try to get the highest score you can. During each round, arrange your four number cards into the largest 2-digit numbers you can and use those numbers to make your ‘teams.’ Find your teams’ partial products and then the total product. Each time you get a larger total, it will become your high score!”

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 partially 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. The materials do not provide opportunities for students to independently engage in non-routine applications of mathematics throughout the grade level.

Examples of students engaging in routine application of mathematics include:

• In Math Masters, Lesson 4-2, Making Reasonable Estimates for Products, Home Link, students use extended multiplication facts to make estimates and check the reasonableness of their answers. In Problem 3, “There are 30 Major League Baseball (MLB) teams and 32 National Football League (NFL) teams. The expanded roster for MLB teams is 40 players and it is 53 for NFL teams. How many more players are in the NFL than the MLB?” (4.OA.3) 
• In Student Math Journal, Lesson 5-3, Adding Fractions, students share strategies to solve fraction addition number stories. Problem 2, “Allie and Cherice both run to stay in shape. On Saturday, Allie ran $$\frac{3}{8}$$ of a mile more than Cherice. Cherice ran $$\frac{7}{8}$$ of a mile. How far did Allie run? a. Fill the whole box. b. Number model with unknown: c. A different way to solve a fraction addition problem: d. Answer.” (4.NF.3d)
• In Math Masters, Lesson 7-3, A Fraction as a Multiple of a Unit Fraction, Home Link, students multiply unit fractions by whole numbers. Problem 3, “Dmitri fixed a snack for 5 friends. Each friend got $$\frac{1}{2}$$ of an avocado. How many avocados did Dmitri use? Multiplication equation: __ Answer:  ___ avocado(s).” (4.NF.4c)
• In Student Math Journal, Lesson 7-10, Solving Multistep Fraction Number Stories, students solve multi-step number stories involving fractions and units of time. They use a table showing how long it takes to burn 100 calories doing a variety of activities. Problem 3, “Teru burned about 1,000 calories downhill skiing before lunch. If she plans to ski for a total of 5 hours today, how many hours will she ski after lunch? ___ hours; Explain how you got your answer.” (4.NF.4c)

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples include:

• In Math Masters, Lesson 1-5, Estimation Strategies, Home Link, students estimate solutions to multistep number stories and explain their estimation strategy. Problem 1, “On the walk home from school, Meg stopped at the library for 22 minutes and at her grandmother’s house for 38 minutes. She spent 17 minutes walking. She left at 3:00 and was supposed to be home by 4:00. Did Meg make it home on time? How did you get your answer?” (4.OA.3)
• In Math Masters, Lesson 5-6, Queen Arlene’s Dilemma, Home Link, students find an error in a fraction problem and write a correct fraction addition equation. “Consider the problem: A king owns land outside of his castle. He has partitioned the land to give as gifts to his 5 sons. (An image of the partitioned land is provided.) What fraction of the land did the king give to each of his sons? Here is Zeke’s solution: Andy got $$\frac{1}{2}$$ , Bill got $$\frac{1}{5}$$, Carl got $$\frac{1}{5}$$, Dirk got $$\frac{1}{8}$$, Evan got $$\frac{1}{8}$$.” Problem 1, “Identify Zeke’s two errors, correct them, and explain why your answer is correct.” Problem 2, “Write a fraction addition equation to represent the correct answers and show the sum of the pieces of land.” (4.NBT.4, 4.NF.3a,3b)
• In Assessment Handbook, Unit 8 Assessment, students apply addition and subtraction with fractions in a multi-step story problem. In Problem 7, “Use the information below to solve the problem. Recipe for Soap Bubbles: 5 ¾ cups water, 16 fluid ounces dishwashing liquid, ¾ cup corn syrup. Combine the water and corn syrup, and then slowly add the dishwashing liquid. a. How many cups do the water and corn syrup equal when combined? b. Will a 2-quart container be big enough to hold all of the ingredients? Explain your answer.” (4.NF.3)

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 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 present in the instructional materials. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding.

