6th Grade - Gateway 1

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

The curriculum is divided into 18 Scopes, and each Scope contains a Standards-Based Assessment used to assess what students have learned throughout the Scope. Examples from Standards-Based Assessments include:

• Scope 2: Integers, Evaluate, Standards-Based Assessment, Question 7, “Coco has $47.15 in her checking account. Part A What does this value tell us? Explain your reasoning by using a positive or negative value. Enter your answer in the box. Part B What does a value of zero mean in this situation? Enter your answer in the box.” (6.NS.5)

• Scope 8: Algebraic Expressions, Evaluate, Standards-Based Assessment, Question 2, “Juan bought light bulbs and leaf bags from a home improvement store. The expression 0.99b+2.15l represents the total cost. Which statements are true about the expression? Select all that apply. The expression is the sum of the cost of two different items.; The total cost is the product of the terms.; Each coefficient of the terms is greater than 1.; Each term represents the cost of the individual items bought.” (6.EE.2)

• Scope 10: Ratios, Rates, and Unit Rates, Evaluate, Standards-Based Assessment, Question 3, “Maggie uses 4 cups of water for every 1 cup of distilled vinegar for a cleaning solution. Which statements are true? Select all that apply. For every 8 cups of water, she uses 2 cups of distilled vinegar.; For every 10 cups of water, she uses 3 cups of distilled vinegar.; For every 20 cups of water, she uses 5 cups of distilled vinegar.; For every 26 cups of water, she uses 7 cups of distilled vinegar.; For every 32 cups of water, she uses 8 cups of distilled vinegar.” (6.RP.2)

• Scope 16: Understand Variability, Evaluate, Standards-Based Assessment, Question 5, “The following questions were asked about houses in a neighborhood. Which of the following are non-statistical questions? Select all that apply. How old is the oldest house on the street?; How many bedrooms does each house on the street have?; How many houses are in the neighborhood?; How many square feet of living space is there in each house?” (6.SP.1)

• Scope 17: Represent and Interpret Data, Evaluate, Standards-Based Assessment, Question 3, students see a histogram with the label Number of Students on the Y-Axis and the label, Sit-Ups and 0-19, 20-29, 30-39, 40-49, and 50+ along the X-Axis. “The following histogram is supposed to represent the number of sit-ups students completed in two minutes. However, there are two errors in the histogram. Identify these errors. Explain your reasoning. Enter your answer below.”(6.SP.4)

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

The materials provide extensive work in Grade 6 as students engage with all CCSSM standards within a consistent daily lesson structure, including Engage, Explore, Explain, Elaborate, and Evaluate. Intervention and Acceleration sections are also included in every lesson. Examples of extensive work to meet the full intent of standards include:

• Scope 5: Coordinate Plane Problem Solving, Explore, Explore 2–Polygons on a Coordinate Plane, engages students in extensive work to meet the full intent 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.) In Procedure and Facilitation Points, students work in groups to use coordinates to draw polygons. “Part I: Pecan Park East, 1. Read the following scenario to students: Pecan Park received funding to upgrade some of their equipment and facilities. The work is being done in two phases. In Phase 1, Pecan Park East is redesigning the layout of the climber dome, slides, baseball fields, and swings. The park chairperson wants to ensure that none of the equipment or facility areas overlap. Each piece of equipment or facility area is represented by a polygon, and the coordinates of the vertices of each area are given. Plot the coordinates of the vertices, and connect them to mark each area on the coordinate plane to determine whether any areas overlap. 2. Give a Student Journal to each student. 3. Give a set of the Part I: Pecan Park East Cards to each group. 4. Students will work cooperatively to read each card and plot the vertices for each piece of equipment or facility area on the coordinate plane on page 1 of the Student Journal. Once the vertices from each piece of equipment or facility area have been plotted, students will draw a polygon by connecting the points. Students will then use the completed park map to determine whether any of the equipment or facility areas overlap. 5. Monitor student collaboration, and use the following guiding questions to assess understanding: a. DOK–1 What polygon is the climbing dome shaped like? b. DOK–1 How are the coordinates for the slide different from the coordinates of the other equipment or areas? c. DOK–1 How do you plot the coordinate (−2.5, −2.5)? d. DOK–1 Why is it important for these areas to not overlap? Part II: Pecan Park West 1. Read the following scenario to the students: Pecan Park West will hold the basketball court, the sandpit, and the picnic area. Each of these facilities requires additional work to complete the upgrade. Each facility is represented by a polygon, and the coordinates of the vertices are given. Plot the coordinates, and connect them to mark each area on the coordinate plane. Then, help the chairperson determine the additional calculations required to complete the upgrades. 2. Give a set of the Part II: Pecan Park West Cards to each group. 3. Have students work in their groups to plot the vertices for each facility area on the coordinate plane on page 2 of the Student Journal. Once the vertices from each facility area have been plotted, students will draw a polygon by connecting the points. Students will then use the completed park map to determine the additional calculations required to complete the upgrades on the Student Journal. 4. Monitor student collaboration, and use the following guiding questions to assess understanding: a. DOK–1 What is the distance between −5 and −2?   b. DOK–1 Why are there 6 grid boxes on the graph between the coordinates (−5, −2.5) and (−2, −2.5)? c. DOK–1 How do you find the area of a rectangle? d. DOK–1 How do you find the perimeter of a rectangle? e. DOK–1 How do you determine when to calculate the area and when to calculate the perimeter?” Exit Ticket, students continue to use coordinates to determine distance on a coordinate plane. “The sandpit at Pecan Park West is so popular that officials have decided to add one additional sandpit. The vertices of the sandpit have coordinates of (−3, 3), (−5, 3), (−5, 5), and (−3, 5). Determine the number of units of wooden border and weather-proof liner needed to complete the upgrade.”

