6th Grade - Gateway 2

The materials reviewed for Math Nation Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Materials develop conceptual understanding throughout the grade level and materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. There are opportunities for students to develop their conceptual understanding in the various parts of each lesson: Warm-up, Exploration Activities, Lesson Synthesis, Cool Down, Check Your Understanding, and Practice Problems. Additionally, students’ conceptual understanding was assessed on Mid-Unit Assessments and End-of-Unit Assessments.

Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Course Guide, “Concepts Develop from Concrete to Abstract, Mathematical concepts are introduced simply, concretely, and repeatedly, with complexity and abstraction developing over time. Students begin with concrete examples, and transition to diagrams and tables before relying exclusively on symbols to represent the mathematics they encounter.” Examples include:

• Unit 2, Lesson 6, 2.6.4 Exploration Activity, students develop conceptual understanding by using ratio and rate reasoning to solve real-world problems (6.RP.3). “Here is a diagram showing Elena's recipe for light blue paint. 1. Complete the double number line diagram to show the amounts of white paint and blue paint in different-sized batches of light blue paint. 2. Compare your double number line diagram with your partner. Discuss your thinking. If needed, revise your diagram. 3. How many cups of white paint should Elena mix with 12 tablespoons of blue paint? How many batches would this make? 4. How many tablespoons of blue paint should Elena mix with 6 cups of white paint? How many batches would this make? 5. Use your double number line diagram to find another amount of white paint and blue paint that would make the same shade of light blue paint. 6. How do you know that these mixtures would make the same shade of light blue paint?” 

• Unit 4, Lesson 6, 4.6.1 Warm-Up, students develop conceptual understanding as they interpret and compute quotients of fractions (6.NS.1). “We can think of the division expression 10\div2\frac{1}{2} as the answer to the question: ‘How many groups of 2\frac{1}{2}’s are in 10? Complete the tape diagram to represent the question. Then find the answer.” 

• Unit 6, Lesson 2, Cool-Down, students develop conceptual understanding by explaining how they know value makes an equation true(6.EE.5). “Explain how you know that 88 is a solution to the equation 18x=11 by completing the sentences: The word ‘solution’ means . . . 88 is a solution to 18x=11 because . . .” 

The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:

• Unit 2, End-of-Unit Assessment (A), Question 1, students develop conceptual understanding as they use ratio language to determine true relationships between quantities (6.RP.1). “Select all the true statements.  A. The ratio of triangles to squares is 2 to 4. B. The ratio of squares to smiley faces is 6:4. C. The ratio of smiley faces to triangles is 6 to 4. D. There are two squares for every triangle. E. There are two triangles for every smiley face. F. There are three smiley faces for every triangle.” Provided is a picture of six smiley faces, two triangles, and four squares. 

• Unit 3, Lesson 11, 3.11.6  Practice Problems, Question 3, students develop conceptual understanding by using ratio reasoning to solve real-world problems (6.RP.3). “At a school, 40% of the sixth-grade students said that hip-hop is their favorite kind of music. If 100 sixth grade students prefer hip hop music how many sixth grade students are at the school? Explain or show your reasoning.” 

• Unit 6, Lesson 1, 6.1.6  Practice Problems, Question 1, students develop conceptual understanding by drawing a tape diagram to represent an equation and then interpret how parts of the equation are represented in the tape diagram (6.EE.6). “Here is an equation x+4=17.  A. Draw a tape diagram to represent the equation. B. Which part of the diagram shows the quantity x? What about 4? What about 17? C. How does the diagram show that x+4 has the same value as 17?”

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

There are opportunities for students to develop their procedural skills and fluency throughout the grade levels in each lesson, these opportunities can be found in the Warm-up, Exploration Activities, and Practice Problems. Examples include:

• Unit 2, Lesson 8, 2.8.4 Exploration Activity, Questions 1-3, students develop procedural skills and fluency solving problems involving unit rate (6.RP.3b). "1. Four bags of chips cost $6. a. What is the cost per bag? b. At this rate, how much will 7 bags of chips cost? 2. At a used book sale, 5 books cost $15. a. What is the cost per book? b. At this rate, how many books can you buy for $21? 3. Neon bracelets cost $1 for 4. a. What is the cost per bracelet? b. At this rate, how much will 11 neon bracelets cost?" 

