The instructional materials reviewed for Everyday Mathematics 4 Grade 6 do not meet expectations for assessing grade-level content. Summative Progress Checks include Unit Self Assessments, Unit Assessments, Open Response Assessments, and Cumulative Assessments. 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 Assessment, Item 13, “A university has a student-faculty ratio of 12:1. Make a ratio/rate table to answer the following questions. a. How many students are there for 2 faculty members? b. How many faculty members are there for 120 students? c. How many students are there for 100 faculty members? d. How many faculty members are there for 5,400 students? e. Explain how you used the ratio/rate table to solve Problem 13d.” (6.RP.3) 
• Mid-Year Assessment, Item 1, “George’s math test scores are 82, 59, 91, and 88. a. Find the mean and median. b. Why is the median higher than the mean? Explain your reasoning. c. In George’s class, you have to have a mean of at least 83 to get a B for the class. What is the lowest score George can get on his last test (the fifth test) in order to get a B? Explain.” (6.SP.5c and 6.SP.3)
• Unit 5 Assessment, Item 1, “Plot and label points A, B, C on the coordinate grid. Connect the points to make a triangle. A: (-4, -5), B: (-4, 3.5), C: (-1, -5). Write a number sentence for calculating the length of each line segment. Length of line AB: ___. Length of line AC: ___.” (6.NS.6, 6.NS.8, 6.G.3) 
• Unit 6 Cumulative Assessment, Item 3, “Write an algebraic expression. a. Samantha is 10 years older than Jess. Jess is m years old. How old is Samantha? b. The school is t blocks from Jim’s house. The library is twice as far as the school is from Jim’s house. How far is the library? c. 38 less than four times the sum of 2 and x.” (6.EE.1)

There are above-grade-level assessment items which cannot be omitted or modified, as they have a significant impact on the underlying structure of the materials. These items refer to solving two-step equations and equations with variables on both sides of the equation. These include:

• Unit 6, Assessment, Item 5a, “Use bar models to solve the problems. Solve 5f + 12 = 3f + 18.” (8.EE.7)
• Unit 6 Assessment, Item 7c, “Solve each equation. Show how you solved it. Check your answer. 2/5x + 3 = 13.” (7.EE.4a)
• Unit 6 Assessment, Item 7d, “Solve each equation. Show how you solved it. Check your answer. 3d +18 = 39 - 4d.” (8.EE.7)
• Unit 7 Assessment, Item 5a, “You have at most $10.00 to spend on lunch. You want to get a sandwich and a few side dishes. The sandwich is $5.00. Each side dish is $1.50. a. Define a variable and write an inequality to represent this situation.” Students write inequalities in the form of px + q > r or px + q < r . (7.EE.4b)
• Unit 8 Assessment, Item 4, “Solve using any method. Show your work. a. 1/2y + 7 = 12 b. 7z - 4 = 3z + 2” (7.EE.4a, 8.EE.7)
• End-of-Year Assessment, Item 25, “Solve the equation using any method you choose, and check your answer. 3d - 5 = 7.” Students solve equations in the form of px + q = r. (7.EE.4a)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 do not meet expectations for spending a majority of instructional time on major work of the grade. 

• There are 8 instructional units, of which 5 units address major work of the grade or supporting work connected to major work of the grade, approximately 63%.
• There are 107 lessons, of which 65.5 address major work of the grade or supporting work connected to the major work of the grade, approximately 61%.
• In total, there are 170 days of instruction (107 lessons, 43 flex days, and 20 days for assessment), of which 78 days address major work of the grade or supporting work connected to the major work of the grade, approximately 46%. 
• Within the 43 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 61% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 2-6, Teacher’s Lesson Guide, reason about and solve one-variable equations (6.EE.7) by using common factors to rewrite expressions using the distributive property (6.NS.4). The teacher displays different representations. The students demonstrate how to combine like terms and justify their steps. The teacher asks, “How do the representations show that the problems are all similar?”
• In Lesson 4-6, Teacher’s Lesson Guide, students apply the properties of operations to generate equivalent expressions (6.EE.3) to find the area of special quadrilaterals (6.G.1). The teacher displays rectangles as students record the matching expression of each rectangle area by looking at patterns. The students create a general rule for each pattern based on how they think the Distributive Property works. The teacher prompt states, “Ask a volunteer to describe how each expression represents the area of the corresponding rectangle. What factor is being distributed? How do you know the equations are true even though the expressions on either side of the equal sign look different?”  
• In Lesson 5-3, Teacher’s Lesson Guide, students find the area of right triangles (6.G.1)  to write, read, and evaluate expressions in which letters stand for numbers (6.EE.2). Students investigate finding the area of triangles. The teacher asks, “How is the area formula for a parallelogram similar to or different from the area of a triangle? How would you write a formula for the area of a triangle?”  
• In Lesson 8-9, Teacher’s Lesson Guide, students fluently divide multi-digit numbers using the standard algorithm (6.NS.2) to understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship (6.RP.2). Mental Math and Fluency teacher prompt states, “On their slates, have students record a unit rate that describes each situation.” Some examples provided for the teacher include, “96 students in 6 classes, 300 miles per 15 gallons, and 75 miles over 6 hours.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 107 lessons. Open Response/Reengagement lessons require 2 days of instruction adding 8 additional lesson days.
• There are 43 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 6 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, “6.NS.5: In Grade 4, students identified lines of symmetry and recognized that when a figure is folded along its line of symmetry, the two parts match.”  
• Unit 4, Teacher’s Lesson Guide, Links to the Past, “6.EE.1: In Grade 4, students informally explored situations that involve whole-number exponents by solving problems that involve multiplying the same factor repeatedly. In Grade 5, students read, wrote, and compared numbers in standard and exponential notations.”  
• Unit 6, Teacher’s Lesson Guide, Links to the Past, “6.EE.4: In Grade 5, students both identified and generated equivalent expressions, including in the context of working with measurements with different units. In Unit 4, students compared equivalent expressions when writing numbers using four 4s. They also wrote and compared equivalent expressions when modeling and solving growing pattern 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, “6.RP.2: Throughout Grade 6, students will continue to explore ratio situations and solve ratio problems. In Grade 7, students will extend their work with ratios to represent proportional relationships with equations. In addition, they will begin a formal exploration of ratios in the context of working with linear equations and slope.”
• Unit 6, Teacher’s Lesson Guide, Links to the Future, “6.EE.5: Throughout Grade 6, students will continue to practice writing equations and inequalities to model and solve problems. In Unit 7, students will write and interpret inequalities to help them identify mystery numbers, to determine the ingredients for a healthy salad, and when using spreadsheets to solve problems. In Unit 8, students will write equations to model and solve various real-world situations.”
• Unit 8, Teacher’s Lesson Guide, Links to the Future, “6.RP.3: In Grade 7, students will continue to practice using proportional relationships to solve multistep ratio and percent problems.”

