The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 1 Assessment, Item 7, “How many cubes would it take to fill this prism? _____ cubes; What is the volume of this prism? _____cubic units.” A rectangular prism partially filled with cubes is shown. (5.MD.3, 5.MD.4)
• Unit 2 Assessment, Item 3, “a. Jesse collects cans for recycling. When he has 1,500 cans, the recycling center will pick them up from his house. Jesse has 120 bags with about 35 cans in each bag. Should he call the recycling center to arrange a pick-up? Explain how you know. b. Did you have to find an exact answer to solve Problem 3a? Explain why or why not.” (5.NBT.5)
• Unit 4 Assessment, Item 19, “Gina is donating money to her neighborhood food pantry. Her aunt agreed to donate two dollars more than Gina donates. The table below shows some of the possible amounts of money they may donate. a. Write the data in the table above as ordered pairs. b. Plot the ordered pairs as points and use a straightedge to connect them.” A table of values is shown. (5.G.2) 
• End-Of-Year Assessment, Item 12, “Reed’s class is painting a giant chessboard on the playground. A chessboard consists of 64 squares arranged in 8 rows and 8 columns. His class is making each square $$\frac{1}{3}$$ m by $$\frac{1}{3}$$ m. a. What will be the length and width of the chessboard in meters? Show your work. b. What will be the area of the completed chessboard? Show your work. Give your answer in square meters. c. How could you use the number of squares on the chessboard to find the area of the chessboard in square meters?” (5.NF.4.b)

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 meet expectations for spending a majority of instructional time on major work of the grade. 

• There are 8 instructional units, of which 6 units address major work of the grade or supporting work connected to major work of the grade, approximately 75%.
• There are 113 lessons, of which 88.5 address major work of the grade or supporting work connected to the major work of the grade, approximately 78%.
• In total, there are 170 days of instruction (113 lessons, 37 flex days, and 20 days for assessment), of which 95.5 days address major work of the grade or supporting work connected to the major work of the grade, approximately 56%. 
• Within the 37 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 78% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 2-7, Teacher’s Lesson Guide, students write simple expressions that record calculations with numbers and interpret numerical expressions without evaluating them (5.OA.2) and fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.5). In the Focus portion of the lesson, teachers guide students through this problem, “Display the work shown to the right. Ask: Do you think 384 is the correct answer for 64 x 15?  How do you know? Interpret the expression 64 x 15 without evaluating it. 64 x 15 is equivalent to a number that is 15 times as large as 64. Ten times as much as 64 is 640, so that means that 64 x 15 is greater than 640. Since 384 is less than 640, it cannot be the correct answer.” 
• In Lesson 3-8, Student Math Journal, students write and interpret numerical expressions (5.OA.A) and apply and extend previous understandings of multiplication and division (5.NF.B). Students use an equal-grouping interpretation of division to explore connections between fractions and division. Problem 1 states, “Write a division expression to model each story, then solve. You can use fraction circles or draw pictures to help. Olivia is running a 3-mile relay race with 3 friends. If the 4 of them each run the same distance, how many miles will each person run?”
• In Lesson 6-4, Teacher’s Lesson Guide, students make a line plot to display a data set of measurements in fractions of a unit (5.MD.2) and add and subtract fractions with unlike denominators (including mixed numbers) (5.NF.1). Students measure the length of their pencils to the nearest 1/4 inch and collect data about the lengths of their peers’ pencils. Then students use this data to make a line plot and answer questions about the data. Teacher prompt states, “What is the difference in length between the longest pencil and the shortest pencil?”  
• In Lesson 8-1, Student Math Journal, students convert like measurements within a given measurement system (5.MD.1) and fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.5). Problem 2 states, “The town has 4 acres of land to use for an athletic center. The land is a rectangle with a length of 160 yards and a width of 121 yards. The town wants the athletic center to have a variety of sports playing surfaces. You have been asked to help decide which playing surfaces should be included and how the surfaces should be arranged. Explain how you and your group created your plan.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 113 lessons. Open Response/Reengagement lessons require 2 days of instruction adding 8 additional lesson days.
• There are 37 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 5 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 present extensive work with grade-level problems. The instructional materials relate grade-level concepts with work in future grades, but there are a few lessons that contain content from future grades that is not clearly identified as such. 

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, “5.OA.1: In Grade 3, students inserted parentheses into number sentences and solved number sentences containing parentheses.”  
• Unit 5, Teacher’s Lesson Guide, Links to the Past,”5.NF.5, 5.NF.5a: In Unit 4, students made conjectures about how an image on a coordinate grid would change based on multiplying one or more of the coordinates. In Grade 4, students interpreted multiplication equations as comparisons.”  
• Unit 7, Teacher’s Lesson Guide, Links to the Past, “5.NF.4, 5.NF.4b: In Unit 1, students used informal strategies to find areas of rectangles with fractional side lengths. In Unit 5, students represented products as rectangular areas as they learned procedures for fraction multiplication. In Grade 4, students applied the area formula for rectangles to solve problems.” 

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, “5.OA.1: In Unit 7, students will use grouping symbols in an expression to model how to solve a multistep problem about gauging reaction time. In Grade 6, students will evaluate expressions and perform operations according to the Order of Operations.”
• Unit 6, Teacher’s Lesson Guide, Links to the Future, “5.NBT.2: In Unit 8, students will multiply and divide numbers by powers of 10 to help them solve rich, real-world problems. In Grade 6, students will write and evaluate numerical expressions with whole-number exponents.”
• Unit 8, Teacher’s Lesson Guide, Links to the Future, “5.MD.1: In Grade 6, students will use ratio reasoning to convert measurement units.”