• Scope 8: Algebraic Expressions, Explore 1 and 4, engages students in extensive work to meet the full intent of 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents.) In Explore 1–Writing Expressions, Math Chat and Exit Ticket, students work with a variable to write expressions that represent given situations. In Math Chat (a table is given with the heading: Questions, and another heading: Sample Student Responses), “DOK–1 Group 1 wrote 3 \cdot x as their expression, while Group 2 wrote 3(x). Which group is correct? DOK–1 Why should we not use an x to represent multiplication? DOK–2 How would the expression change if it were the quotient of 3 and x? DOK–1 Are the expressions \frac{1}{3}x and \frac{x}{3} equivalent?” Exit Ticket, students complete a table, creating algebraic expressions for given scenarios. “Read each statement. Write the algebraic expression that describes each statement. Statement, Algebraic Expression, 4 times x, 4 decreased by x, the quotient of x and 4, 4 times the difference of 4 and x, 4 plus the product of 4 and x, the product of 2 and x.” In Explore 2-Evaluate Expressions, students work in groups to solve algebraic expressions. “Part I, 1. Post the expressions 2\cdot3 and 2(3) on the board. Ask the following question: a. DOK–1 Are these expressions equivalent? 2. Post the expressions 2(3)and 2(x) on the board. Ask the following question: b. DOK–1 Are these expressions equivalent? 3. Post 2(3) and "2(x) when x=3” on the board. Ask the following question: a. DOK–1 Are these expressions equivalent? 4. Explain to students that this is called substitution. Substitution is when you replace a variable in an algebraic expression with a known value. Check for understanding by asking the following questions: a. DOK–1 What is the value of 3(x) when x=4? 3(4)=12 b. DOK–1 What is the value of 5x when x=2? 5x means 5(x). 5(2)=10 5. Read the following scenario to students: A group of students from South Sydney Middle School went to the National Zoo of Australia on a field trip. The students were split into groups, each chaperoned by an adult. At lunchtime, groups examined the menu and compiled their orders into an algebraic expression, which included a tip of $3. Each group had a $40 limit to spend on lunch. Prior to placing the orders, the chaperones wanted to ensure that each group’s order was under budget. Evaluate the expressions representing each group’s order to determine their total and whether they were under or over their $40 limit. 6. Give a Student Journal to each student and an Outback Snack Shack Menu to each group. 7. Explain to students that they will use the prices on the Outback Snack Shack Menu to evaluate the expressions for each group to determine the total cost of lunch. Remind students of the $40 limit. 8. Allow students to collaborate with their groups to complete Part I of the Student Journal. As they work, ask the following questions: a. DOK–1 How do you know which mathematical operation to perform first? b. DOK–1 What does the variable s represent? c. DOK–1 What is the total cost for Group 5’s order? d. DOK–1 How do you determine what to remove from an order if a group is over the $40 budget? Part II, 1. Read the following scenario to students: Groups 7 and 8 lost their lunch orders, but they know how much they spent in relation to Group 6. They read their verbal descriptions to their chaperones to determine how much each group spent on their lunch orders. 2. Explain to students that they will use the verbal descriptions in Part II on the Student Journal to write algebraic expressions to represent the total spent by each group, and then find the total of all three groups. 3. Upon completion of Part II, have students answer the reflection questions on the Student Journal.” 