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

• Lesson 6-3 “introduces the bar model, which can be an effective tool for solving equations with variables on one or both sides of an equation.” Throughout the lesson, students use bar models to solve given equations or equations that arise from word problems. However, the majority of the equations that students solve in the lesson do not align to 6.EE.7. For example, in Student Math Journal, Mathe Message, students solve “3d + 12 = 20 + d”; in Teacher Lesson Guide, Focus, Solving Equations with Bar Models, students solve 2p + 21 = 39 and 3e + 17 = 29 + 2e”; and in Teacher Lesson Guide, Math Masters, Home Link, students solve “4a + 12 = 96 and 6d + 7 = d + 22.”
• In Lesson 6-4, students solve multiple pan-balance problems, but the problems involve comparing the weights of objects and do not align to 6.EE.7. For example, in the Teacher Lesson Guide, Math Message, Problem 1, “Let m be the number of cylinders and n be the number of spheres. Write an equation that shows that the weights of the two sides are equal. 2m = m + 2n”; in Teacher Lesson Guide, Student Math Journal, Problem 3, “One cube weighs as much as ____ marbles.”; and in Math Masters, Home Link, Problem 2, “One ball weighs as much as ____ coin(s).” 
• In Lesson 6-5, students solve multiple pan-balance problems, but the problems involve comparing the weights of objects or equations in forms that do not align to 6.EE.7. For example, in the Teacher Lesson Guide, Math Message, Problem 1, “These two pan balances are each in perfect balance. a. Use the relationships in the pan balances shown above to determine which of the pan balances below are balanced. Circle the ones that are in balance. b. For any pan balance above that you did not circle, add or cross out objects to balance the pans.”, and in Math Masters, Home Link 6-5, students, “Find the value of the missing number that will balance each set of pans below. The same number is missing from both sides of a pan balance.”
• In Lesson 6-8, Student Math Journal, T-Shirt Cost Estimates, Focus, Comparing Models and Strategies, “Students compare and analyze models and strategies they used to solve real-world problems, (6.EE.7).” For example, in Problem 1, “Travis has 64 baseball cards and buys 3 new cards every week. When will Travis have 73 baseball cards? Define a variable and write an equation for Travis’s situation. Let g be the number of weeks. Equation: 64 + 3g = 73.” The equations that students write and solve in this lesson do not align to 6.EE.7.
• In Lesson 7-8, Student Math Journal, Problem 2a, “Complete the table, and write the equation to represent the rule. Rule: 2 * x + 2 = y”, and in Problem 6c, “Record an equation that represents the rule for the number of rhombuses in each step. Rule: 3(x) + 1 = y.” The form of these rules do not align to 6.EE.7.
• In Lesson 8-6, “Students explore how a mobile balances and use the balance formula to solve problems (6.EE.7).” In Student Math Journal, Solving Mobile Problems, Problem 2, “What is the distance from the fulcrum to each of the objects?” Students solve 20 * x = 15 * (x + 3).” In Math Masters, Home Links, Solving Mobile Problems, Problem 2, “What is the distance of each object from the fulcrum?” Students solve 8(x + 4) = 16(x - 4). Solving linear equations with variables on both sides of the equation does not align to 6.EE.7.
• In Lesson 8-8, Anthropometry, Focus, Using the Prediction Line, Teacher’s Lesson Guide, “Explain that these points fall on what is called a prediction line. The prediction line shows the exact values that result from using the formula representing the relationship between height and tibia (6.EE.9, 6.SP.5, 6.SP.5c).” In the Student Math Journal, Problems 2 and 3, “The following rule is sometimes used to predict the height (H) of an adult from the length of the adults tibia (t). Measurements are in inches. H = 2.6t + 25.5. Why do you think this rule might not predict the relationship for everyone? Use the rule above to complete the table. Tibia Length (in.) 11, 14, 19, 17 $$\frac{1}{2}$$: Height Predicted (in.) ?, ?, ?, ?.” Knowing that straight lines are widely used to model relationships between two quantitative variables is aligned to 8.SP.2.

