6th to 8th Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

According to the Grades 6-9 Implementation Guide, the Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, Module Assessments, Pre-Module Assessments, and Benchmark Assessments. “All Eureka Math² assessments are considered formative because they are intended to inform instruction. The assessments may also be considered summative when you choose to use the data to produce a grade or report that becomes part of a student, school, or district record. Exit Tickets are not graded. They are paper-based so that you can quickly review and sort them. Occasionally, an alternative assessment activity may substitute for an Exit Ticket at the end of a lesson. Typical Topic Quizzes consist of 4–6 items that assess proficiency with the major concepts from the topic. Many items allow students to show evidence of one or more mathematical practices. Topic Quizzes, Typical Module Assessments consist of 6–10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Many items allow students to show evidence of one or more mathematical practices. Module Assessments represent the most important content, but they may not assess all the strategies and standards 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:

• Grade 6, Module 1, Module Assessment, Version 1, Problem 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)

• Grade 7, Module 5, Module Assessment, Version 1, Problem 7, “A store owner purchases milk for $2.72 per gallon. She marks up the price by 33%. What is the selling price per gallon of milk? $_______ per gallon.” (7.RP.3)

• Grade 8, Module 6, Module Assessment, Version 1, Problem 3, “Consider the scatter plot. Part A: Which type of association best describes the data in the scatter plot?” Answers provided: “Strong, negative, linear association; Strong, positive, linear association; Weak, negative, linear association; Weak, positive, linear association. Part B: Drag the points to draw a line that fits the data in the scatter plot.” (8.SP.1, 8.SP.2)

The materials reviewed for Eureka Math² Grades 6 through 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed. Materials consistently identify the standards and practices assessed for formal assessments.

The publisher states, “Every formal Eureka Math² assessment item is tagged to at least one CCSS standard, at least one associated Achievement Descriptor, and at least one Mathematical Practice.” Under the overarching Assessments tab in the Great Minds Library, the Standards link for Topic Quizzes and Module Assessments identifies the standards addressed on each assessment. The digital version also identifies the aligned standard below each problem in the Topic Quizzes and Module Assessments; the PDF print versions do not include these standard alignments. The publisher also states, “Across each grade level, all eight SMPs were tagged to multiple items, allowing students to demonstrate the full intent of the grade-level practices.” For each Module Assessment, the digital platform provides a dot chart titled “Mathematical Practice Alignment to Module Assessment Items” that identifies the Mathematical Practices aligned to individual assessment items.

Examples include:

• Grade 6, Module 4, Module Assessment Version 2, Problem 3, Part C states, “Which expression is equivalent to 7 + 5(y + 4) - 2y? A. 27 + 7y, B. 27 + 3y, C. 13 + 3y, D. 11 + 7y” (6.EE.4, MP7)

• Grade 7, Module 6, Module Assessment Version 1, Problem 3 states, “A spinner has regions that are three different colors: red, blue, and yellow. Henry spins the spinner 200 times and records a relative frequency of 0.24 for blue regions. Noor spins the spinner 60 times and records a relative frequency of 0.18 for blue regions. Whose empirical probability of the spinner landing on a blue region would you expect to be closer to the theoretical probability of the spinner landing on a blue region? Why?” (7.SP.7b, MP2, MP3)

• Grade 8, Module 5, Module Assessment Version 2, Problem 6 states, “A streaming service charges $1.99 for each TV episode streamed, and $5.99 for each movie streamed. In June, Ethan streams a combined total of 16 TV episodes and movies. Ethan pays the service $55.89 in streaming charges for June. How many TV episodes and movies does Ethan stream in June? Ethan streams ___ TV episodes and ___movies in June.” (8.EE.8c, MP1, MP2)

The materials reviewed for Eureka Math² Grades 6 through Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The Grades 6-9 Implementation Guide explains that the assessment system uses multiple assessments and sources of data to develop a complete picture of student understanding. The materials state, “On its own, a single assessment does not show a complete picture of student progress” and direct teachers to use “student performance on a combination of assessments and data” to understand student proficiency. The materials further explain that some standards are developed and assessed across multiple modules and that students are not expected to demonstrate full proficiency until all related content has been taught. Across the series, students engage with Quizzes, Module Assessments, Benchmark Assessments, and Performance Tasks that include a variety of item types and modalities, such as constructed response, multiple choice, multiple select, multi-part questions, writing, and modeling. Module and Topic Assessment items also identify associated grade-level content standards and Mathematical Practices. Collectively, these assessment opportunities allow students to demonstrate the full intent of grade-level standards and Mathematical Practices across the series.

