6th Grade - Gateway 2

The materials reviewed for Open Up Resources Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

According to the Course Guide, About These Materials, Design Principles, “Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” 

Materials develop conceptual understanding throughout the grade level. Examples include:

• Unit 2: Introducing Ratios, Section B: Equivalent Ratios, Lesson 4: Color Mixtures, Activity 1: Turning Green, Problem 1, students develop conceptual understanding of ratios and proportions as they use an applet to mix different batches of the same recipe to obtain different shades of the same color. “a. In the left cylinder, mix 5 ml of blue and 15 ml of yellow. This is a single batch of green. b. Suppose you have one batch of green but want to make more. Which of the following would produce the same shade of green? If you’re unsure, try creating the mixture in the right cylinder. Start with the amounts in a single batch (5 ml of blue and 15 ml of yellow) and … 1. add 20 ml of blue and 20 ml of yellow 2. double the amount of blue and the amount of yellow 3. triple the amount of blue and the amount of yellow 4. mix a single batch with a double batch 5. double the amount of blue and triple the amount of yellow For one of the mixtures that produces the same shade, write down the number of ml of blue and yellow used in the mixture. For the same mixture that produces the same shade, draw a diagram of the mixture. Make sure your diagram shows the number of milliliters of blue, yellow, and the number of batches. c. Someone was trying to make the same shade as the original single batch, but started by adding 20 ml of blue and 20 ml of yellow. How can they add more but still create the same shade of green? d. Invent a recipe for a bluer shade of green. Write down the amounts of yellow and blue that you used, and draw a diagram. Explain how you know it will be bluer than the original single batch of green before testing it out.” (6.RP.1)

• Unit 6: Expressions and Equations, Section A: Equations in One Variable, Lesson 3: Staying in Balance, Activity 1: Match Hangers with Equations, Problem 1, students develop conceptual understanding of balanced equations by using hanger models to help them solve missing values in equations. “Match each hanger to an equation. 1. ___ + 3 = 6; 2. 3 · ___ = 6; 3. 6 = ___ + 1; 4. 6 = 3 · ___.” Four hanger models are shown. (6.EE.7)

• Unit 7: Rational Numbers, Section A: Negative Numbers and Absolute Value, Lesson 4: Ordering Rational Numbers, Activity 2: Comparing Points on a Line, Problem 1, students develop conceptual understanding of greater than and less than to describe order and position on a number line. Students are shown a number line with points M and N to the left of zero and points P and R to the right of zero. “Use each of the following terms at least once to describe or compare the values of points M, N, P, R. greater than, less than, opposite of (or opposites), negative number.” (6.NS.6)

Materials allow students to demonstrate conceptual understanding throughout the grade level independently. Examples include:

• Unit 3: Unit Rates and Percentages, Section C: Rates, Lesson 7: Equivalent Ratios Have the Same Unit Rate, Activity 1: Price of Burritos, students independently develop conceptual understanding of equivalent ratios have the same unit rate. “a. Two burritos cost $14. Complete the table to show the cost for 4, 5, and 10 burritos at that rate. Next, find the cost for a single burrito in each case. b. What do you notice about the values in this table? c. Noah bought b burritos and paid c dollars. Lin bought twice as many burritos as Noah and paid twice the cost he did. How much did Lin pay per burrito? d. Explain why, if you can buy b burritos for c dollars, or buy 2 \cdot b burritos for 2 $$\cdot$$ c dollars, the cost per item is the same in either case.” (6.RP.2, 6.RP.3)

• Unit 4: Dividing Fractions, Section B: Meanings of Fraction Division, Lesson 6: Using Diagrams to Find the Number of Groups, Activity 1: Representing Groups of Fractions with Tape Diagrams, Problem 2, students independently develop conceptual understanding of division using tape-diagrams to represent how many in a group. “Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer. a. How many \frac{3}{4}s are in 1? b. How many \frac{2}{3}s are in 3? c. How many \frac{3}{2}s are in 5?” (6.NS.1) 

• Unit 5: Arithmetic in Base Ten, Section A: Warming Up to Decimals, Lesson 1: Using Decimals in a Shopping Context, Warm Up, Snacks from the Concession Stand, students develop conceptual understanding of estimation and computation for all operations with multi-digit decimals. “Clare went to a concession stand that sells pretzels for $3.25, drinks for $1.85, and bags of popcorn for $0.99 each. She bought at least one of each item and spent no more than $10. a. Could Clare have purchased 2 pretzels, 2 drinks, and 2 bags of popcorn? Explain your reasoning. b. Could she have bought 1 pretzel, 1 drink, and 5 bags of popcorn? Explain your reasoning.” (6.NS.3)

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

According to the Course Guide sections “About These Materials”, and “Design Principles,” “Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” Materials develop procedural skills and fluency throughout the grade level. Examples include:

• Unit 1: Area and Surface Area, Section F: Squares and Cubes, Lesson 17: Squares and Cubes, Launch, students develop procedural skill and fluency as they determine whether or not numbers are perfect squares. “The number 9 is a perfect square. a. Find four numbers that are perfect squares and two numbers that are not perfect squares. b. A square has a side length of 7 km. What is its area? c. The area of a square is 64 sq cm. What is its side length?” (6.EE.1) 

• Unit 5: Arithmetic in Base Ten, Section D: Dividing Decimals, Lesson 10: Using Long Division, Activity 1: Lin Uses Long Division, Problem 2, students develop procedural skill and fluency as they complete long division problems using the standard algorithm. “Lin’s method is called long division. Use this method to find the following quotients. Check your answer by multiplying it by the divisor. a. 846 \div 3 b. 1,816 \div 4 c. 768 \div 12.” (6.NS.2)