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

• In Lesson 3-2, Math Journal 1, Zooming in on Number Lines, “In Lesson 1-12, you zoomed in to find fractions between fractions. This process is even easier when you want to find decimals between decimals. a. Look at each number line separately and estimate where Point A is located on each. b. How far apart are the tick marks on each number line?” (6.NS.6)
• In Lesson 4-10, Math Masters, Student Reference Book, students solve inequalities by being the first player in their groups to discard all of their cards. “When it is your turn: Discard any of your number cards that is a solution to the inequality on the Solution Search Card.” (6.EE.5, 6.EE.8) 
• In Lesson 7-5, Math Journal 2, Problem 4, “Ten teaspoons of sugar is 40 grams of sugar. a. Compare the ratio/rate table to find the number of teaspoons of sugar in 1 fluid ounce of each drink type. b. Find the number of teaspoons of sugar in 1 fluid ounce of each drink. Cola: ___ Tsp sugar per fluid ounce, Fruit punch: ___  Tsp sugar per fluid ounce, Sports drink: ___ Tsp sugar per fluid ounce.” (6.RP.3b)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 2-1, “Students find factors and the greatest common factor (GCF) of two or more numbers,” is shaped by 6.NS.B, “Compute fluently with multi-digit numbers and find common factors and multiples.”
• The Lesson Overview for Lesson 4-3, “Students write numerical expressions for special cases of a pattern and learn to generalize a pattern using an algebraic expression,” is shaped by 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions.”
• The Lesson Overview for Lesson 7-8, “Students represent a growing pattern using numbers, symbols, words, and graphs, and make connections between the representations,” is shaped by 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions” and 6.EE.B, “Reason about and solve one-variable equations and inequalities.”
• The Lesson Overview for Lesson 8-3, “On a coordinate grid, students enlarge the dimensions of scale drawings for representations of artwork and use the scale to reproduce the artwork at its original size.” (Teacher’s Lesson Guide, page 732). This is shaped by the cluster heading, 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.” 

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 2-5 connects 6.NS.A with 6.EE.A as students compare models and analyze strategies for fraction multiplication. In the Student Math Journal, Problems 1 and 2, “Use an area or number-line model to show how to find the solution for each problem. 1. $$\frac{2}{3}$$ * $$\frac{6}{8}$$ =      2. $$\frac{6}{8}$$ * $$\frac{2}{3}$$ =    .” In the Teacher’s Lesson Guide, “Have volunteers display their models, and prompt students to describe their models by asking questions like the following: How does your model represent the factors? Where is the answer in your representation?”
• Lesson 4-9 connects 6.EE.A with 6.EE.B as students match number sentence statements to descriptions of inequalities. In the Student Math Journal, Problem 1, “Any number greater than or equal to 5.” Students choose the appropriate inequality statement from an answer bank.
• Lesson 7-9 connects 6.EE.C with 6.RP.A as students use ratio tables to show relationships between dependent and independent variables. In the Student Math Journal, students compare rates in an Ironman Triathlon. They calculate rates in minutes per mile, complete 3 ratio tables relating time and distance, and graph the information on a coordinate grid.

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 (6.RP.A, and 6.EE.3). Examples include: 

• In Student Math Journal, Lesson 2-10, Ratio Models: Tape Diagrams, students develop conceptual understanding of ratios through the use of diagrams. Problem 1, “Gabriel buys a box of 15 apricots. For every 3 apricots he eats, he gives 2 to Beverly to eat. a. Draw a picture to model the problem. b. Write a ratio to show the number of apricots Beverly eats to the number Gabriel eats. c. Write a ratio to show the number of apricots Beverly eats to the total number in the box.” (6.RP.A)
• In Student Math Journal, Lesson 4-6, The Distributive Property and Equivalent Expressions, Problem 3, students use an area model to show the distributive property conceptually. “The area of Rectangle C is 144 square units. a. Write two equations to represent the area of Rectangle C. b. What is the value of x? ” (6.EE.3)
• In Teacher’s Lesson Guide, Lesson 5-7, Solving Surface-Area Problems, Practice, Home Link Math Masters, students draw a net on graph paper to model and find the surface area of a doghouse. In Problem 1c, “Sam is painting the outside of a doghouse dark green...How many square feet is he painting?” (6.G.4) 
• In Teacher’s Lesson Guide, Lesson 6-10, Building and Solving Equations with the Pan-Balance Model, Practice, Home Link, Math Masters page 275, students use pan-balance models and inverse operations to build and solve equivalent equations. For example, Problem 3, “ Find the mistake in the work below: Original pan-balance equation 2x + 10 = 28. Subtract 10. 2x = 38. Divide by 2. x = 19. Describe the mistake and how to correct it.” (6.EE.5).  
• In Student Math Journal, Lesson 7-9, Independent and Dependent Variables, students relate variables to the coordinate plane. Students use tables to discover relationships between dependent and independent variables and graph them appropriately. Problem 8, “Explain how you know which variable is independent and which is dependent.” (6.EE.9)