Examples include:

• Grade 6, Module 1, Module Assessment Version 1, Problem 1 states, “A soup recipe calls for 3 cups of potatoes for every 2 cups of milk. How many cups of potatoes should be used for every 1 cup of milk? A. \frac{2}{3}, B. \frac{3}{2}, C. 2, D. 3” Problem 6, Part A states, “Sasha mows 8 lawns in 4 days. When the rate is expressed in lawns per day, what is the unit rate? A. 8, B. 4, C. 2, D. 1”. Part B states, “What does the unit rate identified in part A mean in this situation?” The materials assess the full intent of 6.RP.2 as students understand the concept of a unit rate a/b associated with a ratio a:b with b≠0, and use rate language in the context of a ratio relationship.

• Grade 7, Module 3, Topic D, Quiz Version 1, Problem 1 states, “Sara has a 30-gallon fish tank. The tank already has 24 gallons of water in it. Sara adds more water at a rate of 0.5 gallons per minute. Part A. Sara wants to determine the range of minutes for which she can add water to the tank before it overflows. If m represents the number of minutes, which inequality models this situation? A. 0.5m+24≤30 B. 0.5m+24≥30 C. 0.5m−24≤30 D. 0.5m−24≥30 Part B. Sara estimates that it will take her 8 minutes to fill the tank before it overflows. Circle an answer choice from each list to make the statement true. After Sara fills the tank for 8 minutes, she (A) adds more water to the tank before it overflows. The range of the number of minutes for which Sara can add water to the tank before it overflows is between (B) and (C) minutes.” The materials assess the full intent of MP1 as students make sense of problems and persevere in solving them.

• Grade 8, Module 5, Topic A, Quiz Version 1, Problem 4 states, “Consider the system of equations. {y=−\frac{3}{4x}-1,-3x+4y= 8} Circle an answer choice from each list to make the statements true. The system of equations has (A). This means the lines that represent the two equations in the system have (B) and (C ).” Topic A, Quiz Version 2, Problem 1 states, “Match the graph of each system of equations with its solution. Write one solution from the given answer choices in each box.” Three graphs are given with six answer choices provided. Topic B, Quiz 1, Problem 1 states, “What is the solution to the system of equations? {2x−4y=−1x=7−3y}” The materials assess the full intent of 8.EE.8b as students analyze graphs of systems of linear equations and identify the solution to each system based on the relationship between the two lines.

The materials reviewed for Eureka Math² Grades 6 through Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide students with consistent opportunities to engage in the full intent of all Grade 6-8 standards. Each lesson consists of four sections (Fluency, Launch, Learn, and Land) that provide extensive work with grade-level problems. Fluency activities provide opportunities for students to practice previously learned content and activate prior knowledge to prepare for new learning. Launch activities build context for the learning goals, while Learn activities present new learning through a series of learning segments. During Land, teachers facilitate discussions that address key questions related to the learning goal. Practice pages provide additional opportunities for students to engage with problems that range from simple to complex. Across the program, these lesson components provide multiple opportunities for students to develop and demonstrate their understanding of the full intent of grade-level standards.

Examples include:

• Grade 6, Module 4, Topic C, Lesson 16, engages students with 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]). Students apply properties of operations, including the distributive property and combining like terms, to generate and identify equivalent expressions, evaluate equivalent expressions for the same value of a variable, and represent the same real-world situation with different expressions. Fluency, Problem 6 states, “Use the distributive property to write an equivalent expression as a sum or difference. \frac{1}{6}(3+5x).” In 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.” In Learn, students combine like terms in algebraic expressions with addition and subtraction, “Display the following expression and ask students to copy it onto 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.” In Classwork, Problem 2, students distribute and combine like terms to create equivalent expressions, “x+5(4x-1).” Classwork Problem 5 states, “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.” In Exit Ticket, Problem 1, students write two expressions that could be used to describe a real-world 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.” In Practice, Problem 4 states, “For problems 1-4, distribute and combine like terms to write an equivalent expression. Show your work. 8(x+3)+5(2x+6).” 

