6th Grade - Gateway 2

The 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, Planning for Rich Math Instruction. This component indicates where conceptual understanding is emphasized within each lesson of the Unit. The Focus portion of each lesson introduces new content, designed to help teachers build their students’ conceptual understanding through exploration, engagement, and discussion. The materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the standards. Examples include: 

• Lesson 2-10, Ratio Models: Tape Diagrams, Focus: using Tape Diagrams, Math Journal 1, Problem 1, students use diagrams to solve ratio problems. “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.” Students develop conceptual understanding of 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.”

• Lesson 4-6, The Distributive Property and Equivalent Expressions, Focus: Representing the Distributive Property, Math Journal 1, 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? ” Students develop conceptual understanding of 6.EE.3, “Apply the properties of operations to generate equivalent expressions.”

• Lesson 6-10, Building and Solving Equations with the Pan-Balance Model, Home Link, Problem 3, students use pan-balance models and inverse operations to build and solve equivalent equations. “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.” Students develop a conceptual understanding of 6.EE.5, “Understand solving an equation or inequality as a process of answering a question; which values from a specified set, if any, make the equation or inequality true?”

• Lesson 7-9, Independent and Dependent Variables, Focus: Comparing Triathlon Rates, Math Journal 2, Problem 8, students use tables to discover relationships between dependent and independent variables and graph them appropriately. “Explain how you know which variable is independent and which is dependent.” Students develop conceptual understanding of 6.EE.9, “Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.”

Home Links, Math Boxes, and Practice provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• Lesson 2-10, Ratio Models: Tape Diagrams, Home Link, Problem 1, “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 used 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.” Students independently demonstrate conceptual understanding of 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.

• Lesson 3-3, Reviewing Decimal Multiplication, Practice: Math Boxes, Problem 2, students complete a table on distance and time from a word problem. “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.” Students independently demonstrate conceptual understanding of 6.RP.3, “use ratio and rate reasoning to solve real-world and mathematical problems.”

• Lesson 4-6, The Distributive Property and Equivalent Expressions, Focus: Applying the Distributive Property, Math Journal 1, Problem 1a, students explore and apply the distributive property to solve problems. “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.” Students independently demonstrate conceptual understanding of 6.EE.3, “Apply the properties of operations to generate equivalent expressions.''

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

All units begin with a Unit Organizer, Planning for Rich Math Instruction. This component indicates where procedural skill and fluency exercises are identified within each lesson of the Unit. The Mental Math Fluency exercises found at the beginning of each lesson develop fluency with basic facts and other skills that need to be automatic while engaging learners. The Practice portion of the lesson provides ongoing practice of skills from past lessons and units through activities and games. Examples include:

• Lesson 3-3, Reviewing Decimal Addition and Subtraction, Focus: Computing with Decimals, Math Journal 1, Problem 1, students use partial-sums addition and column addition to solve problems with decimals. “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. Students develop procedural skills and fluency of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

• Lesson 3-4, Reviewing Decimal Multiplication, Focus: Multiplying Decimals with U.S. Traditional Algorithm, students compare partial products and standard algorithm of multiplication. “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.” Students develop procedural skills and fluency of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

• Lesson 3-5, U.S. Traditional Long Division with Whole Numbers, Focus: Connecting Division Algorithms, Math Journal 1, Problem 1, students compare the partial-quotients method and the standard algorithm for division and practice using the standard algorithm to solve division problems, “$$12\lceil{652}$$.” Students develop procedural skills and fluency of 6.NS.2, “Fluently divide multi-digit numbers using the standard algorithm.”

• Lesson 5-5, Building 3-D Shapes, Name That Number, Practice: Playing Name That Number, Student Reference Book, students play a game naming numbers by using their understanding of equivalent expressions to compare the values of expressions using the order of operations. “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\star2)-8=16 (4 cards used), \frac{10}{(5\star2)}+8+7=16 (all 5 cards used), 5^2-(10-8)-7=16 (all 5 cards used).” Students develop procedural skills and fluency of 6.EE.1, “Write and evaluate numerical expressions involving whole-number exponents.”