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 2-8, Using Reciprocals to Divide Fractions, Home Link 2-8, students use the “Division of Fractions Property” to solve division problems by rewriting them as equivalent multiplication problems. Problem  8, “Phillip went on a 3 $$\frac{1}{2}$$ mile hike. He hiked for 2 hours. About how far did he go in 1 hour? Division number model: _____ Multiplication number model: ______ Solution: ______.” (6.NS.1)
• In Math Masters, Lesson 2-10, Ratio Models: Tape Diagrams, Home Link 2-10, students use tape diagrams to model and solve ratio problems. “Frances is helping her father tile their bathroom floor. They have tiles in two colors: green and white. They want a ratio of 2 green tiles to 5 white tiles. a. They use 30 white tiles. How many green tiles do they use? b. How many white tiles would they need if they use 16 green tiles? c. They use 35 tiles in all. How many are green? d. They use 49 tiles. How many of each color did they use? e. Explain how you used the tape diagram to solve Part d.” (6.RP.A)
• In Student Math Journal, Unit 3-3, Reviewing Decimal Multiplication, Math Boxes, students complete a table on distance and time from a word problem. Problem 2, “A boat traveled 128 kilometers in 4 hours. At this rate, how far did the boat travel in 2 hours and 15 minutes? Use a ratio/rate table to solve the problem. In Problem 5, “Explain how you used the ratio/rate table to help solve Problem 2.” (6.RP.3)
• In Student Math Journal, Lesson 4-6, The Distributive Property and Equivalent Expressions, students explore and apply the distributive property to solve problems.  Problem 1a, “Draw a rectangle like the ones you have been working with in the lesson. Divide one dimension into two parts. Leave the other dimension alone. Label the three lengths that you need to find the area with lengths of your choosing.” Problem 1b, “Use the Distributive Property to write an equation that represents the area of your rectangle. Use the equations at the top of the page to help you.” (6.EE.3)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 3-3, Reviewing Decimal Addition and Subtraction, Focus, students use partial-sums addition and column addition to solve problems with decimals. In the Student Math Journal, Problem 1, “Marilyn owns a tablet that has 75.2 megabytes of available space. First, she downloads a song that takes 4.72 megabytes of space. Then, she downloads an application that uses 62.5 megabytes of space. Estimate how many megabytes of available space Marilyn has after downloading the song and application. Write a number sentence for your estimate.” In Problem 2, students solve the actual problem and show their work. In Problem 3, students analyze the work of an incorrectly solved addition and subtraction decimal problem. This activity provides an opportunity for students to develop fluency of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”
• In Teacher’s Lesson Guide, Lesson 3-4, Reviewing Decimal Multiplication, Focus, “Have students compare their answers for Problem 7 on journal page 122. Display the partial-products method and have a volunteer explain how Martha solved the problem. Display the U.S. traditional multiplication algorithm and have a volunteer explain each step. Have students share how the two methods are similar in structure.” This activity provides an opportunity for students to develop fluency of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”
• In Teacher’s Lesson Guide, Lesson 3-5, U.S. Traditional Long Division with Whole Numbers, Focus, students compare the partial-quotients method and the standard algorithm for division. Then, students practice using the standard algorithm when they solve division problems. In the Student Math Journal, Problem 1, “12⟌652.” This activity provides an opportunity for students to develop fluency of 6.NS.2, “Fluently divide multi-digit numbers using the standard algorithm.”
• In Student Reference Book, Lesson 5-5, Building 3-D Shapes, students play the game Name That Number. Students name numbers using their understanding of equivalent expressions to compare the values of expressions using the order of operations. For example, “Target number: 16, Player 1’s cards: 7, 5, 8, 2, and 10, Some possible solutions: 10 + 8 - 2 = 16 (3 cards used), 10 + (7 * 2) - 8 = 16 (4 cards used), 10 / (5 * 2) + 8 + 7 = 16 (all 5 cards used), 5$$^2$$ - (10 - 8) - 7 = 16 (all 5 cards used).” This activity provides an opportunity for students to develop fluency of 6.EE.1, “Write and evaluate numerical expressions involving whole-number exponents.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level as identified in 6.NS.2, “Fluently divide multi-digit numbers using the standard algorithm,” 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation,” and 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions.” Examples include:

• In Math Masters, Lesson 3-4, Reviewing Decimal Multiplication, Home Link, “For Problems 3-5, record a number sentence to show how you estimated. Then use the U.S. traditional multiplication algorithm to solve. Use your estimate to check your work.” Problem 3, “3.4 * 3.29.” (6.NS.3)
• In Math Masters, Lesson 3-5, U.S. Traditional Long Division with Whole Numbers, Home Link, students solve 6 long division problems using the U.S. traditional method. Problem 1, “38⟌966” (6.NS.2)
• In Math Masters, Lesson 3-6, Exploring Long Division with Decimals, Home Link, students solve multiple division problems including decimals. Problem 2, “Divide and check. 5.976 0.72." Problem 4, “Jamie has 3 cups of berries. Each fruit-and-yogurt parfait he makes contains 0.4 cup of berries. How many parfaits can he make?” (6.NS.3)
• In Math Masters, Lesson 3-8, Introducing Percent, Home Link, students practice subtraction with decimals. Problem 3, “14.7 - 13.2 = .” Problem 4, “4.52 - 3.5 = .” (6.NS.3)
• In Teacher’s Lesson Guide, Lesson 5-7, Solving Surface-Area Problems, Mental Math and Fluency, students evaluate exponential expressions. “On their slates, have students write the number with exponential notation, and then record the number in standard notation. Leveled exercises: “x$$^2$$ when x = 4; when x = 11; when x = 5; x$$^3$$ when x = 1; when x =3; when x = 5; x$$^2$$ when x = $$\frac{1}{2}$$; when x = $$\frac{2}{3}$$; when x = 0.1.” (6.EE.1)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 Student Math Journal, Lesson 2-7, Exploring Relationships in Fraction Division, students match number models to fraction-division situations. “For problems 2-5, circle the best estimate and the correct number model. Then solve.” Problem 2, “Angelina has 2 $$\frac{1}{4}$$ feet of ribbon. She uses $$\frac{3}{4}$$ of a foot of ribbon to wrap each gift. How many gifts can she wrap?” (6.NS.1)
• In Student Math Journal, Lesson 3-12,  Introducing Box Plots, students write number models and use the number models to solve fraction word problems. Problem 5, “Shutter speeds on cameras are measured in seconds and fractions of seconds. A speed of $$\frac{1}{60}$$ is faster than a speed of $$\frac{1}{15}$$. How many times faster is a shutter speed of $$\frac{1}{60}$$ than a shutter speed of $$\frac{1}{15}$$?” (6.EE.7)
• In Student Math Journal, Lesson 7-5, Unit Rate Comparisons, students find unit rates to decide which drink contains the most sugar per unit of volume. Students put drinks in order by volume and predict which drink will have the greatest concentration of sugar. Then they calculate the unit rate and use the unit rates to order the drinks from least to greatest concentration of sugar. “Complete the ratio/rate table to calculate the sugar content for different serving sizes of Thirsty Quench. Use the space below to draw ratio/rate tables for Frosty Cola and Friendly Fruit Punch.” (6.RP.3a)

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 3-3, Reviewing Decimal Addition and Subtraction, students use ratio/rate tables to solve problems. Problem 2, “A boat traveled 120 kilometers in 4 hours. At this rate, how far did the boat travel in 2 hours and 15 minutes? Use a ratio/rate table to solve the problem.” Problem 5, “Explain how you used the ratio/rate table to help you solve problem 2.” (6.RP.3a)  
• In Student Math Journal, Lesson 5-9, Strategies for Finding Volume, Focus, students calculate volume in a real-world context. Students determine the best shipping carton for a music box based on given dimensions and specifications regarding the amount of space needed for the packing peanuts. Problem 2, “What volume of packing peanuts will he need to pack the music box in the carton? Assume he will fill all of the empty space with peanuts. Hint: It may help to label the diagrams above with the dimensions of the music box and the shipping carton you chose.” (6.EE.7, 6.G.2)
• In Math Masters, Lesson 7-5, Unit Rate Comparisons, Home Link, students calculate and use unit rates to compare calories burned in different activities. Problem 4, “On Monday, Edgar ran for 29 minutes and burned 270 calories. On Wednesday he biked for 25 minutes and burned 207 calories. On Friday he played soccer for 13 minutes and burned 124 calories. Which activity burns the most calories per minute?” (6.EE.9) 
• In Math Masters, Lesson 7-11, Mystery Graphs, Home Link, students create graphs to match real-world situations. Problem 1 “Create a mystery graph on the grid below. Be sure to label the horizontal and vertical axes. Describe the situation that goes with your graph on the lines provided.” (6.NS.8)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 2-3, Fraction Multiplication on a Number Line, students use conceptual understanding as they solve a variety of fraction multiplication problems. Problem 8, “Describe any patterns you see in the sets of equations below that might help you predict whether a product will be greater than or less than its factors. 3 * 15 = 45; 4 * 8 = 32; 5 * 25 = 125; $$\frac{1}{3}$$ * 21 = 7; $$\frac{1}{4}$$ * 12 = 3; $$\frac{1}{5}$$ * 10 = 2; $$\frac{1}{3}$$ * $$\frac{1}{6}$$ = $$\frac{1}{18}$$; $$\frac{3}{4}$$ * $$\frac{2}{6}$$ = $$\frac{6}{24}$$; $$\frac{2}{5}$$ * $$\frac{4}{20}$$ = $$\frac{8}{100}$$.” (6.EE.5)
• In Teacher’s Lesson Guide, Lesson 4-1, Parentheses, Exponents, and Calculators, students use procedural skills and fluency as they evaluate products involving powers of 10. For example, “10$$^2$$, 10$$^4$$, 10$$^5$$, 4 * 10$$^3$$, 78 * 10$$^3$$, 60 * 10$$^4$$, 0.26 * 10$$^3$$, 4.5 * 10$$^2$$.” (6.EE.1)
• In Student Math Journal, Lesson 7-6, Running and Measures, Problem 3, students engage in application as they multiply and divide to solve real-world problems, “There are 342 students at Newton Middle School. The principal dives them into 15 homerooms. How will he distribute the students equally as possible?” (6.NS.3)
• In Student Math Journal, Lesson 7-6, Multiplying and Dividing, Problem 1, students engage in application as they determine how much money a person pays in Social Security taxes, “The employee share of Social Security taxes is 6.2%. Mike earned $3,276 as a busboy. How much Social Security tax did he pay?” (6.RP.3c)

