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 , 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.

The materials reviewed for Math Nation Grade 6 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development. 

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Course Overview: A Course Overview (Unit 0) is found at the beginning of each course.  Within each Course Overview there is a Course Narrative, which contains a summary of the mathematical content contained in each course, and a Course Guide. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Resources, Corrections, and Cool-Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the About These Materials section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool-Down. The How to Use These Materials section contains guidance about the three phases of classroom activities (Launch-Work-Synthesize) and utilizing instructional routines. In the Scope and Sequence section, teachers will find a Pacing Guide which contains time estimates for coverage of each of the units.

• Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessments (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Student Task Statements Cool-downs, Practice Problems, Blackline Masters, and My Reflections all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Full Lesson Plan downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.

• Full Lesson Plan: Within each Teacher Edition lesson component, teachers can find a Full Lesson Plan that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Course Guide, Lesson Plans, Lesson Narratives, Overviews, and Warm-up provide useful annotations. Examples include:

• Course Guide, Assessments Overview, “Pre-Unit Diagnostic Assessments At the start of each unit is a pre-unit diagnostic assessment. These assessments vary in length. Most of the problems in the pre-unit diagnostic assessment address prerequisite concepts and skills for the unit. Teachers can use these problems to identify students with particular below-grade needs, or topics to carefully address during the unit. Teachers are encouraged to address below-grade skills while continuing to work through the on-grade tasks and concepts of each unit, instead of abandoning the current work in favor of material that only addresses below-grade skills…What if a large number of students can’t do the same pre-unit assessment problem? Look for opportunities within the upcoming unit where the target skill could be addressed in context…What if all students do really well on the pre-unit diagnostic assessment? Great! That means they are ready for the work ahead, and special attention likely doesn’t need to be paid to below-grade skills.”

• Unit 1, Lesson 9, Full Lesson Plan, 1.9.3 Exploration Activity, “Anticipated Misconceptions The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide the measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the warm-up activity.”

• Unit 5, Lesson 8, Full Lesson Plan, “Lesson Narrative In this culminating lesson on multiplication, students continue to use the structure of base-ten numbers to make sense of calculations (MP7) and consolidate their understanding of the themes from the previous lessons. They see that multiplication of decimals can be accomplished by: thinking of the decimals as products of whole numbers and fractions; writing the non-zero digits of the factors as whole numbers, multiplying them, and moving the decimal point in the product; representing the multiplication with an area diagram and finding partial products; and using a multiplication algorithm, the steps of which can be explained with the reasonings above.”

The materials reviewed for Math Nation Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Course Guide, About These Materials sections, states the following note about standards alignment, “There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as ‘building on.’ When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as ‘building towards.’ When a task is focused on the grade-level work, the alignment is indicated as ‘addressing.’” All lessons in the materials have this correlation information. An example:

• Unit 7, Lesson 4, Full Lesson Plan, Lesson Standards Alignment, Building on 4.NBT.2, 5.NBT.3b; Addressing 6.NS.C, 6.NS.6, 6.NS.6a, 6.NS.7; Building Towards 6.NS.7.

Explanations of the role of the specific grade-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative, and Full Lesson Plan. Examples include:

• Course Guide, Scope and Sequence, Unit 1: Area and Surface Area, “Work with area in grade 6 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students' work with geometric measurement began with length and continued with area…In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths…In grade 8, students will understand ‘identical copy of’ as ‘congruent to’ and understand congruence in terms of rigid motions, that is, motions such as reflection, rotation, and translation. In grade 6, students do not have any way to check for congruence except by inspection, but it is not practical to cut out and stack every pair of figures one sees…”

• Course Guide, Scope and Sequence, Unit 2: Introducing Ratios, “Work with ratios in grade 6 draws on earlier work with numbers and operations. In elementary school, students worked to understand, represent, and solve arithmetic problems involving quantities with the same units. In grade 4, students began to use two-column tables, e.g., to record conversions between measurements in inches and yards. In grade 5, they began to plot points on the coordinate plane, building on their work with length and area…Use of tables to represent equivalent ratios is an important stepping stone toward use of tables to represent linear and other functional relationships in grade 8 and beyond. Because of this, students should learn to use tables to solve all kinds of ratio problems, but they should always have the option of using discrete diagrams and double number line diagrams to support their thinking…The terms proportion and proportional relationship are not used anywhere in the grade 6 materials. A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in grade 7. In high school-after their study of ratios, rates, and proportional relationships-students discard the term ‘unit rate,’ referring to a to b, a:b, and ab as ‘ratios.’”

