6th Grade - Gateway 3

The materials reviewed for Eureka Math² 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 in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. These are found in a variety of sections within the Implementation Guide, including the Overview, Why, Achievement Descriptors Overview, and Lesson Structure. Examples include:

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Overview, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.”

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Why, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.”

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Achievement Descriptors, “The Achievement Descriptors: Overview section is a helpful guide that describes what Achievement Descriptors (ADs) are and briefly explains how to use them. It identifies specific ADs for the module, with more guidance provided in the Achievement Descriptors: Proficiency Indicators resource at the end of each Teach book.”

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Lesson Structure, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 45-minute instructional period. Fluency provides distributed practice with previously learned material. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific lessons. This guidance can be found for teachers within boxes called Differentiation, UDL, and Teacher Notes. The Implementation Guide states, “There are six types of instructional guidance that appear in the margins. These notes provide information about facilitation, differentiation, and coherence. Teacher Notes communicate information that helps with implementing the lesson. Teacher Notes may enhance mathematical understanding, explain pedagogical choices, five background information, or help identify common misconceptions. Universal Design for Learning (UDL) suggestions offer strategies and scaffolds that address learner variance. These suggestions promote flexibility with engagement, representation, and action and expression, the three UDL principles described by CAST. These strategies and scaffolds are additional suggestions to complement the curriculum’s overall alignment with the UDL Guidelines.” Examples include: 

• Module 1, Topic E, Lesson 25: Finding the Whole, Overview, Teacher Note, “Problem 1 promotes critical thinking about percents and wholes through the pictorial representation of a circle graph. If students are unfamiliar with this way of displaying data, support them by asking the following questions. Why are the colored regions of the circle graphs different sizes? What is the total percent shown for each circle graph? Which color balloon do Sana and Tara have the least number of? How do you know? Which color balloon do Sana and Tara have the greatest number of? How do you know? If the circle graphs impede student understanding of and engagement with this task, consider eliminating them and referencing only the tape diagrams.”

• Module 3, Topic B, Lesson 7: Absolute Value, Learn, UDL Engagement, “Digital activities align to the UDL principle of Engagement by including the following elements: Engaging and interesting topics. Students play an interactive ring toss game where the winner is revealed after each round. Opportunities to collaborate with peers. By using the information about the winner after three rounds of the ring toss game, students collaborate to decide how the winner is determined. Immediate formative feedback. Students find the number’s distance from 0 and use the interactive to check for accuracy.”

• Module 5, Topic C, Lesson 9: Properties of Solids, Learn, Differentiation: Challenge, “To challenge students to think further about the properties of right prisms and their faces, pose the following questions. All the rectangular faces of this right regular pentagonal prism are identical. Why do you think that is true? Do you think it must be true for any right pentagonal prism? Is it possible for a right triangular prism to have rectangular faces that are identical? How? What type of right prism has a total of 18 edges? How do you know?”

The materials reviewed for Eureka Math² Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Materials consistently contain adult-level explanations, examples of the more complex grade/ course-level concepts, and concepts beyond the course within Topic Overviews and/or Module Overviews. According to page 7 of the Grade 6-9 Implementation Guide, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.” Page 9 outlines the purpose of the Topic Overview, “Each topic begins with a Topic Overview that is a summary of the development of learning in that topic. It typically includes information about how learning connects to previous or upcoming content.” Examples include:

• Module 1: Ratios, Rates, and Percents, Module Overview, Topic C, “In this topic, students compare ratio relationships in context by using ratios to answer questions such as Which lemonade should have a stronger lemon flavor? Students use a variety of strategies to compare ratio relationships, including making direct comparisons by using a ratio table, by creating equivalent ratios, and by calculating the value of the ratio.”