There are instances where conceptual understanding, procedural skill and fluency, and application are addressed independently throughout the instructional materials. Examples include:

• In Student Math Journal, Lesson 4-6, Introducing Partial-Products Multiplication, students build conceptual understanding as they partition a rectangle and use the partial-products multiplication strategy to solve number stories. Problem 3, “The mayor wants to beautify part of the highway by planting marigolds. She wants to plant 4 marigolds along every foot of highway for an entire mile or 5,280 feet. How many marigolds will she need? Draw a partitioned rectangle to represent the problem. Then use partial-products multiplication to record your work in a similar way.” (4.NBT.5)
• In Math Masters, Lesson 6-4, Partial-Quotients Division, Part 1, students use procedural skill and fluency with partial quotients to divide. Problem 2, “Four sisters love barrettes. They have a value pack that contains 92 barrettes. How many barrettes can each sister have if they share equally?” (4.NBT.6)
• In Math Masters, Lesson 6-5, Fruit Baskets, students engage with application as they use the four operations with whole numbers to solve number stories. Problem 1, “The fourth-grade chess team is planning a fundraiser. They are going to sell fruit baskets. Oscar is in charge of oranges for the baskets. Three students brought oranges. Olivia brought 29 oranges, Ozzie brought 31 oranges, and Olga brought 27 oranges. Each basket must have at least 5 oranges. Some baskets may have 6 oranges if there are any extras after each basket has 5 oranges. a. How many baskets will be needed for the oranges? Show or explain your thinking.” (4.OA.3)

The materials engage students with multiple aspects of rigor simultaneously throughout the materials. Examples include:

• In Math Masters, Lesson 3-1, Equal Sharing and Equivalence, students engage with conceptual understanding and application as they generate equivalent fractions and solve equal sharing number stories. Problem 1, “Four friends shared 5 pizzas equally. How much pizza did each friend get?” (4.OA.4)
• In Math Masters, Lesson 4-8, Money Number Stories, students engage with procedural skills and application as they solve multi-step number stories involving money. Problem 4, “If the cashier only has $10 and $1 bills, what are two ways he could make Mr. Russo’s change?” (4.MD.2)
• In Teacher’s Lesson Guide, Lesson 5-3, Adding Fractions, Focus, Solving Fraction Addition Number Stories, students develop procedural skill, conceptual understanding, and application as they solve fraction addition number stories. “After running $$\frac{3}{4}$$ of a mile, Marisa stopped for a drink of water. Then she ran another $$\frac{3}{4}$$ of a mile. How far did she run in all? What is the whole? Display ‘1 mile’ in a whole box. Does each of the fractions in this problem refer to the same whole? Will your answer be more or less than 1 mile? Encourage strategies such as the following: Use fraction names: just as 3 dogs + 3 dogs = 6 dogs, 3 fourths + 3 fourths = 6 fourths. The unit is fourths. Think about $$\frac{3}{4}$$ as the sum of unit fractions: $$\frac{1}{4}$$ + $$\frac{1}{4}$$ + $$\frac{1}{4}$$. Or more simply with equations: $$\frac{(3+3)}{4}$$ = $$\frac{6}{4}$$. Use the Number-Line Poster: Place a finger on $$\frac{3}{4}$$. Then, beginning at $$\frac{3}{4}$$, count up $$\frac{3}{4}$$ ($$\frac{1}{4}$$ + $$\frac{1}{4}$$ + $$\frac{1}{4}$$) to $$\frac{6}{4}$$.” (4.NF.3, 4.NF.3a, 4.NF.3b, 4.NF.3d)

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

The Implementation Guide states, “The SMPs (Standards for Mathematical Practice) are a great fit with Everyday Mathematics. The SMPs and Everyday Mathematics both emphasize reasoning, problem-solving, use of multiple representations, mathematical modeling, tool use, communication, and other ways of making sense of mathematics. To help teachers build the SMPs into their everyday instruction and recognize the practices when they emerge in Everyday Mathematics lessons, the authors have developed Goals for Mathematical Practice (GMP). These goals unpack each SMP, operationalizing each standard in ways that are appropriate for elementary students.”  