The materials reviewed for Math Nation Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials sections. 

The About These Materials section states the following about the instructional approach of the program, “What is a Problem Based Curriculum? In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student's education. Families, students, teachers, and administrators may need support making this shift. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.”

Examples of materials including and referencing research-based strategies include:

• “The Five Practices Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem…”

• “Supporting English Language Learners This curriculum builds on foundational principles for supporting language development for all students. This section aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional supports and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012).”

• “Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”

Within the Course Guide, How to Use These Materials, a Reference section is included.

The materials reviewed for Math Nation Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.  Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Resources), Teacher Editions, for each lesson, under Lesson Preparation (Required Material(s)), and in Teacher Guides for specific lessons. Examples include:

• Unit 1, Lesson 3, Lesson Preparation, Required Materials: “Blackline master for Activity 3.1, Cool-down, copies of blackline master, geometry toolkits (tracing paper, graph paper, colored pencils, scissors, and an index card)”

• Unit 3, Lesson 3, Lesson Preparation, Required Materials: “Blackline master for Activity 3.2, Cool-down, base-ten blocks, blank paper, cuisenaire rods, gallon-sized jug, graduated cylinders, household items, inch cubes, internet-enabled device, liter-sized bottle, materials assembled from the blackline master, metal paper fasteners, meter sticks, pre-assembled polyhedra, quart-sized bottle, rulers, salt, scale, straightedges, teaspoon, tray”

• Unit 7, Lesson 7, Lesson Preparation, Required Materials: “Cool-down, sticky notes”

The materials reviewed for Math Nation Grade 6 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses. 

Suggestions to teachers for following up with students are provided throughout the materials via the Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.

Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:

• Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses. 

  • Restricted Constructed Response

    • Tier 1 response: Work is complete and correct.

    • Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.

    • Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.

  • Extended Response

    • Tier 1 response: Work is complete and correct, with complete explanation or justification. 

    • Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. 

    • Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. 

    • Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”

• Unit 5, End-of-Unit Assessment (B), Question 6, “A sign in a bakery gives these options: 12 muffins for $25; 24 muffins for $46; 50 muffins for $94 1. Find each unit price to the nearest cent, and show your reasoning. 2. Which option gives the lowest unit price? Solution 1. The unit prices are $2.08, $1.92, and $1.88. Reasoning varies, but long division is a reasonable approach. 2. 50 muffins for $94 has the lowest unit price.  Minimal Tier 1 response: Work is complete and correct. Sample: (Accompanied by work showing long division or other calculation methods.) The unit prices are $2.08, $1.92, and $1.88. 50 muffins for $94 is the best deal. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Acceptable errors: incorrect selection of lowest unit rate comes from errors in calculation of unit rates. Sample errors: one or two errors in long division; incorrect selection of the best deal despite having calculated all three unit rates correctly. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: calculation of unit rates involves a conceptual error, such as dividing the number of muffins by the price; three or more errors in long division; correct selection of the best deal without calculation of each unit rate.”

Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following up with students include:

• Course Guide, Cool-Down Guidance states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”

• Unit 1, Check-Your-Readiness (A), Question 5, “The content assessed in this problem is first encountered in Lesson 5: Bases and Heights of Parallelograms. In this unit, students will find the area of parallelograms and triangles by decomposing them into shapes with perpendicular sides and rearranging the pieces. Students will need to be familiar with perpendicular lines in order to make sense of the ‘height’ of a parallelogram or triangle. If most students struggle with this item, plan to start Lesson 5 Activity 2 by amplifying the term perpendicular for the students. Students may need some visual cues to support this concept.” 

• Unit 6, Lesson 15, Cool-down Guidance, “Support Level 2. Points to Emphasize. Notes If students struggle with using properties of exponents strategically in the cool-down, plan to focus on this idea when opportunities arise over the next several lessons. For example, in the practice problem set for Lesson 17, consider inviting students to reflect on the reasoning behind prompt 3.”

The materials reviewed for Math Nation Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in-classroom. Examples Include:

• Unit 1, Lesson 2, Cool-down, demonstrates the full intent of 6.G.1. “The square in the middle has an area of 1 square unit. What is the area of the entire rectangle in square units? Explain your reasoning.” A figure is given that is composed of many different shapes with a square in the middle of it.