In some lessons, the instructional materials contain content from future grades that is not clearly identified as such. Examples include:

• In Lesson 4-12, Math Masters, Decimal Addition Algorithms, Focus, Extending Whole-Number Addition Algorithms to Decimals, “Students practice using decimal addition algorithms and use estimates to check the reasonableness of their answers (5.NBT.7).” Problem 3, “Find the page in your Student Reference Book that shows how to use your algorithm to add decimals. Write the page number below. Read the example. Then use your algorithm to solve 2.965 + 7.47. Record your work in the space at the right.” Since one of the numbers includes the thousandths place, this problem aligns to 6.NS.3, fluently adding, subtracting, multiplying, and dividing multi-digit decimals using the standard algorithm for each operation.
• In Lesson 4-13, Math Masters, Decimal Subtraction Algorithms, Focus, Extending Whole-Number Subtraction Algorithms to Decimals, “Students practice using decimal subtraction algorithms and use estimates to check the reasonableness of their answers (5.NBT.7).” Problem 2, “Find the page in your Student Reference Book that shows how to use your algorithm to subtract decimals. Write the page number below. Read the example. Then use your algorithm to solve 9.48 - 7.291. Record your work in the space at the right.” Since one of the numbers includes the thousandths place, this problem aligns to 6.NS.3, fluently adding, subtracting, multiplying, and dividing multi-digit decimals using the standard algorithm for each operation.
• In Lesson 8-3, Student Math Journal, Planning an Aquarium, Focus, Choosing a Fish Tank, “Students use area and volume guidelines to choose a tank (5.MD.5).” The image provided shows two fish tanks, and students find the volume of each fish tank. Fish Tank B provides fractional side lengths. Problem 1, “Choose the fish tank that you want for your aquarium. Fish Tank B: Side lengths include 16 in., 6 $$\frac{1}{2}$$ in., 10 in., 10 $$\frac{1}{2}$$ in., 8 in., 20 in.” Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism aligns to 6.G.2.

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

• In Lesson 4-4, Student Math Journal, Comparing and Ordering Decimals, “Darryl and Charity are playing Decimal Top-It. Their record sheet is shown below. Problem 1, Compare their decimals for each round and write >, <, or = in the column. Use the place-value charge above to help you.” (5.NBT.6, 5.NF.3, 5.MD.1)
• In Lesson 5-2, Teacher Lesson Guide, More Strategies for Finding Common Denominators, Focus, Using Factors and Multiples to Find Common Denominators, “Students learn strategies for finding a common denominator for a pair of fractions (5.NF.1).” Have students give factors and multiples of additional numbers. Teachers display the factors and multiples that students name (call out). Suggestions: 12: Factors: 1, 2, 3, 4, 6, 12 Multiples: 12, 24, 36, 48.” Finding factors and multiples is a part of adding and subtracting fractions with unlike denominators. (5.NF.1)

• In Lesson 8-2, Math Journal 2, Problem 3, “Write the numbers from the table in Problem 1 as ordered pairs. Remember to use parentheses and a comma in each ordered pair. Then graph the ordered pairs on the coordinate grid. Draw a line to connect the points.” (5.G.1,2)

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 1-8, “Students relate volume to multiplication and addition by thinking about iterating layers to find the volumes of prisms,” is shaped by 5.MD.C, “Geometric measurement: understand concepts of volume” and “Relate volume to multiplication and to addition.”
• The Lesson Overview for Lesson 3-7, “Students use benchmarks to estimate sums and differences of fractions,” is shaped by 5.NF.A, “Use equivalent fractions as a strategy to add and subtract fractions.”
• The Lesson Overview for Lesson 4-6, “Students are introduced to the coordinate grid and use ordered pairs to plot and identify points,” is shaped by cluster heading for 5.G.A, “Graph points on the coordinate plane to solve real-world and mathematical problems.”
• The Lesson Overview for Lesson 6-9, “Students learn two strategies for solving decimal multiplication problems,” is shaped by 5.NBT.A, “Understand the place value system” and 5.NBT.B, “Perform operations with multi-digit whole numbers and with decimals to hundredths.”

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 4-11 connects 5.NBT.A and 5.NBT.B as students use grids to solve decimal addition and subtraction problems. In the Student Math Journal, Problems 1 and 2, “Shade the grid in one color to show the first addend. Shade more of the grid in a second color to show the second addended. Write the sum to complete the number sentence.” 1. “0.6 + 0.22 = ____?  2. 0.18 + 0.35 = ____?”
• Lesson 6-2 connects 5.NBT.A and 5.NBT.B as students multiply and divide decimals by powers of 10, and compare decimals and whole numbers. In the Focus portion of the lesson, students learn the game “Exponent Ball”. Directions from the game include, “If the card is black, multiply your starting number by the power of 10. If the card is red or blue, divide your starting number by the power of 10. Write an expression to show how to multiply or divide your number. Then find the value of the expression.”
• Lesson 7-1 connects 5.NBT.B and 5.NF.B as students relate operations with fractions to operations with whole numbers. Teachers use the following example with students, “4 $$\frac{2}{3}$$ * 7 = 4 * 7 + $$\frac{2}{3}$$ * 7.”
• Lesson 8-1 connects 5.NBT.B with 5.NF.B as students perform operations with multi-digit whole numbers and apply and extend previous understandings of multiplication and division to multiply and divide fractions. In the Math Message, students convert measurements to find the area of an Olympic beach volleyball court. “The dimensions of an official Olympic volleyball court are 52 feet 6 inches by 26 feet 3 inches. Find the area of the court in square feet.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 standards (5.NF.B, 5.NBT.A, 5.NBT.B). Examples include: 

