6th Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, and Module Assessments. According to the Implementation Guide, “Exit Tickets are not graded. They are paper based so that you can quickly review and sort them. Typical Topic Quizzes consist of 4-6 items that assess proficiency with the major concepts from the topic. You may find it useful to grade Topic Quizzes. Typical Module Assessments consist of 6-10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Module Assessments represent the most important content taught in the module. These assessments use a variety of question types, such as constructed response, multiple select, multiple choice, single answer, and multi-part. There are two analogous versions of each Module Assessment available digitally. Analogous versions target the same material at the same level of cognitive complexity.” Examples of summative Module Assessments items that assess grade-level standards include:

• Module 1, Module Assessment 1, Item 3, “Scott has 4 red shirts, 5 blue shirts, 2 yellow shirts, and 3 orange shirts. Part A: Fill in the blank to complete the statement. The ratio of the number of red shirts to the number of orange shirts is ____. Part B: What could the ratio 5 : 14 represent?” (6.RP.1)

• Module 3, Module Assessment 1, Item 6, “A bank account has a balance of −150 dollars. Part A: Explain what the number −150 means in this situation. Part B: Explain what \lvert-150\rvert means in this situation.” (6.NS.7)

• Module 4, Module Assessment 1, Item 4, “Select all the numbers that make 2x+4<14 a true number sentence.” Answers provided, “7, 5, 3, 1, 0.” (6.EE.5)

• Module 5, Module Assessment 1, Item 3, “Consider the right rectangular prism (prism shown has dimensions 5\frac{1}{2}in, 6in, 4\frac{3}{4}in). Part A: What is the volume of the right rectangular prism in cubic inches? Part B: Consider the cube with an edge length of \frac{1]{4} inch. How many cubes pack the right rectangular prism?” Answers provided, “156\frac{3}{4}, 456, 627 and 10,032.” (6.G.2)

• Module 6, Module Assessment 2, Item 4, “Determine whether each question is a statistical question. Explain your answer. Part A: How long is your favorite movie? Part B: What is the typical length of a song on the radio?” (6.SP.1)

The materials reviewed for Eureka Math² Grade 6 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Each lesson consists of four sections (Fluency, Launch, Learn, and Land) that provide extensive work with grade-level problems and to meet the full intent of grade-level standards. The Fluency section provides opportunities for students to practice previously learned content and activates students’ prior knowledge to prepare for new learning. Launch activities build context for learning goals. Learn activities present new learning through a series of learning segments. During the Land section, teachers facilitate a discussion to address key questions related to the learning goal. Practice pages can be assigned to students for additional practice with problems that range from simple to complex.

Instructional materials engage all students in extensive work with grade-level problems. Examples include:

• Module 1, Topic A, Lesson 2: Introduction to Ratios, Fluency, Problem 4, students multiply or divide by using multiplicative comparisons to prepare for working with ratio relationships, “40 is 4 times as large as what whole number?” Launch, students watch Part 1 of the video “Unfair Tokens” then use multiplicative comparison language to compare the number of tokens each kid receives. “How can we compare the number of tokens the girl receives to the number of tokens the boy receives?” Learn, students begin to use ratio language to compare the number of tokens each child received, “Have students think back to Part 1 of the ‘Unfair Tokens’ video. Display a picture of three cups. Ask the following question. Suppose each cup has 8 tokens in it. How many tokens would each child have? Ask students to choose a different number of tokens that a cup might hold. Then have them determine the number of tokens each child would have. Have students write their examples on a personal whiteboard and hold it up for you to see. Write the word ratios on the board and record several students’ answers by using ratio notation as shown.” Classwork, Problem 3, students select statements that apply to a given relationship, “To make light blue paint, Ryan mixes 2 ounces of white paint with 6 ounces of blue paint. For parts (a)–(e), fill in the blanks. a. A ratio that relates the number of ounces of white paint to the number of ounces of blue paint is ____. b. A ratio that relates the number of ounces of blue paint to the number of ounces of white paint is ____. c. For every ___ ounces of white paint, Ryan mixes 6 ounces of blue paint. d. For every 1 ounce of white paint, Ryan mixes ___ounces of blue paint. d. Ryan uses ___ times as much blue paint as white paint.” Land, students write ratios that relate two quantities as an ordered pair of numbers, “When is it more practical to use ratio language instead of multiplicative comparison language? Give an example of a relationship between two quantities by using ratio language. Describe the meaning of ratio in your own words. What are two quantities that you would love to have in a ratio of 5:2 but would not like to have in a ratio of 2:5?” Exit Ticket, students are given a picture of five blue paint cans and four red paint cans and asked a series of questions regarding the ratio. “a. A ratio that relates the number of cans of blue paint to the number of cans of red paint is___. b. A ratio that relates the number of cans of red paint to the number of cans of blue paint is___. c. There are ___ times as many cans of blue paint as cans of red paint. d. For every ___ cans of blue paint, there are ___ cans of red paint.” Practice, Problem 4, “At an animal shelter, 9 dogs and 15 cats are ready for adoption. Fill in the blanks to make the statement true. a. For every ____ dogs, there are 15 cats. b. For every 3 dogs, there are ____ cats. c. There are ____ times as many cats as dogs.” Students engage in extensive work with grade-level problems of 6.RP.1 (Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities).