• Grade 7, Module 6, Topic B, Lesson 7, engages students with the full intent of 7.SP.7 (Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy). Students determine theoretical and empirical probabilities, conduct repeated chance experiments and analyze relative frequencies, compare observed results to theoretical probability models, and use experimental results to determine whether a probability model is reasonable. In Fluency, Problem 2, students find the theoretical probabilities of events when rolling a fair 6-sided die to prepare for using empirical probability to estimate theoretical probability, “Find the theoretical probability that the event occurs when rolling a 6-sided die with the numbers 1-6 on its sides. 2. Rolling a 4.” In Launch, students use theoretical probability to predict the outcome of a chance experiment, “If you flip your coin two times, how many times would you expect to flip heads? How many times would you expect to flip tails? How would you describe this result by using relative frequencies? Flipping your coin two times is a chance experiment. What is the sample space of this chance experiment? When a coin is flipped two times, what is the theoretical probability of flipping exactly 1 heads? How do you know?” In Learn, Problem 1, students examine the effect of repeated trials on empirical probabilities, “Flip a coin 10 times and record your results in the following table. Write relative frequencies as fractions and as decimals rounded to the nearest hundredth.” In Learn, Problem 3 states, “Summarize your results from problems 1 and 2 in the following table. Write the relative frequency of heads as a fraction and as a decimal rounded to the nearest hundredth.” In Learn, Problem 4 states, “Graph the number of flips and the corresponding relative frequencies from the table in problem 3 as points. Connect consecutive points with line segments.” Land, students discuss using empirical probability to estimate theoretical probability and compare probabilities from a theoretical model to observed relative frequencies, “How can you restate the law of large numbers in terms of the coin flipping experiment? How can we estimate the theoretical probability of an event when the outcomes are not equally likely? How can we use the results of a chance experiment to judge whether a theoretical probability model seems reasonable?” The Exit Ticket, Problem 1 states, “After flipping a penny 30 times, Dylan reports that the relative frequency of heads is about 0.47. a. How many times did the coin land heads up? Explain. b. Dylan flips the coin one more time. Is it possible for the new relative frequency of heads to be 0.55? Explain.” In Practice, Problem 3 states, “Shawn rolls a 6-sided die 36 times and records the results in the table. a. Complete the relative frequency column of the table. Write each relative frequency as a fraction. b. Determine the empirical probability of Shawn rolling a 2. c. Determine the empirical probability of Shawn rolling a 3. d. Determine the empirical probability of Shawn rolling a 6. e. Do you think Shawn’s die is fair? Explain.”

• Grade 8, Module 5, Topic A, Lesson 5; Topic B, Lesson 7 and Lesson 9, engage students with the full intent of 8.EE.8b (Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection). Lesson 5, Practice, Problem 2, students graph a system of equations and estimate the solution, “Consider the system of equations. 5x+4y=20 and y=\frac{5}{2}x-4 a. Graph the system of equations. Estimate the coordinates of the intersection point of the lines. b. Is your estimate from part (a) the solution to the system? Why?” Topic B, Lesson 7, Exit Ticket, students solve a system of equations algebraically, “Solve the system of equations by using the substitution method. Check your solution. 3y-21=2x, y-4x=2.” Lesson 9, Fluency, Problem 3, students identify the number of solutions to a system by inspection, “Determine by inspection whether each system of equations has only one solution, no solution, or infinitely many solutions. y=-4x+6, y=-4x-7.”

The materials reviewed for Eureka Math² Grades 6 through Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade as included in the following grade-level breakdowns. 

Grade 6:

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 4 out of 6, which is approximately 67%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 91 out of 133, which is approximately 68%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 115 out of 167, which is approximately 69%.

Grade 7:

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 4 out of 6, which is approximately 67%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 89 out of 138, which is approximately 64%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 110 out of 170, which is approximately 65%.

Grade 8:

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 5.5 out of 6, which is approximately 92%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 112 out of 129, which is approximately 87%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 141 out of 162, which is approximately 87%.

An instructional day analysis across Grades 6 through Grade 8 is most representative of the instructional materials, as the days include major work, supporting work connected to major work, and the assessments embedded within each chapter. Approximately 69% of the materials in Grade 6, 65% of the materials in Grade 7, and 87% of the materials in Grade 8 focus on major work of the grade.

The materials reviewed for Eureka Math² Grades 6 through Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Materials are designed so that supporting standards are connected to the major standards of the grade. Teachers can locate these connections within the “Achievement Descriptors and Standards” tab in lessons.