• Unit 4, Lesson 4, 4.4.4 Practice Problems, Question 7, students develop procedural skills and fluency as they solve problems with percents (6.RP.3c). “Find each unknown number. a. 12 is 150% of what number? b. 5 is 50% of what number? c. 10% of what number is 300? d. 5% of what number is 72? e. 20 is 80% of what number?” 

• ​​Unit 5, Lesson 11, 5.11.3 Exploration Activity, Question 2, students develop procedural skills and fluency as they solve problems using long-division (6.NS.2). “Use long division to find the value of each expression. Then pause so your teacher can review your work. a. 126\div8; b. 90\div12” 

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

• Unit 4, Lesson 11, 4.11. 7 Check Your Understanding, Question 1, students independently demonstrate procedural skills and fluency while dividing two mixed fractions (6.NS.1). “What is the quotient of 4\frac{3}{8}\div2\frac{1}{4}? (A) \frac{18}{35}; (B) \frac{32}{315}; (C) 1\frac{17}{18}; (D) 9\frac{27}{32}” 

• Unit 6, Lesson 12, 6.12.6 Practice Problems, Question 6, students independently demonstrate procedural skills and fluency by solving one-step equations (6.EE.7). “Solve each equation. A. a-2.01=5.5 B. b+2.01=5.5 C. 10c=13.71 D. 100d=13.71” 

• Unit 7, End-of-Unit Assessment (A), Question 6, students independently demonstrate procedural skills and fluency by positioning pairs of integers on a coordinate plane to draw a polygon (6.NS.6c). “Draw polygon ABCDEF in this coordinate plane, given its vertices A = (-2,-3), B = (0,3), C = (0,1), D = (3,1), E = (3,3), F = (-2,3).”

The materials reviewed for Math Nation Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Materials provide opportunities for students to work with multiple routine and non-routine applications of mathematics throughout the grade level and independently. Applications of mathematics occur throughout a lesson in the exploration activities, practice problems, and assessments.

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

• Unit 4, Lesson 11, 4.11.3 Exploration Activity, Question 2, students solve a routine word problem involving the division of fractions (6.NS.1). “After biking 5\frac{1}{2} miles, Jada has traveled \frac{2}{3} of the length of the trip. How long (in miles) is the entire length of her trip? Write an equation to represent the situation, and find the answer using your preferred strategy.” 

• Unit 5, Lesson 8, 5.8.6 Practice Problems, Question 4, students solve a routine word problem involving adding decimals (6.NS.3). “A pound of blueberries costs $3.98 and a pound of clementines costs $2.49. What is the combined cost of 0.6 pound of blueberries and 1.8 pounds of clementines? Round your answer to the nearest cent.” 

• Unit 6, Lesson 17, 6.17.1 Warm-Up, students solve a non-routine word problem involving unit rate (6.RP.3b). “Lin and Jada each walk at a steady rate from school to the library. Lin can walk 13 miles in 5 hours, and Jada can walk 25 miles in 10 hours. They each leave school at 3:00 and walk 3\frac{1}{4} miles to the library. What time do they each arrive?”

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

• Unit 2, Lesson 9, 2.9.6 Practice Problems, Question 1, students solve a routine word problem independently involving unit rate (6.RP.3b). “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.”

• Unit 4, Lesson 4, 4.4.4 Practice Problems, Question 1, students solve a non-routine word problem by drawing a diagram and writing a multiplication or division equation to represent the situation (6.NS.1). “Consider the problem: A shopper buys cat food in bags of 3 lbs. Her cat eats \frac{3}{4} lb each week. How many weeks does one bag last? a. Draw a diagram to represent the situation and label your diagram so it can be followed by others. Answer the question. b. Write a multiplication or division equation to represent the situation. c. Multiply your answer in the first question (the number of weeks) by \frac{3}{4}. Did you get 3 as a result? If not, revise your previous work.” 

• Unit 6, Lesson 7, 6.7.3 Exploration Activity, students solve a routine word problem involving percentages by writing an equation (6.RP.3c and 6.EE.7). “1. Puppy A weighs 8 pounds, which is about 25% of its adult weight. What will be the adult weight of Puppy A? 2. Puppy B weighs 8 pounds, which is about 75% of its adult weight. What will be the adult weight of Puppy B? 3. If you haven't already, write an equation for each situation. Then, show how you could find the adult weight of each puppy by solving the equation.”

The materials reviewed for Math Nation Grade 6 meet expectations 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 level.