• Module 2, Topic C, Lesson 9: Dividing Fractions by Using Tape Diagrams, Fluency, Problem 4, Students divide a whole number by a unit fraction to prepare for dividing fractions, “4\div\frac{1}{4}” Launch, students sort cards into groups of expressions that have quotients less than 1, equal to 1, and greater than 1 to reason about quotient size in fraction division, “The expressions \frac{6}{4}\div\frac{4}{3} and \frac{5}{6}\div\frac{4}{3} are both in Pile A. Which expression has a greater value? Explain.” Learn, students determine quotients by using tape diagrams and unknown factor equations, “How do you know the tape diagram represents the question \frac{2}{3} is \frac{1}{4} of what number?” Students then solve real-world problems involving fraction division. Exit TIcket, students find the value of a division expression by writing an unknown factor equation and draw a tape diagram to solve, “Consider \frac{2}{3}\div\frac{4}{5}. a. Write \frac{2}{3}\div\frac{4}{5} as an unknown factor equation. b. Draw a tape diagram that represents the unknown factor equation from part (a). c. What is the value of one unit in the tape diagram from part (b)? d. What is \frac{2}{3}\div\frac{4}{5}?” Practice, Problem 12, “It takes \frac{5}{6} gallons of water to fill \frac{1}{3} of a bucket. How many gallons of water fill the whole bucket? Draw a diagram to justify your solution.” Students engage in extensive work with grade-level problems of 6.NS.1 (Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions).

• Module 5, Topic D, Lesson 15: Exploring Volume, Fluency, Problem 5, students multiply and divide fractions to prepare for determining the number of cubes with a fractional edge length that pack a right rectangular prism, “\frac{6}{5}\cdot4\cdot10.” Launch, students engage in a digital platform to determine dimensions of a right rectangular prism with a volume of 48 cubic units, “How many unit cubes are needed to compose the first layer of the right rectangular prism?” Learn, students use an interactive cube to explore various fractional edge lengths that pack a right rectangular prism. The relationship between the volume of the cube and the volume of the right rectangular prism is explored, “Does the volume of the unit cube change when it is broken into cubes with fractional edge lengths? Why?” Exit Ticket, students are given a right rectangular prism and a cube, both with fractional dimensions to calculate the volume and the number of cubes to pack the prism, “Consider the right rectangular prism and cube. Figures are not drawn to scale. a. How many of these cubes pack the right rectangular prism? Explain. b. What is the volume of the cube? Show your work. c. What is the volume of the rectangular prism? Show your work.” The rectangular prism shown has dimensions 1\frac{1}{2} in. by 1 in. by 3\frac{3}{4} in and the cube has \frac{1}{4} in side lengths. Practice, Problem 3, “How many cubes with an edge length of \frac{1}{4} unit pack one unit cubes?” Students engage in extensive work with grade-level problems 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).

