6th Grade - Gateway 2

The materials reviewed for Eureka Math² Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The Learn portion of the lesson presents new learning through instructional segments to develop conceptual understanding of key mathematical concepts. Students independently demonstrate conceptual understanding in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

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

• Module 1, Topic C, Lesson 14: Comparing Ratio relationships, Part 2, Learn, Problem 2 students generate equivalent ratios to compare relationships. “Beekeepers add sugar water to the diet of honeybees. In the spring, the sugar water mixture helps promote colony growth. In the fall, the sugar water mixture helps the bees survive. The ratio tables show the number of cups of water and the number of cups of sugar in the spring sugar water mixture and in the fall sugar water mixture. Based on the tables, which sugar water mixture is sweeter? Explain. How did you determine which sugar water mixture is sweeter?” This activity supports the conceptual understanding of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).

• Module 3, Topic B, Lesson 9: Interpreting Order and Distance in Real-World Situations, Learn, Problem 1, students reason about statements and order of absolute value. “For parts (a)–(e), consider the situation from the video of the fishing boat and the net. a. Use a vertical number line to sketch the situation. Include sea level, the boat, the top of the rope, and the net in your sketch. b. What is the elevation of the top of the rope? c. What is the elevation of the net? d. Interpret the meaning of −38<23 in this situation. e. Interpret the meaning of \lvert-38\rvert>\lvert23\rvert in this situation. What does the absolute value of −38 represent in this situation? What does the absolute value of 23 represent in this situation? We can think of both of these absolute values as representing either the distance between two points, such as the net and sea level, or the length of the rope between two points. Can you determine how much more rope is below sea level than above? Explain. How long is the rope? How do you know? Why is the length of the rope not −61 feet?” This activity supports the conceptual understanding of 6.NS.7 (Understand ordering and absolute value of rational numbers).

• Module 4, Topic D, Lesson 18: Inequalities and Solutions, Learn, students determine whether a number is a solution to an inequality. “Invite students to think–pair–share about the following questions. If \frac{1}{2}x=8, does x=4,  x=8,  or x=16? How do you know? Consider the inequality \frac{1}{2}x<8. What does this inequality mean? Is 16 a solution to this inequality? What about 4 and 8? How do you know? How do you determine whether a number is a solution to an inequality? What do you notice about numbers that are solutions to 12x<8? What do you notice about the numbers that are not solutions?” This activity supports the conceptual understanding of 6.EE.5 (Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true).

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Examples include:

• Module 2, Topic B, Lesson 6: Dividing a Whole Number by a Fraction, Practice, Problem 2, students represent division of a whole number by a fraction using tape diagrams. “Consider 5\div\frac{2}{3}. a. Draw a tape diagram that represents 5\div\frac{2}{3}. b. Use the tape diagram from (a) to evaluate 5\div\frac{2}{3}.” Students independently demonstrate conceptual understanding of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem).

• Module 5, Topic D, Lesson 15: Exploring Volume, Practice, Problems 3 and 4, students find the volume of a right rectangular prism by packing with cubes that have fractional edge lengths. “3. How many cubes with an edge length of \frac{1}{4} unit pack one unit cube? What is the volume of a cube with an edge length of \frac{1}{4} unit? Explain how the volume of one of these cubes relates to the answer in problem 3.” Students independently demonstrate conceptual understanding of 6.G.2 (Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V=lwh and V=bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems).

• Module 6, Topic B, Lesson 9: Variability in a Data Distribution, Exit Ticket, students describe data distribution by using measures of variability. “The dot plots show the numbers of goals scored by the sixth grade hockey team and the seventh grade hockey team in 6 games. a. Find the mean number of goals scored for each team. b. Which data distribution has more variability? Explain. c. For which team does the mean better represent the numbers of goals scored in all games? Explain.” Students independently demonstrate conceptual understanding of 6.SP.3 (Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number).