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 3-3, Reviewing Decimal Addition and Subtraction, Problem 3, students engage with conceptual understanding and procedural skill as they use the standard algorithm while adding and subtracting decimals, “Santoki solved Marilyn’s problem in the following way. Solution: 6.423 megabytes. What mistake(s) did Santoki make? What questions might you ask Santoki to help him see his mistake?” (6.NS.3)
• In Student Math Journal, Lesson 5-2, Area of Parallelograms, Problem 4, students engage with procedural skills and application as they solve problems involving least common multiples and greatest common factors, “Sharon can buy cards in boxes of 12 and stamps in packages of 20. She wants the number of cards and stamps to match exactly. What is the least number of boxes of cards she should buy?” (6.NS.4)
• In Student Math Journal, Lesson 6-3, Using Bar Models to Solve Equations, Focus, Solving Number Stories with Bar Models, students develop procedural skill and conceptual understanding through application as they use bar models to make sense of and solve equations from number stories. “Amy has cats, dogs, and fish. She has twice as many cats as dogs. She has 5 times as many fish as dogs. Let t be the number of dogs. Write expressions to present the number of cats and fish she has. Cats: _____ Fish: ____. Amy has a total of 16 pets. Write an equation to present the situation. How many of each type of pet does she have?” (6.EE.5, 6.EE.7)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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.
• Within the Unit 3 Organizer, MP2, “Reason abstractly and quantitatively,” is addressed. “To be successful problem solvers, students must also make connections among representations. This unit uses number lines, grids, ratio/rate tables, and box plots to aid understanding of rational numbers, percents, and data distributions.” 
• 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 Unit 3, Decimal Operations and Percent, as students persevere in problem solving and check whether answers make sense. In the Teacher’s Lesson Guide, Unit 3 Organizer, “A successful mathematics student is a successful problem solver. Success at problem solving requires students to flexibly engage in a variety of processes including the following: making sense of problems; reflecting on their thinking as they solve problems; persevering when problems are hard; solving problems in multiple ways and comparing strategies. This unit emphasizes the related practice of checking whether answers make sense.”
• MP2 is connected to mathematical content in Unit 1, Data Displays and Number Systems, as students make sense of representations. In the Teacher’s Lesson Guide, Unit 1 Organizer, “Throughout Unit 1, students create and use representations, including dot plots, graphs, and number lines. For example, in Lesson 1-3 students represent data so they can redistribute the points and find the mean.”
• MP4 is connected to mathematical content in Lesson 2-10, Ratio Models: Tape Diagrams, as students discuss and compare ratio models and learn to use tape diagrams. In the Student Math Journal, Problem 4, “Beverly collects stamps. After Gabriel sees her collection, he wants to collect stamps too. So she gives him 1 stamp for every 5 stamps that she keeps. If there are 72 stamps in all, how many does Beverly keep?” Students create and use models to represent real-world information and solve problems. 
• MP7 is connected to mathematical content in Lesson 2-6, Dividing Fractions with Common Denominators, as students find and record division patterns. In the Student Math Journal, students represent situations with a picture or diagram. For instance, “You have 3 large pizzas. Each person gets $$\frac{3}{4}$$ of a pizza. How many people can you serve?” Then, students solve division problems involving fractions, “Students found that their average shoe length was about $$\frac{3}{4}$$ of a foot. They measured their reading rug with a ruler and found it was 6 $$\frac{3}{4}$$ feet long. How many of their ‘average’ shoes could they line up on the rug?” In the Teacher’s Lesson Guide, prompts for the teacher include, “The dividend and the divisor have common denominators. The numerator of the dividend is a multiple of the numerator of the divisor. When you divide the numerators, you get the quotient.”
• MP8 is connected to mathematical content in Lesson 4-7, Applying Properties of Arithmetic, as students use repeated reasoning with the Distributive Property to solve problems in their heads. In the Teacher’s Lesson Guide, “Explain that students have probably been using the Distributive Property to do mental math, even though they did not realize it. For example, by thinking of the number 101 as 100 + 1, students can make a simpler problem that they can do in their heads. 47 x 101 = 47 x 100 + 47 x 1 = 4,700 + 47 = 4,747. Ask: How can you use the Distributive Property to solve 34 x 7 in your head?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 2-14, Graphing Ratios, students make sense of problems as they identify equivalent ratios in order to find possible photo dimensions. Student Math Journal, Problem 1, “Julian took a photography class. For his final project, he took a family portrait. He printed a 6-inch by 9-inch copy that looked beautiful. His grandmother asked for a larger photo. He ordered an 8-inch by 10-inch copy, but his grandmother and aunt were cut out of the photo. What do you think happened?” Teacher’s Lesson Guide, page 206, “Have partners discuss their reasoning and then share with the class why it makes sense that Julian’s grandmother was cut out of the 8-by-10 photo.” 
• MP2: In Lesson 3-14, Comparing Data Representations, students reason abstractly and quantitatively as they match four different kinds of data representations with data sets and make connections among the representations. Student Math Journal, Problem 2, “Match the tables with their titles. Write the table number in the Table column above.” Problem 3, “Describe how you used the numbers to match the tables to the titles.” 
• MP4: In Lesson 5-7, Solving Surface Area Problems, students model with mathematics when they draw nets to find the surface area of a triangular prism. Student Math Journal, Problem 2, “Antonie’s favorite mechanical pencil leads come in a container shaped like a triangular prism. The base of each triangular face is about 2 cm long. The height of each triangular face is about 1.7 cm long. The container is about 6 cm long. Label the container diagram with the measurements. On the grid below, draw a net to represent the container.”
• MP6: In Lesson 4-2, Solving Problems with Order of Operations, students attend to precision as they solve a series of order of operations riddles and reflect on their answers through discussion. Teacher’s Lesson Guide, “Are there any unnecessary parentheses in the displayed expressions?  How would you know if they were unnecessary?”  
• MP7: In Lesson 4-3, Expressions and Patterns, students look for and use structure as they use patterns in numerical expressions to write algebraic expressions. Student Math Journal, Problem 5, “Choose one strategy you used to solve the previous problems. Use this strategy to write expressions for the number of shaded tiles in the tiled areas. Complete the table. Using the same strategy each time you should create a pattern in your expressions.” 
• MP8: In Lesson 8-7, Naming Patterns with Algebraic Expressions, students express regularity in repeated reasoning with expressions as they create geometric patterns. Math Masters, Problem 5, “Create your own growing pattern from one of the initials of your first or last name. Your pattern should grow the same number of squares in each step. Sketch Steps 1-4 of your pattern. Show the squares. Complete the table with numeric expressions. Shade a part of your pattern that remains constant. Circle the part(s) of your pattern that vary.” Problem 6, “Write an algebraic expression for the number of squares in Step n.” Problem 7, “Describe how the parts of your algebraic expression represent the constant part of the pattern and the part that varies.” Problem 8, “Describe how you could shade your pattern differently, and record an algebraic expression that could represent this shading.” Teacher’s Lesson Guide, “What is repeated in all of the numerical expressions in the table, and how is this part of the expressions connected to the algebraic expression shown in Problem 3?” 