Instructional materials provide opportunities for all students to engage with the full intent of grade-level standards. Examples include: 

• Module 3, Topic A, Lesson 1: Positive and Negative Numbers, Fluency, Problem 3, students identify numbers on number lines to prepare for plotting positive and negative numbers on number lines. “Use the number lines to answer the questions. What number is located 10 units to the right of point B?” Launch, students describe temperatures above and below zero degrees. Students are given a picture of two thermometers side by side. One shows 10 degrees and the other shows -10 degrees, “How are the two temperatures alike? How are they different?” Learn, students learn the definition of a negative number and look at various situations that have positive and negative contexts, “A negative number is a number that is less than zero. The negative sign in front of a number indicates that the number is negative. For example, −10 is negative ten. If numbers less than 0, such as −10, are called negative numbers, what do you think numbers that are greater than 0, such as 10, are called? A positive number is a number that is greater than zero. Do you think the number 0 is positive or negative? Why? Have you seen or heard about negative numbers used in other situations? Give some examples.” Classwork, Problem 2, students write integers for written situations, “Write a positive number, a negative number, or 0 to represent the temperature given in each statement. a. Water freezes at 0\degreeC. b. The temperature of the human body is 37 degrees above 0°C. c. The U.S. Food and Drug Administration recommends that freezer temperatures be set at 18 degrees below 0\degreeC.” Classwork, Problem 2, students plot positive and negative numbers on horizontal and vertical number lines, “Plot and label a point on the vertical number line that represents each temperature. a. 7\degreeC b. −4\degreeC c. A temperature that is 2 degrees warmer than 7\degreeC d. A temperature that is 2 degrees colder than −4\degreeC.” Land, students represent quantities in real-world situations by using positive and negative numbers, “How are positive and negative numbers alike? How are they different? How are positive and negative numbers useful in describing real-world quantities? Give an example.” Exit Ticket, “Blake dives into a pool from a diving board 6 feet above the water’s surface. At the deepest part of his dive, Blake is 10 feet below the water’s surface. a. Write a positive number or negative number to represent Blake’s location above the water’s surface before he dives. b. Write a positive number or negative number to represent Blake’s location below the water’s surface at the deepest part of his dive. What does 0 represent in this situation?” Practice, Problem 5, “For parts (a) - (d), write a positive number, a negative number, or 0 to represent each situation. Plot and label a point on the number line that represents each situation. a. A submarine is 400 feet below sea level. b. A bird is 200 feet above sea level. c. A boat is at sea level. d. What does the number 0 represent in parts (a) - (c).” The materials meet the full intent of 6.NS.5 (Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation).

• Module 4, Topic C, Lesson 16: Equivalent Algebraic Expressions, Fluency, Problem 6, students apply the distributive property to prepare for writing equivalent expressions by distributing and combining like terms, “Use the distributive property to write an equivalent expression as a sum or difference. \frac{1}{6}(3+5x).” Launch, students view a picture of four black and eight blue pieces of pipe and write an expression to represent the situation, “Adesh is replacing the pipes in his sprinkler system. The length of each black pipe is x feet and the length of each blue pipe is y feet. What is the total length of all the pipes shown? Write an algebraic expression to represent your answer.” Learn, students combine like terms in algebraic expressions with addition and subtraction, “Display the following expression and ask students to copy it on their whiteboards. 5(2x+3)+2(4x+1). What is the value of this expression when x = 2? Show your thinking. How can we write this expression with fewer terms? Show how you know. What is the value of 18⁢x+17 when x=2? Show how you know. Both expressions evaluate to 53 when 2 is substituted for x. Which expression is easier to evaluate? Explain.” Classwork, Problem 2, students distributive and combine like terms to create equivalent expressions, “x+5(4x-1).” Classwork Problem 5, “Which expressions are equivalent to 3x+6(2x-4)? Choose all that apply. A. 3x+12x+24 B. 3x+12x-24 C. 3x+12x-4 D. 15x-4 E. 15x-24.” Land, students discuss using the properties of operations and combining like terms to create equivalent expressions, Why do we combine like terms in algebraic expressions? What kinds of real-world situations can be represented by expressions that involve the distributive property? Provide an example.” Exit Ticket, Problem 1, students write two expressions that could be used to describe a real life situation, “Sasha starts the week with $50. Each week, she mows the lawn and does x chores. She earns $10 for mowing the lawn and $3 for each chore she does. Write two expressions that each represent the total amount of money Sasha earns by the end of 4 weeks.” Practice, Problem 4, “For problems 1-4, distribute and combine like terms to write an equivalent expression. Show your work. 8(x+3)+5(2x+6).” The materials meet the full intent of 6.EE.4 (Identify when two expressions are equivalent [i.e., when the two expressions name the same number regardless of which value is substituted into them]).