Math Boxes, Home Links, Games, and Daily Routines provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade. Examples include:

• Lesson 3-4, Reviewing Decimal Multiplication, Home Link, Problem 3, students estimate and multiply using the traditional algorithm., “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. 3.4\star3.29.” Students independently demonstrate procedural skill and fluency of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

• Lesson 3-5, U.S. Traditional Long Division with Whole Numbers, Home Link, Problem 1, students solve long division problems using the U.S. traditional method, “$$38\lceil{966$$.” Students independently demonstrate procedural skill and fluency of 6.NS.2, “Fluently divide multi-digit numbers using the standard algorithm.”

• Lesson 3-8, Introducing Percent, Home Link, Problem 3, students practice subtraction with decimals, “$$14.7-13.2=$$” and Problem 4, “$$4.52-3.5=$$.” Students independently demonstrate procedural skill and fluency of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

• Lesson 5-7, Solving Surface-Area Problems, Warm-Up: 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.” Students independently demonstrate procedural skill and fluency of 6.EE.1, “Write and evaluate numerical expressions involving whole-number exponents.”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Focus activities introduce new content, provide routine exercises, review recent learning, and provide challenging problem-solving tasks that help build conceptual understanding, procedural skill and fluency, and application of mathematics. Open Response lessons provide challenging problems that involve more than one strategy or solution. Home-Links relate to the Focus activity and provide informal mathematics activities for students to do at home. Examples of routine and non-routine applications of the mathematics include:

• Lesson 2-7, Exploring Relationships in Fraction Division, Home-Link, Problem 2, students match number models to fraction-division situations. “The area of a rectangle is 10\frac{1}{2} square feet. The length is 5\frac{1}{4} feet. How wide is the rectangle?” This activity provides the opportunity for students to apply their understanding of 6.NS.1, “Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.”

• Lesson 3-7, Exploring Peruvian Flutes Day 1, Focus: Solving the Open response Problem, Problem 4, students solve real-world problems as they calculate the cost of making a Peruvian flute. “Even if Paola uses the least expensive option and least amount of bamboo for one flute, she will still have some left over. Depending on which pipes she cuts from the yard-long or foot-long piece of bamboo, she will have different lengths left over. Explain how she should cut the bamboo and make the pipes so that she can use her left over lengths to make more pipes for another flute in the future. Describe your reasoning.” This activity provides the opportunity for students to apply their understanding of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

• Lesson 5-8, Comparing Areas Day 1, Focus: Solving the Open Response Problem, Problems 4-7, students order and compare six polygons based on areas. “4a. How do the areas of Polygons B and E compare? b. Draw pictures to model how you compared the areas of Polygons B and E. c. Describe how your drawing represents what you did. 5. Work with your partner. List your partner’s polygon order below. 6. Compare your order with your partner’s. Describe how your orders are different. If they are the same, compare your strategies for ordering a difficult polygon pair. 7. Work with your partner to agree on the order of the polygons. Describe at least one change you or your partner made. Explain why the change was made.” This activity provides the opportunity for students to apply their understanding of 6.G.1, “Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.”

• Lesson 7-5, Unit Rate Comparisons, Focus: Comparing “Sweetness” Rates, Math Journal 2, Problem 5, students solve real-world problems using unit rates to decide which drink contains the most 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.” This activity provides the opportunity for students to apply their understanding of 6.G.1, “Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.”

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Independent Problem Solving provides “additional opportunities for children to apply the content they have learned during the section to solve non-routine problems independently. These problems often feature: applying math in the real world, multiple representations, drawing information or data from pictures, tables, or graphs, and opportunities for children to choose tools to support their problem solving.” Examples of independent demonstration of routine and non-routine applications of the mathematics include:

• Independent Problem Solving 2a, Problem 2 “to be used after Lesson 2-8”, students solve a word problem using division of fractions by fractions. “Papedia is packing lunches for her hiking club. The club purchased reusable pint bottles for their drinks. One pint is \frac{1}{8} of a gallon. The club purchased 2\frac{1}{2} gallons of lemonade on sale. Papedia had to figure out how many of their reusable bottles she could fill with the lemonade. She used the number line at the right to solve the problem. She decided she could fill 20 bottles. Do you agree or disagree with Papedia? Agree If you agree, explain or show why her diagram is correct. If you disagree, explain her mistake. Write a number sentence to show how you can solve the problem.” This activity provides the opportunity for students to independently demonstrate an understanding of 6.NS.1, “Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.”