• Unit 3, Lesson 6, Cool-down, demonstrates the full intent of 6.RP.3 and MP2. “Two pounds of grapes cost $6. 1. Complete the table showing the price of different amounts of grapes at this rate. 2. Explain the meaning of each of the numbers you found.” A table is provided with two columns one labeled “grapes (pounds)” and the other labeled “price (dollars)”.

• Unit 7, End-of-Unit Assessment (A), Question 4, demonstrates the full intent of 6.NS.7. “Given $x = -2$, mark and place these expressions on the same number line. $x$, $-x$, $|-1.5|$, $-4$, $|5|$, $|-6|$”

The materials reviewed for Math Nation Grade 6 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Course Guide, How to Use These Materials, Supporting Students with Disabilities sections states the following: “The philosophical stance that guided the creation of these materials is the belief that with proper structures, accommodations, and supports, all children can learn mathematics. Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students. While the suggested supports are designed for students with disabilities, they are also appropriate for many children who struggle to access rigorous, grade-level content. Teachers should use their professional judgment about which supports to use and when, based on their knowledge of the individual needs of students in their classroom.” Suggested supports are identified for teachers in the Full Lesson Plan to support learners of all levels. Lesson and activity-level supports, identified as “Support for Students with Disabilities,” are aligned to an area of cognitive functioning and are paired with a suggested strategy aimed to increase access and eliminate barriers. Supports are classified under the following categories: eliminate barriers, processing time, peer tutors, assistive technology, visual aids, graphic organizers, and brain breaks. Examples include:

• Eliminate Barriers: “Eliminate any barriers that students may encounter that prevent them from engaging with the important mathematical work of a lesson. This requires flexibility and attention to areas such as the physical environment of the classroom, access to tools, organization of lesson activities, and means of communication.” Unit 2, Lesson 15, Full Lesson Plan, 2.15.3 Exploration Activity, “Support for Students with Disabilities Executive Functioning: Eliminate Barriers. Chunk this task into more manageable parts (e.g., presenting one question at a time), which will aid students who benefit from support with organizational skills in problem solving”.

• Peer Tutors: “Develop peer tutors to help struggling students access content and solve problems. This support keeps all students engaged in the material by helping students who struggle and deepening the understanding of both the tutor and the tutee. For students with disabilities, peer tutor relationships with non-disabled peers can help them develop authentic, age-appropriate communication skills, and allow them to rely on a natural support while increasing independence.” Unit 6, Lesson 6, Full Lesson Plan, 6.6.3 Exploration Activity, “Support for Students with Disabilities… Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors…”

• Processing Time: “Increased time engaged in thinking and learning leads to mastery of grade level content for all students, including students with disabilities. Frequent switching between topics creates confusion and does not allow for content to deeply embed in the mind of the learner. Mathematical ideas and representations are carefully introduced in the materials in a gradual, purposeful way to establish a base of conceptual understanding. Some students may need additional time, which should be provided as required.” Unit 4, Lesson 12, Full Lesson Plan, 4.12.1 Warm Up, “Support for Students with Disabilities Memory: Processing Time. Provide sticky notes or mini whiteboards to aid students with working memory challenges. Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.”

There are several accessibility options (accessed via the wrench icon in the lower left-hand corner of the screen) available to help students navigate the materials. Examples include:

• Tools Menu allow students to change the language, and access a Demos Scientific and Graphing Calculator.

• Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, screen reader support. 

• UserWay, allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), Saturation (4 settings). 

Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.

The materials reviewed for Math Nation Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. Every extension problem is made available to all students with the heading ‘Are You Ready for More?’ These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts at grade level or that are outside of the standard K-12 curriculum. They are not routine or procedural, and intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems and it is not expected that any student works on all of them. Are You Ready for More? problems may also be good fodder for a Problem of the Week or similar structure.” If individual students would complete these optional activities, then they might be doing more assignments than their classmates.

Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:

• Unit 1, Lesson 15, 1.15.4 Exploration Extension: Are you Ready for More?, “1. Figure C shows a net of a cube. Draw a different net of a cube. Draw another one. And then another one. Show your work. 2. How many different nets can be drawn and assembled into a cube?” 

• Unit 2, Lesson 1, 2.1.4 Exploration Extension: Are you Ready for More?, “Use two colors to shade the rectangle so there are 2 square units of one color for every 1 square unit of the other color. 2. The rectangle you just colored has an area of 24 square units. Draw a different shape that does not have an area of 24 square units, but that can also be shaded with two colors in a 2:1 ratio. Shade your new shape using two colors.”

• Unit 8, Lesson 13, 8.13.3 Exploration Extension: Are you Ready for More?, “Invent a data set with a mean that is significantly lower than what you would consider a typical value for the data set.”

