6th to 8th Grade - Gateway 1

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

Program assessments include Diagnostic Assessments (Check Your Readiness), Cool-downs, Checkpoints, Mid-Unit Assessments, and End-of-Unit Assessments which are summative. Materials for Grade 6 contain nine units. Five of these units include Mid-Unit Assessments, and eight units contain End-of-Unit Assessments, both of which are summative. Unit 9 consists of 11 optional lessons and does not include an End-of-Unit Assessment, as there is no course-level assessment. According to the Assessment Guidance, “At the end of each unit is the End-Of-Unit Assessment. These assessments have a specific length and breadth, with problem types that gauge students’ understanding of the key concepts of the unit while also preparing students for new-generation standardized exams. Problem types include multiple-choice, multiple response, short answer, restricted constructed response, and extended response. Problems vary in difficulty and depth of knowledge.”

A Grade 6 example of an End-of-Unit Assessment includes:

• Unit 3, Unit Rates and Percentages, End-of-Unit Assessment Option B, Problem 4 states, “It takes Andre 4 minutes to swim 5 laps. Type your answers in the boxes. a. How many laps per minute is that? b. How many minutes per lap is that? c. At that rate, how long does it take Andre to swim 22 laps?” (6.RP.2, 6.RP.3b)

Program assessments include Diagnostic Assessments (Check Your Readiness), Cool-downs, Checkpoints, Mid-Unit Assessments, and End-of-Unit Assessments which are summative. Materials for Grade 7 contain nine units. Two of these units include Mid-Unit Assessments, and eight units contain End-of-Unit Assessments, both of which are summative. Unit 9 consists of 12 optional lessons and does not include an End-of-Unit Assessment, as there is no course-level assessment. “At the end of each unit is the End-Of-Unit Assessment. These assessments have a specific length and breadth, with problem types that gauge students’ understanding of the key concepts of the unit while also preparing students for new-generation standardized exams. Problem types include multiple-choice, multiple response, short answer, restricted constructed response, and extended response. Problems vary in difficulty and depth of knowledge.”

A Grade 7 example of an End-of-Unit Assessment includes:

• Unit 5, Rational Number Arithmetic, End-of-Unit Assessment Option A, Problem 7 states, “A water tank can hold 30 gallons when completely full. A drain is emptying water from the tank at a constant rate. When Jada first sees the tank, it contains 21 gallons of water. After draining for 4 more minutes, the tank contains 15 gallons of water. a. At what rate is the amount of water in the tank changing? Use a signed number, and include the unit of measurement in your answer. Type your answer in the box. b. How many more minutes will it take for the tank to drain completely? Type your answer in the box. Explain your reasoning. c. How many minutes before Jada arrived was the water tank completely full? Type your answer in the box. Explain your reasoning.” (7.NS.2 and 7.NS.3)

Program assessments include Diagnostic Assessments (Check Your Readiness), Cool-downs, Checkpoints, Mid-Unit Assessments, and End-of-Unit Assessments which are summative. Materials for Grade 8 contain nine units. Two of these units include Mid-Unit Assessments, and eight units contain End-of-Unit Assessments, both of which are summative. Unit 9 consists of 6 optional lessons and does not include an End-of-Unit Assessment, as there is no course-level assessment. “At the end of each unit is the End-Of-Unit Assessment. These assessments have a specific length and breadth, with problem types that gauge students’ understanding of the key concepts of the unit while also preparing students for new-generation standardized exams. Problem types include multiple-choice, multiple response, short answer, restricted constructed response, and extended response. Problems vary in difficulty and depth of knowledge.”

A Grade 8 example of an End-of-Unit Assessment includes:

• Unit 6, Associations in Data, End-of-Unit Assessment Option B, Problem 4 states, “a. Draw a scatter plot that shows a negative, linear association and has one clear outlier on the grid. Circle the outlier. b. Draw a scatter plot that shows a positive association that is not linear.” (8.SP.1)

The materials reviewed for Imagine IM Grade 6 through Grade 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

Formal assessments, including Mid-Unit Assessments and End-of-Unit, consistently align with grade-level content standards. 