All three aspects of rigor are present independently throughout each grade level. Examples include:

• Unit 1, Lesson 15, Cool Down, students develop conceptual understanding as they solve problems with three-dimensional figures (6.G.4). “In this net, the two triangles are right triangles. All quadrilaterals are rectangles. What is its surface area in square units? Show your reasoning. 2. If the net is assembled, which of the following polyhedra would it make?”

• Unit 3, Lesson 1, 3.1.6 Practice Problems, Question 5, students develop procedural skills and fluency as they use unit rates to answer questions about sandwiches (6.RP.2). “A sandwich shop serves 4 ounces of meat and 3 ounces of cheese on each sandwich. After making sandwiches for an hour, the shop owner has used 91 combined ounces of meat and cheese. a. How many combined ounces of meat and cheese are used on each sandwich? b. How many sandwiches were made in the hour? c. How many ounces of meat were used? d. How many ounces of cheese were used?”

• Unit 6, Lesson 12, 6.12.2 Exploration Activity, students apply their understanding of exponents to write equivalent expressions and evaluate numerical expressions with whole-number exponents (6.EE.1). ”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. 1. The number of coins on the third day will be 2⋅2⋅2. Write an equivalent expression using exponents. 2. What do 2^5 and 2^6 represent in this situation? Evaluate 2^5 and 2^6 without a calculator. Pause for discussion. 3. How many days would it take for the number of magical coins to exceed $50,000? 4. Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.”

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

• Unit 2, Lesson 13, 2.13.6 Practice Problems, Question 2, students build conceptual understanding as they become more fluent using ratio and rate reasoning to find equivalent ratios on double number lines and ratio tables (6.RP.3).“A bread recipe uses 3 tablespoons of olive oil for every 2 cloves of crushed garlic. a. Complete the table to show different-sized batches of bread that taste the same as the recipe. b. Draw a double number line that represents the same situation. c. Which representation do you think works better in this situation? Explain why.” 

• Unit 5, Lesson 10, 5.10.2 Exploration Activity, students discuss with a partner the similarities and differences between different division methods and then divide numbers using one of the methods (6.NS.2). “Lin has a method of calculating quotients that is different from Elena’s method and Andre’s method. Here is how she found the quotient of 657\div3: 1. Discuss with your partner how Lin’s method is similar to and different from drawing base-ten diagrams or using the partial quotients method. a. Lin subtracted 3\cdot2, then 3\cdot1, and lastly 3\cdot9. Earlier, Andre subtracted 3\cdpt200, then 3\cdot10, and lastly 3\cdot9. Why did they have the same quotient? b. In the third step, why do you think Lin wrote the 7 next to the remainder of 2 rather than adding 7 and 2 to get 9? 2. 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\div3 b. 1,816\div4 c. 768\div12” 

• Unit 8, Lesson 10, 8.10.5 Practice Problems, Question 1, students use the real-world problem of walking to school to build conceptual understanding as they fluently calculate means (6.SP.3). “On school days, Kiran walks to school. Here are the lengths of time, in minutes, for Kiran’s walks on 5 school days. 16 11 18 12 13 A. Create a dot plot for Kiran’s data. B. Without calculating, decide if 15 minutes would be a good estimate of the mean. If you think it is a good estimate, explain your reasoning. If not, give a better estimate and explain your reasoning. C. Calculate the mean for Kiran’s data. D. In the table, record the distance of each data point from the mean and its location relative to the mean. E. Calculate the sum of all distances to the left of the mean, then calculate the sum of distances to the right of the mean. Explain how these sums show that the mean is a balance point for the values in the data set.”

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative, and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide).   

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to analyze and make sense of problems, work to understand the information in problems, and use a variety of strategies to make sense of problems. Examples include:

• Unit 1, Lesson 12, 1.12.1 Warm-Up, students estimate the surface area of a cabinet. “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?” This activity attends to the full intent of MP1 as students need to make sense of the problem and persevere in solving it as they are given no specific techniques for how to calculate surface area ahead of time.

• Unit 2, Lesson 14, 2.14.2 Exploration Activity, students engage in an Info-Gap activity with a partner. “Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the problem card: a. Read your card silently and think about what you need to know to be able to answer the questions. b. Ask your partner for the specific information that you need. c. Explain how you are using the information to solve the problem. d. Solve the problem and show your reasoning to your partner. If your teacher gives you the data card: a. Read your card silently. b. Ask your partner ‘What specific information do you need?’ and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. c. Have them explain ‘Why do you need that information?’ before telling them the information. d. After your partner solves the problem, ask them to explain their reasoning, even if you understand what they have done. Both you and your partner should record a solution to each problem.” This activity attends to the full intent of MP1 as students need to make sense of the problems by determining what information is necessary and know what questions to ask to receive the information they need to solve the problem.