• Scope 10: Ratios, Rates, and Unit Rates, Explore, Explore 1-3 engage students in extensive work to meet the full intent of 6.RP.1 (Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.) In Explore 1 - Ratios, Procedure and Facilitation Points, students use a Fruit Stand Scenario to understand a ratio. “1. Read the following scenario: Trixie is picking fruit from her fruit farm to sell at the local farmers’ market fruit stand. Each week, she picks different fruits, depending on that week’s demand. She wants to keep a log of each week’s demand by representing the numbers of each type of fruit in the form of a ratio. Help her determine the ratio between each type of fruit that she brings to the market each week. 2. Display the Fruit Stand Card for week 1. a.  Discuss the following questions with the students:How many blueberries are there for week 1? There are 6 blueberries. b. How many strawberries are there for week 1? There are 8 strawberries. c.We want to show the relationship of strawberries to blueberries in week 1. (Write out, “There are ___ strawberries for every ___ blueberries.To model the relationship between strawberries and blueberries, what would we need to fill in the blanks with? In the first blank we would need to put 8 because there are 8 strawberries. In the second blank we will need to write 6 because there are 6 blueberries. d. Explain to class: Mathematicians call this relationship a ratio. Mathematicians write ratios in the order that the ratio asks.  In this scenario, we want to represent strawberries to blueberries, therefore we write the number for strawberries in the first spot and the number for blueberries in the second spot. We can also write the ratio for the information shown in week 1’s Fruit Stand Card in the following ways: “For every 8 strawberries, there are 6 blueberries”; “8:6”; “8 to 6”; or “There are 8 strawberries for every 6 blueberries.’ e. How can we draw a tape diagram to represent the relationship between strawberries and blueberries? Draw one tape diagram with 8 spaces to represent the strawberries. Below draw a tape diagram that is only 6 spaces to represent the 6 blueberries.Model for students how to draw the tape diagram model for 8 strawberries to 6 blueberries. f. As you work through each week’s scenario, we will learn other ways to represent ratio relationships as well. 3. Give one Student Journal to each student and one set of Fruit Stand Cards to each group. 4. Students will work collaboratively to represent the information on each Fruit Stand Card. They will record the number of each type of fruit, draw a tape diagram to show the number of each type of fruit, and then write the ratio describing their models in two different ways. 5. Provide linking cubes to students who are struggling. Have students model the ratio with linking cubes by linking the correct amount together for each fruit in the scenario. Then students can draw a model of their linking cubes as a tape diagram on the Student Journal. In Explore 2 - Ratio Tables and Graphs, Procedure and Facilitation Points PART II,  students use ratio language to describe a ratio relationship between two quantities. “1. Read the following scenario: Trixie wants to use the ratios of fruit bushes in her current farm to determine the number of bushes she should plant in the expansion. Using the Purchasing Fruit Bushes Cards – Part II, determine the ratios for each comparison to help Trixie find equivalent ratios of fruit bushes that she could plant at the farm. 2. Give a Purchasing Fruit Bushes Card – Part II to each group. 3. Students will work collaboratively to use the information on the Purchasing Fruit Bushes Cards – Part II to determine the ratio for raspberry bushes to blueberry bushes and the ratio for strawberry bushes to blackberry bushes that Trixie has in her garden. Then they will use these ratios to complete the ratio tables for each comparison and answer the questions that follow on the Student Journal.

• Scope 11: Percents, engages students in extensive work to meet the full intent of 6.RP.3c (Find a percent of a quantity as a rate per 100…; solve problems involving finding the whole, given a part and the percent.) Explain, Show What You Know - Part 1: Represent Percents Using a 100s Grid, students need to determine the equivalent fractions by using 100s grid. For example, “Solve each problem. Show your work using a 100s grid and/or by determining equivalent fractions. \frac{6}{20} of the students walk to school each day. What percent of students walk to school? Workspace with a 100s grid. Solution Statement: ___” Show What You Know - Part 2: Solving Percent Problems Using Benchmark Fractions and Percents, students complete missing information for a given scenario, “Scenario: Vera poured 500 mL of water into her bottle at the start of the day. By noon, she had drunk 25% of the water. How many mL of water had Vera drunk by noon?”