• Independent Problem Solving 2b “to be used after Lesson 2-13”, Problem 1, students use rates and ratio reasoning to solve real-world problems. “Nakia decided to make an aquaponic garden. This kind of garden requires tanks with fish and garden beds with plants. The fish waste fertilizes the garden. Nakia wants to figure out how many fish to buy. She has three garden beds. Each bed is a cubic meter. Nakia is going to buy a fish called tilapia. Nakia’s research turned up a few facts that will help her with her calculations. To provide enough nutrients for her plants, Nakia must feed her fish 45 grams of food per day. The fish will grow for the first six months. The fish will eat a total of 45 grams of food per day. If she buys the right number of fish, they will stop growing after six months. As tilapias grow (for six months), every two pounds of fish food produces one pound of fish. (For conversions: 1000 grams = 1 kilogram and 1 kilogram = 2.2 pounds.) To be the ideal size for her fish tank, the grown tilapias should be between 1 and 1\frac{1}{2} pounds. Based on the number of pounds of fish food the fish will consume while they grow (for six months), how many fish should she buy? Explain your reasoning. Show all of your work.” This activity provides the opportunity for students to independently demonstrate understanding of 6.RP.3, “Use ratio and rate reasoning to solve real-world and mathematical problems.”

• Independent Problem Solving 4b “to be used after Lesson 4-13” Problem 1, students write inequalities to represent real world problems. “Josiah plays the computer game Fortnite with his friend Keoke. They use V-Bucks to buy skins (outfits) for the game. Josiah likes to change his skin every week. Skins cost 800, 1200, 1500, or 2000 V-Bucks, depending which kind of skin you want. Josiah wants to earn enough money to buy one skin per week. One thousand V-Bucks costs $7.99. Josiah’s mom said that he can do chores to earn money. She will pay him $3 per hour. The rule is that he has to work a full hour to get paid. How many hours might Josiah work in a week? a. Define a variable. b. Record two inequality number models for how many hours Josiah might work to earn money to buy a skin each week. c. Explain why your inequalities make sense.” This activity provides the opportunity for students to independently demonstrate an understanding of 6.EE.8, “Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.”

• Independent Problem Solving 6b “to be used after Lesson 6-11” Problem 1, students use expressions and unit rates to solve real-world problems. “Imala was from the Tlingit tribe from the northwest coast of Canada. She explained to her classmates that families in her tribe have a tradition of making button blankets. The blankets tell stories about their family history. Imala’s class decided to make button blankets to tell their own family stories. Imala’s grandmother visited the class to help them figure out how to make representations of their own stories for their button blankets. The class was divided into 3 teams. Each team had to figure out how many pearl buttons they needed. See below. A table chart shows: Team 1, 2, 3: Large buttons: 60, 25, 125: Small buttons: 85, 140, 50. a. Let m be the cost per large button. Write an expression to find the total cost for large buttons. Simplify your expression.b. Let p be the cost per small button. Write an expression to find the total cost for small buttons. Simplify your expression. c. Use your simplified expressions to help you decide which store has a better deal, Kateri’s Beautiful Buttons or Nahale’s Button Barn. Show or explain how you solved the problem. Kateri's Beautiful Buttons: Large Buttons $0.32 each: Small Buttons $0.08 each. Nahale's Button Barn: Large Buttons $0.25 each: Small Buttons $0.11 each.”  This activity provides the opportunity for students to independently demonstrate understanding of 6.EE.6, “Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout Grade 7. Examples where materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Lesson 2-3, Fraction Multiplication on a Number Line, Focus: Using a Number-Line Model, Math Journal 1, Problem 8, students solve a variety of fraction multiplication problems. “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\star15=45; 4\star8=32; 5\star25=125; \frac{1}{3}\star21=7; \frac{1}{4}\star12=3; \frac{1}{5}\star10=2; \frac{1}{3}\star\frac{1}{6}=\frac{1}{18}; \frac{3}{4}\star\frac{2}{6}=\frac{6}{24}; \frac{2}{5}\star\frac{4}{20}=\frac{8}{100}.” Students extend their conceptual understanding of 6.EE.5, “Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.” 