Examples include:

• Grade 6, Module 5, Topic B, Lesson 5, Classwork connects the supporting work of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems) to the major work of 6.NS.8 (Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate). Students use the coordinate plane to plot points, draw line segments to form polygons, and use the coordinates of the vertices to determine the lengths of horizontal and vertical sides while solving mathematical problems involving points in all four quadrants of the coordinate plane. Plotting Polygon states, “For problems 2-8, use the coordinate plane to answer the questions. 2. Plot points A(1,4) and B(6,4) Draw \overline{AB}. What is the length of \overline{AB}? 3. Plot point C(6,-3). Draw \overline{BC}. What is the length of \overline{BC}? 4. Plot point D(1,-3). Draw \overline{CD}. What is the length of \overline{CD}? 5. Draw \overline{DA}. What is the length of \overline{DA}? 6. What type of polygon is ABCD? What is its perimeter? What is its area? 7. Plot points E(-4.5,-3), F(-4.5,0), and G(1,0). Draw DE, EF, and FG. What are the lengths of \overline{DE}, \overline{EF}, and \overline{FG}? 8. Polygon ABCDECG is formed by composing rectangles ABCD and DEFG. What is the perimeter of polygon ABCDEFG? What is the area?” Students are provided a coordinate plane.

• Grade 7, Module 6, Topic A, Lesson 4, Classwork connects to the supporting work of 7.SP.7 (Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy) to the major work of 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems) Students develop a probability model, use it to determine the probabilities of events, and apply proportional reasoning to interpret and calculate theoretical probabilities. Problem 2 states, “A chance experiment consists of flipping a nickel and a penny. E. If you flipped both coins 12 times, how many times would you expect the coins to both land tails up? Explain. f. If you flipped both coins 12 times, how many times would you expect exactly one coin to land heads up? Explain.”

• Grade 8, Module 1, Topic E, Lesson 22, Teach Book, Learn connects the supporting work of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers, show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number) to the major work of 8.EE.2 (Use square roots and cube root symbols to represent solutions to equations in the form x^2=p and x^3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational). Students evaluate square roots of perfect and non-perfect squares and determine whether the resulting numbers are rational or irrational by reasoning about their decimal representations. Teacher notes state, “Are all square roots irrational? Invite a variety of opinions and then display the following examples. \sqrt{2}=1.41421356… and \sqrt{25}=5. Give students time to Think-Pair-Share about the following discussion questions: Are these numbers rational or irrational? How do you know? Are all square roots irrational? How do you know? After pairs share their answers with the class, instruct students to add each number, \sqrt{2} and \sqrt{25}, to the Examples row of the table in problem 9.”

The instructional materials for Eureka Math² Grade 6 through Grade 8 meet expectations for including problems and activities that connect two or more clusters in a domain, or two or more domains in a grade. 

Connections among the major work of the grade appear throughout the materials where appropriate. Each lesson includes Achievement Descriptors that describe what students should be able to do by the end of the lesson and identify the associated standards. The Teach books include a Module Achievement Descriptors by Lesson resource at the end of each module that identifies the Achievement Descriptors addressed in each lesson, allowing teachers to trace the standards addressed across lessons and modules. 

Examples include:

• Grade 6, Module 4, Topic B, Lesson 9, Exit Ticket, 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). Students precisely define variables and write algebraic expressions involving addition and subtraction to represent real-world situations. Problem 1 states, “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.”

• Grade 7, Module 1, Topic B, Lesson 11, Learn, Problem 6, connects the major work of 7.RP.A (Analyze proportional relationships and use them to solve real-world and mathematical problems) to the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations). Students determine a unit rate, interpret its meaning in context, write an equation to represent a proportional relationship, define a variable, and use the equation to find an unknown value. In Problem 6, “You count 20 heartbeats in 15 seconds. a. What is the unit rate, and what does it mean in this situation? b. Write an equation to represent this situation. Define the variable in your equation. c. At this rate, how many times does your heart beat in a minute?”

• Grade 8, Module 1, Topic D, Lesson 19, Teach Book, Learn, connects the major work of 8. EE.A (Work with radicals and integer exponents) to the major work of 8. G.B (Understand and apply the Pythagorean Theorem). Students apply the Pythagorean Theorem to find the length of a hypotenuse by evaluating the square root of the sum of the squares of the legs. Problems 3 and 4 read, “For problems 1-4, find the length of the hypotenuse c.” In Problem 3, students are given a right triangle with side lengths of 0.3 and 0.4 units and determine that the length of the hypotenuse is 0.5 units. In Problem 4, students are given a right triangle with side length of 3 and the hypotenuse of \frac{5}{4} units and determine that the length of the missing side is \frac{13}{4} units.