• Module 3: Rational Numbers, Module Overview, Why, “I notice expressions involving adding and subtracting the absolute values of numbers. Do students perform operations with integers and negative rational numbers in grade 6? No. Students do not perform operations with integers and negative rational numbers in grade 6. However, they do make observations about the distances between rational numbers and 0 and between nonzero rational numbers. This lays a foundation for students’ later work, both in solving problems in the coordinate plane and in grade 7, when students begin to compute with rational numbers. In topic B, students are asked to reason about the distances between rational numbers in context, such as a sailor on a boat 4 feet above sea level and the ocean floor at 15 feet below sea level. By using sea level, or 0, as a reference point, students realize that the distance between these two numbers can be thought of as the sum of each number’s distance from 0, or its absolute value. As an extension, students may notice that the distance between the two numbers can be expressed as \lvert4\rvert+\lvert-15\rvert. Students have the opportunity to make similar observations about the distances between two numbers that are both negative or both positive, realizing that the difference of the numbers’ absolute values is the distance between these points. Students later apply this reasoning in topic D, when they determine the distances between the endpoints of horizontal and vertical line segments in the coordinate plane, including line segments with endpoints in the same quadrant and endpoints in different quadrants. All of these observations are grounded in pictorial representations, the number line and the coordinate plane, and students can verify their observations by physically counting units. This merely lays a foundation for later computational work.”

• Module 5: Area, Surface Area, and Volume, Module Overview, Why, “Why are students writing expressions and equations in this module? Geometry contexts are ideal opportunities for students to practice and apply their understanding of expressions and equations from grade 6 module 4. Because the lessons in this module encourage using multiple strategies to determine area, students write and compare different numerical expressions that represent the area of a given polygon. They apply mathematical properties, such as the distributive property, to recognize when expressions for the area of a polygon are equivalent. In addition, students apply their understanding of solving single-variable equations when they encounter geometry problems such as finding an unknown measurement in a figure. For example, finding the height of a triangle that has an area of 20 square inches and a base of 4 inches leads to the equation 20=\frac{1}{2}(4)h, similar to the equations that students solve in module 4 lesson 22.”

The materials reviewed for Eureka Math² Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

The Achievement Descriptors, found in the Overview section, identify, describe, and explain how to use the standards. The lesson overview includes content standards addressed in the lesson. Additionally, a Proficiency Indicators resource at the end of each Teach book, helps assess student proficiency. Correlation information and explanations are present for the mathematics standards addressed throughout the grade level in the context of the series. Examples include:

• Module 2: Operations with Fractions and Multi-Digit Numbers, Achievement Descriptors and Standards, “6.Mod2.AD1 Solve word problems by dividing multi-digit numbers by using the standard algorithm. (6.NS)”

• Module 3, Topic C, Lesson 12: Reflections in the Coordinate Plane, Achievement Descriptors and Standards, “6.Mod3.AD4 Identify relationships between the signs of the numbers in an ordered pair and the point’s location in a coordinate plane. (6.NS.C.6.b)”

• Module 4: Expressions and One-Step Equations, “In module 4, students work with numerical and algebraic expressions and equations. First, they learn that exponents represent repeated multiplication, evaluate powers with whole number, fraction, and decimal bases, and use the order of operations to evaluate numerical expressions. Then, students learn why and how to use variables to represent unknown numbers and quantities. They write and evaluate algebraic expressions and use properties of operations to generate equivalent expressions. Students reason about and solve single-variable, one-step equations, and they understand the meaning of a solution to an equation or inequality. At the end of the module, they revisit ratio relationships and write and graph equations in two variables, identifying independent and dependent variables in real-world situations.”

The materials reviewed for Eureka Math² Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

Materials explain the instructional approaches of the program. According to the Grades 6-9 Implementation Guide, “Eureka Math² features a set of instructional routines that optimize equity by increasing access, engagement, confidence, and students’ sense of belonging. The following is true about Eureka Math² instructional routines: Each routine presents a set of teachable steps so students can develop as much ownership over the routine as the teacher. The routines are flexible and may be used in additional math lessons or in other subject areas. Each routine aligns to the Stanford Language Design Principles (see Works Cited): support sense-making, optimize output, cultivate conversation, maximize linguistic and cognitive meta-awareness.” Examples of instructional routines include:

• Instructional Routine: Always Sometimes Never, students make justifications and support their claims with examples and nonexamples. Implementation Guide states, “Present a mathematical statement to students. This statement may hold true in some, all, or no contexts, but the goal of the discussion is to invite students to explore mathematical conditions that affect the truth of the statement. Give students an appropriate amount of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim. Encourage use of the Talking Tool. Conclude by bringing the class to consensus that the statement is [always/sometimes/never] true [because …].”