• Scope 17: Summarize Numerical Data, Explore 1-Mean and Median and Show What you Know-Part 1: Mean and Median, engage students with extensive work to meet the full intent of 6.SP.2 (Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.) Explore 1, Procedure and Facilitation Points, students work in groups to find the center and create dot plots. “Part I: Mean as Balance Point 1, Read the following scenario to the class: José is tracking his total number of home runs and total number of hits for several series of baseball games he has played in. José wants to determine how many home runs and how many total hits he would have to have in each series to have balanced out his hits and home runs evenly between each series. Help José find the mean as a balance point to determine the number of home runs and the number of total hits he would have to have in each series to balance the home runs and total hits. 2. Give a Student Journal to each student. 3.Give a set of José’s Baseball Scenario Cards and linking cubes to each group. 4. Have students discuss what the term balance point means. Explain to students that the balance point is the mean of a data set. 5.Have students use José’s Home Runs scenario card to record the number of home runs hit in each series on the table in the Student Journal. Students will also create a dot plot to represent the number of home runs hit in each series. They will use the linking cubes to represent the data points for José’s home runs. Note: One linking cube represents one data point (table). For example, in series 1 José has 3 home runs = 3 linking cubes. 6. Have students stack the linking cubes to represent each series. Note: Series 1 should have 3 cubes, series 2 should have 1 cube, series 3 should have 5 cubes, series 4 should have 5 cubes, and series 5 should have 1 cube. Once the correct number of linking cubes is stacked for each series, students should figure out how many cubes each series should have to have an equal number of home runs hit by redistributing the stacks so that each stack has an equal amount of cubes. Students will only use cubes for José’s Home Runs scenario card. 7. Actively monitor students as they are working with their groups using the cubes to understand mean as the balance point. Ask questions such as the following: a. DOK–1 How can you use the linking cubes to determine the balance point? b. DOK–2 How do you think the balance point would be affected if we added more data points above the balance point? c. DOK–2 How can you interpret mean as the balance point using a dot plot? 8. Allow time for students to input the data from both of José’s Baseball Scenario Cards and create their dot plots. They will analyze each dot plot by answering questions and then answer the Part I reflection questions. Part II: Finding the Center and Shape of Data 1. Read the following scenario to the class: Sammy wants to track the total number of times he is up to bat and the total number of hits that he has for the past seven series of baseball games he has played in. Sammy will use his data to determine the middle value and the average number of times that he is up to bat as well as the middle value and average number of hits he has per series. Help Sammy find the center and shape of data using the histogram. 2. Give a set of Sammy’s Baseball Scenario Cards to each group. 3. Encourage students to discuss their observations with their groups as they work through Sammy’s Baseball Scenario Cards. 4. Explain to students that they will use Sammy’s Baseball Scenario Cards to record Sammy’s total times at bat and Sammy’s total hits in the frequency tables on the Student Journal. They will use the frequency tables to create histograms. 5. Monitor and assess students as they are working by asking the following questions: a. DOK–2 Why do you have to arrange the numbers from least to greatest? b. DOK–1 How do you find the middle value if there are two middle numbers? c. DOK–1 How can you determine the mean in a data set? d. DOK–1 How can you identify the shape of the data distribution? 6. Ask students, “What other term might you use for the middle value?” Accept all reasonable answers. Inform students that mathematicians call the middle value the median.” On the Exit Ticket, students are given a table of numbers representing the number of hits in recent games. “Jerry recorded the number of hits he has had in his most recent baseball games. Use the information in the table to create a dot plot that shows the number of hits Jerry has had. 30, 31, 33, 33, 34, 34, 34, 35, 35, 35, 35, 35, 36, 36, 36, 36, 37, 37, 38, 40, Title:, 1. Determine the mean of the data set. 2. Determine the median of the data set. 3. What is the shape of the data?

The materials reviewed for STEMscopes Math Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major cluster of each grade.

The instructional materials devote at least 65% of instructional time to the major clusters of the grade:

• The approximate number of scopes devoted to major work of the grade (including assessments and supporting work connected to the major work) is 12 out of 18, approximately 67%.

• The number of lesson days and review days devoted to major work of the grade (including supporting work connected to the major work) is 118 out of 155, approximately 76%.

• The number of instructional days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 132 out of 180, approximately 73%.

An instructional day analysis is most representative of the instructional materials because this comprises the total number of lesson days, all assessment days, and review days. As a result, approximately 73% of the instructional materials focus on the major work of the grade.