• Unit 3, Lesson 9, 3.9.6 Practice Problems, Question 2, students solve a unit rate problem about a copy machine. “A copy machine can print 480 copies every 4 minutes. For each question, explain or show your reasoning. a. How many copies can it print in 10 minutes? b. A teacher printed 720 copies. How long did it take to print?” This problem attends to the full intent of MP1 as students must make sense of the problem in order to answer the questions about copies and time.

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to represent situations symbolically, attend to the meaning of quantities, and understand relationships between problem scenarios and mathematical representations. Examples include:

• Unit 2, Lesson 4, 2.4.6 Practice Problems, Question 3, students use reasoning to explain how to apply the ratio concepts learned to mix varying shades of blue paint. “To make 1 batch of sky blue paint, Clare mixes 2 cups of blue paint with 1 gallon of white paint. a. Explain how Clare can make 2 batches of sky blue paint. b. Explain how to make a mixture that is a darker shade of blue than the sky blue. c. Explain how to make a mixture that is a lighter shade of blue than the sky blue.” This problem attends to the full intent of MP2 as students need to reason abstractly and quantitatively about how to make the different batches of paint.

• Unit 7, Lesson 13, 7.13.2 Exploration Activity, students reason quantitatively about a graph and answer questions based on the information in the graph. “The graph shows the balance in a bank account over a period of 14 days. The axis labeled b represents account balance in dollars. The axis labeled d represents the day. 1. Estimate the greatest account balance.  On which day did it occur.  2. Estimate the least account balance.  On which day did it occur? 3.  What does the point (6,-50) tell you about the account balance?  4.  How can we interpret |-50| in the context?” This problem intentionally develops the full intent of MP2 as students need to reason about the meaning of quantities given the context of the problem.

• Unit 8, Lesson 14, Cool-Down, students reason quantitatively and abstractly regarding the measures of center for given data sets. “For each dot plot or histogram: 1. Predict if the mean is greater than, less than, or approximately equal to the median. Explain your reasoning. 2. Which measure of center—the mean or the median— better describes a typical value for the following distributions? 1. Heights of 50 NBA basketball players 2. Backpack weights of 55 sixth-grade students 3. Ages of 30 people at a family dinner party.” This problem attends to the full intent of MP2 as students reason quantitatively and abstractly about the means and medians of data.

The materials reviewed for Math Nation 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. The materials provide opportunities for students to construct viable arguments and critique the reasoning of others in whole class and small group settings (i.e. exploration activities) and independent work settings (i.e. practice problems and assessments). 

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to construct mathematical arguments by explaining/justifying their strategies and thinking, performing error analysis of provided student work/solutions, listening to the arguments of others and deciding if it makes sense, asking useful questions to better understand, and critiquing the reasoning of others. Examples include:

• Unit 1, Lesson 2, 1.2.6 Practice Problems, Question 5, students explain why a statement about area is incorrect. “A student said, ‘We can't find the area of the shaded region because the shape has many different measurements, instead of just a length and a width that we could multiply.’ Explain why the student's statement about area is incorrect.” This question intentionally develops MP3 as students critique the reasoning of others and explain why the statement is incorrect.

• Unit 2, Lesson 8, 2.8.1 Warm-Up, students mentally find a quotient and then explain their strategy. “Find the quotient mentally. 246\div12” Full Lesson Plan, Teacher Guidance: “Invite students to share their strategies. Record and display student explanations for all to see. Ask students to explain if or how the dividend or divisor impacted their choice of strategy and how they decided to write their remainder. To involve more students in the conversation, consider asking: • ‘Who can restate ___’s reasoning in a different way?’ • ‘Did anyone solve the problem the same way but would explain it differently?’ • ‘Did anyone solve the problem in a different way?’ • ‘Does anyone want to add on to _____’s strategy?’ • ‘Do you agree or disagree? Why?’” This activity intentionally develops MP3 as students explain their strategy for agreeing or disagreeing with their classmates' strategies. 