The materials reviewed for Eureka Math² Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The Learn portion of the lesson presents new learning through instructional segments to develop procedural skill of key mathematical concepts. Students independently demonstrate procedural skill in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials develop procedural skills and fluency throughout the grade level. Examples include:

• Module 2, Topic E, Lesson 17: Partial Quotients, Learn, students divide 3-digit numbers by 1-digit and 2-digit numbers by using partial quotients. “Display the division problem showing 885\div15. Allow students to turn and talk about any similarities and differences they see from the model used in problem 2. Then complete the problem as a class. Guide students in dividing 885 by 15 as shown in the solution for problem 3. As you divide, consider asking the following questions to promote students’ thinking: Can we distribute 100 to each of the 15 groups? How do you know? Can we distribute 10 to each of the 15 groups? What about 20? 30? 40? 50? 60? Why? What number do we subtract from 885? Why? How many 15’s are in 135? How do you know?” Students develop procedural fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm).

• Module 2, Topic E, Lesson 20: Real-World Division Problems, Learn, students create and solve division word problems. Teachers, “Use the Numbered Heads routine. Organize students into an even number of groups with 2–3 students per group. Assign each student a number, 1 through 3. Display the following constraints. A. 3-digit or 4-digit dividend, 2-digit divisor with a whole-number quotient. B. 3-digit or 4-digit dividend, 1-digit divisor with a decimal quotient. Assign each group one of the constraints. Give students 5–7 minutes to create a division word problem that meets that constraint and to solve their problem. Have students record their word problem and solution in their books for problem 11. Then give each group an index card. Write the letter that you were assigned on the front of the index card. Write the word problem your group created on the back of the index card. Put your completed card in the correct stack: A or B.” Students develop procedural fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• Module 4, Topic D, Lesson 19: Solving Equations with Addition and Subtraction, Learn, Problem 2b, students solve equations using tape diagrams, an equation, 9+b=15. “Have students complete problem 2(b) in pairs (9+b=15). Solve the equation by using tape diagrams. Then solve the equation algebraically. Use substitution to check your solution. Circulate as students work. Ask the following questions as needed to support students: What do we know? What can we draw? What can we label? Should one tape be longer or shorter than the other? What is the same in both tapes? What does the tape diagram show us? How can we determine the value of the unknown section?” Students develop procedural fluency of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x+p=q and px=q  for cases in which p, q, and x are all nonnegative rational numbers).

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

• Module 1, Topic A, Lesson 5: Equivalent Ratios, Exit Ticket, Problem 1, “Show that the ratio   5 : 6 is equivalent to the ratio 35 : 42.” Students independently demonstrate procedural skill of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• Module 4, Topic A, Lesson 1: Expressions with Addition and Subtraction, Practice, Problem 3, students evaluate expressions using addition and subtraction. “For problems 2−9, evaluate. Show your work. 21+5-10-13+1” Students independently demonstrate procedural skill of 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents).

• Module 5, Topic D, Lesson 16: Applying Volume Formulas, Practice, Problem 2, students find the volume of a rectangular prism. “Consider the right rectangular prism. a. Compute the area of the base of the prism. b. Compute the volume of the prism.” A prism with dimensions 3.2 in, 1.2 in, and 5 in is pictured. Students independently demonstrate procedural skill of 6.G.2 (Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems).

• Each lesson begins with Fluency problems that provide practice of previously learned material. The Implementation Guide states, “Fluency provides distributed practice with previously learned material. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. Fluency activities are included with each lesson, but they are not accounted for in the overall lesson time. Use them as bell ringers, or, in a class period longer than 45 minutes, consider using the facilitation suggestions in the Resources to teach the activities as part of the lesson.” For example, Module 1, Topic D, Lesson 17: Rates, Fluency, Problem 1 students complete tables of equivalent ratios, “Directions: Complete each ratio table.” Students are provided a table with missing values. Number of Cups of Bananas 1, ___, 2 and Number of Cups of Grapes, ___, 1, 6. Students practice fluency of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

The materials reviewed for Eureka Math² Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The Learn portion of the lesson presents new learning through instructional segments to develop application of mathematical concepts. Students independently demonstrate routine application of the mathematics in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