• Unit 6: Expressions and Equations, Section C: Expressions with Exponents, Lesson 15: Equivalent Exponential Expressions, Activity 2: Exponent Experimentation, students develop procedural skill and fluency as they evaluate numerical expressions involving whole-number exponents. “Find a solution to each equation in the list that follows. a. 64 = x^2 b. 64 = x^3 c. 2^x = 32 d. x = (\frac{2}{5})^3 e. \frac{16}{9} = x^2 f. 2 \cdot 2^5 = 2^x g. 2x = 2^4 h. 4^3 = 8^x.” (6.EE.1)

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Unit 2: Introducing Ratios, Section C: Representing Equivalent Ratios, Lesson 6: Introducing Double Number Line Diagrams, Practice Problems, Problem 2, students independently demonstrate procedural skill and fluency as they use double number lines to solve ratio and rate problems. “This double number line diagram shows the amount of flour and eggs needed for 1 batch of cookies. a. Complete the diagram to show the amount of flour and eggs needed for 2, 3, and 4 batches of cookies. b. What is the ratio of cups of flour to eggs? c. How much flour and how many eggs are used in 4 batches of cookies? d. How much flour is used with 6 eggs? e. How many eggs are used with 15 cups of flour?” (6.RP.3)

• Unit 5: Arithmetic in Base Ten, Section E: Let’s Put it to Work, Lesson 14: Using Operations on Decimals to Solve Problems, Practice Problems, Problem 1, students independently demonstrate procedural skill and fluency as they add, subtract, multiply, and divide multi-digit decimals using the standard algorithm. “A roll of ribbon was 12 meters long. Diego cut 9 pieces of ribbon that were 0.4 meter each to tie some presents. He then used the remaining ribbon to make some wreaths. Each wreath required 0.6 meter. For each question, explain your reasoning. a. How many meters of ribbon were available for making wreaths? b. How many wreaths could Diego make with the available ribbon?” (6.NS.3)

• Unit 7: Rational Numbers, Section B: Inequalities, Lesson 10: Interpreting demonstrates procedural skills and fluency throughout the grade level independently from skill and fluency as they create inequalities from scenarios. “ a. Jada is taller than Diego. Diego is 54 inches tall (4 feet, 6 inches). Write an inequality that compares Jada’s height in inches, j, to Diego’s height. b. Jada is shorter than Elena. Elena is 5 feet tall. Write an inequality that compares Jada’s height in inches, j, to Elena’s height.” (6.EE.5)

The materials reviewed for Open Up Resources Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics.

According to the Course Guide sections “About These Materials” and “Design Principles,” “Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts.”

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 4: Dividing Fractions, Section C: Algorithm for Fractional Division, Lesson 11: Using an Algorithm to Divide Fractions, Activity 2: All in Order, Problem 3, students engage in a non-routine application problem as they explore relationships between dividends, divisors, and quotients. “Without computing, estimate each quotient and arrange them in three groups: close to 0, close to 1, and much larger than 1. Be prepared to explain your reasoning30 \div \frac{1}{2}, 9 \div 10, 18 \div 19, 15,000 \div 1,500,000, 30 \div 0.45, 9 \div 10,000, 18 \div 0.18, 15,000 \div 14,500 ." (6.NS.1)

• Unit 6: Expressions and Equations, Section B: Equal and Equivalent, Lesson 7: Revisit Percentages, Activity 1: Representing a Percentage Problem with an Equation, Problem 3, students engage in a non-routine application problem as they represent basic percentage problems using an equation. “Write an equation to help you find the value of each variable. Solve the equation. a. 60% of c is 43.2 b. 38% of e is 190.” (6.EE.7)

• Unit 6: Expressions and Equations, Section D: Relationships Between Quantities, Lesson 17: Two Related Quantities (Part 2), Activity 1: The Walk-a-Thon, Problem 1, students engage in a routine application problem as they calculate unit rates. “Complete the table to show how far each participant walked during the walk-a-thon.” Students are given a table with missing values. Columns are titled: time in hours, miles walked by Diego, miles walked by Elena, and miles walked by Andre. (6.RP.3)

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 2: Introducing Ratios, Section F: Let’s Put it to Work, Lesson 17: A Fermi Problem, Activity 2: Researching Your Own Femi Problem, students independently engage in a non-routine application problem as they use ratio reasoning to solve problems. “a. Brainstorm at least five Fermi problems that you want to research and solve. If you get stuck, consider starting with “How much would it cost to …?” or “How long would it take to …?” b. Pause here so your teacher can review your questions and approve one of them. c. Use the graphic organizer to break your problem down into sub-questions. d. Find the information you need to get closer to answering your question. Measure, make estimates, and perform any necessary calculations. If you get stuck, consider using tables or double number line diagrams. e. Create a visual display that includes your Fermi problem and your solution. Organize your thinking so it can be followed by others.” (6.RP.3)

• Unit 4: Dividing Fractions, Mid-Unit Assessment: Version A, Problem 1, students independently engage in a routine application problem as they divide fractions, “Jada made 6 cups of blueberry jam and divided the jam equally among 4 containers. How much jam went into each container? a. \frac{2}{3} of a cup b. 1 cup, c. \frac{3}{2} of a cup, d. 24 cups.” (6.NS.1) 

• Unit 6: Expressions and Equations, Section A: Equations in One Variable, Lesson 4, Practice Solving Equations and Representing Situations with Equations: Cool Down: More Storytime, students independently engage in a non-routine application problem as they create a word problem for a given equation. “ a. Write a story to match the equation x + 2$$\frac{1}{2}$$ = 10. b. Explain what x represents in your story. c. Solve the equation. Explain or show your reasoning.” (6.EE.7)

The materials reviewed for Open Up Resources 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 each grade level. Examples include:

• Unit 2: Introducing Ratios, Section C: Representing Equivalent Ratios, Lesson 9: Constant Speed, Practice Problems, Problem 1, students develop procedural skill and fluency as they calculate distances using unit rate. “Han ran 10 meters in 2.7 seconds. Priya ran 10 meters in 2.4 seconds. a. Who ran faster? Explain how you know. b. At this rate, how long would it take each person to run 50 meters? Explain or show your reasoning.” (6.RP.3)

• Unit 4: Dividing Fractions, Section B: Meanings of Fraction Division, Lesson 5: How Many Groups (Part 2), Cool Down: Bags of Tangerines, students apply their understanding of fraction multiplication and division to write expressions that represent a situation. “A grocery store sells tangerines in $$\frac{2}{5}$$ kg bags. A customer bought 4 kg of tangerines for a school party. How many bags did he buy? a. Select all equations that represent the situation. A. 4 $$\cdot$$ $$\frac{2}{5}$$ = ? ; B. ? $$\cdot$$ $$\frac{2}{5}$$ = 4 ; C. \frac{2}{5} \div 4 = ? ; D. 4 \div \frac{2}{5} = ? ; E. ? \div \frac{2}{5} = 4.” (6.NS.1)

• Unit 8: Data Sets and Distributions, Section A: Data, Variability, and Statistical Questions, Lesson 2: Statistical Questions, Activity 1: What’s in the Data? Problem 1, students demonstrate conceptual understanding as they reason abstractly and quantitatively about numerical data sets to match them with questions that are likely to produce the data. “Ten sixth-grade students at a school were each asked five survey questions. Their answers to each question are shown here. Match each of the following questions to a data set that could represent the students’ answers. Explain your reasoning. a. Question 1: Flip a coin 10 times. How many heads did you get? b. Question 2: How many books did you read in the last year? c. Question 3: What grade are you in? d. Question 4: How many dogs and cats do you have? e. Question 5: How many inches are in 1 foot?” A data table with 5 data sets and 10 numbers in each set is shown. (6.SP.1)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. Examples include:

• Unit 2: Introducing Ratios, Section E: Part-Part-Whole Ratios, Lesson 16, Solving More Ratio Problems, Activity 2: Salad Dressing and Moving Boxes, students build conceptual understanding while applying ratio reasoning to solve problems. “Solve each problem, and show your thinking. Organize it so it can be followed by others. If you get stuck, consider drawing a double number line, table, or tape diagram. a. A recipe for salad dressing calls for 4 parts oil for every 3 parts vinegar. How much oil should you use to make a total of 28 teaspoons of dressing? b. Andre and Han are moving boxes. Andre can move 4 boxes every half hour. Han can move 5 boxes every half hour. How long will it take Andre and Han to move all 72 boxes?” (6.RP.3)

• Unit 4: Dividing Fractions, Section D: Fractions in Length, Areas, and Volumes, Lesson 15: Volume of Prisms, Activity 1: Cubes with Fractional Edge Lengths, Problem 2, students develop procedural skill and fluency as they apply fraction reasoning to find volume. “Lin and Noah are packing small cubes into a larger cube with an edge length of 1$$\frac{1}{2}$$ inches. Lin is using cubes with an edge of $$\frac{1}{2}$$ inch, and Noah is using cubes and edge length of $$\frac{1}{4}$$ inch. a. Who would need more cubes to fill the 1$$\frac{1}{2}$$ inch cube? Be prepared to explain your reasoning. b. If Lin and Noah each use their small cubes to find the volume of the larger 1$$\frac{1}{2}$$ inch cube, will they get the same answer? Explain or show your reasoning.” (6.G.2)

• Unit 6: Expressions and Equations, Section B: Equal and Equivalent, Lesson 11: The Distributive Property (Part 3), Activity 2: Writing Equivalent Expressions Using the Distributive Property, Are You Ready For More? Students build conceptual understanding and develop procedural skill and fluency as they write equivalent expressions using the distributive property. “This rectangle has been cut up into squares of varying sizes. Both small squares have side length 1 unit. The square in the middle has side length x units. a. Suppose that x is 3. Find the area of each square in the diagram. Then find the area of the large rectangle. b. Find the side lengths of the large rectangle assuming that x is 3. Find the area of the large rectangle by multiplying the length times the width. Check that this is the same area you found before. c. Now suppose that we do not know the value of x. Write and expression for the side lengths of the large rectangle that involves x.” (6.EE.2).

The materials reviewed for Open Up Resources 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. 

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is 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 teacher's support and independently throughout the units. Examples include:

• Unit 1: Area and Surface Area, Section E: Surface Area, Lesson 12: What is Surface Area?, Lesson Narrative, “Students begin exploring surface area in concrete terms, by estimating and then calculating the number of square sticky notes it would take to cover a filing cabinet. Because students are not given specific techniques ahead of time, they need to make sense of the problem and persevere in solving it (MP1).” Warm Up: Covering the Cabinet (Part 1), students analyze and make sense of problems as they estimate surface area. “Your teacher will show you a video about a cabinet or some pictures of it. Estimate an answer to the question: How many sticky notes would it take to cover the cabinet, excluding the bottom?”

• Unit 2: Introducing Ratios, Section D: Solving Ratio and Rate Problems, Lesson 14: Solving Equivalent Ratio Problems, Lesson Narrative, “The purpose of this lesson is to give students further practice in solving equivalent ratio problems and introduce them to the info gap activity structure. The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1).” Warm Up: What Do You Want to Know?, students reflect on and revise their problem solving strategies as they solve ratio problems. “Consider the problem: A red car and a blue car enter the highway at the same time and travel at a constant speed. How far apart are they after 4 hours? What information would you need to be able to solve the problem?”

