6th to 8th Grade - Gateway 1

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

According to the Implementation Handbook, the assessment system provides a variety of opportunities for students to independently demonstrate their mastery of grade-level standards through formal summative assessments, which include the following:  Chapter Test, Big Idea Task, Multi-Chapter Test, Quarter DAP (Diagnostic Adaptive Progression—online only), End-of-Course Test, Post-Course DAP, and Alternative Chapter Assessment. These assessments vary in format, providing different ways to accurately describe student performance at a particular point-in-time. All assessments are aligned with grade-level content standards as represented in the “Assessment Correlation (by Standard)” and an “Assessment Correlation (by Course)”. Both include a breakdown identifying content standards for each assessment item.

The materials in Grades 6–8 are organized into ten chapters, each containing a Chapter Test and a Big Idea Task. In addition, the materials include four Multi-Chapter Tests, four Quarter DAP Tests, one End-of-Course Test, and one Post-Course DAP assessment. Examples include:

• Grade 6, Chapter 3, Chapter Test, Exercise 12 states, “You bake cookies for a bake sale. The ratio of chocolate chip cookies to oatmeal cookies is 4:3. You bake 12 oatmeal cookies. How many chocolate chip cookies do you bake?” (6.RP.3)

• Grade 7, Multi-Chapter Test 2, Exercise 6 states, “At 5 P.M., the total snowfall is 2 centimeters At 10 P.M., the total snowfall is 10 centimeters. What is the mean hourly snowfall? Write your answer in simplest form as a fraction or mixed number.” (7.NS.2)

• Grade 8, Chapter 1, Big Idea Tasks, Exercise 3 states, “The total costs (in dollars) to rent an electric bicycle for minutes from two different companies are given. Company A: T = 0.3m + 1 Company B: T = 0.25m +1.5 a. Solve and interpret. 0.3m +1 = 0.25m +1.5 b. You want to rent an electric bicycle. When should you rent an electric bicycle from Company A? Company B? Construct mathematical arguments to justify your answers. c. Solve both equations for m. Explain when it is more convenient to use these forms of the equations. d.  A group of friends rents electric bicycles from one of the companies. The group rides the electric bicycles for 25  minutes and spends a total of $38.75. Which company did the group rent electric bicycles from? How many friends are in the group? Explain your reasoning.” (8.EE.7)

The materials reviewed for Math & YOU Grades 6 though Grade 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

CCSS standards are identified on the digital assessments for each item on the following formal assessments: Chapter Performance Task, Big Idea Task, Chapter Test, Multi-Chapter Test, and End-of-Course Test. The print materials do not always identify the CCSS or Practices; however, the problems on the print assessments are identical to the problems in the digital assessments. Two correlation resources demonstrate assessment items alignment to the standards. The Assessment Correlation (by Course) is organized by assessment and identifies the standards addressed on each assessment. The Assessment Correlation (by Standard) lists every assessment item in which a specific standard is addressed. 

The Digital Teaching Experience and the Teacher Toolkit: Course Essentials identify the Standards for Mathematical Practice (SMPs) for Big Idea Tasks and Chapter Performance Tasks; however, the materials do not identify the SMPs consistently for each item on other formal assessments.

Examples include:

• Grade 6, Chapter 6, Big Ideas Task, Exercise 5 states, “A bag contains pennies, nickels, dimes, and quarters. The bag has a total of 455 coins. There are four times as many nickels as quarters, five times as many dimes as nickels, twice as many pennies as dimes. Find the total value of the coins in the bag. Show your work and verify that your solution satisfies the conditions in the problem.” (6.EE.6, 6.EE.7, MP1)

• Grade 7, End-of-Course Test, Exercise 38 states, “Describe the cross section that is formed by cutting a basketball in half.” Answer choices include: square, rectangle, circle, triangle. (7.G.3)

• Grade 8, Chapter 8, Chapter Test, Exercise 2 states, “Evaluate the expression. (-5)^{2}+18\div3= ____.” (8.EE.1)

The materials reviewed for Math & YOU 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.

Students are exposed to a variety of question types, modalities, and complexity levels to develop and demonstrate their understanding of course content. The modalities and question types provide different contexts and settings, ensuring students have opportunities to demonstrate their understanding through tasks that require them to reason and communicate in a variety of ways. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, short answer, and extended response.