• In Student Math Journal, Lesson 2-7, U.S. Traditional Multiplication, Part 1, Problem 5, students use the U.S. traditional multiplication to multiply 2-digit numbers by 2-digit numbers and estimation to determine whether their products make sense. “Complete the area model. Explain how it relates to your work for Problem 3. Area model.” Students create an area model for 87 x 46 by breaking 46 into 40 and 6. (5.NBT.5)
• In Teacher’s Lesson Guide, Lesson 3-9, Introduction to Adding and Subtracting Fractions and Mixed Numbers, Math Message, students explore strategies and tools for adding and subtracting fractions and mixed numbers to develop conceptual understanding of addition and subtraction of fractions and mixed numbers. “Deena and her brother are making two kinds of bread. Both recipes call for $$\frac{2}{3}$$ cup of flour. How much flour do they need in all? You may use fraction circles or the Fraction Number Lines Poster to help you.” In the Teacher’s Lesson Guide, sample strategies are provided for teachers to discuss with students. For instance, “Use fraction circle pieces with the red circles as the whole. Use 2 orange pieces to show $$\frac{2}{3}$$ and 2 more orange pieces to show another $$\frac{2}{3}$$. Put 3 orange third pieces together to make a whole. The total is 1 whole and 1 third, or 1 $$\frac{1}{3}$$.” (5.NF.1)
• In Teacher’s Lesson Guide, Lesson 4-1, Decimal Place Value, Focus, Extending Place-Value Patterns to The Thousandths Place, students extend place-value patterns to decimals, and practice reading and writing decimals to thousandths. Students determine the value of a 1-by-10 rectangular strip and 10-by-10 square when a small square is defined as 1. Next, students discuss times 10 and $$\frac{1}{10}$$ of place value relationships and extend the patterns through the thousandths place. (5.NBT.1) 
• In Student Math Journal, Lesson 4-2, Representing Decimals through Thousandths, Problem 4, students develop conceptual understanding of representing decimals to the thousandths place using three different ways. “Use words, fractions, equivalent decimals, or other representations to write at least three names for each decimal in the name-collection box. Then shade the grid to show the decimal 0.8.” (5.NBT.A)
• In Student Math Journal, Lesson 4-11, Addition and Subtraction of Decimals with Hundredths Grids, students use grids to represent and solve decimal addition and subtraction problems. For example: “Problem 2, 0.18 + 0.35 =? Problem 5, Choose one of the problems above. Clearly explain how you solved it.” (5.NBT.7)
• In Student Math Journal, Lesson 5-4, Subtraction of Fractions and Mixed Numbers, Practice, students compare decimals and shade grids to represent the decimals. For example, Problem 1, “Shade the first grid to represent one tenth. Shade the second grid to represent ninety-nine thousandths. Write <, >, or = to make a true number sentence. 0.1 ____ 0.099.” (5.NBT.3a,b) 
• In Teacher’s Lesson Guide, Lesson 5-7, Fractions of Fractions, Focus, Finding Fractions of Fractions, students find a fraction of a fraction. “Remind students that when they solve fraction-of problems, such as $$\frac{2}{3}$$ of 6, they start by thinking about what $$\frac{1}{3}$$ of 6 would be. Ask them to think similarly about this problem to find $$\frac{2}{3}$$ of $$\frac{1}{2}$$, they should first think about what $$\frac{1}{3}$$ of $$\frac{1}{2}$$ would look like. Ask: How could we use shading to show $$\frac{1}{3}$$ of $$\frac{1}{2}$$ of our paper?” (5.NF.4a,4b,.6)
• In Student Math Journal, Lesson 5-9, Understanding an Algorithm for Fraction Multiplication, Problem 8, students find how many total parts and how many shaded parts in an area model while using patterns to derive a fraction multiplication algorithm. “Choose one of the above problems. Draw an area model for the problem. Explain how it shows that your answer is correct.” Students can choose from these problems: “$$\frac{1}{2}$$ * $$\frac{3}{6}$$, $$\frac{2}{3}$$ * $$\frac{1}{4}$$, $$\frac{3}{5}$$ * $$\frac{1}{6}$$, $$\frac{3}{4}$$ * $$\frac{3}{8}$$, $$\frac{2}{5}$$ * $$\frac{4}{10}$$, or $$\frac{7}{4}$$ * $$\frac{2}{12}$$.” (5.NF.4)