An example in Grade 6 includes:

• Unit 7 End-of-Unit Assessment Option B answer key specifies the standards addressed for each problem. Problem 2 aligns with 6.NS.4: “Select all the numbers that are a common multiple of 8 and 12. A. 96 B. 80 C. 48 D. 32 E. 24 F. 20 G. 4”

An example in Grade 7 includes:

• Unit 8 Mid-Unit Assessment Option A answer key specifies the standards addressed for each problem. Problem 4 aligns with 7.SP.8b: “A toy store sells different types of action figures based on a movie. The figures come in 3 sizes: small, medium, and large. There are also 3 characters to choose from: the main character, the friend, or the villain. How many different action figures are available if any combination could be made? Type your response in the space below.”

An example in Grade 8 includes:

• Unit 5 Mid-Unit Assessment Option A answer key specifies the standards addressed for each problem. Problem 2 aligns with 8.F.5: “This graph shows the temperature in Diego’s house between noon and midnight one day. Select all the true statements. A. Time is a function of temperature. B. The lowest temperature occurred between 4:00 and 5:00. C. The temperature was increasing between 9:00 and 10:00. D. The temperature was 74 degrees Fahrenheit twice during the 12-hour period. E. There was a four-hour period during which the temperature did not change.”

According to the Teacher Course Guides in Grades 6 through 8 (Flipbook), Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. Some instructional routines are generally associated with certain MPs. For example: The Card Sort routine often asks students to reason abstractly and quantitatively (MP2) and to look for and make use of structure (MP7). The Information Gap routine often requires students to make sense of problems and persevere in solving them (MP1) as well as attend to precision (MP6) in their language as they ask questions of their partner. The Math Talk routine offers opportunities to look for and make use of structure (MP7) and look for and express regularity in repeated reasoning (MP8) as students explain the strategies they use and apply strategies as they develop fluency. The Which Three Go Together? routine also offers opportunities for attending to precision when describing why something doesn’t belong (MP6). The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs.” Examples include:

• “MP3: I Can Construct Viable Arguments and Critique the Reasoning of Others. I can explain or show my reasoning in a way that makes sense to others. I can listen to and read the work of others and offer feedback to help clarify or improve their work. I can explain my reasoning for why something is true.”

• “MP4: I Can Model with Mathematics​​​​​. I can think of mathematical questions to ask about a situation. I can identify the questions and information that are essential to solve a problem. I can collect data or explain how to collect it. I can model a situation, using a representation such as a drawing, an equation, a table, or a graph. I can use a mathematical model to draw conclusions about a situation. I can interpret and report on the results of a model, in the context of the situation.”

• “MP7: I Can Look for and Make Use of Structure​​​​. I can connect problems I have solved to new problems. I can compose and decompose numbers, expressions, and figures to make sense of the parts and the whole. I can make connections between multiple mathematical representations. I can make use of patterns to solve a problem. I can make use of structure to create a different representation to solve a problem.”

The materials reviewed for Imagine IM Grade 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.

Formative assessment opportunities include Activities, Cool-Downs, Practice Problems, and Checkpoints in each section of each unit. Summative assessments include Mid-Unit Assessments and End-of-Unit Assessments. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types including multiple choice, multiple response, short answer, restricted constructed response, and extended response. 

An example of a Summative assessment item in Grade 6 includes:

• Unit 8, Data Sets and Distributions, Mid-Unit Assessment Option B, develops the full intent of 6.SP.4, displaying numerical data in plots on a number line, including dot plots, histograms, and box plots. Problem 6 states, “a. Draw two dot plots, each with 7 or fewer data points, so that: both dot plots display data with the same mean; the data displayed in Dot Plot B has a much larger IQR (interquartile range) than the data displayed in Dot Plot A. b. How can you tell, visually, that one dot plot displays data with a larger MAD than another?”

An example of a Summative assessment item in Grade 7 includes:

• Unit 3, Measuring Circles, End-of-Unit Assessment Option A, develops the full intent of 7.G.4, know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Problem 5 states, “For each quantity, decide whether circumference or area would be needed to calculate it. Explain or show your reasoning. a. The distance around a circular track. b. The total number of equally-sized tiles on a circular floor. c. The amount of oil it takes to cover the bottom of a frying pan. d. The distance your car will go with one rotation of the wheels.”