Formative assessments occur at the lesson level through structures such as the Prerequisite Skills Test, Chapter Performance Task, In-Class Practice, and Connecting Big Ideas. These assessments provide opportunities for students to demonstrate their understanding of grade-level content standards through a variety of item types, including Drag and Drop, Fill in the Blank, Matching, Multi-Select, Response Matrix, Short Response, and Single Select.

Examples include: 

• Grade 6, Chapter 8 Test, Exercise 1 states, “Write a positive or negative integer that represents the situation. You have a debt of 41. An integer that represents the situation is ___.” Exercise 15 states, “The table shows the changes in the value of a toy on a website over a period of three weeks. During which week does the value of the toy change the most?” The table includes positive and negative dollar amounts. End-of-Course Test, Exercise 34 states, “Determine the type of number that can be used for each scenario. Pounds of strawberries, Owe 7 to a friend, Ground floor of a parking garage, Deep sea diving, Altitude of a bird.” Students can select positive integer, 0, or negative integer. The materials assess the full intent of 6.NS.5 as students understand that positive and negative numbers are used together to describe quantities having opposite directions or values.

• Grade 7, Chapter 5 Test, Exercise 10 states, “You mix \frac{2}{3} quart of blue paint for every \frac{1}{4} quart of red paint to make 5\frac{1}{2} quarts of purple paint. How much blue paint and how much red paint do you use?” Chapter 6, Big Idea Tasks, Exercise 2 states, “The balances in your bank account for three months are shown. a. What was the percent of change in the balance from Month 1 to Month 2? Month 2 to Month 3? Be sure to indicate whether the percent of change was a percent of increase or a percent of decrease. b. Determine the balance in Month 4 that makes each statement true. Justify your answer. i. The percent of change from Month 1 to Month 4 is 0%. Ii. The percentage of change from Month 1 to Month 4 is a 20% decrease. Iii. The percent of change from Month 1 to Month 4 is a 15% increase. c. For each balance in Month 4 that you found in part (b), determine the change from Month 3 to Month 4. i. The percent of change from Month 1 to Month 4 is 0%. Ii. The percent of change from Month 1 to Month 4 is a 20% decrease. iii. The percent of change from Month 1 to Month 4 is a 15% decrease. d. Your friend says that the percent of change from Month 1 to Month 2 plus the percent of change from Month 2 to Month 3 plus the percent of change from Month 3 to Month 4 is equal to the percent of change from Month 1 to Month 4. Do you agree? Mathematically justify your reasoning.“ The materials assess the full intent of 7.RP.3 as students use proportional reasoning to solve multistep ratio and percent problems, apply reasoning flexibly across real-world contexts, and justify their conclusions mathematically.

• Grade 8, Chapter 2, Big Ideas Task, Exercise 4 states, “You want to make a drawing that you have in your sketchbook into a large painting. The rectangular pages of your sketchbook are 9 inches by 12 inches. a. You want the painting to have similar dimensions as your drawing. You can buy a 21-inch by 28-inch, 45-inch by 60-inch, or 66-inch by 68-inch canvas. Which canvas would you choose for your painting? Justify your choice. b. Find the perimeters of the canvas you chose and the sketchbook page. Using the perimeters, how do you know the canvas and sketchbook page are similar? Explain your thinking. c. You decide that you want to move and dilate a 2-inch by 2-inch object that you already painted. The object is near the bottom left corner. You want to dilate the object by a scale factor of 3 with respect to the bottom left corner and move it to place it near the upper right corner. Describe a similarity transformation between the two objects. What tools can you use to demonstrate this transformation?” The materials assess the full intent of MP.1 as students must understand the purpose of the multi-step problem and persevere in selecting canvases and tools that support solving the task.

The materials reviewed for Math & YOU 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 begins with an opening activity, Dig In or Motivate, followed by student-centered explorations in Investigate. Learning continues with Key Concept and concludes with opportunities to make real-world connections in Connect to Real Life. The Student Practice workbooks offer additional opportunities for students to reinforce the knowledge and skills developed through each lesson.

Three correlation resources demonstrate alignment to the standards and show that students engage with the full intent of the standards throughout the course. The Standards Correlation (by Course) is organized by course component and identifies the standards addressed in each lesson. The Standards Correlation (by Standard) lists every lesson in which a specific standard is addressed. The Standards-Based Practice Correlation connects each Standards-Based Practice activity in the Practice Workbook to a content standard and identifies lessons where that standard is reinforced. Across the program, students have multiple opportunities to independently demonstrate their understanding of the full intent of the standards.