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 3-9, Introduction to Adding and Subtracting Fractions and Mixed Numbers, Home Link 3-9, students draw pictures and use number lines to solve mixed-number addition and subtraction number stories. Problem 2, “Ethel had 4 feet of ribbon. She used 1$$\frac{1}{2}$$ feet for a craft project. How many feet of ribbon does she have left?” (5.NF.2)
• In Student Reference Book, Lesson 5-1, Using Equivalent Fractions to Find Common Denominators, Practice, students play the game, Decimal Top-It: Subtraction, to practice subtracting decimals. The directions state: “1. Shuffle the deck and place it number-side down on the table. 2. Each player turns over 6 cards and, using the counter as a decimal point, forms 2 numbers with digits in the ones, tenths, and hundredths places. Players may put their cards in any order. 3. Each player finds the sum of his or her numbers. Players then compare their sums. The player with the larger sum takes all the cards. Players can check their answers with a calculator. 4. The game ends when there are not enough cards left for each player to have another turn. The player with the most cards wins.” (5.NBT.A,B)
• In Math Masters, Lesson 5-11, Explaining the Equivalent Fraction Rule, Home Link 5-11, students use the multiplication rule to find equivalent fractions. Problem 3, “Addison wanted to find a fraction equivalent to $$\frac{3}{8}$$ with 16 in the denominator. He thought: “8 * 2 = 16, so I need to multiply $$\frac{3}{8}$$ by 2.” He got an answer of $$\frac{3}{16}$$. a. Is $$\frac{3}{16}$$ equivalent to $$\frac{3}{8}$$? How do you know? b. What mistake did Addison make?” (5.NF.4a,5b)
• In Student Math Journal, Lesson 6-5, Working with Data in Line Plots, Math Boxes, students shade a rectangle to represent a multiplication problem. Problem 1, “Shade a rectangle to represent $$\frac{2}{3}$$ * $$\frac{5}{6}$$ = ?” (5.NF.4)
• In Student Math Journal, Lesson 7-7, Playing Property Pandemonium, Math Boxes, Problems 4 and 5, “4. Which expressions have a value equal to 6? Check all that apply. 6 * $$\frac{2}{2}$$, 6 * $$\frac{8}{7}$$, $$\frac{3}{2}$$ * $$\frac{6}{1}$$, 6 * $$\frac{9}{10}$$, 6 * 1. 5. Explain how you solved Problem 4 without multiplying.” (5.NF.5a,5b)

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 Teacher’s Lesson Guide, Lesson 2-7, U.S. Traditional Multiplication, Part 3, Focus, students discuss how to solve 2-digit numbers multiplied by 2-digit numbers, “Suppose you already figured out that 54 * 8 = 432. How could you use that information to help you solve 54 * 18?” This activity provides an opportunity for students to develop fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”
• In Teacher’s Lesson Guide, Lesson 4-1, Decimal Place Value, Practice, students practice multiplying fractions and whole numbers by playing the game Fraction Of. In the Student Reference Book, “Players take turns. On your turn, draw 1 card from each deck. Place the cards on your gameboard to create a fraction-of problem. The fraction card shows what fraction of the whole you must find. The whole card offers 3 possible choices. Choose a whole that will result in a fraction-of problem with a whole-number answer.” This activity provides an opportunity for students to develop fluency of 5.NF.4, “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”
• In Teacher’s Lesson Guide, Lesson 4-7, Playing Hidden Treasure, Practice, students estimate and solve problems using U.S. traditional multiplication. The Student Math Journal includes 6 multi-digit multiplication problems where students first estimate the product and then solve using U.S. traditional multiplication. For example, Problem 6, “511*219.” (5.NBT.5) 
• In Teacher’s Lesson Guide, Lesson 6-9, Multiplication of Decimals, Focus, students multiply decimals as if they were whole numbers and use estimation to place decimal points in the products. “Display the following problems: 1.2 * 0.8 = ?; 1.2 * 8 = ?; 12 * 8 = ? What do you notice about the factors in all three problems? Do you think this pattern is true for all multiplication problems with factors that have the same digits in the same order?” This activity provides an opportunity for students to develop procedural skill and fluency of 5.NBT.7, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used,” and 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”
• In Teacher’s Lesson Guide, Lesson 6-10, Fundraising, Focus, students determine and discuss strategies for choosing the correct product without actually calculating the exact answer. In the Student Math Journal, “For each set of problems, circle the number sentence with the correct product. Do not calculate the exact answer. Explain how you knew which product was correct without finding the exact answer.” In the Teacher’s Lesson Guide, “As students share strategies, consider listing them on the Class Data Pad or chart paper. Display the list for reference when students solve the open response problem.” This activity provides an opportunity for students to develop procedural skill of 5.NBT.7, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level as identified in 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.” Examples include:

• In Teacher’s Lesson Guide, Lesson 2-9, One Million Taps, Focus, students estimate the amount of time it takes to address 10 and 100 envelopes based on the amount of time it takes to address 1 envelope. Student Math Journal, Problem 1, “Zoey is mailing invitations for a fifth-grade party. It takes her about 30 seconds to address 1 envelope. About how many seconds would it take Zoey to address 10 envelopes? Show your work. " (5.NBT.2,5)
• In Math Masters, Lesson 3-1, Connecting Fractions and Division, Part 1, Home Link, students make an estimate and solve problems using the standard algorithm of multiplication. Problem 4, “2,598 x 3”; Problem 5, “417 x 63.” (5.NBT.5)
• In Student Math Journal, Lesson 4-7, Playing Hidden Treasure, students solve problems using the standard algorithm for multiplication such as Problem 1, “25 x11” and Problem 4, “Another student estimated and began solving problems 4-6 using U.S. traditional multiplication. Finish solving the problems.” (5.NBT.5)
• In Math Masters, Lesson 4-11, Addition and Subtraction of Decimals with Hundredths Grids, Home Link, students use the standard algorithm of multiplication to solve problems involving multi-digit numbers. Problem 5, “Make an estimate. Then solve using U.S. traditional multiplication. 412 * 176.” (5.NBT.5) 
• Students have the opportunity to independently demonstrate procedural skill and fluency for 5.NBT.5 in the online game, Multiplication Top It. In this game, students multiply 2-digit numbers by 2 or 3-digit numbers and compare the products. Directions: “Players multiply the numbers shown on gems, then compare the products. The player with the greater product takes all of the gems. Players earn points for correctly multiplying the numbers on their gems, correctly comparing the products, and having the greater product.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 5-5, Connecting Fraction-Of Problems to Multiplication, Home Link, students solve fraction-of problems with non-unit fractions. Problem 4, “Erica’s garden has an area of 24 square feet. She will use $$\frac{3}{4}$$ of the space for vegetables and $$\frac{1}{4}$$ of the space for flowers. How much space will she use for vegetables?” (5.NF.6)
• In Student Math Journal, Lesson 5-13, Fraction Division, Part 1, students solve real-world problems involving division of unit fractions by whole numbers. Problem 1, “Two students equally share $$\frac{1}{4}$$ of a stick of clay. How much of the stick will each student get?” (5.NF.7c)
• In Teachers Lesson Guide, Lesson 5-14, Fraction Division, Part 2, Focus, students use fraction circle pieces to divide a whole number by a fraction. Math Message, “Jane has 2 pieces of pita bread to share with her friends. If she cuts each pita into $$\frac{1}{4}$$s, how many pieces of pita bread will she have to share?” (5.NF.7c)
• In Student Math Journal, Lesson 8-3, Planning an Aquarium, students multiply whole numbers and mixed numbers to determine area and volume. Problems 1 and 2, “Glenn is buying a fish tank for his goldfish, Swimmy. Glenn learned that a 1-inch goldfish needs at least 230 cubic inches of water to be healthy. Swimmy is 1 inch long. Find the volume in cubic inches of each fish tank shown below. Which fish tank should Glenn buy for Swimmy?” (5.NF.6).