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

• Module 1, Topic B, Lesson 6: Ratio Tables and Double Number Lines, Learn, Problem 5, students solve problems about quantities in equivalent ratios by using ratio tables and double number lines. “Leo buys fabric at a craft store. Every 2 yards of fabric costs $7.00. a. Create a double number line to show the relationship between possible amounts of fabric in yards and the total cost in dollars. b. If the total cost of the fabric is $21.00, how many yards of fabric does Leo buy? c. If Leo buys 1 yard of fabric, what is the total cost of the fabric?” In this routine problem, students apply the mathematics of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• Module 2, Topic C, Lesson 10: Dividing Fractions by Using the Invert and Multiply Strategy, Learn, Problem 3, students determine if errors exist in fraction problems. “Riley explains to her friend that \frac{5}{6}\div\frac{3}{4}=\frac{15}{24} because \frac{5}{6}\times3=\frac{15}{6} and \frac{15}{6}\times\frac{1}{4}=\frac{15}{24}. Explain any errors Riley made.” In this routine problem, students apply the mathematics of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions). 

• Module 5, Topic A, Lesson 2: The Area of a Right Triangle, Learn, Problem 8, students determine the height or the base of a right triangle given its area. “The area of a right triangle is \frac{9}{16} square feet and the base of the triangle is \frac{3}{4} feet. Write an equation that relates the area of the triangle to its base and its height h in feet. Solve the equation to determine the height of the triangle.” In this non-routine problem, students apply the mathematics of 6.G.1 (Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes: apply these techniques in the context of solving real-world and mathematical problems).

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

• Module 3, Topic D, Lesson 17: Problem Solving with the Coordinate Plane, Practice Problem 3-7, students graph points on the coordinate plane. “The coordinate plane shows various locations in a town. Each square on the grid represents 1 block. Each line on the grid represents a street. Use the coordinate plane to complete problems 3–7. 3. Write the ordered pair of the point that represents each location: Post office, Grocery store, Gas station, and School 4. Plot and label a point that represents an auto shop at (0, 5). 5. Plot and label a point that represents a pet store that has the same x-coordinate as the grocery store. What could be the ordered pair of the point for the pet store? 6. Plot and label a point that represents a playground that is in Quadrant IV and is the same distance from the y-axis as the school. What could be the ordered pair of the point for the playground? 7. Jada walks on streets from the school directly to the grocery store. What is the total number of blocks Jada walks?” In these routine problems, students independently apply the mathematics of 6.NS.8 (Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane).

• Module 4, Topic B, Lesson 11: Modeling Real-World Situations with Expressions, Learn, Slide 14, students write an algebraic expression for a situation involving going out with their friends. “After the arcade you have time for one more activity. Describe a situation that could be represented by 10+6.5n. Be sure to define n.” In this non-routine problem, students independently apply the mathematics of 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem).

• Module 4, Topic E, Lesson 25: The Statue of Liberty, Practice, Problem 3, students engage in non-routine problems as they use variables to represent two quantities in a real-world problem, that change in relationship to one another, and write an equation to express one quantity (6.EE.9). “Sana uses a fitness watch to track the total number of steps that she takes throughout the day. One day, Sana hikes uphill at a constant rate. The table shows the amount of time in minutes Sana hikes and the total number of steps she takes on this day. a. Complete the table. b. Write a description of the relationship between the amount of time Sana hikes and the total number of steps she takes on this day. c. Identify the independent variable and the dependent variable. d. Write an equation that represents the relationship between the total number of steps Sana takes and the amount of time she hikes. Define the variables. e. Sana reaches the peak of a hill after hiking at this rate for 42 minutes. Determine the total number of steps Sana has taken when she reaches the peak of the hill. Use the equation you wrote in part (c). Show your work. f. Can you determine the number of minutes Sana has been hiking when her fitness watch shows that she has taken a total of 6,000 steps? Explain.” In this non-routine problem, students independently apply the mathematics of 6.EE.9 (Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation).

The materials reviewed for Eureka Math² Grade 6 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

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

• Module 2, Topic C, Lesson 11: Applications of Fraction Division, Practice, Problem 3, students solve real-world problems dividing fractions and mixed numbers. “One batch of pancakes needs 1\frac{1}{4} cups of pancake mix. One box has 9 cups of pancake mix. How many batches of pancakes can be made with one box of pancake mix?” Students attend to the application of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions).

• Module 4, Topic D, Lesson 19: Solving Equations with Addition and Subtraction, Exit Ticket, students solve equations. “Solve each equation algebraically. Use substitution to check your solution. a. h-8=5 b. f+2.8=7.3.” Students attend to the procedural skill of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the from x+p=q and px=q for cases in which p, q, and x are all nonnegative rational numbers).