Examples include:

• Grade 6, Chapter 7, Lessons 1-3, engages students with the full intent of 6.G.1(Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.) Students apply their understanding of area to find missing dimensions and calculate the areas of triangles, parallelograms, and trapezoids, then extend this reasoning to a real-world context by approximating and comparing the population densities of two counties. Lesson 1, Practice, Exercise 10 states, “Find the missing dimensions of the parallelogram discussed. 10. b = 6 ft  h =___ft  A = 54ft^{2}.” Lesson 2, Practice, Exercise 3, “Find the area of the triangle.” A triangle with a base of 3 cm and a height of 4 cm is shown. Lesson 3, Practice, Exercise 12, “A trapezoid has a height of 12 yards and a base lengths of 5 yards and 7 yards. What is the area (in square feet) of the trapezoid?” Answer choices include: 72ft^{2}, 216ft^{2}, 648ft^{2}, 1260ft^{2}. Practice, Exercise 19 states, “You can use a trapezoid to approximate the shape of Inyo County, California. The population is about 19,000. About how many people are there per square mile? Compare your results to the number of people per square mile in Kern County, California.”

• Grade 7, Chapter 6, Lessons 2, 5, Practice Test, engages students with the full intent of 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.) Students use proportional reasoning to solve multistep ratio and percent problems. Lesson 2, Practice, Exercise 9 states, “You earn $18.75 per hour. This is 125% of the hourly rate you earned last year. How much did you earn per hour last year?” Lesson 5, Closure, Entry Ticket states, “The original price of a book is $10. The book is on sale for 50% off. The sale price is discounted an additional 50%. Does this mean that the book is free? Explain.” Practice Test with CalcChat, Exercise 14 states, “You estimate that there are 1,152 pennies in a jar. The actual number of pennies is 1,200. a. Find the percent error. b. What other estimate gives the same percent error?”  

• Grade 8, Chapter 7, Lesson 3, engages students with the full intent of 8.F.3 (Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.) Students interpret the equation y = mx + b as defining a linear function whose graph forms a straight line, using the rate of change (m) and initial value (b) to describe and analyze real-world relationships. Investigate, Exercise 1 states, “The table shows the cost of buying several movie tickets online. a. Represent the situation graphically. What do you notice? b. Does the situation represent a linear function? Explain. c. Interpret the rate of change and initial value in the situation.” Chapter 7 STEAM Performance Task, Exercises 1-4 state, “The Heat Index is a measure of how hot it feels on a warm day. When the humidity (the amount of moisture in the air) is high, sweat does not dry as quickly. So, the air feels hotter than it does during times of low humidity. When the relative humidity is 80% every 1° increase in temperature from 83\degreeF to 8\degreeF causes a 3\degree temperature increase in Heat Index. How can you use a function to represent this relationship? 1. Complete the table assuming the relative humidity is 80%. 2. Plot the ordered pairs and draw a line through the points. Then describe the pattern. 3. Write a linear function for this data. Explain your reasoning. 4. The Heat Index is an important indicator of dangerous temperatures. With 80% relative humidity, the Heat Index category changes from “danger” to “extreme danger” at a temperature of 94\degreeF. a. Using your model, what is the Heat Index in this situation? b. Research the actual Heat Index for this temperature. According to your research, does the relationship between temperature and the Heat Index when the relative humidity is 80% represent a linear function? Explain.”

The materials reviewed for Math & YOU 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 chapters devoted to major work of the grade (including assessments and related supporting work) is 8 out of 10, approximately 80%. 

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 36 out of 56, approximately 64%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 115 out of 163, approximately 71%.

Grade 7:

• The approximate number of chapters devoted to major work of the grade (including assessments and related supporting work) is 8 out of 10, approximately 80%. 

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 37 out of 52, approximately 71%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 110 out of 161, approximately 68%.

Grade 8:

• The approximate number of chapters devoted to major work of the grade (including assessments and related supporting work) is 8 out of 10, approximately 80%. 

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 45 out of 53, approximately 85%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 126 out of 157, approximately 80%.

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 71% of the materials in Grade 6, 68% of the materials in Grade 7, and 80% of the materials in Grade 8 focus on major work of the grade.

The materials reviewed for Math & YOU 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/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers within the Standards Correlation (by Course). 