• Unit 7: Rational Numbers, Section C: The Coordinate Plane, Lesson 18: Using Common Multiples and Common Factors, Lesson Narrative, “In this lesson, students apply what they have learned about factors and multiples to solve a variety of problems. In the first activity, students to use what they have learned about common factors and common multiples to solve less structured problems in context (MP1).” Activity 1: Factors and Multiples, Problem 1, students use a variety of strategies to make sense of common factor and common multiple problems. “Party. Elena is buying cups and plates for her party. Cups are sold in packs of 8 and plates are sold in packs of 6. She wants to have the same number of plates and cups. a. Find a number of plates and cups that meets her requirement. b. How many packs of each supply will she need to buy to get that number? c. Name two other quantities of plates and cups she could get to meet her requirement.”

There is 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 teacher's support and independently throughout the units. Examples include:

• Unit 4: Dividing Fraction, Section A: Making Sense of Division, Lesson 2: Meanings of Division, Lesson Narrative, “As they represent division situations with diagrams and equations to interpret division equations in context, students reason quantitatively and abstractly (MP2).” Activity 1: Bag of Almonds, Problem 1, students represent division situations symbolically. “A baker has 12 pounds of almonds. She puts them into bags, so that each bag has the same weight. Clare and Tyler drew diagrams and wrote equations to show how they were thinking about 12 \div 6. How do you think Clare and Tyler thought about 12 \div 6? Explain what each diagram and the parts of each equation could mean about the situation with the bag of almonds. Make sure to include the meaning of the missing number.” Students are given 2 diagrams, Clare’s diagram has 12 divided into 2 equal parts, each labeled 6. Her equation is __ $$\cdot$$ 6 = 12. Tyler’s diagram has 12 divided into 6 equal parts, each labeled 2. His equation is 6 $$\cdot$$ __ = 12. 

• Unit 5: Arithmetic in Base Ten, Section C: Multiplying Decimals, Lesson 8: Calculating Products of Decimals, Instructional Routine, “The application invites students to use MP2, deciding what mathematical operations to perform based on context and then using context to understand how to deal with the result of complex calculations.” Activity 1: Calculating Products of Decimals, Problem 1, students explain the meaning of numbers and symbols in a multiplication expression. “A common way to find a product is to calculate a product of whole numbers, then place the decimal point in the product. Here is an example for (2.5) $$\cdot$$ (1.2). Use what you know about decimals and place value to explain why the decimal point of the product is placed where it is.”

• Unit 8: Data Sets and Distributions, Section D: Median and IQR, Lesson 17: Using Box Plots, Lesson Narrative, “In the previous lesson, students analyzed a dot plot and a box plot in order to study the distribution of a data set. They saw that, while the box plot summarizes the distribution of the data and highlights some key measures, it was not possible to know all the data values of the distribution from the box plot alone. In this lesson, students use box plots to make sense of the data in context (MP2), compare distributions, and answer statistical questions about them.” Warm Up: Using Box Plots, students understand the relationship between five-number summaries and box plots. “Ten sixth-grade students were asked how much sleep, in hours, they usually get on a school night. Here is the five-number summary of their responses. Minimum: 5 hours, Median: 7.5 hours, Maximum: 9 hours, First Quartile: 7 hours, and Third quartile: 8 hours. a. On the grid, draw a box plot for this five-number summary. b. What questions could be answered by looking at this box plot?”

The materials reviewed for Open Up Resources 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.

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

Students construct viable arguments in connection to grade-level content as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 2: Introducing Ratios, Section C, Representing Equivalent Ratios, Lesson 10, Comparing Situations by Examining Ratios. Lesson Narrative, “In each case, the numbers are purposely chosen so that reasoning directly with equivalent ratios is a more appealing method than calculating how-many-per-one and then scaling. The reason for this is to reinforce the concept that equivalent ratios describe the same rate, before formally introducing the notion of unit rate and methods for calculating it. However, students can use any method. Regardless of their chosen approach, students need to be able to explain their reasoning (MP3) in the context of the problem.” Activity 1: Concert Tickets, “Diego paid $47 for 3 tickets to a concert. Andre paid $141 for 9 tickets to a concert. Did they pay at the same rate? Explain your reasoning.”

• Unit 6: Expressions and Equations, Section D: Relationships Between Quantities, Lesson 16: Two Related Quantities (Part 1), Instructional Routine, “The purpose of this Warm Up is for students to remember that unit price can be used to figure out which price option is a better deal and also how to compute unit price. When students explain their reasoning, they may engage in constructing arguments and critiquing the reasoning of their classmates (MP3). The question, “Which one would you choose?” is purposefully asked because there is not one correct answer. While there is a choice that is a better deal, that is not the question. In defending their reasoning, students may have other reasons for their choice based on how they make sense of the context. For example, students might reason that a 5-gallon container is easier to store, or that 3 1-gallon containers are easier to share, or they might reject both options because they don’t like honey.” Warm Up: Which One Would You Choose?, “Which one would you choose? Be prepared to explain your reasoning. A 5-pound jug of honey for $15.35. Three 1.5-pound jars of honey for $13.05.”

• Unit 8: Data Sets and Distributions, Section B: Dot Plots and Histograms, Lesson 8: Describing Distributions on Histograms, Instructional Routines, “This Warm Up encourages students to make sense of histograms in terms of center and spread. It prompts students to hold mathematical conversations and explain their reasoning (MP3), and gives the teacher the opportunity to hear how students compare data sets represented by histograms.” Warm Up: Which One Doesn’t Belong: Histograms, “Which histogram does not belong? Be prepared to explain your reasoning.” Students are given 4 histograms to examine. Histogram A has five rectangles and a range from 75-125, with the largest distribution from 95-105 at 30. Histogram B has five rectangles and a range from 55-105, with the largest distribution from 75-85 at just over 30. Histogram C has 9 rectangles with a range from 55-155, with the largest distribution from 95-105 somewhere between 25-30. Histogram D has five rectangles and a range from 75-125, with the largest distribution from 95-105 at 25. 