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

• In Student Math Journal, Lesson 3-2, Connecting Fractions and Division, Part 2, students write division number stories by working backward from a fractional answer. Problem 4, “Write a division number story with an answer of $$\frac{12}{8}$$.” (5.NF.3)
• In Math Masters, Lesson 3-13, Fraction-Of Problems, Part 1, Home Link, students solve fraction multiplication problems. Problem 5, “A teacher had 20 ounces of water in her water bottle. She drank $$\frac{1}{5}$$ of the water. How many ounces did she drink?” (5.NF.6)
• In Math Masters, Lesson 5-14, Fraction Division, Part 2, Home Link, students divide whole numbers by unit fractions. Problem 1, “Charity is packing a 2-pound container of trail mix into bags for a camping trip. Each bag holds $$\frac{1}{8}$$ pound of trail mix. If Charity uses all 2 pounds of trail mix, how many $$\frac{1}{8}$$ pound bags will she have?” (5.NF.7c) 
• In Math Masters, Lesson 6-5, Working with Data in Line Plots, Home Link, students plot 8 milkshake amounts (a visual of 8 glasses filled with milkshake using various mixed numbers is provided) and use the line plot to answer questions. A Milkshake Problem, “Rachel is having a slumber party with 7 friends. Her mom made a big batch of milkshakes. Rachel’s little brother tried to help by pouring the milkshakes in glasses, but he had trouble pouring the same amount in each glass.” Problem 2a, “How many total ounces of milkshake did Rachel’s mom make? _____ ounces” (5.NF.2)

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 Math Masters, Lesson 2-7, U.S. Traditional Multiplication, Part 3, students demonstrate procedural skill and fluency as they use the standard algorithm to multiply multi-digit whole numbers. Problem 4. “82 * 11.” (5.NBT.5)
• In Student Math Journal, Lesson 3-1, Connecting Fractions and Division, Part 1, students use conceptual understanding as they draw pictures to solve division stories with non-circular wholes. Problem 3, “A school received a shipment of 4 boxes of paper. The school wants to split the paper equally among its 3 printers. How much paper should go to each printer?” (5.NF.3)
• In Math Masters, Lesson 4-12, Decimal Addition Algorithms, students engage in application as they use addition to solve number stories involving decimals. Problem 4, “At the 2012 Summer Olympics in London, Usain Bolt won the men’s 100-meter race with a time of 9.63 seconds and the men’s 200-meter race with a time of 19.32 seconds. How long did it take the sprinter to run the two races combined?” (5.NBT.7) 

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single lesson throughout the materials. Examples include:

• In Student Math Journal, Lesson 2-3, Applying Powers of 10, students engage with conceptual understanding and application as they estimate with powers of 10 to solve multiplication problems while checking the reasonableness of products. Problem 1, “A hardware store sells ladders that extend up to 12 feet. The store’s advertising says: Largest inventory in the country? If you put all our ladders end to end, you could climb to the top of the Empire State Building! The company has 295 ladders in stock. The Empire State Building is 1,453 feet tall. Is it true the ladders would reach the top of the building? Explain how you solved the problem.” (5.NBT.2)
• In Student Math Journal, Lesson 4-4, Comparing and Ordering Decimals, students engage with procedural skills and application as they solve division number stories and interpret the remainders. Problem 2, “Elisabeth is 58 inches tall. What is her height in feet? Number model: Quotient: Remainder: Answer: Elisabeth is ___ feet tall. Circle what you did with the remainder. Ignored it, Reported it as a fraction, Rounded the quotient up; Why?” (5.NF.3)
• In Teacher’s Lesson Guide, Lesson 5-7, Fractions of Fractions, Focus, Finding Fractions of Fractions, students use procedural skill and conceptual understanding to solve application problems as they find fractions of whole numbers and fractions of fractions as they make sense of and solve problems. “Larry has $$\frac{1}{2}$$ of a fruit bar. He wants to give $$\frac{1}{2}$$ of what he has to his brother. What part of a whole fruit bar will Larry give to his brother? Tell students they will use a sheet of paper to represent Larry’s fruit bar. Have them fold it in half vertically and then unfold it. Demonstrate to students how to orient their sheet so that the fold line is vertical, and ask them to shade in $$\frac{1}{2}$$. Ask: If the paper is an entire fruit bar, what does the shaded part of the paper model represent? What part of the $$\frac{1}{2}$$ fruit bar are we trying to find? Have students fold their sheets in half in the opposite direction, unfold them, and orient the sheets so that the new fold is horizontal...” Teachers finish the activity by having students shade in the $$\frac{1}{2}$$ of a $$\frac{1}{2}$$ to demonstrate how much of the whole Larry is giving his brother. (5.NF.4a,4b,6)