An example of a connection in Grade 6 includes:

• Chapter 8, Lesson 6, Student Edition 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 draw a diagram of the rectangular floor of a tree house in the coordinate plane using three given vertices, and then determine the area of the floor. In-Class Practice, Exercise 7 states,  “You design a tree house using a coordinate plane in which the coordinates are measured in feet. Three vertices of the rectangular floor are (-2,-3), (-2,4), and (5,4). Draw a diagram of the situation. Then find the area of the floor.”

An example of a connection in Grade 7 includes:

• Chapter 7, Lesson 2, Student Edition connects 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 use a theoretical probability and a proportion to find the number of marbles in a bag. Practice, Exercise 19 states, “Each donor at a charity raffle randomly draws a marble from a bag and replaces it. A blue marble represents a win, and a red marble represents a loss. The theoretical probability of drawing a winning marble is \frac{3}{10}. The bag contains 9 winning marbles. a. How many marbles are in the bag? b. Out of 20 donors, how many do you expect to win?”

An example of a connection in Grade 8 includes: 

Chapter 9, Lesson 5, Student Edition 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 root and cube root symbols to represent solutions to equations of 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 use the Pythagorean Theorem to approximate the distance between two points to the nearest tenth. In-Class Practice, Self Assessment, Exercise 6 & 7 state,“Approximate the distance between the points to the nearest tenth. 6. (-3, -1) and (-2, -2). 7. (-7, 10) and (3, -5).”

The instructional materials for Math & YOU 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 are present throughout the materials where appropriate. These connections are listed for teachers Standards Correlation (by Course) within the Digital Teaching Experience under Teacher Toolkit: Course Essentials and may appear in one or more phases of a typical lesson: Example, In-Class Practice, and Practice Exercises. 

An example of a connection in Grade 6 includes:

• Chapter 6, Lesson 3, Student Edition, connects the major work of 6.NS.A (Apply and extend previous understandings of multiplication and division to divide fractions by fractions) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities). Students solve a one variable equation by dividing a fraction by another fraction. Practice Exercise 5 states,  “Solve the Equation. \frac{2}{3}=\frac{1}{4}k.”

An example of a connection in Grade 7 includes:

• Chapter 5, Lesson 4, Student Edition, 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 write and solve proportions to determine the number of calories that could be burned in a problem. In-Class Practice Exercise 8 states,  “You burn 35 calories every 3 minutes running on a treadmill. You want to run for at least 15 minutes, but no more than 30 minutes. Use proportions to find the possible numbers of calories that you can burn.”

An example of a connection in Grade 8 includes:

• Chapter 7, Lesson 2, Student Edition, connects the major work of 8.F.A (Define, evaluate, and compare functions) to the major work of 8.F.B (Use functions to model relationships between quantities). Students represent the relationship between miles driven and fuel remaining by writing a function and graphing it to show how the number of gallons decreases as distance increases. In-Class Practice Exercise 8 states,  “Your car burns 4 gallons of gasoline for every 100 miles you drive. You currently have 14 gallons of gasoline in your tank. Write and graph a function that describes the number of gallons of gasoline remaining in the tank after you drive x miles.”

The instructional materials reviewed for Math & YOU Grades 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. 

The materials provide multiple features to support coherence across grade levels. Co-Author Notes, Insight Videos, and Coherence details in chapters and lessons explain how current learning builds from prior learning and extends to future learning. Nick’s Notes Overview in each chapter provides coherence perspectives through “What we’re doing,” “Why we’re doing it,” and “Essential background” insights. Standards for Content and Mathematical Practice provides COHERENCE Throughout the Grades presents chapter charts that show learning progressions, available in both the Teacher Edition and the Digital Teaching Experience, with Common Core standard codes for reference to prior, current, and future learning. Each lesson includes COHERENCE Through the Chapter section in the overview that summarizes the lesson focus within the broader learning progression with preparing, learning, and extending for content standards.

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

• Chapter 6, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Throughout the Grades, connects 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers) to the operational thinking in Grade 7, where students build upon this learning and write and solve one- and two-step equations (7.EE.4a). Nick’s Note Overview states, ​“In this chapter, students will reason about different types of equations and solve one-variable equations. They will begin by writing equations in one variable and creating equations to represent real-life problems. Students will learn that a solution of an equation is a value of the variable that makes the equation true. Students will also discover that they can use inverse operations to solve equations. They will use properties of equality to solve one-step equations involving nonnegative rational numbers. You want students to understand that recording their steps is part of the equation-solving process. Although addition, subtraction, multiplication, and division are used to solve the equations in this chapter, you do not want students to think they must use an inverse operation to solve all equations. For example, multiplying by the reciprocal is a more efficient method for solving an equation with a fractional coefficient. Because students are working with one-step equations in this chapter, they may question why they need to show their work when they can solve an equation using mental math. Acknowledge their thinking and explain that the equation-solving process will become more complex as they extend the process to more complicated equations. In Grade 7, students will write and solve simple equations, including equations involving negative rational numbers. They will also learn to solve two-step equations and compare algebraic solutions to arithmetic solutions. In Grade 8, students will further extend their understanding of equations to write and solve multi-step equations and equations with variables on both sides.”