• Instructional Routine: Critique a Flawed Response, students communicate with one another to critique others’ work, correct errors, and clarify meanings. Implementation Guide states, “Present a prompt that has a partial or broken argument, incomplete or incorrect explanation, common calculation error, or flawed strategy. The work presented may either be authentic student work or fabricated work. Give students an appropriate amount of time to identify the error or ambiguity. Invite students to share their thinking with the class. Then provide an appropriate amount of time for students to solve the problem based on their own understanding. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the prompt given. Then facilitate a class discussion by inviting students to share their solutions with the whole group. Encourage use of the Talking Tool. Lead the class to a consensus about how best to correct the flawed response.”

• Instructional Routine: Stronger, Clearer Each Time, students revise and refine their written responses. Implementation Guide states, “Present a problem, a claim, or a solution path and prompt students to write an explanation or justification for their solution path, response to the claim, or argument for or against the solution path. Give students an appropriate amount of time to work independently. Then pair students and have them exchange their written explanations. Provide time for students to read silently. Invite pairs to ask clarifying questions and to critique one another’s response. Circulate and listen as students discuss. Ask targeted questions to advance their thinking. Direct students to give specific verbal feedback about what is or is not convincing about their partner’s argument. Finally, invite students to revise their work based on their partner’s feedback. Encourage them to use evidence to improve the justification for their argument.”

Materials include and reference research-based strategies. The Grades 6-9 Implementation Guide states, “In Eureka Math² we’ve put into practice the latest research on supporting multilanguage learners, leveraging Universal Design for Learning principles, and promoting social-emotional learning. The instructional design, instructional routines, and lesson-specific strategies support teachers as they address learner variance and support students with understanding, speaking, and writing English in mathematical contexts. A robust knowledge base underpins the structure and content framework of Eureka Math². A listing of the key research appears in the Works Cited for each module.”

The materials reviewed for Eureka Math² Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Each module and individual lesson contains a materials list for the teacher and student. The lesson preparation identifies materials teachers need to create or assemble in advance. Examples include:

• Module 2, Topic D, Lesson 16: Applications of Decimal Operations, Materials, “Teacher: None. Students: Interlocking cubes, 1 cm (20 per group of 3 or 4 students). Lesson Preparation: None.”

• Module 4, Topic A, Lesson 3: Exploring Exponents, Materials, “Teacher: None. Students: Computers or student devices (1 per student pair). Lesson Preparation: None.”

• Grade 6, Module 6: Statistics, Module Overview, Materials, “The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher. Basketball (1), Pencils (35), Calculators (25), Personal whiteboards (24), Chart paper, sheets (18), Personal whiteboard erasers (24), Computer with internet access (1), Projection device (1), Dry-erase markers (24), Rulers, plastic (24), Eureka Math² measuring tapes (8), pool of string (1), Interlocking cubes 1 in (1), Sticky notes pads (12), Learn books (2), Student computers or devices (12), Letter size paper, sheets (2), Tape roll, masking (1), Marker sets (6), Teach book (1).”

The materials reviewed for Eureka Math² Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level/ course-level standards and practices across the series.

Assessments identify standards and include opportunities for students to demonstrate the full intent of grade-level/course-level standards. Examples include: 

• Module 1, Module Assessment 1, Item 8, Part A, students select correct representations of a percentage. “At a middle school, 40% of the students participate in after-school activities. Part A: Which representations of 40% are correct. Select all that apply.” Answer choices: “40, \frac{40}{100}, 0.04, 0.4, (a hundreds grid with 40 shaded is shown), (a hundreds grid with 4 shaded is shown), \frac{4}{5},  and (a fraction bar with \frac{2}{5} shaded). (6.RP.A.3.b and 6.RP.A.3.d)”