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

• In Lesson 1-11, Building a Number Line Using Fraction Strips, Teacher’s Lesson Guide, students use fraction strips to place fractions on a number line. “Have students use their fraction strips to name and order fractions.” 
• In Lesson 2-3, Fraction Multiplication on a Number Line, students represent and solve fraction-multiplication problems using a given number line. Student Math Journal, Problem 6, “There was $$\frac{3}{4}$$ of a sandwich left over from lunch. Vana ate $$\frac{1}{2}$$ of the leftover part for dinner. What fraction of the whole sandwich did Vanna eat for dinner?” 
• In Lesson 7-1, Inequalities and Mystery Numbers, students use given number lines to model a strategy of finding mystery numbers. Student Math Journal, Problem 3, “Margaret represents her mystery number with the variable p.” Problem 3a. “Clue: One-third of p is less than or equal to 1.4. The number p is greater than 4.” Problem 3b includes a number line including 3, 4, and 5 for students to use as a tool to solve the inequality. Problem 3c, “Three numbers that p could be.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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, Introducing the Mean, Student Math Journal, students analyze the work of others as they compare two strategies for equally splitting a lunch bill. “Six friends went out to lunch. The graph shows how much each person paid. Maria and Jalen used it to figure out how much the friends would have paid had they split the bill equally. Maria imagined moving dollars around so each person had the same number. Jalen found the total dollars spent and divided it by the number of people. They both got the same answer, $8, but whose method will work for splitting any bill equally? Explain.” 
• In Lesson 1-9, Analyzing Data, Student Math Journal, students analyze histograms showing final exam scores and use a graph to support their explanation. Problem 6, “If you wanted to argue that the students did pretty well on the exam, which graph would you use to support your position? Explain.”
• In Lesson 6-8, T-shirt Cost Estimates, Math Masters, students develop a plan to solve a real-world problem. In Problem 2, “Everyone loves the T-shirts, so the Citizenship Club decides to sell T-shirts to raise money to buy books for the community library. Ms. Miller says that school organizations usually charge between $12 and $15 for T-shirts. Based on past experience, they know that the more they charge, the fewer they will sell. If they charge $12, they will likely sell about 160 T-shirts. If they charge $15, they will likely sell about 130 T-shirts. Develop a plan that shows the following: which company the club should use, how many T-shirts they should buy, how much money they should charge for the T-shirts in order to make a donation of at least $700 to the library.”
• In Lesson 7-7, Water-Saving Plan, Math Masters, students develop a plan to cut water usage while meeting certain criteria. Problem 3, “Consider how Olivia could convince her family that her plan meets their requirements. Describe Olivia’s plan using clear mathematical language. Label any rates and measurements with appropriate units.” 

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

• In Lesson 2-5, Comparing Strategies for Multiplying Fractions, Focus, Student Math Journal, Problem 6, “Vera started to use Mara’s method to solve $$\frac{2}{3}$$ * $$\frac{3}{2}$$. Here is what she wrote: $$\frac{2}{3}$$ * $$\frac{3}{2}$$ = (2 * $$\frac{1}{3}$$) * (3 * $$\frac{1}{2}$$) = (2 * $$\frac{1}{2}$$) * (3 * $$\frac{1}{3}$$) = 1 * 1 = 1. Explain how Vera’s strategy is similar to and different from Mara’s strategy. Why does it make sense for Vera to regroup her factors the way she did?” Students analyze their strategies for properties of multiplication problems. 
• In Lesson 2-9, Introducing Ratios, Focus, Math Message, Student Math Journal, Problem 1, Students justify their reasoning on plant growth, “Two weeks ago, Morton and Juliana measured the heights of two tomato plants. Plant A was 2 inches and Plat B was 8 Inches. Now Plant A measures 4 inches and Plant B measures 10 inches. Morton said that Plant A grew more. Juliana said that both plants grew the same amount. How could both Morton and Juilana justify their claims?”
• Lesson 6-7, Generating Equivalent Express and Equations, Practice, Math Boxes, Student Math Journal, Problem 5, “Maila says the answer to Problem 4 is 7:00 P.M, because you multiply 15 by 20 to get 300 minutes. Explain how you know Malia did not find the next time the clocks chime together.