• Module 6, Topic C, Lesson 16: Interpreting Box Plots, Fluency, students practice finding the five-number summary from a box plot to prepare for comparing two data distributions displayed in box plots. Students are shown a box plot for Beeswax Candles Sold and find the minimum value, first quartile, median, third quartile, and maximum value. Launch, Problem 1, students write statistical questions that can be answered using summary measures, “The chief justice of the US Supreme Court is the lead judge of the court. Someone serving as chief justice can remain in this position for the rest of their life. Some people think this means that the chief justice typically serves for a very long time. The table shows the numbers of years that past US Supreme Court chief justices served, rounded up to a whole year. The information shown is as of the year 2020. The box plot summarizes the numbers of years that the chief justices served. a. Use the box plot to find the five-number summary, range, and interquartile range of the data distribution. b. Is the data distribution approximately symmetric, skewed to the left, or skewed to the right? Explain how you know.” Learn, Problem 5, students compare the spread and the shape of two data distributions by using dot, “The dot plots show the average amount of precipitation each month of the year in two cities. Use the dot plots for parts (a)–(d).” A dot plot of Weather in City A and B is provided with average Monthly Precipitation (inches) shown. “a) How many data values are in each dot plot? What does each data value represent? b) Which city looks like it has more variability in the average monthly amounts of precipitation? Why? c) Find and interpret the interquartile range of each data distribution. d) How do the interquartile ranges of the distributions compare? What does this tell you?” Land, Summarize a data distribution by using a box plot, the median, and the interquartile range. “How can box plots, the median, and the interquartile range help us answer statistical questions? How do box plots help us to compare two data distributions? What are the disadvantages of using box plots to compare two data distributions?” Exit Ticket, “Two classes took the same test. The box plots summarize the test scores for the classes. a. Compare the median score for the classes. What does this tell you about the typical scores for the classes? Compare the interquartile ranges for the two classes. What does this tell you about the variability of the scores for the classes? c. Which class scored better on the test? Explain.” Practice, Problem 4, “The box plots summarize the numbers of years that past US Supreme Court associate justices and chief justices served, rounded up to a whole year. The information shown is as of the year 2020. a. Describe the shape of each distribution. b. Compare the centers of the distributions. What does this tell you? c. Compare the spreads of the distributions. What does this tell you?” The materials meet the full intent 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) and 6.SP.4 (Display numerical data in plots on a number line, including dot plots, histograms, and box plots).

The materials reviewed for Eureka Math² Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade.

• There are 6 instructional modules, of which 5 modules address major work of the grade or supporting work connected to major work of the grade, approximately 83%.

• There are 132 instructional lessons, of which 94 lessons address major work of the grade or supporting work connected to major work of the grade, approximately 71%.

• There are 166 instructional days, of which 122 address major work of the grade or supporting work connected to the major work of the grade, approximately 73%. Instructional days include 132 instructional lessons, 28 topic assessments, and 6 module assessments.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work and supporting work connected to major work. As a result, approximately 71% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² Grade 6 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Each lesson contains Achievement Descriptors that provide descriptions and about what the students should be able to do after completing the lesson and lists standards. Materials do not provide information about connections between standards in lessons.

Materials connect learning of supporting and major work to enhance focus on major work. Examples include:

• Module 2, Topic B, Lesson 8: Dividing Fractions by Making Common Denominators, Learn Problem 9, students divide fractions using a common unit, “Consider \frac{3}{5}\div\frac{2}{3}. a. Is the quotient greater than 1 or less than 1? Explain. b. How can we rewrite \frac{3}{5}\div\frac{2}{3} so the fractions have a common denominator? c. Draw a tape diagram to model \frac{3}{5}\div\frac{2}{3}.” This connects the supporting work of 6.NS.4 (Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12) to the major work of 6.NS.1 (Interpret and compute quotients of fractions and solve word problems involving division of fractions).