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

Materials are designed so supporting standards/clusters are connected to the major standards/ clusters of the grade. Examples of connections include:

• Scope 7: Equivalent Numerical Expressions Explain, Show What You Know–Part 1: Greatest Common Factors, connects the supporting work of 6.NS.4 (Find the greatest common factor or two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12…) to the major work of 6.EE.3 (Apply the properties of operations to generate equivalent expressions...) and 6.EE.4 (Identify when two expressions are equivalent…) Students use the greatest common factor to generate equivalent expressions. “A florist is making mixed bouquets with roses and carnations. The florist has 36 roses and 48 carnations. Each bouquet must have the same combination of flowers, and all flowers must be used. What is the greatest number of mixed bouquets that the florist can make? A table shows blanks for factors for Roses and Carnations, Common Factors, and Greatest Common Factor. ___ bouquets can be made. ___ roses and ___ carnations in each bouquet.”

• Scope 8: Algebraic Expressions, Explore, Explore 4–Evaluate Expressions, Exit Ticket, connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers). Students evaluate expressions given decimal numbers to determine the cost of a meal. “Outback Snack Shack, Exit Ticket, 1. Based on the condensed menu to the right, the following expression was ordered: 5(h+s)+2f+3 Use substitution to determine the total cost of the order.” Hamburger, h, $4, Salad, s, $4.25, Fries, f, $3, and Apple Slices, a, $3.50.

• Scope 14: Area and Volume, Explore, Explore 1–Discovering Area Formulas, Exit Ticket, connects the supporting work of 6.G.1 (Find the area of right triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes…) to the major work of 6.EE.3 (Apply the properties of operations to generate equivalent expressions.) Students find the area of a parallelogram by making it into a rectangle and subtracting the area of two triangles. Students provide the formula and then evaluate the expression.  “X-traordinary Landscaping Company is creating a blueprint of the Tranquility Garden and wants to know the area of the garden. Decompose and rearrange the garden figure to create a new garden in the shape of a rectangle. Label the base and the height of each garden. Find the area in square units of both gardens using the area formula. Each grid square is 1 square unit. Tranquility Garden, b = ___ units, h = ___ units, How can you find the area of this garden? Formula: Area:, New Garden, b = ___ units, h = ___ units, How can you find the area of this garden? Formula: Area”

The materials for STEMscopes Math Grade 6 meet expectations that materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Materials are coherent and consistent with the Standards. These connections are sometimes listed for teachers in one or more of the three sections of the materials: Engage, Explore and Explain. Examples of connections include:

• Scope 6: Positive Rational Number Operations, Explore, Explore 3–Division of Fractions, Procedure and Facilitation Points connects The Number System domain to the Ratios & Proportional Relationships domain. Students use visual and algebraic representations to show their division of fractions. “1. Read the following scenario to students: Demarcus needs to determine if there will be enough fabric to create tablecloths for the tables at the fundraiser. Model division of fractions using number lines to determine how many tablecloths can be made using each color of fabric. 2. Give one set of Tablecloth Fabric Cards to each group and one Student Journal to each student. 3. Ask the class the following questions: a. DOK-1 How can you model division of fractions using the pattern we discovered in the previous Explore activity? We can multiply the first numerator by the second denominator and then multiply the first denominator by the second numerator. b. Explain new vocabulary to the class: Mathematicians call this the reciprocal or multiplicative inverse. 4. Have students work with their groups to determine the amount of tablecloths that can be made with each color of fabric. Students should start with the blue fabric card before moving on to the remainder of the Tablecloth Fabric Cards.”

• Scope 9: Equations and Inequalities, Explain, Show What You Know–Part 1: Write, Model, and Solve Equations, 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 create a model based on a scenario, then write an algebraic equation and solve it. For example, “Complete the missing information. Scenario: Four candy bars of the same type are lined up end to end and measure 20 inches. What is the length of each candy bar? Model ___; Equation ___; Solution Statement ___”.

• Scope 10: Ratios, Rates, and Unit Rates, Explain, Show What You Know –Part 3: Rates and Unit Rates, connects the Ratios & Proportional Relationships domain to the Expressions & Equations domain. Students find the unit rate based on the given scenario. For example, “Complete the missing information. Scenario: A train travels at a constant rate. It travels 360 miles in 6 hours. What is the train’s rate of speed per hour? Strategy ___; Unit Rate ___.”