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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.
• The Unit 4 Organizer identifies MP2, Reason abstractly and quantitatively. “Mathematically proficient students reason abstractly and quantitatively. In other words, they make sense of quantities and their relationships in problem situations. To do this in Unit 4, students create, make sense of, and make connections among representations of decimals.”
• 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 8-2, Applying the Rectangle Method for Area, as students discuss how a rectangle can be used to find the area of a triangle and find areas of non-rectangular shapes by relating them to rectangles. Teacher’s Lesson Guide, “How could we use what we know about the area of the rectangle in Problem 1a to help us find the area of the triangle in Problem 1b? Then ask them to compare this answer to the answer they got by counting squares.”
• MP2 is connected to mathematical content in Lesson 4-6, Introduction to the Coordinate System, as students locate cities on a map using a letter-number pair and connect this type of representation to a coordinate grid using ordered pairs.  Teacher’s Lesson Guide, “Find the city of Tralee on the map. Be prepared to explain how you found it.” Students continue to locate cities on the map using a letter-number pair. Later in the lesson, “Point out the difference between the letter-number pairs used on maps and the ordered pairs used on a coordinate grid, which consist of two numbers.” Students are then asked to locate a point on the coordinate grid and explain how to plot the point.
• MP4 is connected to mathematical content in Lesson 1-11, Volume Explorations, as students use a rectangular prism to model a real-world volume situation. Teacher’s Lesson Guide, “You have a suitcase that is 14 inches long, 8 inches wide, and 20 inches tall. What is the volume of the suitcase? If you lay the suitcase on its side, does it still have the same volume?” Guidance for teachers includes, “Sketch a rectangular prism and label it with the suitcase dimensions. Point out that the sketch is a mathematical model of the suitcase. The model does not look exactly like a suitcase because it does not include details like pockets or handles. Explain that when an object is about the same shape as a geometric figure, mathematicians often adopt a simplified picture as a model and use the model to help them solve problems.”  
• MP7 is connected to mathematical content in Lesson 6-2, Playing Exponent Ball, as students compare expressions by applying what they know about decimal point patterns. In the Teacher’s Lesson Guide, students solve, “Put the following expressions in order from least to greatest: 3 x 10$$^4$$, 4 x 10$$^3$$, 3 divided by 10$$^4$$, 4 divided by 10$$^3$$.” Guidance for teachers includes, “Be sure to discuss the following strategy: Study the expressions. Notice that all of the expressions start with single-digit whole numbers and involve powers of 10. Two expressions involve multiplication and two involve division. Think about what you know about multiplication and division. Multiplication results in a product greater than both factors if both factors are greater than 1. Both powers of 10 in the multiplication expressions are greater than 1, so both expressions will result in numbers greater than 3 and 4. Division results in a quotient less than the dividend when the divisor is greater than 1. Both powers of 10 in the division expressions are greater than 1, so both expressions will result in quotients less than 3 and 4.”
• MP8 is connected to mathematical content in Unit 7 as students look for regularity and repeated reasoning to create rules and shortcuts. Teacher’s Lesson Guide, Unit 7 Organizer, “In Unit 7 students have many opportunities to engage in Standard for Mathematical Process and Practice 8 as they look for regularity, or patterns, and use what they notice to create rules and shortcuts. In Lesson 7-4 students look for patterns in number sentences such as $$\frac{15}{5}$$ ÷ $$\frac{1}{5}$$ = 15 and $$\frac{12}{3}$$ ÷ $$\frac{1}{3}$$ = 12 and discover that renaming numbers with a common denominator and dividing the numerators is a shortcut for dividing fractions.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 8-4, A Treasure Hunt, students make sense of problems as they decide if changing the size of a juice box affects the volume of juice inside the juice box. Student Math Journal, “Clark has a juice box that is not completely filled with juice. The hole at the top of the box is open. Clark grabs the juice box and squeezes it, but the juice doesn't come out of the box.” Problem 1, “What do you think happened to the shape of the juice box when Clark squeezed it?” Problem 2, “What happened to the amount of juice in the box and the height of the juice when Clark squeezed it?” 
• MP2: In Lesson 4-1, Decimal Place Value, students reason abstractly and quantitatively by making sense of representations to model decimal numbers. Teacher’s Lesson Guide, “In Lesson 4-1, students consider a picture of a large square (representing 1) and then 10 strips or 100 small squares that compose the large square, using language like 10 times and $$\frac{1}{10}$$ of to describe the relative sizes of the squares and strips. Abstracting from these ideas, they represent decimals on a place-value chart and with numerals.” 
• MP4: In Lesson 5-7, Fractions of Fractions, students model finding fractions of fractions by folding paper and shading to illustrate understanding. Student Math Journal, Problem 2, “Fold paper to help you solve this number story. Carolyn had $$\frac{1}{3}$$ liter of water. She drank $$\frac{3}{4}$$ of the water. What part of a liter did she drink? Fold and shade paper to show $$\frac{1}{3}$$. Then fold and double-shade it to show $$\frac{1}{4}$$ of $$\frac{1}{3}$$. Add double-shading to your paper to show $$\frac{3}{4}$$ of $$\frac{1}{3}$$. What part of a liter of water did Carolyn drink?” 
• MP6: In Lesson 6-6, Applying Volume Concepts, students clearly and precisely explain their mathematical thinking when they explain which estimating strategy is more efficient. Student Math Journal, students estimate the volume of a tower. Problem 2, “Describe the strategy your group used to estimate the volume of Willis Tower. Explain your strategy as clearly as you can.” Problem 3, “Do you think your group could have used a more efficient strategy? Explain at least one way your strategy could have been more efficient.” 
• MP7: In Lesson 7-5, A Hierarchy of Triangles, students use structure to classify triangles based on their properties. Student Math Journal, Problem 4, “In each statement below, replace the underlined category with a different category from the triangle hierarchy so that the new statement is still true.” Problem 4b, “All isosceles triangles have at least two sides the same length.” 
• MP8: In Lesson 3-2, Connecting Fractions and Division, Part 2, students express regularity in repeated reasoning when they work backwards to solve division number stories. Teacher’s Lesson Guide, “Do all of your number models follow the pattern we noticed earlier, where the dividend is the same as the numerator of the answer and the divisor is the same as the denominator of the answer? How do you know?”