• Lesson 4-1, Warm-Up, students evaluate products involving powers of 10, “$$10^2$$, 10^4, 10^5, 4\star10^3, 78\star10^3, 60\star10^4, 0.26\star10^3, 4.5\star10^2.” Students develop procedural skills and fluency of 6.EE.1, “Write and evaluate numerical expressions involving whole-number exponents.”

• Lesson 7-6, Running and Measures, Practice: Multiplying and Dividing, Math Journal 2, Problem 1, students 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?” Students engage in application of 6.RP.3c, “Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.”

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

• Lesson 3-4, Reviewing Decimal Multiplication, Home-Link, Problem 6, students solve real-world decimal multiplication problems. “Dr. Goode prescribes 0.2 gram of cold medicine for Donald. This medicine comes in tablets that are 0.05 gram. Should Donald take 4 of the 0.5 gram tablets or 4 of the 0.05 gram tablets? How do you know?” Students use all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they reason with real-world decimal multiplication problems using the standard algorithm of 6.NS.3, “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

• Lesson 5-2, Area of Parallelograms, Focus: Developing an Area Formula, Math Journal 2, Problem 4, students 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?” Students engage with procedural skills and application of 6.NS.4, “Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.”

• Lesson 6-3, Using Bar Models to Solve Equations, Focus: Solving Number Stories with Bar Models, Math Journal 2, Problem 3, students 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?” Students develop conceptual understanding and procedural skills and fluency of 6.EE.5, “Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.”, and 6.EE.7, “Solve real-world and mathematical problems by writing and solving equations of the form x+p=q and px=q for cases in which p, q and x are all nonnegative rational numbers.”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 4-10, Finding and Graphing Solution Sets of Inequalities, Focus: Investigating Inequalities, Math Journal 1, Problem 7, students make sense of problems as they graph inequalities. “a. Name four solutions to t+4>7. b. Name four numbers that are NOT solutions to t+4>7.” The Teacher’s Lesson Guide, “Why does it make sense that the inequalities can be true or false?”

• Lesson 6-5, Solving Simple Equations with a Pan Balance, Part 2, Practice: Math Boxes, Math Journal 2, Problem 3, students analyze and make sense of unit rate problems. “Cindy ran 5 km in 26.5 minutes. Deirdre ran 15 km in 78 minutes. Whose average rate was faster?” Problem 5, “Explain two ways you can determine who was faster in Problem 3.”

• Lesson 7-2, Making Healthy Choices, Focus: Making Salad Choices, Math Journal 2, Problem 2, students analyze and make sense of the information as they graph inequalities. “After more discussion, Archie and Natalia decide to charge $7.99 per pound for the salad bar. Let x be the number of pounds. Record and graph the solution set to the nearest hundredth. How many pounds of bacon bits sold separately could you buy for up to $6? a. Record the solution set for x using set notation. b. Record inequalities to represent the solution set. c. Graph the solution set for x that makes both inequalities true.” 

• Independent Problem Solving 3a, “to be used after Lesson 3-6”, Problem 1, students use a variety of strategies to solve decimal addition problems. “Daniel’s mom made up a math maze for him so that he could practice adding decimals. She challenged him to find a path from Start to Finish with a sum of exactly 1. The only rule is that he cannot travel over the same section twice. Trace a path that Daniel might follow to get a sum of 1. What hints would you give someone else that would help them find a path with a sum of 1?”

Materials provide intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 3-14, Comparing Data Representations, Focus: Analyzing Data Representations, Math Journal 1, Problem 2, students make connections between different representations as they match four different kinds of data representations to data sets. “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.” The Teacher’s Lesson Guide, “As they match the graphs and box plots to the title and data tables, have students record their reasoning on journal page 160. For example, ask them to note what features they used to match the graphs and how they used the IQR (box length) and the length of line segments outside the box, the whiskers.”

• Lesson 5-3, Area of Triangles, Home-Link, Problem 3, students understand the relationships between problem scenarios and mathematical representations as they match geometric solids with their corresponding nets. “Use the net and its corresponding geometric solid in Problem 2. a. Which polygons make up the faces of your solid? How many are there of each kind? b. Which faces are parallel? c. Which faces are congruent? d. How many edges are there? How many vertices?”