Students critique the reasoning of others in connection to grade-level content as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 2: Introducing Ratios, Section A: What are Ratios?, Lesson 2: Representing Ratios with Diagrams, Instructional Routines, “Writing and using ratio language requires attention to detail. This task further develops students’ ability to describe ratio situations precisely by attending carefully to the quantities, their units, and their order in the ratio. Students work in pairs to match ratios of sauce ingredients to discrete diagrams and to explain reasoning (MP3).” Activity 3: Card Sort, Spaghetti Sauce, “Your teacher will give you cards describing different recipes for spaghetti sauce. In the diagrams: a circle represents a cup of tomato sauce, a square represents a tablespoon of oil, a triangle represents a teaspoon of oregano. a. Take turns with your partner to match a sentence with a diagram. For each match that you find, explain to your partner how you know it’s a match. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement. b. After you and your partner have agreed on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches. c. There were two diagrams that each matched with two different sentences. Which were they? Diagram ___ matched with both sentences ___ and ___. Diagram ____ matched with both sentences ___ and ____.”

• Unit 6: Expressions and Equations, Section C: Expressions with Exponents, Lesson 14: Evaluating Expressions with Exponents, Instructional Routines, “In this activity, students use the order of operations to evaluate expressions with exponents. They engage in MP3 as they listen and critique their partner’s reasoning when they do not agree on the answers.” Activity 2: Expression Explosion, “Evaluate the expressions in one of the columns. Your partner will work on the other column. Check with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error.”

• Unit 8: Data Sets and Distributions, Section B: Dot Plots and Histograms, Lesson 5: Using Dot Plots to Answer Statistical Questions, Instructional Routines, “During the partner discussion, - the teacher will identify two students and another one who agrees with Clare and another who agrees with Tyler - to share during the whole-class discussion (MP3).” Warm Up: Packs on Backs, “This dot plot shows the weights of backpacks, in kilograms, of 50 sixth-grade students at a school in New Zealand. a. The dot plot shows several dots at 0 kilograms. What could a value of 0 mean in this context? b. Clare and Tyler studied the dot plot. Clare said, ‘I think we can use 3 kilograms to describe a typical backpack weight of the group because it represents 20%—or the largest portion—of the data.’ Tyler disagreed and said, 1I think 3 kilograms is too low to describe a typical weight. Half of the dots are for backpacks that are heavier than 3 kilograms, so I would use a larger value.’ Do you agree with either of them? Explain your reasoning.”

The materials reviewed for Open Up Resources 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.

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is intentional development of MP4 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 teacher's support and independently throughout the units. Examples include:

• Unit 3: Unit Rates and Percentages, Section E: Let’s Put it to Work, Lesson 17: Painting a Room, Lesson Narrative, “Students determine the area of the walls of a bedroom, estimate the amount of paint needed to paint them, and determine the cost associated with the project (MP4). Along the way, they reason about areas of two-dimensional figures, convert units of measurements, solve ratio and rate problems, and work with percentages. Though there is a single correct measure for the total area of the walls to be painted, the amount of paint needed will depend on some assumptions and decisions students make about the work involved.” Activity 1: How Much it Costs to Paint, students use the math they know to estimate needed supplies and their associated costs. “Here is the floorplan for a bedroom: Imagine you are planning to repaint all the walls in this room, including inside the closet. The east wall is 3 yards long. The south wall is 10 feet long but has a window, 5 feet by 3 feet, that does not need to be painted. The west wall is 3 yards long but has a door, 7 feet tall and 3 feet wide, that does not need to be painted. The north wall includes a closet, 6.5 feet wide, with floor-to- ceiling mirrored doors that do not need to be painted. There is, however, a smaller wall between the west wall and the closet that does need to be painted on all sides. The wall is 0.5 feet wide and extends 2 feet into the room. The ceiling in this room is 8 feet high. All of the corners are right angles. a. If you paint all the walls in the room, how many square feet do you need to cover? b. An advertisement about the paint that you want to use reads: “Just 2 quarts covers 175 square feet!” If you need to apply two coats of paint on all the walls, how much paint do you need to buy? c. Paint can only be purchased in 1-quart, 1-gallon, and 5-gallon containers. How much will all supplies for the project cost if the cans of paint cost $10.90 for a quart, $34.90 for a gallon, and $165.00 for 5 gallons? d. You have a coupon for 20% off all quart-sized paint cans. How does that affect the cost of the project?” 

• Unit 4: Dividing Fractions, Section E: Let’s Put it to Work, Lesson 17: Fitting Boxes into Boxes, Lesson Narrative, “In this three-part culminating activity, students use what they have learned to determine the most economical way to ship jewelry boxes using the United States Postal Service (USPS) flat-rate options. In Part 1, students make sense of the task, outline what they will need to know and do to answer the question, and map out their plan. In Part 2, they model the problem, calculate the number of jewelry boxes that will fit into each shipping box, and determine the associated costs. Students experiment with different orientations for the jewelry boxes to optimize space and minimize cost. In Part 3, they present, reflect, and discuss. Students explain their strategies and reasoning (MP3) and evaluate the decisions about how to fit all 270 jewelry boxes so they ship at the lowest cost (MP4). As a class, students reflect on how the orientation of the jewelry boxes and the size of the shipping boxes affected the unit cost for shipping each box of jewelry.” Activity 3: Determining Shipping Costs (Part 3), students check to see if they used appropriate model, and revise calculations as needed. “a. Share and discuss your work with the other members of your group. Your teacher will display questions to guide your discussion. Note the feedback from your group so you can use it to revise your work. b. Using the feedback from your group, revise your work to improve its correctness, clarity, and accuracy. Correct any errors. You may also want to add notes or diagrams, or remove unnecessary information. c. Which shipping boxes should the artist use? As a group, decide which boxes you recommend for shipping 270 jewelry boxes. Be prepared to share your reasoning.” 