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 1-4, Introducing the Mean as a Balance Point, Teacher’s Lesson Guide, teachers guide students in constructing viable arguments in Problem 4 (in the Student Math Journal, page 12). In Problem 4a, “Are the two sides balanced around the balance point? Explain.” Problem 4b, “Draw a dot to balance the two sides.” Problem 4c, “Explain how you could add two more dots and still keep the two sides balanced.” Problem 4d, “Explain how you could add three more dots and still keep the two sides balanced.” Prompts for teachers include, “When most have finished, ask volunteers to pose arguments for why the two sides are not balanced in Part a.” 
• In Lesson 1-9, Analyzing Data, Teacher’s Lesson Guide, teachers prompt students to justify arguments, “Have students share how they justified their argument that the students did pretty well on the exam. Ask: Which histogram better supports this point of view? Which features of the graph helped you make your argument?”  
• In Lesson 2-5, Comparing Strategies for Multiplying Fractions, Student Math Journal, students complete a series of problems in which they must compare fraction multiplication models and analyze methods. Teachers facilitate a discussion after students have completed the journal pages. In the Teacher’s Lesson Guide, page 150, “When most students have finished the pages, review their reasoning about the strategies using prompts and questions like the following: How does Mara formulate her unit fractions? Describe how and why Mara uses the Commutative and Associative Properties. Compare the strategies Jonah and Mara used. How was Jonah’s strategy similar to Mara’s? How did Jonah’s strategy differ from Mara’s? What is different about Vera’s problem?” 
• In Lesson 3-3, Reviewing Decimal Addition and Subtraction, Teacher’s Lesson Guide, “Display Santoki’s strategy from Problem 3. Have small groups discuss why this strategy did not work. Ask questions like the following: Does Santoki’s strategy make sense? Would moving the decimal point give you the right answer? Explain. Display students’ questions for Santoki. Have students explain how their questions might help him understand his mistakes.” 
• In Lesson 4-8, The Banquet Table, Teacher’s Lesson Guide, guides teachers in setting expectations for discussion and critiquing the arguments of others. “A significant part of the Day 2 reengagement portion of the lesson is a class discussion about how students used tables, pictures and expression to explore the patterns in the underlying relationship between number of people and number of tables. To promote a cooperative environment, consider revisiting the class guidelines for discussion that you developed in Unit 1. Revisit some of the sentence frames to model and encourage students to use appropriate language when discussing other students’ work. For example: I noticed, I agree because, Could you explain, I disagree because, I don’t understand, and I wonder why.”
• In Lesson 5-5, Building 3-D Shapes with Nets, Math Masters, students work with partners and compare their conjectures and reasoning on which patterns (nets) make a cube. “Use the patterns on this page. Make a conjecture. Which patterns can be folded into cubes?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 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 1-13, Locating Negative Rational Numbers on the Number Line, Student Math Journal, the term “opposites” is in bold print, and an example is provided within the sentence: “The numbers -7 and 7 are opposties. The numbers 1/2 and -1/2 are also opposites. Write a definition of opposite of a number.” 
• In Lesson 2-1, Focus: The Greatest Common Factor, Teacher’s Lesson Guide, “Ask a volunteer to explain what a prime number is.”
• In Lesson 4-4, Focus: Representing Unknown Quantities, Teacher’s Lesson Guide, “In the expression 8n, is called the coefficient. A coefficient is the constant factor in a product that involves numbers and variables.”
• In the Students Reference Book, “Rates are ratios that compare two quantities with unlike units.” The example that follows shows a picture of 3 Honeycrisp apples for 1.89. “This ratio is a rate because different units are being compared: a number of apples is being compared to an amount of money. This is the same rate as 1 apple for $0.63 (since 1.89 / 3 = 0.63).” 
• In the Student Reference Book, “An independent variable is one whose value does not rely on any other variable. A dependent variable is one whose value depends on the value of another variable. Identify the independent and dependent variables in the equations above. The formula for the area A of a square with a given side length s is A = s$$^2$$. An equation for the number of miles m a car travels in a given number of hours h at a constant speed of 45 miles per hour is 45 * h = m. An equation for the “What’s My Rule?” function machine at the right is y = x + 5.” 

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

• In the Student Reference Book, “You can use counting-up subtraction to find the difference between two numbers by counting up from the smaller number to the larger number. There are many ways to count up. One way is to start by counting up to the nearest multiple of 10, then continue counting by 10s and 100s.” 
• In the Student Reference Book, “One 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, How much will 6 pens cost if the price is 74 cents per pen? The digit in the highest place value in 74 cents is the 7 in the tens place. Use 70 cents. Calculate: 6 * 70 cents = 420 cents, or $4.20. Estimate: the 6 pens will cost a little more than $4.20.”
• 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 1-4, Focus: Finding Balance Points, Teacher’s Lesson Guide, “Explain that the platform’s sides are balanced around what is called a balance point. When the two people weigh the same and they are the same distance from the balance point, the sides will balance.”