• Module 4, Topic C, Lesson 12: Applying Properties to Multiplication and Division Expressions, Learn, Problem 3, students write and evaluate algebraic expressions by using multiplication and division, “Jada has 24 boxes of toys with t toys in each box. She gives one-third of the boxes to a charity. Write an expression to represent the total number of toys that Jada gives to the charity. If each box has 9 toys, how many toys does Jada give to the charity?” This connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers).

• Module 5, Topic B, Lesson 6: Problem Solving with Area in the Coordinate Plane, Learn, Problem 3, students determine the areas of triangles that are not right triangles and are graphed in the coordinate plane, “Write and evaluate two numerical expressions to find the area of the trapezoid.” Students are shown two trapezoids graphed on the coordinate plane with areas of 45 square units. This connects the supporting work of 6.G.1 (Find the area of right 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) to the major work of 6.EE.3 (Apply the properties of operations to generate equivalent expressions).

The materials reviewed for Eureka Math² Grade 6 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Each lesson contains Achievement Descriptors that provide descriptions and about what the students should be able to do after completing the lesson and lists standards. Materials do not provide information about connections between standards in lessons.

Materials provide connections from major work to major work throughout the grade-level when appropriate. Examples include.

• Module 4, Topic B, Lesson 9: Addition and Subtraction Expressions from Real-World Situations, Exit Ticket, students precisely define variables and write algebraic expressions involving addition and subtraction to represent real-world situations, “Tara has 4 fewer goldfish than Julie. a. Define a variable that represents one of the quantities in this situation. b. Use the variable from part (a) to write an algebraic expression that represents the other quantity in this situation.” This connects the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities).

• Module 4, Topic E, Lesson 22: Relationships Between Two Variables, Learn, Problem 7, students write two equations about ratios to represent a situation, “A lemonade recipe calls for 3 tablespoons of lemon juice for every 1 cup of water. a. Complete the table. b. Let j represent the number of tablespoons of lemon juice. Use j to write an expression that represents the number of cups of water. c. Let w represent the number of cups of water. Write an equation that represents the ratio relationship between the number of cups of water and the number of tablespoons of lemon juice. Use w and the expression you wrote in part (b). Identify the independent variable and the dependent variable. d. Complete the table. e. Use w to write an expression that represents the number of tablespoons of lemon juice. f. Write an equation that represents the ratio relationship between the number of tablespoons of lemon juice and the number of cups of water. Use j and the expression you wrote in part (e). Identify the independent variable and the dependent variable.” This connects the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the major work of 6.EE.C (Represent and analyze quantitative relationships between dependent and independent variables).

• Module 5, Topic C, Lesson 13: Surface Area in Real World Situations, Practice, Problem 4, students write and solve an equation to find the surface area of a pyramid and solve a related unit rate problem using their solution, “A company makes play tents for children. The tent is shaped like a square pyramid with the measurements shown. Fabric is used to enclose the entire tent, including the bottom. a. How many square feet of fabric are needed to enclose the tent? b. The cost of fabric to make one tent is $0.40 per square foot. What is the total cost of the fabric to make one tent?” An image of a pyramid is shown with a square base having side lengths of 3\frac{1}{2ft. and a height of 5 ft. This connects the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions) to the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems).

Materials provide connections from supporting work to supporting work throughout the grade-level when appropriate. Examples include:

• Module 5, Topic B, Lesson 7: Areas of Trapezoids and Other Polygons, Learn, Problem 1 students decompose polygons and identify unknown measurements that are used to find the area of the polygon, “Circulate as students work, asking the following questions: What strategy are you using to determine the area of that figure? What calculations do you need to make to use that strategy? Do you have the information you need to use that strategy? Determine the area of the trapezoid. Show your work.” Students are shown a trapezoid with bases 0.8 in. and 2.4 in. and a height of 1 in. This connects the supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples) to the supporting work of 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume).