• Module 5, Topic D, Quiz 2, Item 4, students use volume of right rectangular prisms to solve real world problems. “Mr. Sharma builds a set of steps. The bottom level of the steps is in the shape of a right rectangular prism and made of bricks. Each of the top two levels of the steps is in the shape of a right rectangular prism and made of concrete. Each step has a height of 0.5 feet. A diagram of the steps is shown. How many cubic feet of bricks does Mr. Sharma use to build the bottom level of the steps? 1. ___ cubic feet. How many cubic feet of bricks and concrete does Mr. Sharma use to build the steps? 2. ___ cubic feet. (6.G.A.2)”

• Module 6, Module Assessment 2, Item 4, students explain their reasoning when determining if a question is a statistical question. “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.A.1)”

Assessments do not identify mathematical practices in either teacher or student editions. Although assessment items do not clearly label the MPs, students are provided opportunities to engage with the mathematical practices to demonstrate full intent. Examples include:

• Module 2, Module Assessment 1, Item 3, “Match each tape diagram to the division expression it represents. Drag one division expression into each box.” This item addresses MP4, model with mathematics.

• Module 4, Module Assessment 2, Item 3, “Consider 7+5(y+4)-2y. Riley's work to write an equivalent expression is shown. Part A: Select from the drop-down list to make the statement true. Riley made a mistake in step ____. Part B: Explain Riley's mistake. Part C: Which expression is equivalent to 7+5(y+4)-2y.” This item addresses MP3, construct viable arguments and critique the reasoning of others.

• Module 4, Topic C, Quiz 1, Item 1, “Which expressions are equivalent to 4(n+2)? Select all that apply. 4n+2, 4n+8, 2(n+2)+2(n+2), 2(n+2)+2n, and n+n+n+n+8.” This item addresses MP7, look for and make use of structure.

The materials reviewed for Eureka Math² 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/series mathematics.

Materials provide strategies, supports, and resources for students in special populations to support their regular and active participation in grade-level mathematics. According to the Implementation Guide, “There are six types of instructional guidance that appear in the margins. These notes provide information about facilitation, differentiation, and coherence.” Additionally, “Universal Design for Learning (UDL) is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain. Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. Eureka Math2 lessons are designed with these principles in mind.” Examples include:

• Module 1, Topic E, Lesson 25: Finding the Whole, Learn, Differentiation: Support, “Some students may benefit from additional practice using mental math to calculate the whole. Before students complete problem 2, consider doing a Whiteboard Exchange using the following sample sequence. 30 is 50% of what number? 30 is 25% of what number? 30 is 20% of what number? 30 is 10% of what number? 30 is 5% of what number? 30 is 1% of what number? After the Whiteboard Exchange, ask students to explain their strategies for calculating the whole, or 100%.”

• Module 4, Topic C, Lesson 15: Combining Like Terms by Using the Distributive Property, UDL: Representation, “Digital activities align to the UDL principle of Representation by including the following elements: Scaffolds that connect new information to prior knowledge. Students apply their prior knowledge of the areas of rectangles and the distributive property as they write equivalent algebraic expressions by combining like terms.Strategies that emphasize essential patterns, relationships, and key ideas. Students use their knowledge of the commutative property of addition and the distributive property to write equivalent expressions and to add and subtract like terms.”

• Module 6, Topic C, Lesson 13: Using the Interquartile Range to Describe Variability, Learn, Language Support, “If students wonder why there are only three quartiles instead of four, have them think about a piece of paper. Ask them how many times they need to tear it to make it into four pieces. Connect the

The materials reviewed for Eureka Math² Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Materials do not require advanced students to do more assignments than their classmates. Instead, students have opportunities to think differently about learning with alternative questioning, or extension activities. Specific recommendations are routinely highlighted as Teacher Notes within parts of each lesson, as noted in the following examples: 

• Module 1, Topic E, Lesson 26: Solving Percent Problems, Learn, Differentiation: Challenge, “Extend the Cafeteria Calculations problem by asking students to also create a breakfast menu. Tell students that the breakfast menu should have a total number of calories that is no fewer than 400 and no more than 550. Allow students to use internet access or information from their school cafeteria to look up the nutrition data on common breakfast foods.”

• Module 3, Topic A, Lesson 4: Rational Numbers in Real-World Situations, Launch, Differentiation: Challenge, “For more practice, have students calculate the fortune for several more transactions. Sell 1 sheep. Buy 1 pig; Sell 2 cows. Buy 1 sheep; Sell 2 cows and 1 pig. Buy 2 sheep.”