• Unit 3, Lesson 6, 3.6.6 Practice Problems, Question 4, students calculate unit rate for a real-world situation. “A large art project requires enough paint to cover 1,750 square feet. Each gallon of paint can cover 350 square feet. Each square foot requires \frac{1}{350} of a gallon of paint. Andre thinks he should use the rate \frac{1}{350} gallons of paint per square foot to find how much paint they need. Do you agree with Andre? Explain or show your reasoning.” This question intentionally develops MP3 as students critique the reasoning of others while constructing arguments to justify their conclusions.

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide). 

Examples where and how the materials use MPs 4 and/or 5 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

• Unit 6, Lesson 1, 6.1.1 Warm-Up, students identify equations that match diagrams and draw a diagram that represents equations. “Here are two diagrams. One represents 2+5=7. The other represents 5\cdot2=10. Which is which? Label the length of each diagram. Draw a diagram that represents each equation. 1. 4+3=7; 2. 4\cdot3=12" This activity intentionally develops (MP4), model with mathematics, and (MP5) use appropriate tools strategically as students choose which tools to use as they create their model.

• Unit 6, Lesson 6, 6.6.6 Practice Problems, Question 1, students write and evaluate expressions with numbers and variables. “Instructions for a craft project say that the length of a piece of red ribbon should be 7 inches less than the length of a piece of blue ribbon. How long is the red ribbon if the length of the blue ribbon is: 10 inches? 27 Inches? x inches? How long is the blue ribbon if the red ribbon is 12 inches?” Students are given the option to represent expressions with tape diagrams. This activity intentionally develops (MP4), model with mathematics, and(MP5) use appropriate tools strategically as students can choose to use an appropriate tool to set-up the expressions based on the scenario.

• Unit 7, Lesson 9, 7.9.1 Warm-up, Question 1, given a number line with several points students complete blank inequality statements with the points to make the inequality true. “1. Fill in each blank with a letter so that the inequality statements are true A. ___ > ___ B. ___ < ___  2. Jada says that she found three different ways to complete the first question correctly. Do you think this is possible? Explain your reasoning. 3. List a possible value for each letter on the number line based on its location.“ This activity intentionally develops (MP4), model with mathematics, and (MP5) use appropriate tools strategically as students use the number line strategically in order to answer the questions. 

• Unit 8, Lesson 3, 8.3.1 Warm-Up, students create a statistical question about a given scenario and explain their reasoning. ”Clare collects bottle caps and keeps them in plastic containers. Write one statistical question that someone could ask Clare about her collection. Be prepared to explain your reasoning.” This activity intentionally develops MP4, model with mathematics.

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide). 

There is intentional development of MP6 to meet its full intent in connection to grade-level content and the instructional materials attend to the specialized language of mathematics. Students communicate using grade-level appropriate vocabulary and conventions and formulate clear explanations when engaging with course materials. Students must calculate accurately and efficiently, specify units of measure, and use and label tables and graphs appropriately when engaging with course materials. Teacher guidance very clearly develops the specialized language of mathematics as teachers are explicitly prompted when to introduce content-related vocabulary and use accurate definitions when communicating mathematically. Examples include:

• Unit 1, Lesson 5, 1.5.1 Warm-Up, students compare and contrast two strategies for finding the area of a parallelogram. “Elena and Tyler were finding the area of this parallelogram: Move the slider to see how Tyler did it: [An applet shows you how Tyler decomposed his parallelogram] Move the slider to see how Elena did it: [An applet shows you how Elena decomposed his parallelogram How are the two strategies for finding the area of a parallelogram the same? How are they different?” Full Lesson Plan, Teacher Guidance: “The two measurements that we see here have special names. The length of one side of the parallelogram—which is also the length of one side of the rectangle—is called a base. The length of the vertical cut segment—which is also the length of the vertical side of the rectangle—is called a height that corresponds to that base.” This activity attends to the specialized language of mathematics as students learn and are encouraged to use the correct terms to describe Elena and Tyler's strategies.

• Unit 2, Lesson 5, 2.5.6 Practice Problems, Question 1, students explain why given ratios are equivalent. “Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a diagram that shows why they are equivalent ratios. 1. 4:5 and 8:10; 2. 18:3 and 6:1; 3. 2:7 and 10,000:35,000,” This activity attends to the full intent of MP6 and the specialized language of mathematics as students communicate using grade-level appropriate vocabulary and conventions.