• Module 5, Topic A, Lesson 1: The Area of A Parallelogram, Land, students compose parallelograms into rectangles to derive the formula for the area of a parallelogram. “How does knowing how to find the area of a rectangle help you find the area of a parallelogram? What is the relationship among the base, height, and area of a parallelogram? How are the base and the height of a parallelogram related? How is the height of a parallelogram different from the height of a rectangle?” Students attend to the conceptual understanding of 6.G.1 (Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems).

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

• Module 1, Topic D, Lesson 16: Speed, Practice, Problem 1, students interpret and represent rates in context. “Karl Benz drove the first car in Mannheim, Germany, in 1886. The car traveled at a top speed of 10 miles per hour. Assume the car kept that constant speed. a. Interpret the meaning of the car’s speed. b. Use your answer from part (a) to complete the ratio table. c. Create a double number line to represent the situation.” Students engage in conceptual understanding and application of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• Module 4, Topic B, Lesson 9: Addition and Subtraction Expressions from Real-World Situations, Practice, Problem 2, students write algebraic expressions involving addition and subtraction to represent real-world situations. “In the last year, Riley’s height increased by 1\frac[1}{4} inches. a. Define the variable in this situation. b. Write an expression to represent Riley’s current height in inches.” Students engage in procedural skill and application of 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set).

• Module 5, Topic B, Lesson 8: Areas of Composite Figures in Real-World Situations, Learn, Problem 3, students use composite figures to determine the area of a wall that needs to be painted. “Consider if you were to paint one of the walls of your classroom. a. Create an accurate drawing of the wall that needs to be painted. Label the measurements of the wall on your drawing. b. Calculate the area of the wall that needs to be painted. c. One quart of paint covers 80 square feet and costs $16.99. How much does the paint required to paint the wall cost?” Students engage in procedural skill, conceptual understanding, and application of 6.G.1 (Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems).

The materials reviewed for Eureka Math² Grade 6 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

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

• Module 1, Topic D, Lesson 20: Solving Rate Problems, Learn, Station 2, students use a variety of strategies to solve a multi-step rate problem. “Each part provides the number of miles and the number of minutes for the beginning of each family’s drive. a. The Evans family drives 21 miles in the first 20 minutes of their trip. What is their speed in miles per hour during this time? Create a representation to show your work. b. The Perez family drives 16.5 miles in the first 15 minutes of their trip. What is their speed in miles per hour during this time? Create a representation to show your work. c. The Chan family drives 35 miles in the first 30 minutes of their trip. What is their speed in miles per hour during this time? Create a representation to show your work.” Teacher margin note states, “Ask the following questions to promote MP1: What steps can you take to start solving the problem? How can you simplify the problem? Does your answer make sense? Why?”

• Module 2, Topic B, Lesson 7: Dividing a Fraction by a Whole Number, Learn, Problem 3, students monitor and evaluate their progress to determine if their answers make sense. “Six friends share \frac{2}{3} of a pan of lasagna equally. What fraction of the pan of lasagna does each friend get? Show how you know.” Teacher margin note states, “Ask the following questions to promote MP1: How can you explain the problem in your own words? What steps can you take to start solving this problem? Does your answer make sense? Why?”

• Module 3, Topic C, Lesson 13: Constructing the Coordinate Plane, Learn, students work to understand information provided in the problem to construct a coordinate plane. “Suppose we have the same-size grid as the grid in problem 2. We want to plot the point (3.5, −5). What interval length can we choose for each axis if we want the point to lie on the intersection of grid lines? What interval length can we choose for each axis if we want to plot the point (-5,−1\frac{1}{3}) so that it lies on the intersection of grid lines? What if we want to plot the point (−150, 200) so that it lies on the intersection of grid lines? Can you think of an interval length that would work? Sometimes we can make different choices for the scales of the axes that are appropriate and that accurately show the locations of the points. In some cases, we will not be able to choose a scale so that all the points we plot fall exactly on the intersections of grid lines, and we need to estimate their locations between grid lines.” Teacher margin note states, “The probing questions guide student planning when they construct the coordinate plane. Consider elaborating on student responses related to how students determine the number of grid lines they need. For example, reinforce the value of monitoring one’s own progress and changing course as needed by modeling a think-aloud: When I am constructing the coordinate plane, I ask myself questions. Do I have enough grid lines to show all the points I need to plot? Did I label the grid lines correctly? Is there another interval length that might work better?”

