7th Grade - Gateway 3

The materials reviewed for Eureka Math² Grade 7 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 2, Topic A, Lesson 3: Adding Integers Efficiently, Learn, Teacher Note, “Do not give students more than 2–3 minutes to solve the Making a Purchase problem. If students run out of time to solve the problem because they were creating a number line with values up to 54, this is a great opportunity to talk about efficiency. Consider asking a student to share why the time given was not enough to complete the problem. Use this to initiate a class discussion about the need for a more efficient strategy.”

• Module 4, Topic C, Lesson 9: Constructing a Circle, Launch, Differentiation: Challenge, “Consider deepening comprehension by discussing the couple of places where the interactive plots a point but does not form a triangle. Where are these points? What is happening at these points?”

• Module 6, Topic D, Lesson 18: Comparing Population Means, Learn, UDL: Action and Expression, “To support students in monitoring their own progress, consider providing questions that guide self-monitoring and reflection. For example, post the following for students to refer to as they work independently: How is this problem like the jigsaw puzzle problem? How have we used mean absolute deviation in previous problems similar to this?”

The materials reviewed for Eureka Math² Grade 7 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 2: Operations with Rational Numbers, Module Overview, Why, “I notice that this module does not encourage the use of manipulatives. Why not? The use of manipulatives can support student engagement and provide differentiation and equity. Manipulatives can promote student thinking and aid in communicating about the mathematics being learned. Manipulatives often bridge learning from the conceptual stage to the pictorial or abstract stages of learning. However, students may lose the chance to deepen their understanding of concepts if manipulatives are used in isolation of mathematical connection. Colored chips are a widely used manipulative to engage students in understanding the ‘rules’ for integer arithmetic. Although the chips can be helpful in representing opposites and communicating the associative and commutative properties of addition, some difficulties may occur when students employ these manipulatives: 1. Students are tasked with assigning a number to a set of colored chips and remembering which color represents a positive number and which color represents a negative number. 2. Use of colored chips connects easily to addition, but some learners struggle to conceptualize when subtracting a negative number or multiplying when the first factor is negative. Some interpretations of division cannot be modeled by using the chips. 3. Chips do not represent non-integer rational numbers and cannot be used to model arithmetic with rational numbers. The number line is a coherent model in Eureka Math², and it is a representation that builds understanding of sums and differences of rational numbers. Addition is represented by concatenation of lengths on the number line, beginning in grade 2. This extends through grade 4, where students add fractions with the same unit on the number line, and it further extends to adding, subtracting, multiplying, and dividing fractions in grade 6. Students continue building fluency with addition and subtraction of positive rational numbers on the number line in grade 7, extending this understanding to addition and subtraction of negative rational numbers. In grade 8 and in high school, students work with vectors, and their work with concatenation allows them to understand this as one-dimensional vector addition. For this reason, we have chosen to use the number line instead of colored chips.”

• Module 3: Expressions, Equations, and Inequalities, Module Overview, Why, “I notice that this module includes standards for geometry. Why are these standards addressed in this module? Students understand and apply angle relationships to determine unknown angle measures. These relationships necessitate equivalence. To determine whether angles are complementary, students understand that the two angle measures must sum to 90°. To determine whether angles are supplementary, students understand that the two angle measures must sum to 180°. A natural approach to determine unknown angle measures in these and other cases is to solve for the unknown by using an equation. Determining unknown angle measures drives the need to solve equations. Students use equations to show why angles are equal in measure.”

• Module 5: Percent and Applications of Percent, Module Overview, Why, “Students link prior knowledge of part–whole relationships to the learning of percents. Using a proportion is helpful in determining unknown values in proportional relationships. In this module, students connect percents to proportional relationships and represent them by using equations in the forms y=k⁢x and ab=cd. A common error that students make is placing numbers in incorrect locations in the proportion because they lack understanding of what each value represents. Using the procedure of cross multiplication to solve for an unknown value in a proportion also compromises that understanding. This module’s Math Past examines methods of solving proportions, starting with the early Egyptians and continuing through modern times. In this Math Past, cross multiplication is discussed, along with why it works.”