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

• In Lesson 1-6, Exploring Nonstandard Volume Units, students use three different kinds of pattern blocks to measure the volume of a rectangular prism. Teacher’s Lesson Guide, “Provide each student with several square pattern blocks. Instruct students to use the blocks to measure the length of the rectangle they drew for the Math Message. Remind them that their measurement unit is the side length of the pattern block square, not the entire square.” 
• In Lesson 3-4, Fractions on a Number Line, students are directed to use number lines to solve a problem. Teacher’s Lesson Guide, “Pose the following problem: Gary ran 1 $$\frac{2}{3}$$ miles and Lena ran $$\frac{7}{6}$$ miles. Who ran farther? Have students partition and label the number lines in Problems 2a and 2b on journal page 80 to help them solve the problem.” 
• In Lesson 3-10, Exploring Addition of Fractions with Unlike Denominators, students use fraction circle pieces to find fraction sums. Teacher’s Lesson Guide, “Use your fraction circle pieces. The red circle is the whole. Place a $$\frac{1}{4}$$ piece next to a $$\frac{1}{8}$$ piece. What is $$\frac{1}{4}$$ + $$\frac{1}{8}$$? Talk with a partner about how you could use your fraction circle pieces to find the sum.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 1-3, Quilt Area, Student Math Journal, students determine the solution to a game using operations. Problem 4 & 5, “Two friends were playing a game and recorded their score below. Player 1: 42 + 51 points. Player 2: 4 + (42 + 51) points. Who has more points? Did you have to calculate the scores to find out who had more points in Problem 3? Why or Why not?”
• In Lesson 3-8, Renaming Fractions and Mixed Numbers, Student Math Journal, students compare mixed fractions by breaking apart wholes and then explain their solutions. Problem 8b, “Manny the monkey has 4 whole bananas and 1 half banana. Do Mojo and Manny have the same amount of bananas? Explain how you know.” Problem 8c, “Marcus the monkey has the same amount of banana as Mojo. He only has half-bananas. How many half-bananas does he have? Explain your answer.”
• In Lesson 6-8, Estimating Decimal Products and Quotients, Teacher’s Lesson Guide, Math Message, students make a conjecture about a decimal product based on its factors. “You know that 2.4 * 1 = 2.4. Will 2.4 * 1.8 be greater than or less than 2.4? How do you know? Share your conjecture and argument with a partner.” 
• In Lesson 8-11 Pendulums, Part 1, Student Math Journal, students make a conjecture about the effect of the length of a pendulum on swing time. “Imagine that you have two pendulums, one longer than the other. Do you think the two pendulums would take the same amount of time to swing back and forth, or would one have a longer swing time than the other? Explain your answer.” 