The materials reviewed for Math Nation Grade 6 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The Course Guide, How to Use These Materials section states the following: “The framework for supporting English language learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” The four design principles are, support sense-making, optimize output, cultivate conversation, and maximize meta-awareness. Each design principle has an explanation that goes into more detail about how teachers can use it to support students. The routines are the Mathematical Language Routines (MLRs), the materials state, “The mathematical language routines (MLRs) were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize uses of language that is meaningful and purposeful, rather than about just getting answers. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. Each MLR facilitates attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessment for all learners. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others' ideas.” These design principles and routines are referenced under Instructional Routines, in the Full Lesson Plan for lesson, to assist teachers with lesson planning. The “Supports for English Language Learners” section within the Full Lesson Plan contains explanations of how to implement the MLRs. 

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Unit 1, Lesson 18, Full Lesson Plan, 1.18.3 Exploration Activity, “Support for English Language Learners Conversing: MLR 3 Clarify, Critique, Correct. Present an incorrect response such as, ‘If the cube has edge length s, then the area of each face is 2s because ‘$$s×s=2s$$.’ Ask students to identify the error and to offer a correct argument to write an expression for the area of each face. This will help students to use symbolic representations while generalizing calculations related to surface area. Design Principle(s): Optimize output (for generalization); Maximize meta-awareness”

• Unit 2, Lesson 16, Full Lesson Plan, 2.16.2 Exploration Activity, “Support for English Language Learners Speaking: MLR 8 Discussion Supports. Provide sentence frames for students to state their reasoning (e.g., ‘I liked this method of solving the problem because _______’; ‘This way worked best because ________’; ‘The _____ strategy is the same as / different from the _____ strategy because _____ ‘). The helps students place extra attention on the language used to engage in mathematical communication and reasoning. Design Principle(s): Maximize meta-awareness; Optimize output (for generalization)”

• Unit 6, Lesson 4, Full Lesson Plan, 6.4.3 Exploration Activity, “Support for English Language Learners Reading: MLR 6 Three Reads. Demonstrate this routine with the first situation to support reading comprehension. Use the first read to help students understand context. Ask, ‘What is this situation about?’ (e.g., Clare and Mai each have a different number of books). After the second read, ask students ‘What are the quantities in the situation?’ (e.g., the number of books Mai has, the number of books Clare has). After the third read, ask students to brainstorm possible strategies to connect the situation with the appropriate equation(s). Encourage students to repeat this routine themselves for each situation. This helps students connect the language in the word problem with the equation(s) while keeping the intended level of cognitive demand in the task. Design Principle(s): Support sense-making”

The materials reviewed for Math Nation Grade 6 meet expectations for providing manipulatives, both virtual and physical, that are representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Unit 1, Lesson 4, 1.4.2 Exploration Activity, students use an applet to find the area of parallelograms by decomposing a parallelogram into pieces to find the area. “1. Find the area of each parallelogram. Show your reasoning. 2. Change the parallelogram by dragging the green points at its vertices. Find its area and explain your reasoning. 3. If you used the polygons on the side, how were they helpful? If you did not, could you use one or more of the polygons to show another way to find the area of the parallelogram?” Two Geogebra applets are provided with seven parallelograms on each, which students can manipulate to find the area.

• Unit 4, Lesson 14, 4.14.3 Exploration Activity, Questions 1 and 2, students use physical cubes or an applet of snap cubes to model the volume of a rectangular prism. “Use cubes or the applet to help you answer the following questions. 1. Here is a drawing of a cube with an edge length of 1 inch. a. How many cubes with an edge length of  $\frac{1}{2}$inch are needed to fill this cube? b. What is the volume, in cubic inches, of a cube with an edge length of $\frac{1}{2}$inch? Explain or show your reasoning. 2. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of $\frac{1}{2}$inch. Sketch the prism, and find its volume in cubic inches.” A GeoGebra applet is available that allows students to make prisms by manipulating the block length, width, and height. 

• Unit 6, Lesson 8, 6.8.6 Practice Problems, Question 1, students use an applet to show and explain when two equations are equivalent. A. Draw a diagram of $x + 3$ and a diagram of 2x when x is 1. B. Draw a diagram of $x + 3$ and 2x when x is 2. C. Draw a diagram of $x + 3$ and 2x when x is 3. D. Draw a diagram of $x + 3$ and 2x when x is 4…” Students have access to an applet with a grid where they can draw the diagrams.