• Module 5, Topic B, Lesson 7: Areas of Trapezoids and Other Polygons, Learn, Differentiation: Challenge, “If students finish problems 1–5 early, ask them to write a formula to describe the area of a trapezoid. As needed, guide students to use one of the strategies to find the area of trapezoid TRAP and then to use variables to write a general formula for the area of any trapezoid. Once students have a formula, compare it to the standard trapezoid area formula, A=\frac{1}{2}(b_1+b_2)h, and discuss the meanings of b_1 and b_2.”

The materials reviewed for Eureka Math² 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.

Support for active participation in grade-level mathematics is consistently included within a Language Support Box embedded within parts of lessons. The Implementation Guide explains supports for language learners, “Multilingual learners, or learners who speak a language other than English at home, require specific learning supports for gaining proficiency with the English needed to access the mathematics. Research suggests that best practices for these learners include opportunities and supports for student discourse and for using precise terminology. In addition to precise domain-specific terminology, high-impact academic terminology that supports learners across learning domains is explicitly introduced and used repeatedly in various contexts to build familiarity and fluency across the grade levels. Eureka Math² is designed to promote student discourse through classroom discussions, partner or group talk, and rich questions in every lesson.” 

Examples include:

• Module 2, Topic D, Lesson 16: Applications of Decimal Operations, Launch, Language Support, “To support the terms cost, revenue, profit, and loss, build background knowledge by using a familiar context. For example, have students consider making and selling cups of lemonade. Use the following prompts to promote students’ understanding: What supplies would you need for a lemonade sale? How much would those items cost?; How much would you sell each cup of lemonade for? If you sold 25 cups of lemonade for $1.00 per cup, how much revenue would you earn?; Which number would you want to be greater: the amount you spend on supplies or the amount you earn from sales?; The difference between the amount you earn, or revenue, and the amount you spend, or cost, could be a profit or a loss. When your total revenue is greater than your total cost, you make a profit. On the other hand, when your total cost is greater than your total revenue, you experience a loss.”

• Module 5, Topic C, Lesson 14: Designing a Box, Launch, Language Support, “This is the first instance of the verb report in the curriculum. Consider previewing the meaning of the word before students see it in print in the Thinking Outside the Box task guidelines. Highlight a synonym for report that students can use in conjunction with the word, such as tell or describe.”

• 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 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 providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives provide accurate representations of mathematical objects. Examples Include:

• Module 4, Topic A, Lesson 1: Expressions with Addition and Subtraction, Learn, students use interlocking cubes to model expressions involving addition and subtraction. “Provide each student with 20 cubes. Have each student do the following: Build a stack of 8 cubes. Add 7 more cubes to the stack. Write an expression on their whiteboards to represent how the number of cubes in the stack changed. Have students show you their whiteboards. Check that they have written 8+7. Then, have students do the following: Add 2 cubes to the stack. Revise the expression written on their whiteboards to represent how the number of cubes in the stack changed. Have students show you their whiteboards. Check that they have written 8+7+2.”

• Module 5, Topic A, Lesson 3: The Area of a Triangle, Launch, students use precut triangles to compose parallelograms. “Students begin this lesson by working with a partner to determine different ways to compose parallelograms from two identical triangles. Through a guided discussion, students conclude that because any two identical triangles can compose a parallelogram, the area of any triangle is determined by \frac{1}{2}bh, half the area of a parallelogram.” 

• Module 6, Topic B, Lesson 7: Using the Mean to Describe the Center, Learn, “Pair students and distribute 30 cubes to each student pair. Prompt one student in each pair to remove the Empty Plates page from their book. Have students arrange the cubes on the Empty Plates removable to match the plates of tacos in Julie’s picture. Then direct students to use the cubes to show what it looks like when all students have an equal share of the tacos. Following this work, have partners complete parts (d) and (e). D. If each student got the same number of tacos, how many tacos would each student have? Draw a picture and explain how you got your answer. E. Do you think this value is a better measure of the center than Scott’s value? Why?”