• Unit 8: Data Sets and Distributions, Section E: Let’s Put It to Work, Lesson 18: Using Data to Solve Problems, Lesson Narrative, “In this lesson, students compare the center and spread of different distributions. They determine what these different measures (mean and MAD or median and IQR) represent in context. They select an appropriate representation for the distribution based on the structure of the data, an appropriate set of measures of center and spread, and interpret their meaning in the context (MP4).” Activity 3: Will the Yellow Perch Survive?, students identify important information as they use a histogram to determine appropriate measures of center and variability, and draw conclusions about certain fish populations. “Scientists studying the yellow perch, a species of fish, believe that the length of a fish is related to its age. This means that the longer the fish, the older it is. Adult yellow perch vary in size, but they are usually between 10 and 25 centimeters. Scientists at the Great Lakes Water Institute caught, measured, and released yellow perch at several locations in Lake Michigan. The table shows a summary that is based on a sample of yellow perch from one of these locations. Problem 1, Use the data to make a histogram that shows the lengths of the captured yellow perch. Each bar should contain the lengths shown in each row in the table. Problem 2, How many fish were measured? How do you know? Problem 3, Use the histogram to answer the following questions. a. How would you describe the shape of the distribution? b. Estimate the median length for this sample. Describe how you made this estimate. c. Predict whether the mean length of this sample is greater than, less than, or nearly equal to the median length for this sample of fish? Explain your prediction. d. Would you use the mean or the median to describe a typical length of the fish being studied? Explain your reasoning.” 

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

• Unit 2: Introducing Ratios, Section C: Representing Equivalent Ratios, Lesson 7: Creating Double Line Diagrams, Lesson Narrative, “Double number lines are included in the first few activity statements to help students find an equivalent ratio involving one item or one unit. In later activities and lessons, students make their own strategic choice of an appropriate representation to support their reasoning (MP5).” Activity 1: Just a Little Green, students use double number lines as they reason about equivalent ratios, “The other day, we made green water by mixing 5 ml of blue water with 15 ml of yellow water. We want to make a very small batch of the same shade of green water. We need to know how much yellow water to mix with only 1 ml of blue water. a. On the number line for blue water, label the four tick marks shown. On the number line for yellow water, draw and label tick marks to show the amount of yellow water needed for each amount of blue water. b. How much yellow water should be used for 1 ml of blue water? Circle where you can see this on the double number line. c. How much yellow water should be used for 11 ml of blue water? d. How much yellow water should be used for 8 ml of blue water? e. Why is it useful to know how much yellow water should be used with 1 ml of blue water?” 

• Unit 6: Expressions and Equations, Section B: Equal and Equivalent, Lesson 6: Write Expressions Where Letters Stand for Numbers, Lesson Narrative, “This lesson is a shift from previous work in this unit. Up until now, we were focused on writing and solving equations. Starting in this lesson, we begin to focus on writing expressions to represent situations. Students write expressions that record operations with numbers and with letters standing in for numbers. Students can choose to represent expressions with tape diagrams if they wish (MP5).” Activity 1: Lemonade Sales and Heights, students use appropriate tools and strategies as they write expressions to represent situations. “Problem 1: Lin set up a lemonade stand. She sells the lemonade for $0.50 per cup. a. Complete the table to show how much money she would collect if she sold each number of cups. b. How many cups did she sell if she collected $127.50? Be prepared to explain your reasoning. Problem 2: Elena is 59 inches tall. Some other people are taller than Elena. a. Complete the table to show the height of each person. b. If Noah is 64$$\frac{3}{4}$$ inches tall, how much taller is he than Elena?”

• Unit 8: Data Sets and Distributions, Section B: Dot Plots and Histograms, Lesson 4: Dot Plots, Lesson Narrative, “In this lesson, students continue to choose appropriate representation (MP5) to display categorical and numerical data, reason abstractly and quantitatively (MP2) by interpreting the displays in context, and study and comment on features of data distributions they show. Here they begin to use the everyday meaning of the word “typical” to describe a characteristic of a group. They are also introduced to the idea of using center and spread to describe distributions generally. Planted here are seeds for the idea that values near the center of the distribution can be considered “typical” in some sense. These concepts are explored informally at this stage but will be formalized over time, as students gain more experience in describing distributions and more exposure to different kinds of distributions.” Activity1: Pizza Toppings (Part 2), students use appropriate tools and strategies as they represent information graphically. “a. Use the tables from the Warm Up to display the number of toppings as a dot plot. Label your drawing clearly. b. Use your dot plot to study the distribution for the number of toppings. What do you notice about the number of toppings that this group of customers ordered? Write 2–3 sentences summarizing your observations.”

The materials reviewed for Open Up Resources Grade 6 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities’ Narratives for some lessons.

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Students attend to precision as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 3: Unit Rates and Percentages, Section C: Rates, Lesson 8: More About Constant Speed, Activity 2: Swimming and Biking, Instructional Routines, “This problem has less scaffolding than the previous activity. There are many different unit rates students may choose to calculate while solving this problem. Specifying the units and explaining the context for a rate gives students an opportunity to attend to precision (MP6).” Activity 2: Swimming and Biking, students attend to precision as they use rates to solve problems involving constant speed. “Jada bikes 2 miles in 12 minutes. Jada’s cousin swims 1 mile in 24 minutes. Problem 1: Who is moving faster? How much faster? Problem 2: One day Jada and her cousin line up on the end of a swimming pier on the edge of a lake. At the same time, they start swimming and biking in opposite directions. a. How far apart will they be after 15 minutes? b. How long will it take them to be 5 miles apart?”