• Lesson 7-8, Connecting Equations, Tables, and Graphs, Home-Link, Problem 1, students consider units involved and attend to the meaning of quantities to represent patterns in different ways. “Use the pattern below to answer the questions. In the space below, sketch and label Step 4 and Step 5 of the sequence.” Problem 2, “Complete the table, and record an equation to represent the rule for finding the perimeter.” The table provides, “Step Number (x): 1, 2, 3, 4, 5, 10. Perimeter (y) (units): 5, 8, 11, 14, 17, 32.”

• Independent Problem Solving 1a, “to be used after Lesson 1-7”, Problem 1, students represent situations symbolically as they use graphs to find the measure of center. “Monarch butterflies migrate from Canada and the U.S. to hibernate in Mexico every winter. Some travel 3,000 miles. One way that scientists estimate the number of hibernating butterflies each year is to calculate the size of the area of the forest where the butterflies hibernate. The two graphs below show the area of the forest for two different eight-year periods. Explain how you might use one or both of the graphs to support the claim in each case. Use measures of center (mean and median) and measures of spread (maximum, minimum, and range) in your explanation. Case 1: Monarch butterflies are in trouble. Fewer of them make it to Mexico each year. Case 2: The Monarch butterfly population is making a comeback.”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide support for the intentional development of MP3 by providing opportunities for students to construct viable arguments in connection to grade-level content. Examples include:

• Lesson 1-4, Data Displays and Number Systems, Focus: Balancing to Find the Mean, Math Journal 1, Problem 4a, students justify their strategies and thinking as they reason that the balancing point is the mean. “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.” Teacher’s Lesson Guide, “When most have finished, ask volunteers to pose arguments for why the two sides are not balanced in Part a.”

• Independent Problem Solving 2b, “to be used after Lesson 2-13”, Problem 2, students construct arguments as they use ratio reasoning to justify their answers. “In 2021, Texas was hit with a record winter storm. The storm knocked out power supplies across the state causing a shortage of electricity. Electricity is measured in kilowatt-hours. Customers are charged according to how many kilowatt-hours they use. An average household uses just over 30 kilowatt-hours per day. Some customers pay a small monthly fee ($10) for a variable rate contract. This means that when electricity is cheap, they have a lower rate and pay less. When the cost goes up, they have a higher rate and pay more. Before the storm hit, customers who had a variable rate were paying on average about 12 cents per kilowatt-hour. Because of the shortage caused by the storm, some customers had their variable rates go up to as much as 9 dollars per kilowatt-hour. If the high prices last for five days, why might some customers claim their bills are not fair? Make a mathematical argument. Explain your thinking. Show your work.”

• Independent Problem Solving 4a, “to be used after Lesson 4-7”, Problem 2, students construct mathematical arguments as they work with expressions. “Rob decided to raise panther chameleons. A female can lay anywhere from 20 to 40 eggs every three or four months. It takes six to nine months for the eggs to hatch. Rob has a deal with a local pet store. They will buy his chameleons for $35 per chameleon. (The pet store will resell them for anywhere from $200 to $700 depending on the color of the chameleon!) a. Write an algebraic expression that Rob can use to figure out his earnings based on how many chameleons he sells. b. Define your variable. c. Use your expression to calculate how much Rob will earn if he sells 32 chameleons to the pet store. d. Rob’s chameleon cost $150. The mealworms and crickets she eats cost about $5 per month. Do you think Rob is charging the pet store enough for the chameleons he is raising? Make a mathematical argument for your answer.”

Materials provide support for the intentional development of MP3 by providing opportunities for students to critique the reasoning of others in connection to grade-level content. Examples include:

• Lesson 3-3, Reviewing Decimal Addition and Subtraction, Focus, Estimating to Add and Subtract, Math Journal 1, Problem 3, students critique the reasoning of others as they identify the mistake in adding and subtracting decimals. “Santoki solved Marilyn’s problem in the following way: Step1, 4.72+62.5=1.097, Step 2, 75.20-1.097=6.423. Solution: 6.423 megabytes. What mistake(s) did Santoki make? What questions might you ask Santoki to help him see his mistake?”