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

• Module 4, Topic B, Lesson 10: Multiplication and Division Expressions from Real-World Situations, Learn, Problem 1, students represent, write, and interpret algebraic expressions symbolically to represent real-world situations. “The number of cups of seltzer water in a punch recipe is 1\frac{1}{2} times the number of cups of lemonade. Complete the table.” The table provided shows the number of cups of lemonade and the number of cups of seltzer water. Teacher margin note states, “Ask the following questions to promote MP2: How does this expression represent this situation? What does this coefficient mean in this situation? Does the expression you wrote make sense mathematically?”

• Module 5, Topic A, Lesson 4: Areas of Triangles in Real-World Situations, Learn, Problem 5 students attend to the meaning of quantities to solve real-world problems involving areas of triangles. “Mr. Perez builds a fence to enclose a play area for his dog. The enclosed area is in the shape of a triangle with a base of 48 meters and a height of 32 meters. a. What is the size of the play area for Mr. Perez’s dog in square meters? b. Ryan says the play area can be a right triangle, an acute triangle, or an obtuse triangle. Do you agree? Explain.” Teacher margin note states, “Ask the following questions to promote MP2: What does the situation tell you about the base and the height of the triangle? What does the area of the triangle mean in this situation? Does the solution you found make sense mathematically?”

• Module 6, Topic B, Lesson 7: Using the Mean to Describe the Center, Learn, Problem 2, students work to understand relationships between the problem scenario and the mathematical representation. “Julie asks 10 of her classmates how many mini tacos they got on taco day. She draws the following picture to represent the data set she collected. a. Create a dot plot to represent this data set. b. Scott always chooses the most common value to estimate the center of a data distribution. What would Scott choose as the center of Julie’s data distribution? c. Do you think Scott’s choice represents a typical value in Julie’s data distribution? Why?” Teacher margin note states, “Ask the following questions to promote MP2: What does each cube represent in this context? What does equal sharing tell you about the center of a data distribution?”

The materials reviewed for Eureka Math² Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

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

• Module 1, Topic E, Lesson 25: Finding the Whole, Exit Ticket, students construct viable arguments as they calculate the whole when given a part and a percent. “A team has raised $300.00 for new uniforms, which is 60% of the total amount of money they need to raise. What is the total amount of money the team needs to raise? Justify your answer.”

• Module 2, Topic D, Lesson 13: Decimal Addition and Subtraction, Practice, Problem 13, students create their own conjectures as they compare sums of decimals. “The price of frozen yogurt is based on the weight of the yogurt. The table shows the prices for different weights of yogurt. Riley’s yogurt weighs 8.641 ounces. Sasha’s yogurt weighs 5.77 ounces. To pay a lower price, should Riley and Sasha pay for their yogurt together or separately? Justify your answer.”

• Module 4, Topic A, Lesson 3: Exploring Exponents, Learn, students explain their thinking as they compare repeated addition and multiplication. “Play the Money, Money, Money video. In the video, students are presented with two options for acquiring money. Option 1: Receive $2.00 on day 1. Each day for the next 14 days, receive $2.00 more than the amount of money received the previous day. Option 2: Receive $0.02 on day 1. Each day for the next 14 days, receive double the amount of money received the previous day. Circulate and make sure students understand the two options. Encourage students to take their time making their decisions. Ask the following questions to promote students’ thinking as needed: How does the amount of money in option 1 grow? How does the amount of money in option 2 grow? Is your choice a guess or do you know for sure? How do you know for sure? Why does your strategy for choosing the better option work? Convince your partner.” 