• Unit 4: Dividing Fractions, Section B: Meanings of Fraction Division, Lesson 5: How Many Groups? (Part 2), Lesson Narrative, “In this lesson, students continue to work with division situations involving questions like ‘how many groups?’ or ‘how many of this in that?’ Unlike in the previous lesson, they encounter situations where the quotient is not a whole number, and they must attend to the whole when representing the answer as a fraction (MP6).” Activity 2: Drawing Diagrams to Show Equal-Sized Groups, students attend to precision as they use diagrams and equations to represent situations. “For each situation, draw a diagram for the relationship of the quantities to help you answer the question. Then write a multiplication equation or a division equation for the relationship. Be prepared to share your reasoning. a. The distance around a park is \frac{3}{2} miles. Noah rode his bicycle around the park for a total of 3 miles. How many times around the park did he ride? b. You need \frac{3}{4} yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for? c. The water hose fills a bucket at \frac{1}{3} gallon per minute. How many minutes does it take to fill a 2-gallon bucket?” 

• Unit 7: Rational Numbers, Section D: Common Factors and Common Multiples, Lesson 16: Common Factors, Lesson Narrative, “In this lesson, students use contextual situations to learn about common factors and the greatest common factor of two whole numbers. They develop strategies for finding common multiples and least common multiples. They develop a definition of the terms common factor and greatest common factor for two whole numbers (MP6).” Activity 1: DIego’s Bake Sale, students attend to precision as they find greatest common factors in context to put equal amounts of baked goods into bags. “Diego is preparing brownies and cookies for a bake sale. He would like to make equal-size bags for selling all of the 48 brownies and 64 cookies that he has. Organize your answer to each question so that it can be followed by others. a. How can Diego package all the 48 brownies so that each bag has the same number of them? How many bags can he make, and how many brownies will be in each bag? Find all the possible ways to package the brownies. b. How can Diego package all the 64 cookies so that each bag has the same number of them? How many bags can he make, and how many cookies will be in each bag? Find all the possible ways to package the cookies. c. How can Diego package all the 48 brownies and 64 cookies so that each bag has the same combination of items? How many bags can he make, and how many of each will be in each bag? Find all the possible ways to package both items. d. What is the largest number of combination bags that Diego can make with no left over? Explain to your partner how you know that it is the largest possible number of bags.”

Students attend to the specialized language of mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1: Area and Surface Area, Section D: Polygons, Lesson 11: Polygons, Instructional Routines, “This activity prompts students to develop a working definition of polygon [sic] that makes sense to them, but that also captures all of the necessary aspects that make a figure a polygon (MP6).” Activity 1: What Are Polygons, students use the specialized language of mathematics as they define polygons and categorize them by type. “Here are five polygons: Here are six figures that are not polygons: a. Select the figures that are polygons. b. What do the figures you circled have in common? What characteristics helped you decide whether a figure was a polygon?”

• Unit 2: Introducing Ratios, Section A: What are Ratios? Lesson 2: Representing Ratios with Diagrams, Lesson Narrative, “Students used physical objects to learn about ratios in the previous lesson. Here they use diagrams to represent situations involving ratios and continue to develop ratio language. The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems. Both the visual and verbal descriptions of ratios demand careful interpretation and use of language (MP6).” Activity 1: A Collection of Snap Cubes, students use the specialized language of mathematics as they read ratio information and represent it in a diagram. “Here is a collection of snap cubes. 1. Choose two of the colors in the image, and draw a diagram showing the number of snap cubes for these two colors. 2. Trade papers with a partner. On their paper, write a sentence to describe a ratio shown in their diagram. Your partner will do the same for your diagram. 3. Return your partner’s paper. Read the sentence written on your paper. If you disagree, explain your thinking.”

• Unit 3: Unit Rates and Percentages, Section D: Percentages, Lesson 10: What Are Percentages? Lesson Narrative, “This lesson is the first of two that introduce students to percentages as a rate per 100 (MP6) and the ways they are used to describe different types of situations.” Activity 2: Coins on a Number Line, students use the specialized language of mathematics as they reason about percents of 1 dollar. “A $1 coin is worth 100% of the value of a dollar. Here is a double number line that shows this. a. The coins in Jada’s pocket are worth 75% of a dollar. How much are they worth (in dollars)? b. The coins in Diego’s pocket are worth 150% of a dollar. How much are they worth (in dollars)? c. Elena has 3 quarters and 5 dimes. What percentage of a dollar does she have?”

The materials reviewed for Open Up Resources 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.

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and use structure as they work independently with the teacher's support throughout the units. Examples include:

• Unit 1: Area and Surface Area, Section A: Reasoning to Find Area, Lesson 3: Reasoning to Find Area, Lesson Narrative, “This lesson is the third of three lessons that use the following principles for reasoning about figures to find area: If two figures can be placed one on top of the other so that they match up exactly, then they have the same area. If a figure is composed from pieces that don’t overlap, the sum of the areas of the pieces is the area of the figure. If a given figure is decomposed into pieces, then the area of the given figure is the sum of the areas of the pieces. Following these principles, students can use several strategies to find the area of a figure. They can: Decompose it into shapes whose areas they can calculate. Decompose and rearrange it into shapes whose areas they can calculate. Consider it as a shape with one or more missing pieces, calculate the area of the shape, then subtract the areas of the missing pieces. Enclose it with a figure whose area they can calculate, consider the result as a region with missing pieces, and find its area using the previous strategy. Use of these strategies involves looking for and making use of structure (MP7); explaining them involves constructing logical arguments (MP3). For now, rectangles are the only shapes whose areas students know how to calculate, but the strategies will become more powerful as students’ repertoires grow. This lesson includes one figure for which the ‘enclosing’ strategy is appropriate, however, that strategy is not the main focus of the lesson and is not included in the list of strategies at the end.” Activity 1: On The Grid, students analyze the problem and look for an approach as they find areas of regions. “Each grid square is 1 square unit. Find the area, in square units, of each shaded region without counting every square. Be prepared to explain your reasoning.” Four composite shapes are pictured.