• Lesson 6-7, Equivalent Expressions and Solving Equations, Practice: Math Boxes, Math Journal 2, Problem 5, students critique the reasoning of others as they use the least common multiple to solve a problem. “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.”

• Independent Problem Solving 8a, “to be used after Lesson 8-5”, Problem 2, students critique the reasoning of others as they solve real-world problems. “To the right is a diagram of the seating area in the Tivoli Theater. To socially distance, Jim the manager blocked some seats with tape. The government, at the time, said that each person requires 1 square yard of space. This is a population density of 1 person per square yard. (Note that some chairs are together for people who attend with a friend.) a. The theater seating area is 34 feet across and 44 feet deep. What is the population density (people per square yard) of Jim’s plan for theater seating? Show or explain your work. b. Laith argues that they can safely seat about 150 people and still have a population density that will be less than 1 person per square yard. Explain why you agree or disagree with Laith. Describe other considerations besides population density that might affect how many people can be seated in the theater.”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP4 to meet its full intent in connection to grade-level content. Students model with mathematics to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 5-7, Solving Surface-Area Problems, Focus: Finding the Surface Area of a Cube, Math Journal 2, Problem 2, students model the situation with an appropriate representation and use an appropriate strategy to solve a real-world problem when they draw nets to find the surface area of a triangular prism. “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.”

• Independent Problem Solving 2a, “to be used after Lesson 2-8”, Problem 1, students create a plan (model) and explain how the plan (model) will work. “Llewellyn’s mom built a planting shed in their backyard. She had two planks of wood leftover. Both planks are 9 inches wide. Llewellyn thought that 9 inches is a perfect width for shelves that hold books. Lewellyn’s mom said he could use the planks to make a set of shelves. One plank is 72 inches long and the other plank is 108 inches long. Llewellyn realized that he could use all of the wood and make his shelves all the same length. Make a plan for Llewellyn so that he uses all of the wood and his shelves are all the same length. Explain how you know your plan will work. The plan and how I know it works. In the space below, draw a diagram showing your plan for how to cut the planks. ” 

• Independent Problem Solving 8a, “to be used after Lesson 8-5”, Problem 1, students use the math they know to solve problems and everyday situations as they draw a floor plan using ratio reasoning. “In 2010, the Guinness Book of World Records declared that the Continental Giant rabbit named Darius was the longest rabbit in the world. On one cool April day in 2021, Darius disappeared from his garden home. Imagine Darius is found and you are charged with building him a new home. Draw the floor plan for Darius’ new home. Use the information below: Rabbits need space to hop 3 times in a row. Each hop is as long as its width (stretched out). Rabbits need separate areas for exercise, food, and going to the bathroom. Darius has a width of about 4 feet stretched out. He weighs about 50 pounds. A large rabbit has a width of about 3 feet stretched out. Its home must be at least 32 square feet. Use this fact to figure out how much space Darius needs. a. Use the grid below to draw a floor plan for Darius’ new home. b. Explain how you used the information to design Darius’ new home.” 

Materials provide intentional development of MP5 to meet its full intent in connection to grade-level content. Students choose appropriate tools strategically as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 2-5, Comparing Strategies for Multiplying Fractions, Home-Link, Problem 4, students choose and use appropriate tools and strategies as they solve fraction multiplication problems. “Use any model or strategy to solve Problems 3-4. Write a number sentence. A room measures 8\frac{1}{2} feet by 10\frac{2}{3} feet. What is the area of the room?”

• Independent Problem Solving 3b, “to be used after Lesson 3-13”, Problem 1, students choose and use appropriate tools and strategies as they solve problems with percentages. “Jada decided to read one book per week for the entire school year (September 1st through June 7th). The school year is 40 weeks long. After six weeks (and six books), she wanted to calculate what percent of her goal she had reached. Choose and use an appropriate tool to solve the problem. Explain what tool you used and why.”