The materials reviewed for Eureka Math² Grade 7 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 1: Ratios and Proportional Relationships, Achievement Descriptors and Standards, “7.Mod1.AD1 Compute unit rates associated with ratios of fractions given within contexts. (7.RP.A.1)”

• Module 3, Topic B, Lesson 7: Angle Relationships and Unknown Angle Measures, Achievement Descriptors and Standards, “7.Mod3.AD12 Write and solve equations to find unknown angle measures by using known facts about angle relationships. (7.G.B.5)”

• Module 4: Percents and Application of Percent, “In module 5, connection to the learning from previous modules drives the need for students to use percents. Students realize the equation \frac{a}{b}=\frac{c}{d} represents proportional relationships and use proportions and rate language to examine percent as a rate per 100. They identify part, whole, and percent and use proportional reasoning to solve percent problems in real-world contexts, understanding that the unknown could either be part of 100 or more or less than 100%.”

The materials reviewed for Eureka Math² Grade 7 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 7 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 1, Topic B, Lesson 11: Constant Rates, Materials, “Teacher: None. Students: None. Lesson Preparation: Review the Math Past resource to support the delivery of Launch.”

• Module 2: Operations with Rational Numbers, 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. Chart paper, tablet (1), Personal whiteboards (24), Colored pencils (24), Personal whiteboard erasers (24), Dry-erase markers (24), Projection device (1), Highlighters (24), Scientific calculators (24), Eureka Math²™ Integer Cards, set of 12 decks, Sticky notes, pads (8), Learn books (24), Student computers or devices (12), Markers (9), Teach book (1), Paper, blank sheets (15), Teacher computer or device (1), Pencils (24).”

• Module 5, Topic A, Lesson 2: Racing for Percents, Materials, “Teacher: None. Students: Computers or devices (1 per student pair). Lesson Preparation: None.”

The materials reviewed for Eureka Math² Grade 7 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, Topic B, Quiz 3, Item 3, students use proportional relationships to solve real world multi-step problems. “A battery-powered car moves at a constant speed. The table shows the distance the car is from its starting position measured at various times after the car starts moving. How far is the car from its starting position after 24 seconds? (7.RP.A.3)” A table identifying time in seconds and distance from start in meters is shown. 

• Module 3, Topic D, Quiz 1, Item 4, students solve real-world problems using inequalities. “Ethan has $4.50 to spend on 2 bags of chips that cost $0.75 each and some gumballs. Each gumball costs $0.25. Part A: Which statement defines a variable for Ethan’s purchase? g represents the price of 1 gumball Ethan buys. g represents the amount Ethan spends on 2 gumballs. g represents the total number of gumballs Ethan buys. g represents the amount of money Ethan spends on gumballs. Part B: Using the variable defined in part A, enter an inequality that shows that range of the number of gumballs that Ethan can buy. Part C: What is the maximum number of gumballs Ethan can buy? Ethan can buy a maximum of ___ gumballs. (7.EE.B.4.b)”

• Module 5, Module Assessment 2, Item 4, students digitally create a scale drawing of a given figure. “Modify figure B to be a scale drawing of figure A with a scale factor of 175%. (7.G.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 1, Module Assessment 1, Item 1, “Pedro opens a savings account. He deposits $15 into the account each month. Which equation represents the number of dollars d in Pedro's savings account after m months?” This item addresses MP4, model with mathematics.

• Module 3, Module Assessment 2, Item 2, “For each expression, enter an equivalent expression by using the fewest terms possible. -3(0.75x-2y)+6(0.5x-2y).” This item addresses MP7, look for and make use of structure.

• Module 6, Topic B, Quiz 1, Item 3, “Dylan knows that 20% of students at his school walk to school. He designs a simulation to approximate the theoretical probability of exactly one student walking to school when four students are randomly selected. Part A: Dylan makes a spinner with 10 equal-size regions. How many regions should he shade to represent students who walk to school? Part B: Dylan spins the spinner four times and records the result by using S for shaded and N for not shaded. Dylan conducts the simulation 40 times. The results of the simulation are shown. Based on the results of the simulation, what is the empirical probability that exactly one student walks to school when four students are randomly selected?” This item addresses MP1, make sense of problems and persevere in solving them.