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

• Module 1, Topic E, Lesson 24: Finding a Part, Learn, Problem 2, students critique the reasoning of others as they calculate percent of a number. “Lisa, Julie, and Toby each calculated 16% of 40 but in different ways. Use their work to answer parts (a)–(g). a. Explain how Lisa calculated 16% of 40. b. Use Lisa’s method to calculate 31% of 50. c. What is another percent of 40 you could calculate by using Lisa’s method? Calculate that percent. d. Explain how Julie calculated 16% of 40. f. How are Lisa’s and Toby’s methods similar? How are they different?” Teachers are prompted to ask, “What parts of Toby’s method do you question? When do you think Lisa’s method works? Can you find a situation other than 10% of 40 where Noah’s method in problem 3 works?”

• Module 2, Topic F, Lesson 22: Dividing a Decimal by a Decimal Greater than 1, Learn, Problem 11, students perform error analysis of others as they analyze division errors, “Toby calculates 0.084\div2.4 as 0.35 by doing the following work. Explain Toby’s mistake.” Toby’s standard algorithm division is shown.

• Module 5, Topic B, Lesson 5: Perimeter and Area in the Coordinate Plane, Practice 5, Problem 6, students critique the reasoning of others as the find the area of a rectangular figure on the coordinate plane. “Jada says that to find the perimeter of polygon ALFEKB, you should add the perimeters of rectangles JLFE and AJKB. She says that the perimeter of polygon ALFEKB is 48 units because 22+26=48 a. Explain Jada’s mistake. b. What is the perimeter of polygon ALFEKB? Explain.”

The materials reviewed for Eureka Math² Grade 6 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

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

• Module 1, Topic D, Lesson 21: Solving Multi-Step Rate Problems, Learn, Problem 3, students solve rate problems using math models. "Choose a method of travel from the list. The distance from Earth to the moon is about 238,855 miles. How many weeks would it take to travel from Earth to the moon? Assume the chosen traveler or mode of transportation moves at the given constant speed. As needed, assume there is actually a road from Earth to the moon.” Teacher prompts include, “What math model can you draw to help you understand the moon problem? What key ideas in the moon problem do you need to make sure you include in your model? How can you improve your model to better represent the moon problem?”

• Module 3, Topic C, Lesson 14: Modeling with the Coordinate Plane, Learn, Problem 2, students create a table and graph to model the elevation of a vehicle over time as it descends toward the ocean floor. “The Mariana Trench is in the Pacific Ocean and is the deepest oceanic trench on Earth. An oceanic trench is like a valley on the ocean floor. The deepest place in the bottom of the Mariana Trench is called Challenger Deep. It is farther from sea level than the summit of Mt. Everest is from sea level. How long do you think it would take a vehicle to descend from sea level to Challenger Deep? Record your data in the table. Then graph your time and elevation data for the dive.” Teachers' prompts include, “How could you improve your graph to better represent the time it takes the vehicle to dive to the bottom of the trench? What assumptions could you make to help you approximate the time it takes the vehicle to dive to the bottom of the trench? What do you wish you knew in order to determine how long it would take the vehicle to reach the bottom of the trench?”

• Module 4, Module Assessment 2, Item 8, students use defined variables and equations to represent given problems. “Toy A costs $6.00 more than toy B. What is the total cost of toy A and toy B? Can each pair of defined variables and equations be used to represent this problem? Select Yes or No.” Students select yes or no from a table with the following statements, “Let b represent the cost of toy B in dollars. Let c represent the total cost of toy A and toy B in dollars. c=b+(b+6) Let b represent the cost of toy B in dollars. Let c represent the total cost of toy A and toy B in dollars. c=b+(b+6) Let a represent the cost of toy A in dollars. Let c represent the total cost of toy A and toy B in dollars. c=a+(a+6) Let a represent the cost of toy A in dollars. Let c represent the total cost of toy A and toy B in dollars. c=a+(a-6).”

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

• Module 1, Topic C, Lesson 12: Multiple Ratio Relationships, Learn, students recognize both the insight to be gained from different tools/strategies and their limitations. “Here are tables with ratios equivalent to the ratios you created for mixture 1 and mixture 2 to be the same shade of paint. Use the sketch tool to circle rows in the tables that show the same number of parts blue paint.” Teachers are prompted to ask, “What tools can help you compare two paint mixtures? Which tool would be the most efficient to determine whether two paint mixtures are the same shade? Why?” 