• Independent Problem Solving 7b, “to be used after Lesson 7-11”, Problem 2, students choose and use appropriate tools and strategies to solve ratio problems. “Choose and use appropriate tools to solve the following set of rate problems related to the bicycle trip described in Problem 1. Show your work. a. What is the approximate value today in U.S. dollars of the 2000 Indian rupees each man had at the beginning of the trip? (When this problem was written, 1 rupee was worth about 0.013 U.S. dollars.)”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure throughout the units as they describe, and make use of patterns within problem-solving as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 2-6, Application: Unit Conversions, Focus: Finding a Division Pattern, Math Message, Math Journal 1, students look for and explain the structure within mathematical representations as they identify patterns. “You have 3 large pizzas. Each person gets \frac{3}{4} of a pizza. How many people can you serve?” Teacher’s Lesson Guide, “When most students have finished, have them share their representations and discuss the patterns they found with their partners. Record and display volunteer’s pattern descriptions and discuss their common features. Highlight the following patterns: The dividend and 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.”

• Lesson 4-3, Representing Decimals in Expanded Form, Focus: Writing Expressions, Math Journal 1, Problem 5, students look for patterns or structures to make generalizations and solve problems to write algebraic expressions. “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.”

• Independent Problem Solving 8b, “to be used after Lesson 8-9”, Problem 1, students make use of the structure as they use expressions and equations to solve problems with rows and seats. “The sixth graders at the Sara Armin Middle School are having a Young Authors’ night. Each student wrote a short story that they will read aloud to a small audience. The sixth graders will use four corners in the cafeteria for four groups. The art teacher made four small stages where students will stand to read their stories. Taraji created a diagram to show how seats can be arranged in rows around a stage. In her diagram, the circles are chairs. a. Add a picture to Taraji’s diagram that shows a stage with four rows of chairs. b. They plan to have 10 students in a group with up to 4 guests per student. Taraji’s friend Trai said that each group needs 6 rows for the students and their guests. Explain why you agree or disagree with Trai. If you disagree, describe how many rows you think each group needs. c. Write an expression you could use to find the number of seats for n rows. d. Taraji learned that it might rain on Young Authors’ Night. If it rains, the roof leaks and they can only use half of the cafeteria. Then they will combine 4 small groups into 2 large groups. Use your expression and the information in Part b to figure out the minimum number of rows they need for each large group. Show or explain your reasoning.”

Materials provide intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning throughout the units to make generalizations and build a deeper understanding of grade level math concepts as they work with the support of the teacher and independently throughout the units. Examples include

• Lesson 4-7, Playing Hidden Treasure, Focus: Using the Distributive Property, students evaluate the reasonableness of their answers and thinking about the Distributive Property. “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\times101=47\times100+47\times1=4,700+47=4,747. Ask: How can you use the Distributive Property to solve 34\times7 in your head?” Practice, Home-Link, Math Masters, Problem 3, “Show how to solve the problems mentally. a. 85\times101= ___. b. 156\star9= ___. c. 48\star24= ___.”

• Lesson 6-9, Multiplication of Decimals, Focus: Reversing Steps in Number Tricks, Math Journal 2, Problem 3, students notice repeated calculations to understand algorithms and to solve number tricks using inverse steps. “Marcy took her secret number, added 5, doubled the sum, and subtracted 2. Her result was 58. a. What was Marcy’s secret number? b. What steps did you complete to find Marcy’s secret number? c. How did you know what steps to complete and in what order to complete them?” Problem 4, “Juan has a number trick, but he forgot the last two steps. Figure out the last two steps so that his final number is the same as his beginning number. Pick a number. Add 8. Multiply by \frac{1}{3}.”

• Independent Problem Solving 7a, “to be used after Lesson 7-4”, Problem 1, students explain a general formula to find the test scores average. “Ms. Winningham uses spreadsheets to calculate her grades. Below is a spreadsheet with math quiz scores for one group of students in her class. a. Each quiz has a different number of possible points for the score. Use the information shown in the spreadsheet to figure out the number of possible points for each quiz. Record the total number of possible points for each quiz.Explain your strategy for figuring out the total possible points for Quiz 1. b. Enter the missing values in the spreadsheet cells. c. Ms. Winningham uses a formula to calculate the % correct for each quiz. What formula could Ms. Winningham use in cell E5: d. Explain how you figured out the formula for Part c. e. What formula could Ms. Winningham use in cell J5 to find the average percent for Jeannene’s four quizzes.”