The materials reviewed for Eureka Math² Grade 7 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 C, Lesson 16: Using a Scale Factor, Learn, UDL: Representation, “Have students color-code the sides with the colors of the measurements written in the table. This will help them determine which measurements correspond when they are creating the scale drawing.”

• Module 3, Topic C, Lesson 17: Using Equations to Solve Problems, Learn, Differentiation Support, “If students need more support with solving the equation, suggest that students create a double number line. This grade 6 strategy could help them visualize and make sense of the problem.”

• Module 4, Topic D, Lesson 20: Surface Area of Right Pyramids, Learn, Language Support, “Consider having students highlight the line segments that are 5 cm in one color and write Slant Height in that same color. Then, in a different color, have students highlight the height of 3 cm and in that color write Height of Pyramid.”

The materials reviewed for Eureka Math² Grade 7 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 C, Lesson 20: Creating Multiple Scale Drawings, Differentiation: Challenge, “Challenge groups of students to predict whether a second drawing will be the same if they complete their scale drawings in reverse order. Have students verify their predictions by going back to the original figure their group was assigned and completing the scale drawings again, this time using the scale factors in reverse order. Students should determine that when the scale drawings are completed in reverse order, the final scale drawing still looks the same.”

• Module 3, Topic B, Lesson 9: Solving Equations to Determine Unknown Angle Measures,  Launch, Differentiation: Challenge, “If time permits, challenge groups to solve the equations and determine each angle measure. Print the Challenge answer key and allow students to check their answers after they determine all the angle measures.”

• Module 4, Topic D, Lesson 21: Surface Area of Other Solids, Learn, Differentiation: Challenge, “Challenge students by having them sketch a composite figure that has a given surface area, such as 100 square units.”

The materials reviewed for Eureka Math² Grade 7 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 1, Topic C, Lesson 16: Using a Scale Factor, Learn, Language Support, “As the class discussion unfolds, consider providing students with sentence frames and key terminology as a scaffold for peer conversations. For example, post the key terms scale drawing, scale factor, enlargement, and reduction on the board along with a sentence frame such as ‘The scale drawing is a _____ because _____.’”

• Module 2, Topic B, Lesson 8: Subtracting Integers Part 1, Launch, Language Support, “In this lesson and throughout topic B, model correct and consistent use of the subtraction terms minuend, subtrahend, and difference. To support student use of the subtraction terminology, consider displaying a model equation with each part labeled and color-coded with the correct term.” An example is shown, “8-5=3, Minuend - Subtrahend = Difference.”

• Module 3, Topic D, Lesson 19: Using Equations to Solve Inequalities, Learn, Language Support, provides teachers with guidance to understand a key term for engaging in content. “Students may need support with understanding the word restrictions. Pose the following questions to help students make connections between how the word is used in real life and how the word might be used in a math context: How have you heard the word restrictions used in daily life? What do you think the word restrictions means here? Highlight student responses that identify the restrictions featured in the sample student response.”

The materials reviewed for Eureka Math² Grade 7 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 2, Topic B, Lesson 10: Subtracting Rational Numbers Part 1, Launch, UDL: Representation, “It also may be helpful to encourage students to draw a number line numbered −5 to 5, scaled by 0.5. Have them use this number line to model the balance of the account after one charge. Then have them use the number line to model the balance of the account after the second charge.”

• Module 4, Topic A, Lesson 3: Side Lengths of a Triangle, Launch, students use pieces of spaghetti to investigate if a triangle can be formed with given side lengths. “Break apart the pieces of spaghetti to form the specific lengths shown. Then manipulate the pieces of spaghetti and determine whether a triangle can be formed with those side lengths. Record your observations in the table.” 

• Module 6, Topic A, Lesson 2: Empirical Probability, Learn, Problem 2, students are given a bag containing 20 cubes of varying colors to identify empirical probability. “You and your partner have 20 cubes, which were randomly pulled from a large bucket of cubes. These 20 cubes represent the outcomes of 20 trials where a cube was randomly pulled from the large bucket and then replaced.” Using a data table, students record how many cubes of each color are drawn. After conducting multiple trials of this game of chance and collecting data, students calculate the empirical probability for an event.