• Module 4, Topic E, Lesson 23: Graphs of Ratio Relationships, Learn, students use appropriate tools strategically and compare their effectiveness when they choose from a table, a graph, or an equation to help find total earnings in a real-world situation. “You decide to buy gasoline at the rate of $2.50 per gallon. Complete the table.” Students are shown a table with Number of Gallons of Gasoline g, Calculation, and Total Cost c in dollars. Teachers are prompted to ask, “What tools could help you solve this problem? How can you estimate your total earnings? Does your estimate sound reasonable? Which tool would be the most helpful to find your exact earnings? Why?”

• Module 6, Topic A , Lesson 5: Comparing Data Displays, Learn, Problems 3, students use appropriate tools strategically when they choose among bar graphs, frequency histograms, and relative frequency histograms to display different data sets. “For problems 1–5, determine whether a bar graph or a histogram is appropriate for summarizing the data set. The amount of time between when students wake up and when they arrive at school.” Teachers are prompted to ask, “What graph could help you find the most common zip code of homes in a city? Which graph would be most helpful to determine the percent of club members who wear a certain hat size? How can you estimate the percent of breakfast cereals that have fewer than 125 calories per serving? Does your estimate sound reasonable?”

The materials reviewed for Eureka Math² 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. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. Margin Notes, Language Support, provide suggestions for student-to-student discourse, support of new and familiar content-specific terminology or academic language, or support of multiple-meaning words.

Materials provide intentional development of MP6 to meet its full intent in connection to grade-level content. Examples include:

• Module 1, Topic A, Lesson 3: Ratios and Tape Diagrams, Practice, Problem 3, students attend to precision as they consider the order of numbers in ratios and use correct ratio language and notations to describe the relationship between two quantities. “Tyler has 24 quarters and 6 dimes. a. Write and explain the meaning of two ratios that could represent this situation. b. Tyler puts all the coins in bags. Every bag has the same number of quarters and the same number of dimes. How many bags of coins can Tyler make? How many quarters and how many dimes are in each bag? c. Use ratio language to describe the relationship between the number of quarters and the number of dimes in each bag from part (b).”

• Module 2, Topic A, Lesson 4: The Least Common Multiple, Learn, Problem 6, students attend to precision as they identify the least common multiple of two numbers by using prime factorization. “Consider the numbers 11 and 12. a. What is the least common multiple of 11 and 12? b. Write the prime factorizations of 11 and 12.” Teachers are prompted to ask, “When using the prime factorizations of two numbers to find the least common multiple, what steps do you need to be extra careful with? Why? Where is it easy to make mistakes when finding the least common multiple? In the last lesson, we used prime factorization to find the greatest common factor of two numbers. What is the greatest common factor of 6 and 10? How do you know?”

• Module 5, Topic C, Lesson 10: Discovering Nets of Solids, Learn, Learn Problem 1, students attend to precision as they match nets with the solids. “Determine which two-dimensional figure can be folded to form each solid in the table. Then name the solid.” Six nets are shown. Teachers are prompted to ask, “How can we represent this solid by using a two-dimensional figure? Where is it easy to make mistakes when matching a two-dimensional figure with the solid it creates when folded?”

The instructional materials attend to the specialized language of mathematics. Examples include:

• Module 2, Topic A, Lesson 3: The Greatest Common Factor, Learn, Teacher Note, “The act of speaking the whole term greatest common factor, rather than using the acronym GCF, supports students with this new term. In mathematics, there are other acronyms that refer to the same concept, such as HCF (highest common factor), LCF (largest common factor), GCD (greatest common divisor), and LCD (largest common divisor). When only the acronym is used to refer to this new concept, a common misconception for students is to confuse the greatest common factor with the least common multiple (LCM).”

• Module 4, Topic A, Lesson 5: Exploring Order of Operations, Learn, Teacher Note, “The curriculum continues to build the definition of term by showing that a term can be the product of two numbers. Students formally define term when variables are introduced in topic B. At this point, informally model correct use of term as you speak to students.”

• Module 6, Topic B, Lesson 10: The Mean Absolute Deviation, Learn, Language Support, “Consider displaying an anchor chart to help students keep track of the terms associated with the spread of a data distribution. Variability is how much the data values in a data set differ from one another. Range is the difference between the maximum and minimum values in a data set. Mean absolute deviation is the average distance between a data value and the mean of a data distribution.”