• Unit 4, Lesson 5, 4.5.6 Practice Problems, Question 3, students use a ruler to write multiplication and division equations. ”Use a standard inch ruler to answer each question. Then, write a multiplication equation and a division equation that answer the question. How many \frac{1}{2}s are in 7?  How many \frac{3}{8}s are in 6? How many \frac{5}{16}s are in 1\frac{7}{8}?” This activity attends to the full intent of MP6 and the specialized language of mathematics as students communicate using grade-level appropriate vocabulary such as halves, eighths…etc.

• Unit 6, Lesson 2, 6.2.1 Warm-Up, students determine if an equation is true or false when substituting in chosen values. “The equation a+b=c could be true or false. a. If a is 3, b is 4, and c is 5, is the equation true or false? b. Find new values of a, b, and c that make the equation true. c. Find new values of a, b, and c that make the equation false.” This activity attends to the full intent of MP6 as students would need to attend to precision to get the correct value.

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide). 

There is intentional development of MP7 to meet its full intent in connection to grade-level content.  Materials provide opportunities for students to look for patterns or structures to make generalizations and solve problems, look for and explain the structure of mathematical representations, and look at and decompose “complicated” into “simpler.”  Examples include:

• Unit 1, Lesson 3, 1.3.2 Exploration Activity, students find the area of the shaded region in different diagrams by decomposing, rearranging, subtracting, and enclosing figures. “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. (Students are given four figures.)”  This activity attends to the full intent of MP7 as students use the structure of the shape to decompose them into simpler ones. 

• Unit 4, Lesson 8, 4.8.1 Warm-Up, students interpret a division statement and write a question in which the equation represents the scenario. “1. Think of a situation with a question that can be represented by 12\div\frac{2}{3} = ? Describe the situation and the question. 2. Trade descriptions with your partner, and answer your partner's question.” This activity intentionally develops MP7 as students must look at the structure of the equation in order to write a problem that represents it.

• Unit 5, Lesson 5, 5.5.1 Warm-up, students compare the same variable in each equation to determine which value is the largest, multiplying by 10 to consider the effect on how the decimal point moves. “1. In which equation is the value of x the largest? A. x\cdot10=810 B. x\cdot10=81 C. x\cdot10=8.1 D. x\cdot10=0.81 2. How many times the size of 0.81 is 810?” This activity intentionally develops MP7 as students use the structure of the equations to multiply other decimal products.

There is intentional development of MP8 to meet its full intent in connection to grade-level content. 

Materials provide opportunities for students to notice repeated calculations to understand algorithms and make generalizations or create shortcuts, evaluate the reasonableness of their answers and their thinking, and create, describe, or explain a general method/formula/process/algorithm. Examples include:

• Unit 1, Lesson 9, 1.9.1 Warm-Up, students identify base-height measurements of triangles, use them to determine area, and look for a pattern in their reasoning to help them write a general formula for finding area. “Study the examples and non-examples of bases and heights in a triangle. Answer the questions that follow. These dashed segments represent heights of the triangle.  These dashed segments do not represent heights of the triangle. Select all the statements that are true about bases and heights in a triangle.” The following are the statements:  Any side of a triangle can be a base; There is only one possible height; A height is always one of the sides of a triangle; A height that corresponds to a base must be drawn at an acute angle to the base; A height that corresponds to a base must be drawn at a right angle to the base; Once we choose a base, there is only one segment that represents the corresponding height; and A segment representing a height must go through a vertex. This warm-up attends to the full intent of MP8 as students create, describe, and explain a general formula, process, method, algorithm, model, etc. This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students must repeatedly reference the information given to determine which of the statements are correct. 

• Unit 3, Lesson 7, 3.7.2 Exploration Activity, Question 1, students calculate the unit price of a burrito and then generalize to find the cost for any number of burritos. “Two burritos cost $14.00. Complete the table to show the cost of 4, 5, and 10 burritos at that rate. Next, find the cost for a single burrito in each case.” This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students use repeated calculations to fill out the table in order to create a general method.

• Unit 6, Lesson 10, 6.10.3 Exploration Activity, students complete a table representing the width, length, and area of several rectangles. “For each rectangle, write expressions for the length and width of two expressions for the total area. Record them in the table. Check your expressions in each row with your group and discuss any disagreements.” Students are given six rectangles labeled A - F. This activity attends to the full intent of MP8 as students notice repeated calculations to understand algorithms and make generalizations or create shortcuts. This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students repeatedly write the expressions for area of the rectangle, they formulate a general method for the distributive property.