• Module 6, Topic A, Lesson 3: Creating a Dot Plot, Learn, Problem 1, students create a dot plot and describe data distributions. “Mr. Sharma is curious whether his sixth graders get enough sleep during the school week. He decides to explore the statistical question, How many hours do my sixth graders typically sleep on a school night? He asks all 15 students in his class how many hours of sleep they get on a school night. He collects the following responses. 7, 8, 8, 9, 9, 9, 12, 7, 10, 10, 11, 9, 8, 6, 9 Create a dot plot by using this data set.” Teachers guide the class in describing the data distribution by asking questions, “What number of hours of sleep can we use to describe the center of the data distribution? Why? How many students get less than 7 hours of sleep? One student gets 11 hours of sleep. How does this compare with other students in this class? Think back to Mr. Sharma’s statistical question. How many hours do his sixth graders typically sleep on a school night? How do you know?” This connects the supporting work of 6.SP.B (Summarize and describe distributions) to the supporting work of 6.SP.A (Developing understanding of statistical variability).

• Module 6, Topic B, Lesson 8: The Mean as the Balance Point, Practice, Problem 3, students find the distance from each box to the balance point and notice that the sum of the distances to the left of the balance point is equal to the sum of the distances to the right of the balance point, “The dot plot shows the prices in dollars of 5 different game downloads. The price for a sixth game is missing. The mean price for all 6 games is $3.80. Use the balancing process to determine the price of the sixth game. Explain how you got your answer.” A dot plot is shown of Game Downloads with decimal prices in dollars. This connects the supporting work of 6.SP.B (Summarize and describe distributions) to the supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples).

The materials reviewed for Eureka Math² Grade 6 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

Each Module Overview contains Before This Module and After This Module looking forward and back respectively, to reveal coherence across modules and grade levels. The Topic Overview includes information about how learning connects to previous or future content. Some Teacher Notes within lessons enhance mathematical reasoning by providing connections/explanations to prior and future concepts.

Content from future grades is identified and related to grade-level work. Examples include:

• Module 2: Operations with Fractions and Multi-Digit Numbers, Module Overview, After This Module, Grade 6 Module 4, “Students revisit the greatest common factor in module 4 when they use the distributive property and apply their understanding of factors to generate equivalent expressions. Later in module 4, students apply their understanding of fraction division, multi-digit number division, and fraction and decimal operations when they solve one-step equations that include rational number constants and coefficients.”

• Module 3, Topic B: Ordering and Magnitude, Topic Overview, “Students apply their understanding of ordering and magnitude in topics C and D when they extend their work with number lines to the coordinate plane and in module 5 when they study coordinate geometry.”

• Module 4, Topic A, Lesson 1: Expressions with Addition and Subtraction, Learn, Teacher Note, “In topic B, the word term will be defined as each single number, variable, or product of numbers and variables in an expression. This lesson introduces an informal understanding of terms in numerical expressions to prepare students to identify terms in algebraic expressions. In later grades, students understand that subtracting a number is equivalent to adding the opposite of that number. a-b=a+(-b) Once students have that understanding, they identify the terms of the expression a-b as a and -b.”

Materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples include:

• Module 1: Ratio, Rates and, and Percents, Module Overview, Before this Module, Grade 4, Module 2 and Grade 5, Module 6, “In grade 4, students solve problems involving multiplicative comparisons, such as Blake has 4 times as many stickers as Adesh. This prior work provides a foundation for students’ understanding of ratios as multiplicative comparisons of two numbers. In grade 5, students work with the first quadrant of the coordinate plane as they plot points to represent ordered pairs of numbers.”

• Module 2, Topic A: Factors, Multiples, and Divisibility, Topic Overview, “In grade 5, students learn how to divide a whole number by a unit fraction and a unit fraction by a whole number. In grade 6 module 2 topic B, students extend this understanding of division to divide a whole number by a non–unit fraction, a non–unit fraction by a whole number, and a fraction by a fraction. Throughout the topic, students make connections between multiplication and division and write division expressions as unknown factor equations. In addition, students build flexibility in their thinking as they learn a variety of methods to divide fractions. In each lesson, students divide with mixed numbers as well as fractions.”

• Module 4, Topic C, Lesson 13: The Distributive Property, Learn, Teacher Note, “In earlier grades, students write the product of a number and a numerical sum as the sum of two products. They also write the product of a number and a numerical difference as the difference of two products. Encourage students to use the same understanding when they write equivalent algebraic expressions by using the distributive property.”