The materials reviewed for Eureka Math² Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

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

• Module 1, Topic A, Lesson 8: Addition Patterns in Ratio Relationships, Learn, Problem 1, students look for patterns or structures to make generalizations as they examine addition patterns in tables and graphs of ratio relationships. “The graph represents the ratio relationship between the number of cups of orange juice and the number of cups of pineapple juice in batches of a citrus punch. Use the graph to complete the ratio table.” Teachers are prompted to ask, “How are the ratio table and graph of the ratio relationship related? How can that help you find equivalent ratios? What is another way you can complete a ratio table to help you find equivalent ratios? How can what you know about equivalent ratios help you find points on the graph of a ratio relationship?”

• Module 3, Topic B, Lesson 6: Ordering Rational Numbers, Learn, Problem 4, students make use of structure to order variables using a number line. “Points A, B, C, and D each represent a different rational number. Use the number line to order A, B, C, and D from least to greatest. Explain how visualizing these numbers on a number line helps you know that your answer is correct.” Students are provided a number line where B and A are on ticks of the number line and C and D are on spaces between the ticks. 

• Module 6, Topic C, Lesson 14: Using a Box Plot to Summarize a Distribution, Learn, Problem 3, students analyze a problem and look for more than one approach as they create a box plot from a given data set. “The gardening club plants 10 cherry tomato plants. They record the following numbers of cherry tomatoes harvested from the plants. 38, 20, 31, 25, 35, 42, 21, 44, 40, 21 a. Create a box plot to represent the data distribution. b. Use the box plot to describe the typical number of cherry tomatoes harvested from these plants. c. Use the box plot to describe and interpret the spread of the numbers of cherry tomatoes harvested from these plants.” Teachers are prompted to ask, “What is another way you can organize the numbers of cherry tomatoes harvested that will help you find the five-number summary? How does what you know about the lengths of the box and box plot help you describe the spread of the data distribution?”

Materials provide intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning to make generalizations and build a deeper understanding of grade-level math concepts. Examples include:

• Module 2, Topic A, Lesson 1: Factors and Multiples, Learn, students define, model, and explain as they find common factors of two numbers using a visual model. “What mathematical question might the person in this picture need to answer to tile this region?” Students are shown a picture of someone using square tiles to tile a countertop. Teachers are prompted to ask, “What patterns do you notice when you find squares to tile the rectangle? What is the same about how you do each example? When you look for squares that can cover a rectangle, does anything repeat? How can that help you find the side lengths of the squares that can cover a rectangle more efficiently?”

• Module 3, Topic C, Lesson 12: Reflections in the Coordinate Plane, Learn students create, define, and explain reflections as they graph points and their reflections across the axes. They recognize when the coordinates of two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. “Point B is a reflection of Point A across the y-axis. How is the distance from the y-axis to Point A related to the distance of the y-axis to Point B?” Teachers are prompted to ask, “What patterns do you notice when you plot a point and its reflection across an axis? How do you know that the reflection of (a,b) across the x-axis will always be (a,−b)?”

• Module 5, Topic D, Lesson 17: Problem Solving with Volume, Learn, Problem 3, students notice repeated calculations to understand algorithms and make generalizations as they calculate and compare volumes of right rectangular prisms when one, two, or three edge lengths are each multiplied by a number. “Prism P is a right rectangular prism with edge lengths l, w, and h. a. Write an algebraic expression to represent the volume of prism P. b. Prism S is a right rectangular prism. The volume of prism S is 3 times the volume of prism P. Which of the following could represent the edge lengths of prism S? Choose all that apply. A. 3l, 3w,3h B. 3l,2,h C. 3l,w,3h D. l,w,3h E. l,3w,h F. l,\frac{1}{3}w,9h c. Prism T is a right rectangular prism. Each edge length of prism T is 3 times the corresponding edge length of prism P. Write an algebraic expression to represent the volume of prism T by using as few factors as possible.” Teachers are prompted to ask, “What patterns did you notice when you compared the edge lengths and volumes of the prisms to the edge lengths and volumes of prisms A and B? What is the same about how the volume was affected when edge lengths were each multiplied by 2 and when edge lengths were each multiplied by 12?”