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

• In Lesson 1-3, Quilt Area, Focus, Math Masters, Problems 1 and 2, “Allyson and Justin are working together to sew a quilt. Justin wrote down the length and width of the quilt and started to sketch a plan for the design. He showed Allyson his sketch and told her they will use 54 square feet of fabric. Allyson disagrees and says they will only use 13 $$\frac{1}{2}$$ square feet of fabric. Why might Justin think they will use 54 square feet of fabric? Do you agree or disagree with Justin's answer? Why? Why might Allyson think they will use 13 $$\frac{1}{2}$$ square feet of fabric? Do you agree or disagree with Allyson/s answer? Why?” Students explain why they agree or disagree with how students find the area of a quilt. 
• In Lesson 3-6, Fraction Estimation with Number Sense, Home Link, Math Masters, Problem 1 “Josie calculated $$\frac{1}{5}$$ + $$\frac{1}{2}$$ and said the answer was $$\frac{2}{7}$$. Explain how you know that Josie’s answer does not make sense. Did you need to calculate an exact answer to know that Josie’s answer doesn’t make sense? Tell someone at home why you did or didn’t need to calculate an exact answer.” Students use their understanding of benchmark fractions and make arguments to support their reasoning. 
• In Unit 3, Open Response Assessment, students analyze the mathematical reasoning of others. The problem states, “Claire was training for a running race. She decided to run ⅜ mile from school to the park. Later she left the park and ran $$\frac{1}{2}$$ mile home. She told her brother the distances she ran. Her brother said, ‘You ran a total of $$\frac{2}{8}$$ mile.’ Do you agree with Claire’s brother? Use pictures, words, number sentences, or other representations to explain why you agree or disagree.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 1-3, Quilt Area, Teacher’s Lesson Guide, teachers facilitate a discussion between students after they complete their Open Response Problem. “Show students’ solutions for Problems 1 and 2 in which they used a picture to show why Justin answered 54 square feet and Allyson answered$$13\frac{1}{2}$$ square feet, such as in Students A’s and B’s work. Ask: How did this student make sense of Justin’s or Alysons’s mathematical thinking using the picture? Who is correct? Why? Does this student agree with Justin's solution of 54 square feet of fabric? How much fabric does Student C say there is in the quilt? How do you think this student arrived at that answer?”  
• In Lesson 3-7, Fraction Estimation with Benchmarks, Teacher’s Lesson Guide, students solve fraction number stories and teachers facilitate a discussion afterward. “Have partners share their thinking and invite a few students to share their partner’s thinking with the whole class. Explain that discussing how they solve problems will help students refine their mathematical communication skills. In addition, when they restate a partner’s ideas in their own words, students learn to make sense of others’ ideas and expand their understanding of problem-solving strategies. Encourage students to ask questions to make sure they understand their partner’s thinking.”  
• In Lesson 3-7, Fraction Estimation with Benchmarks, Teacher’s Lesson Guide, teachers prompt students to justify their arguments using benchmarks to estimate their sums and differences of fractions. “Pose the following problems orally. Have students share their arguments and justify their solutions. Is $$\frac{12}{13}+\frac{7}{8}$$ closer to 1 or 2? Is $$\frac{1}{2}-\frac{7}{17}$$ closer to $$\frac{1}{2}$$ or 0? Is $$\frac{18}{19}+\frac{23}{12}$$ closer to 2 or 3?” 
• In Lesson 4-8, Decimal Concepts; Coordinates Grids, Teacher’s Lesson Guide, students use rules to create and plot ordered pairs. “How do you think New Sailboat 1 will look compared to the Original Sailboat? Why? Repeat the process for New Sailboat 3. When it is clear that students understand the two rules, have them work in partnerships to complete journal pages 134 and 135.”
• In the Unit 5 Organizer, Teacher’s Lesson Guide, “In Lesson 5-5, for example, students are asked to construct an argument for why ⅗ of a number will (or will not) always be 3 times ⅕ of the same number. Some students may draw a picture, noting that the shading $$\frac{3}{5}$$ of a rectangle is the same as shading $$\frac{1}{5}$$ of the rectangle 3 times. Others may state their argument in terms of number sense, pointing out the $$\frac{3}{5}$$ is the same is $$\frac{1}{5}+\frac{1}{5}+\frac{1}{5}$$ and adding 3 copies of $$\frac{1}{5}$$ is the same as multiplying by 3.”
• In Lesson 6-8, Investigations in Measurement; Decimal Multiplication and Division, Teacher’s Lesson Guide, students make a conjecture about the relative size of a decimal product based on its factors. “You know that 2.4 * 1 = 2.4. Will 2.4 * 1.8 be greater than or less than 2.4? How do you know? Share your conjecture and argument with a partner. Ask several students to share their conjectures and arguments. Would 2.4 * 0.6 be greater than 2.4 or less than 2.4? How do you know?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 2-1, Focus: Understanding Place Value, Teacher’s Lesson Guide, “Another way to represent a number is by writing it in expanded form. Expanded form represents a number as the sum of values of each digit.”
• In Lesson 4-6, Focus: Introduction to the Coordinate System, Teacher’s Lesson Guide, “Explain that pairs of numbers used to locate points on a coordinate grid are called ordered pairs, and as the name implies, the order of the numbers is very important.”
• In the Student Reference Book, “Grouping symbols, such as parentheses ( ), brackets [ ], and braces { } make the meaning of number sentence or expression clear. When there are grouping symbols in a number sentence or expression, the operations inside the grouping symbols are always done first. Example: Evaluate the following expression: (17 - 4) * 3, The parentheses tell you to subtract 17 - 4 first. Then multiply by 3. The answer is 39.” 
• In the Student Reference Book, “One way to identify factors of a counting number is to make a rectangular array, an arrangement of objects in rows in columns that form a rectangle. Each row has the same number of objects. Each column has the same number of objects. A multiplication number model can represent a rectangular array.”

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

• In Lesson 4-13, Decimal Subtraction Algorithms, Student Math Journal, Problem 7, students must select an algorithm term trade-first subtraction, counting-up subtraction, or the U.S. traditional subtraction and answer questions. “Choose one problem. Think about the algorithm you used. Answer the questions below. How did your choice of algorithm help you get an accurate answer? Was your choice of algorithm the most efficient choice? Why or why not?”
• In the Student Reference Book, “A common way to produce an estimate is to keep the digit in the highest place value and replace the rest of the digits with zeros. This is called  front-end estimation. Example: A girl saved $219. The digit in the highest place value in $219 is 2 in the hundreds place. So the front-end estimate is $200. The girl says, “I saved more than 2 hundred dollars.”
• In the Student Reference Book, “In previous grades, you may have used Frames-and-Arrows diagrams to show number patterns. In a Frames-and-Arrows diagram, the frames hold the numbers, and the arrows show the path from one frame to the next. Each diagram has a rule box. The rule in the box tells how to get from one frame to the next. The numbers in the frames are the terms.”