• Scope 18: Summarize Numerical Data, Explain, Show What You Know–Part 4: Comparing Different Representations of the Same Data, connects the supporting cluster 6.SP.B (Summarize and describe distributions) to the supporting work of 6.SP.A (Developing understanding of statistical variability). Students answer some statistical questions based on the data provided in different graphs within the context. For example, “A convenience store kept track of how many bottles of sunscreen were sold each day. The results are shown on a box plot, a dot plot, and a histogram. Analyze each representation, and answer the questions on the second page. Justify each response by referencing and explaining a specific data representation. Three graphs are provided Graph 1 is a box plot, with min 1 max 8, Q1 at 3, Q2 at 4, and Q3 at 7; Graph 2 is a dot plot, one dot at 1, one dot at 2, 3 dots at 3, 3 dots at 4, 2 dots at 5, 2 dots at 7, and 2 dots at 8; Graph 3 is a histogram, horizontal axis labeled x, step 2, vertical axis labeled y and frequency, the first bar has frequency 2, the 2nd bar has frequency 6, the third bar has a frequency 2, and the fourth bar has frequency 6. Justify each response by referencing and explaining a specific data representation. Question 1: For how many weeks did the convenience store collect the data?; Question 2: What is the approximate center or typical amount of bottles sold each day?; Question 3: What is the median or middle amount of bottles sold?; Question 4: What is the range?; Question 5: Are there any gaps in the data?; Question 6: Between which two values did 50% of all sales fall?”

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

Prior and future connections are identified within materials in the Home, Content Support, Background Knowledge, as well as Coming Attractions sections. Information can also be found in the Home, Scope Overview, Teacher Guide, Background Knowledge and Future Expectations sections. 

Examples of connections to future grades include:

• Scope 2: Integers, Home, Content Support, Coming Attractions, connects 6.NS.C (Apply and extend previous understandings of numbers to the system of rational numbers.) to work in grade 7. “Seventh-grade students extend their understanding of positive and negative numbers to understand subtraction of rational numbers as adding the additive inverse, p-q=p+(-q). They will relate the distance between two rational numbers as the absolute value of their difference and apply this principle in real-world contexts.”

• Scope 7: Equivalent Numerical Expressions, Home, Scope Overview, Teacher Guide, Future Expectations, connects 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions.) with future work in grade 7. “Visual representations and concrete models can help students develop understanding as they move toward using abstract symbolic representations in seventh grade. Students build on their understanding of multiples, factors, and mathematical properties to generate and use the arithmetic of rational numbers.”

• Scope 16: Understand Variability, Home, Content Support, Coming Attractions, connects 6.SP.1 (Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers…) to the future work. “Students in seventh grade use random sampling to draw inferences about populations, they investigate chance processes, and they develop, use, and evaluate probability models. In eighth grade, students investigate patterns of association in bivariate data. This work extends into high school, where students continue to interpret categorical and quantitative data, and then explore conditional probability and the rules of probability.”

Examples of connections to prior grades include:

• Scope 4: Coordinate Planes, Home, Content Support, Background Knowledge connects 6.NS.C (Apply and extend previous understandings of numbers to the system of rational numbers.) to work done in previous grades. “In previous grades, students understood positive rational numbers as points on a number line. They have represented real-world and mathematical problems by graphing and interpreting points in the first quadrant of the coordinate plane.”

• Scope 11: Percents, Home, Scope Overview, Teacher Guide, Background Knowledge, connects 6.RP.3c (Find a percent of a quantity as a rate per 100…; solve problems involving finding the whole, given a part and the percent.) to previous work in grades 4 and 5. “Students are introduced to the concept of rate and multiplicative comparisons beginning in grade four. In fourth grade, students use two-column tables to record conversions between measurements. In Grade 5, students built on multiplicative comparisons to interpret multiplication as scaling and to apply an understanding of fractions as decimals. Fifth-grade students plotted points on the coordinate plane, and they generated and graphed numerical patterns. Prior to this scope, sixth-grade students studied the concept of a ratio as an association between two quantities, and they used ratio language to describe ratio relationships. Tape diagrams, double number lines, tables, and graphs are featured strategies applied when solving mathematical and real-world problems. Experiences with ratio and unit conversion tables provide the foundation for understanding percents, which begins in grade six.” 

• Scope 13: Dependent and Independent Variables, Home, Scope Overview, Teacher Guide, Background Knowledge connects 6.EE.C (Represent and analyze quantitative relationships between dependent and independent variables.) to work prior to the 6th grade. “In previous grades, students have generated numerical patterns involving positive rational numbers. Students generated two corresponding numerical patterns and analyzed the relationship between them using an input/output table and by graphing the corresponding ordered pairs on the coordinate plane."