• Unit 3: Unit Rates and Percentages, Section D: Percentages, Lesson 11, Percentages and Double Number Lines, Lesson Narrative, “Students continue to have double number lines as a reasoning tool to use if they want. In several cases the double number line is provided. There are two reasons for this. First, the equal intervals on the provided double number line are useful for reasoning about percentages. Second, using the same representation that was used earlier for other ratio and rate reasoning reinforces the idea of a percentage as a rate per 100 (MP7). It is perfectly acceptable, however, for students to use strategies other than double number lines for solving percentage problems.” Activity 2: Puppies Grow Up, Problem 1, students look for patterns or structure as they 100% of quantities given other percentages. “Jada has a new puppy that weighs 9 pounds. The vet says that the puppy is now at about 20% of its adult weight. What will be the adult weight of the puppy?” Problem 2, “Andre also has a puppy that weighs 9 pounds. The vet says that this puppy is now at about 30% of its adult weight. What will be the adult weight of Andre’s puppy?”  

• Unit 5: Arithmetic in Base Ten, Section B: Adding and Subtracting Decimals, Lesson 4: Adding and Subtracting Decimals with Many Non-Zero Digits, Instructional Routines, “Students deepen their understanding of regrouping by tackling problems that are more challenging and that prompt them to notice and use structure (MP7). Students build on both their work with whole-number differences (such as 1000 - 256) to find differences such as 1 - 0.256. To add and subtract digits, they may think in terms of bundling and unbundling base-ten units, but there are also other opportunities to use structure here. Let’s take the example 1,000 - 256. Since 1,000 = 999 + 1, students could calculate 1,000 - 256 by first finding 999 - 256 = 743, and then adding 1 to get 744. They could use the same reasoning to find sums and differences of decimals.” Activity 2: Missing Numbers, students look for structure in expression as they regroup addition and subtraction problems. “Write the missing digits in each calculation so that the value of each sum or difference is correct. Be prepared to explain your reasoning. a. 0.404 + ___ = 1 b. 9.8765 + ___ =10 c. 0.7 - ___ = 0.012 d. 7 - ___ = 3.4567 e. 70 - ___ = 0.0089” 

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning as they work independently with the teacher's support throughout the units. Examples include:

• Unit 2: Introducing Ratios, Section D: Solving Ratio and Rate Problems, Lesson 12: Navigating a Table of Equivalent Ratios, Lesson Narrative, “Students see that a table accommodates different ways of reasoning about equivalent ratios, with some being more direct than others. They notice (MP8) that to find an unknown quantity, they can: Find the multiplier that relates two corresponding values in different rows (e.g., “What times 5 equals 8?”) and use that multiplier to find unknown values. (This follows the multiplicative thinking developed in previous lessons.) Find an equivalent ratio with one quantity having a value of 1 and use that ratio to find missing values.” Activity 2: Hourly Wages, students use the method of calculating unit rate to find equivalent ratios. “Lin is paid $90 for 5 hours of work. She used the following table to calculate how much she would be paid at this rate for 8 hours of work. a. What is the meaning of the 18 that appears in the table? b. Why was the number \frac{1}{5} used as a multiplier? c. Explain how Lin used this table to solve the problem. d. At this rate, how much would Lin be paid for 3 hours of work? For 2.1 hours of work?”

• Unit 4: Dividing Fractions, Section B: Meanings of Fraction Division, Lesson 4: How Many Groups (Part 1), Lesson Narrative, “This lesson is the first in a group of six lessons that trace out a gradual progression of learning—from reasoning with specific quantities, to using a symbolic formula for division of fractions (MP8).” Warm Up: Equal-sized Groups, students generalizations of multiplication as representing equal-sized groups and the relationship between multiplication and division. “Write a multiplication equation and a division equation for each statement or diagram. a. Eight $5 bills are worth $40. b. There are 9 thirds in 3 ones.” Part c. shows a bar diagram equally divided into 5 sections each labeled \frac{1}{5}."

• Unit 6: Expressions and Equations, Section C: Expressions with Exponents, Lesson 12: Meaning of Exponents, Instructional Routines, “The purpose of this task is to show a simple context where exponent notation is naturally useful. The task lends itself to connecting repeated calculations with an expression involving exponents (MP8). This motivates creating a shorthand notation that can be used to answer the questions.” Activity 1: The Genie’s Offer, students create shortcuts by using exponents to represent repeated multiplication. “You find a brass bottle that looks really old. When you rub some dirt off of the bottle, a genie appears! The genie offers you a reward. You must choose one: $50,000; or a magical $1 coin. The coin will turn into two coins on the first day. The two coins will turn into four coins on the second day. The four coins will double to 8 coins on the third day. The genie explains the doubling will continue for 28 days. a. The number of coins on the third day will be 2 \cdot 2 \cdot 2. Can you write another expression using exponents for the number of coins there will be on the third day? b.What do 2^5and 2^6represent in this situation? Evaluate 2^5 and 2^6 without a calculator. c. How many days would it take for the number of magical coins to exceed $50,000? d. Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.”