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

• Chapter 2, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Throughout the Grades, connects 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem) to the previous work in Grade 5, where students divide whole numbers by unit fractions and divide unit fractions by whole numbers (5.NF.7). Nick’s Note Overview states, ​“This chapter begins with a section on adding and subtracting fractions and mixed numbers. Although students completed this learning progression in Grade 5, this section provides an opportunity for students to revisit representing fractions and mixed numbers using area models, number lines, and tape diagrams. Then these models can be used to support students as they multiply and divide fractions and mixed numbers. In the second half of the chapter, students will perform operations with decimals. They will begin by adding and subtracting decimals. Be sure to reinforce the concept of working with like place values, just as you do with whole numbers. As students revisit multiplying decimals, they should look for patterns and use strategies based on place value to determine where to place the decimal point within a product. Students should discover that multiplying two decimals will result in the same digits as multiplying two corresponding whole numbers, regardless of the positions of the decimal points. It is important for students to investigate these relationships before using the standard algorithm to multiply decimals. Before dividing decimals, students will use the standard algorithm to divide multi-digit whole numbers. Then they will learn that dividing two decimals will result in the same digits as dividing two corresponding whole numbers, regardless of the positions of the decimal points. Students should explore this relationship before formalizing the standard algorithm for dividing decimals. In previous grades, students were introduced to many of the ideas that they will explore and formalize in this chapter. Decimal operations were introduced in Grade 5, but fluency is expected by the end of Grade 6. Understanding place value, how to write decimals as fractions, and models that represent fractions and decimals are essential to understanding the concepts in this chapter.”

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

• Chapter 10, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 7.G.6 (Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms) to the future work in Grade 8, where students solve problems involving the volume of cylinders, cones, and spheres (8.G.9). Nick’s Notes Overview states, “Three-dimensional objects exist in the world around us. This chapter provides experiences for students to make sense and notice the geometric beauty that exists in our day-to-day life. Gather three-dimensional models for students to engage with and discuss the attributes of each real-life object. In future grades, students will continue their work with three-dimensional figures and volumes to include volumes of cones, cylinders, and spheres.”

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

• Chapter 1, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram) to the previous work in Grade 6, where students learned to graph integers and rational numbers and their opposites on a number line (6.NS.6). Nick’s Notes Overview states, “In the previous grade, students completed their operations work with positive fractions by extending previous understandings to divide fractions by fractions. They sought to achieve fluency with dividing multi-digit numbers and performing operations with multi-digit decimals. Students have also explored the concept of rational numbers and their locations on a number line. They learned that the negative side of a number line operates as a mirror-like reflection of the positive side and that the absolute value of a number is the distance between the number and 0 on a number line.”

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

• Chapter 7, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 8.F.2 (Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) to future work in Algebra, as students learn to use function notation to evaluate, interpret, and graph functions and begin to graph linear and quadratic functions in different forms (HSA.CED.2). Nick’s Notes Overview states, “Understanding how functions can be used to model relationships between quantities will better prepare students for creating equations of linear, exponential, and quadratic functions in higher mathematics courses. In future courses, students will use function notation to evaluate, interpret, and graph functions. They will learn to graph linear and quadratic functions in different forms.”

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

• Chapter 1, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 8.EE.7 (Solve linear equations in one variable) to previous work in Grade 7, where students applied properties of operations to simplify, add and subtract linear expressions with rational coefficients (7.EE.1). Nick’s Notes Overview states, “In this chapter, students will extend their understanding of expressions and equations from prior grades. Writing and solving an equation with infinitely many solutions or no solution is new for students. In the first two sections, students will revisit writing and solving simple and multi-step equations. They will begin by modeling equations with algebra tiles and then extend their understanding to equations with rational-number coefficients. Then students will explore equations with variables on both sides and learn to identify whether an equation has one solution, no solution, or infinitely many solutions. In the previous grade, students wrote and solved equations of the form px+q=r and p(x+q)=r.”