All MPs are clearly identified throughout the materials, with few or no exceptions. Examples include:

• In the Teacher’s Lesson Guide, Unit Organizer, Mathematical Background: Process and Practice, provides descriptions for how the Standards for Mathematical Practices are addressed and what mathematically proficient students should do.
• In the Unit 1 Organizer, MP2, “Reason abstractly and quantitatively,” is addressed. “Students reason about numbers by representing them in various ways. Students create different representations, make sense of them, and learn to see the connections that exist between them.” 
• Lessons identify the Math Practices within the Warm Up, Focus, and Practice sections.

The MPs are used to enrich the mathematical content. Examples include:

• MP1 is connected to mathematical content in Lesson 4-5, Walking Away with a Million Dollars, as students estimate how many dollar bills will cover a page. In the Student Math Journal, an image of a $1 bill is provided and students are given the following prompt, “This picture of a dollar bill is about the same size as an actual dollar bill. All United States bills are the same size and weight.” Problem 1, “How many bills does it take to cover your book?” Teacher’s Lesson Guide, page 349, “For Problem 1, did anyone cover the book completely without gaps or overlaps?” 
• MP2 is connected to mathematical content in the Unit 1 Organizer, Teacher’s Lesson Guide, “Much of the work in Unit 1 focuses on the standards strand Numbers and Operations in Base Ten. Students reason about numbers by representing them in various ways, as required by Mathematical Process and Practice 2. In Unit 1 students create these different representations, make sense of them, and learn to see the connections that exist between them.” 
• MP4 is connected to mathematical content in Lesson 3-1, Equal Sharing and Equivalencies, Teacher’s Lesson Guide, as students share drawings for equal sharing scenarios. Students solve, “Two brothers go to lunch and share three 8-inch pizzas equally. How much pizza does each brother get? Be prepared to share a drawing you could use to solve the problem.” Support for teachers includes, “Have students share the drawings they used to model and solve the problem. Provide plenty of time for discussion. If no one mentions any of the strategies below, be sure to include them in the discussion.”
• MP7 is connected to mathematical content in Lesson 6-1, Extended Division Facts as students explore the connections between multiplication and division fact families. Teachers support a discussion around patterns in extended division facts. Teacher’s Lesson Guide, “Tell students that, as with multiplication, extended facts can help solve division problems. Display the problem 35 divided by 7, 350 divided by 7, and 3,500 divided by 7. Have partners discuss strategies for solving. If no one mentions it, guide students in a discussion of using basic facts and knowledge of place value to solve extended division facts: Identify the basic fact, solve the basic fact, note place value.”  
• MP8 is connected to mathematical content in Lesson 7-9, Generating and Identifying Patterns, as students generate and analyze patterns. Teacher’s Lesson Guide, “Guide students to discover a rule for generating rectangular numbers. Display arrays from the Math Message and ask: What multiplication equations can we write to represent these arrays? Display equations next to the corresponding arrays, making connections between the numbers in the equations and the corresponding parts of the arrays.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials attend to the full meaning of most of the MPs, but they do not attend to the full meaning of MP5 as students do not get to choose tools strategically.

Examples of the materials attending to the full meaning of most MPs include:

• MP1: In Lesson 4-8, Money Number Stories, students make sense of problems as they conceptualize the conversions between units of money to help them solve the problems. Student Math Journal, Problem 1a, “Colleen, Emilia, and Theresa are going home to San Diego. They buy 3 train tickets using two $100 bills. How much change should they get from the cashier?” Problem 1b, “The cashier wants to use the least number of bills when she gives the girls change and has only $10 and $1 bills. How many $10 and $1 bills could she give them?” 
• MP2: In Lesson 2-8, Multiplicative Comparisons, students reason abstractly and quantitatively as they use symbols to represent an unknown in an equation, solve the unknown, and then interpret number stories in context. Student Math Journal, Problem 5, “Sally is 21 years old. Tonya is 3 times as old as Sally. How old is Tonya? a. Equation with unknown. b. Answer: ___ years old.” Problem 6, “Write a comparison number story using the equation 8 * 5 = 40.” 
• MP4: In Lesson 8-10, Fractions and Liquid Measures, students model with mathematics as they solve liquid measurement problems with fractions using drawings, measurement scales, and equations. Students must determine if the statement is true or false and explain. In the Student Math Journal, students analyze a punch recipe and use the recipe to determine if statements are true or false and provide an explanation. In Problem 4, “There is more than twice as much orange juice as apple juice in the recipe.” In Problem 7, “The combined amount of juice in the recipe is 1 $$\frac{3}{4}$$ cups more than the amount of soda.”
• MP6: In Lesson 4-2, Making Reasonable Estimates for Products, students attend to precision by making an estimate and then evaluating if their answer is reasonable. Student Math Journal, Problem 2, “The best cows give about 400 cups of milk every day. The best goats give about 8 cups of milk every day. About how many more cups of milk will a cow give in 1 year than a goat? Is your answer reasonable? How do you know?”  
• MP7: In Lesson 7-9, Generating and Identifying Patterns, students use structures to solve problems and answer questions as they build arrays. Teacher’s Lesson Guide, Math Message, “Use centimeter cubes to build the following arrays: 1-by-2, 2-by-3, 3-by-4. Be prepared to discuss any patterns you notice.”
• MP8: In Lesson 8-3, Pattern-Block Angles, students decompose angles and make generalizations about their results. Teacher’s Lesson Guide, “How did you use known angle measures to find unknown angle measures?”

Examples of the materials not attending to the full meaning of MP5 because students do not get to choose tools strategically include:

• In Lesson 1-10, U.S. Customary Units of Length, students are instructed to use a measurement scale relating yards to feet. Teacher’s Lesson Guide, “Have students work in partnerships to complete Problems 5-9 on journal pages 24 & 25. Remind them to use a measurement scale to help them see the relationship between the units of length.” 
• In Lesson  2-7, Units of Time, students are directed to use a measurement scale to convert hours to minutes and minutes to seconds. Teacher’s Lesson Guide, “How can we use the measurement scale to find how many minutes are in 3 hours?” Math Masters, page 70, “Use the measurement scales to fill in the tables and answer the questions.” 
• In Lesson 5-1, Fraction Decomposition, students are instructed to use fraction circles to decompose fractions and mixed numbers. Teacher’s Lesson Guide, “How many dark greens did you use to show $$\frac{4}{5}$$? Ask: Can you think of another way to decompose $$\frac{4}{5}$$ into the sum of two or more fractions that have 5 as the denominator? Encourage students to use fraction circle pieces to justify each decomposition.” 
• In Lesson 7-4, Multiplying Fractions by Whole Numbers, students complete problems using suggested tools. Student Math Journal, Problem 3, “Maria walks the same distance at every meeting. Find out how far she will walk after 7 meetings.” Teacher’s Lesson Guide, instructions for the teacher states, “Have partners complete journal pages 238 & 239, encouraging them to use visual fraction models such as fraction circles and number lines to help them solve the problems.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments. Examples include:

• In Lesson 2-8, Little and Big, Student Math Journal, students construct a viable argument as they decide if rules for a table are correct and make arguments to support their decisions. “Mr. Cheng’s class is trying to figure out the rules for the table below. For the rule to be correct, the rule must work for all rows.” The “in” column shows 1, 2, 4; the “out” column shows 1, 3, 7. “In the table below, the first column shows educated guesses, or conjectures, for rules that Mr. Cheng’s students made. Some rules are correct and some are not. Circle yes or not to tell whether the rule is correct. Then write an explanation or argument for why you think the rule is correct or not.” 
• In the Unit 3 Open Response Assessment, Assessment Handbook, students analyze the thinking of others as they compare fractional portions of paper, Problem 1, “Ten members of the Student Council are each making a poster for the school picnic. Each student is going to cut out shapes from pink or yellow paper to paste on poster board with the picnic information. The sheets of paper are the same size for both colors. Ms. Fondy, the Student Council advisor, gave the group of students 2 sheets of pink paper and 3 sheets of yellow paper. Six students chose pink and 4 chose yellow. Libby and Marcus complained that the students getting the pink sheets and the ones getting the yellow sheets were not getting equal shares. Ms. Fondy asked Libby and Marcus to explain why they thought that everyone was not getting the same amount of paper. Libby and Marcus began by agreeing about the following: Each of the 6 students who want pink will get $$\frac{2}{6}$$ of a sheet. Each of the 4 students who want yellow will get $$\frac{3}{4}$$ of a sheet. Marcus said, ‘I think that people who chose yellow will get more paper, $$\frac{2}{6}$$ is less than $$\frac{1}{2}$$, and $$\frac{3}{4}$$ is greater than $$\frac{1}{2}$$. That means that $$\frac{3}{4}$$ is greater than $$\frac{2}{6}$$.’ Use drawings or fraction strips, fraction circles, or a number line to show how Marcus used a benchmark fraction in his thinking. Explain how the drawing shows his thinking.” 
• In Lesson 3-7, Comparing and Ordering Fractions, Teacher’s Lesson Guide, Math Message, students construct an argument when they compare two fractions with different denominators and numerators. “Which fraction is smaller: $$\frac{3}{8}$$ or $$\frac{1}{5}$$? Or are they equivalent? Record your answer on your slate using one of the symbols >, =, or <. Be prepared to justify your conclusion.” 
• In Lesson 6-5, Fruit Baskets, Math Masters, students explain their thinking as they solve multi-step problems about distributing oranges into baskets and putting baskets into boxes. Problem 1a, “Each basket must have at least 5 oranges. Some baskets may have 6 oranges if there are any extras after each basket has 5 oranges. How many baskets will be needed for the oranges? Show or explain your thinking.” 

Student materials consistently prompt students to analyze the arguments of others. Examples include:

• In Lesson 5-3, Adding Fractions, Student Math Journal, “Cassie said, ‘I think 10.6 is less than 10.06 since it doesn’t have any hundredths.’ Is she correct? Explain your answer.” Students compare decimal place value when they order the times from slowest to fastest. 
• In Lesson 5-11, Unit Iteration for Angles, Practice, Playing Fraction Top-It, Student Reference Book, “Players compare cards. The player with the larger fraction wins the round and takes all of the cards. Players may check who has the larger fraction by turning over the cards and comparing the amounts shaded or the distance from 0 on the number line. Players should justify their comparison. Each player records the comparison on his or her Top-It Record Sheet. If the fractions are equivalent, each player plays another card. The player with the larger fraction takes all the cards from both plays.” Students justify their fractions are greater by formulating arguments. 
• In Lesson 8-9, More Fraction Multiplication Number Stories, Focus, Student Math Journal, “Ella bought 3 yards of fabric from a bolt that is 45 inches wide. She said, ‘I have 135 square inches of fabric for my project.’ Do you agree with Ella? Explain why or why not.” 

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The Teacher’s Lesson Guide assists teachers in engaging students in constructing viable arguments and/or analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate and explain their reasoning to each other. Examples include:

• In Lesson 2-6, Little and Big, Teacher’s Lesson Guide, students construct a viable argument as they analyze someone’s conjecture. “Have students turn to Student Reference Book pages 10-11. Read the pages together and remind students that today’s Math Message asked them to decide whether someone’s conjectures about the rules for the “What’s My Rule?” table are correct. Use the Student Reference Book pages to discuss the meaning of conjecture and argument as a class.  Refer students to the Standards for Mathematical Practice Poster for GMP3.1.”  
• In Lesson 3-6, Comparing Fractions, Teacher’s Lesson Guide, students compare fractions and justify conclusions. “Ask students to write a number model on their slates using the symbols >, =, or < to record the results of the comparison $$\frac{4}{5}$$ and $$\frac{3}{5}$$. Then invite students to justify their conclusions.” Possible responses are provided for the teacher.  
• In Lesson 4-5, Walking Away with a Million Dollars, Teacher’s Lesson Guide, students analyze the reasoning of others through the use of a rubric. “Have partners review and discuss other students’ work using the student-friendly rubric on Math Masters page 156. Choose sample work from three students who all correctly agree that one million dollars in $1 bills cannot fit in the box, but it can fit using $100 bills. Show a range of calculations, estimations, and drawings. Use work in which students showed the connections between the drawings, calculations, and estimations.” 
• In Lesson 5-11, Unit Iteration for Angles, Teacher’s Lesson Guide, students compare fractions and justify conclusions. “To practice comparing fractions and justifying comparisons, have students play Fraction Top-It.” Student Reference Book page 265, “Players compare cards. The player with the larger fraction wins the round and takes all of the cards. Players may check who has the larger fraction by turning over the cards and comparing the amounts shaded or the distance from 0 on the number line. Players should justify their comparison. Each player records the comparison on his or her Top-It Record Sheet. If the fractions are equivalent, each player plays another card. The player with the larger fraction takes all the cards from both plays.”
• In Lesson 7-6, Three Fruit Salad, Teacher’s Lesson Guide, teachers support students in analyzing the work of others. “Display a response that clearly shows how a student used an appropriate tool such as in Student A’s work. Ask: Does this recipe follow the two rules? What tools did this student use? Were the tools used effectively? Why do you think this student described a salad that weighs more than 5 pounds? How could this work be improved?” 
• In Lesson 8-3, Pattern-Block Angles, Teacher’s Lesson Guide, teachers are guided to support students to analyze the work of others. “Display work in which a student sketched a correct answer for Problem 2 but did not provide a complete response, as in Student B’s work. Ask: How does this student’s diagram show how to find the measure of a large angle of the white rhombus?  What else does this student need to do to complete the answer?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 4 partially meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols, but there are instances when the materials use mathematical language that is not precise or appropriate for the grade level.

The Section Organizer provides a vocabulary list of words to be used throughout lesson discussions. Each lesson contains a vocabulary list, Terms to Use, and vocabulary words appear in bold print in the teacher notes. Some lessons incorporate an Academic Language Development component that provides extra support for the teacher and students. Additionally, the Teacher’s Lesson Guide contains a detailed glossary with definitions and images where appropriate. Examples of explicit instruction on how to communicate mathematical thinking include:

• In Lesson 2-2, Focus: The Area Formula for Rectangles, Teacher’s Lesson Guide, “Tell students that in today’s lesson, the class will identify a rule, or formula for finding the area.”

• In Lesson 4-3, Focus: Partitioning Rectangles, Teacher’s Lesson Guide, “If no one mentions it, illustrate how finding the area of smaller parts - or partitions - of the total area of May’s floor makes the problem easier to solve.”
• In the Student Reference Book, “Parallel Lines, are lines on a flat surface that never cross or meet. Example: Think of a straight railroad track that goes on forever. The two rails are parallel lines. The rails never meet or cross, and they are always the same distance apart.” 
• In the Student Reference Book, “The word fraction, comes from the Latin word frangere, which means “to break.” A common use for fractions is to name parts of wholes, where the whole is “broken” into equal-size pieces. Fractions are also used to represent other mathematical ideas such as division and ratios.” 

Examples of the materials using mathematical language that is not precise or appropriate for the grade level include:

• In the Student Reference Book, “A Frames-and-Arrows diagram, is one way to show a number pattern. This type of diagram has three parts: a set of frames that contains numbers; arrows that show the path from one frame to the next frame; and a rule box with an arrow below it. The rule tells how to change the number in one frame to get the number in the next frame.”
• In the Student Reference Book, “To use trade-first subtraction, compare each digit in the top number with each digit below it and make any needed trades before subtracting.”
• In Lesson 2-13, Focus: Applying Rules, Teacher’s Lesson Guide, “Remind students how a function table works. A number (the input) is dropped into the machine. The machine changes the number according to a rule. A new number (the output) comes out the other end.”