The instructional materials reviewed for Eureka Math² Grade 6 through Grade 8 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 Teach Book begins with a Module Overview that includes a topic-by-topic summary of the development of learning throughout the module. The Module Overview also contains Before This Module and After This Module coherence sections that look backward and forward to show the module's place in the development of learning within and across grade levels. In addition, The Why Behind the Lesson Design explains the organization of the module's content and structure, including how the module connects to previous and future grade-level learning. Each Topic Overview summarizes the development of learning within the topic and typically describes how the content connects to previous or upcoming learning, including learning from prior or future grade levels. Some Teacher Notes within lessons also explain connections between current learning and prior and future mathematical concepts.

An example of a connection to future grades in Grade 6 includes:

• Module 4, Topic A, Lesson 1, Teach Book, connects 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions) to future work in Grade 7, where students interpret subtraction as adding the opposite of a number (7.NS.1). The Teacher Note states, “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.”

An example of a connection to prior knowledge in Grade 6 includes:

• Module 2, Topic A, Topic A Overview, connects 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. Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor) to previous work in Grade 4, where students identify factors and multiples, determine whether numbers are prime or composite, and recognize that a whole number is a multiple of each of its factors (4.OA.4). The Topic Overview states, “In grades 4 and 5, students identify factors and multiples in the range of 1–100 and determine whether numbers are prime or composite. Students recognize that a whole number is a multiple of each of its factors. They use this understanding to determine whether a whole number is a multiple of a given one-digit number. In topic A of grade 6, students extend their understanding about factors, multiples, and divisibility and apply what they know to solve problems involving the greatest common factor and least common multiple.”

An example of a connection to future grades in Grade 7 includes:

• Module 6, Topic D, Topic Resources, Topic D Overview, connects 7.SP.A (Use random sampling to draw inferences about a population) and 7.SP.B (Draw informal comparative inferences about two populations) to future work in Grade 8, where students investigate bivariate data and fit linear models to represent relationships between two quantitative variables (8.SP.1-3). The Topic Overview states, “An understanding of random sampling and informal statistical inference is the foundation for understanding bivariate data and linear regression in grade 8. It is important in this topic to reinforce the difference between sample statistics and population characteristics, along with the reasonableness of using sample statistics to make inferences about population characteristics.”

An example of a connection to prior knowledge in Grade 7 includes:

• Module 1, Module Resources, Module 1 Overview, connects 7.RP.2 (Recognize and represent proportional relationships between quantities) to previous work in Grade 6, where students understand ratio relationships and use ratios, rates, and unit rates to describe relationships between quantities (6.RP.1-2). The Coherence: Before This Module section states, “Students apply knowledge of multiplicative comparisons to understand ratio relationships. They represent the two values in a ratio as a quotient—known as the value of the ratio—and then use that value to determine rates and unit rates of ratio relationships. Throughout the coursework of grade 6, students apply ratio reasoning to work with percents, equations, graphs, geometry, and statistics. Grade 7 module 1 elevates the work of grade 6 by introducing the terms proportional relationships and scale factor.”

An example of a connection to future grades in Grade 8 includes:

• Module 2, Module Resources, Module 2 Overview, connects 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software) to future work in Algebra I, where students identify the effects of transformations on the graphs of functions (HSF-BF.3), and to Geometry, where students recognize rigid motions as functions of the plane and use translations, reflections, and rotations to justify triangle congruence theorems (HSG-CO.2, HSG-CO.6-8). The Coherence: After This Module states, “In Algebra I, students apply rigid motions to describe transformations of functions and to graph functions in the coordinate plane. In Geometry, students combine their knowledge of rigid motions and functions to recognize a rigid motion as a function of the plane. They also use the properties of translations, reflections, and rotations to justify triangle congruence theorems.”

An example of a connection to prior knowledge in Grade 8 includes:

• Module 5, Topic C, Lesson 14, Teach Book, connects 8.EE.C (Analyze and solve pairs of simultaneous linear equations) to previous work in Grade 8, where students graph lines and write equations of lines (8.EE.6). The Teacher Note states, “If students need support with writing the equation for line p, consider having them graph the points (5, 1) and (−1, 1). Students may also refer to module 4 topic F for more practice.”