6th Grade - Gateway 2

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 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 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?”