An example of a Summative assessment item in Grade 8 includes:

• Unit 3, Linear Relationships, End-of-Unit Assessment Option B, develops the full intent of 8.EE.5, Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Problem 5 states, “Three different airplanes take off from an airport, and each maintains a constant speed until they near their destination. The equation d = 9.5t represents the distance of the first airplane (in miles), d, from the airport after t minutes. The second airplane's information is in the table” A table displays time (in minutes) in the first column with values 2, 10, 35, and 60. The second column shows the distance from the airport (in miles) with values 16, 80, 280, and 480. “The graph shows the distance from the airport (in miles) of the third plane with respect to the time in minutes.” A graph displays a proportional relationship, showing a line that increases by 100 miles every 10 minutes. “Which airplane is flying the slowest? Explain how you know.”

The materials reviewed for Imagine IM Grade 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 extensive work in Grade 6 through Grade 8 by including in every lesson a Warm-Up, one to three instructional Activities, and Lesson Synthesis. In Grade 6 through Grade 8, students engage with all CCSS standards. 

An example of extensive work in Grade 6 includes:

• Unit 6, Expressions and Equations, Lesson 11, engage students in extensive work with 6.EE.3 (Apply the properties of operations to generate equivalent expressions.). Students write equivalent expressions using the distributive property. Activity 2, Student Task Statement states, “The distributive property can be used to write equivalent expressions. On the next card is a table. In each row, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.” A two-column table labeled product and sum or difference is shown with different expressions. One row shows, 3(3+x), and students must find the sum or difference. They alternate between these operations. The next row shows 4x - 20 for the sum or difference, and students must find the product. Cool-Down states, “Use the distributive property to write an expression that is equivalent to each expression. If you get stuck, consider drawing a diagram. 1. (3r - 1)8 2. p(6 + 2t + 5y) 3. 12 + 4x.” 6.6.11 Practice Problem, Problem 3 states, “Select all the expressions that are equivalent to 16x +36. A. 16(x+20) B. x(16+36) C. 4(4x+9) D. 2(8x+18) E. 2(8x+36)" 

An example of extensive work in Grade 7 includes:

• Unit 4, Proportional Relationships and Percentages, Lesson 6, engage students in extensive work with 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems.) Students use proportional reasoning and percentages to describe and solve problems involving increases and decreases. Activity 1, Student Task Statement states “1. A shampoo bottle says that now it contains 20% more. Originally, it came with 18.5 fluid ounces of shampoo. How much shampoo does the bottle come with now? 2. The price of a shirt is $18.50, but you have a coupon that lowers the price by 20%. What is the price of the shirt after using the coupon?” Activity 2, Problem 1 states, “Match each situation to a diagram. Be prepared to explain your reasoning (two tape diagrams with 25% shaded area shown, one diagram is longer on the bottom, the second has equal parts on top and bottom). a. Compared with last year’s strawberry harvest, this year’s strawberry harvest is a 25% increase. b. This year’s blueberry harvest is 75% of last year’s. c. Compared with last year, this year’s peach harvest decreased 25%. d. This year’s plum harvest is 125% of last year’s plum harvest.” Two diagrams with this year and last year situations are shown. 7.4.6 Practice Problem, Problem 3 states, “Write each percent increase or decrease as a percentage of the initial amount. The first one is done for you. a. This year, there was 40% more snow than last year. The amount of snow this year is 140% of the amount of snow last year. b. This year, there were 25% fewer sunny days than last year. c. Compared to last month, there was a 50% increase in the number of houses sold this month. d. The runner’s time to complete the marathon was 10% less than the time to complete the last marathon.”

An example of extensive work in Grade 8 includes:

• Unit 4, Linear Equations and Linear Systems, Lesson 9, engage students in extensive work with 8.EE.7 (Solve linear equations in one variable.). Students write and interpret one variable equations to represent situations with two conditions. Activity 2, Student Task Statement states, “A building has two elevators that both go above and below ground. At a certain time of day, the travel time, in seconds, that it takes Elevator A to reach height h in meters is given by the equation t=0.8h+16 seconds. The travel time for Elevator B is given by the equation t=-0.8h+12. 1. What is the height of each elevator at this time? 2. How long does it take each elevator to reach ground level at this time? 3. If the two elevators travel toward one another, at what height do they pass each other? How long does it take?” Are you ready for more? “1. In a two-digit number, the ones digit is twice the tens digit. If the digits are reversed, the new number is 36 more than the original number. Find the number. 2. The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 less than the original number. Find the number. 3. The sum of the digits in a two-digit number is 8. The value of the number is 4 less than 5 times the ones digit. Find the number.” Cool-Down states, “To own and operate a home printer, it costs $100 for the printer and an additional $0.05 per page for ink. To print out pages at an office store, it costs $0.25 per page. Let p represent number of pages. 1. What does the equation 100+0.05p=0.25p represent? 2. The solution to that equation is p = 500. What does the solution mean?” 8.4.9. Practice Problem, Problem 4 states, “For what value of x do the expressions \frac{2}{3}x+2 and \frac{4}{3}x-6 have the same value?”

The materials provide opportunities for students to engage in the full intent of the standards in Grade 6 through Grade 8 by including in every lesson a Warm-Up, one to three instructional Activities, and Lesson Synthesis. In Grade 6 through Grade 8, students engage with all CCSS standards. 

An example in Grade 6 includes:

• Unit 2, Introducing Ratios, Lesson 1, meets 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.). Students describe two quantities at the same time. Warm-Up, students see a variety of color cubes connected together and brainstorm various ways to sort them. In Activity 1, Student Task Statement, Problem 2, students use ratio language to describe dinosaurs that are shown in a picture. “Write at least two sentences that describe ratios in the collection. Remember, there are many ways to write a ratio: The ratio of one category to another category is to . The ratio of one category to another category is : . There are of one category for every of another category.” In Activity 2, students write ratios to describe items from their own collection that were brought from home. “2. Write at least two sentences that describe ratios in the collection. Remember, there are many ways to write a ratio.” In the Cool-Down, students see a picture of dogs, mice and cats, “Here is a collection of dogs, mice, and cats: Write two sentences that describe a ratio of types of animals in this collection.”

An example in Grade 7 includes:

• Unit 2, Introducing Proportional Relationships, Lesson 8, meets the full intent of 7.RP.2 (Recognize and represent proportional relationships between quantities.). Students determine whether relationships are proportional. Activity 3, Student Task Statement states, “1. Here are six different equations. y=4+x, y=\frac{x}{4}, y=4x, y=4^{x}, y=\frac{4}{x}, y=x^{4}. Predict which of these equations represent a proportional relationship. 2. Complete each table using the equation that represents the relationship. 3. Do these results change your answer to the first question? Explain your reasoning. 4. What do the equations of the proportional relationships have in common?” The Cool-Down states, “Andre is setting up rectangular tables for a party. He can fit 6 chairs around a single table. Andre lines up 10 tables end-to-end and tries to fit 60 chairs around them, but he is surprised when he cannot fit them all. 1. Write an equation for the relationship between the number of chairs and the number of tables when: the tables are apart from each other: the tables are placed end-to-end: (images of tables are provided). 2. Is the first relationship proportional? Explain how you know. 3. Is the second relationship proportional? Explain how you know.” 7.2.8. In the Practice Problem, Problem 2, states, “Decide whether or not each equation represents a proportional relationship. a. The remaining length (L) of a 120-inch rope after x inches have been cut off: 120-x=L. b. The total cost (t) after 8% sales tax is added to an item’s price (p): 1.08p=t c. The number of marbles each sister gets (x) when m marbles are shared equally among four sisters: x=\frac{m}{4}. d. The volume (V) of a rectangular prism whose height is 12 cm and base is a square with side lengths s cm: V=12x^{2}."

An example in Grade 8 includes:

• Unit 8, Pythagorean Theorem and Irrational Numbers, Lesson 8, meets the full intent of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.) Students calculate the unknown side length of a triangle using the Pythagorean Theorem. In Activity 2, students find the missing lengths of a side of a right triangle. Student Task Statement states, “Find the unknown side lengths in these right triangles.” Students are given a diagram of two right triangles. The first triangle has side lengths of 2 and 5. The second triangle has a hypotenuse of 8 and a side length of \sqrt{32}. Activity 3, states, “Your teacher will give your group a sheet with 4 figures. Cut out the 5 shapes in Figure 1. 1. Arrange the 5 cut out shapes to fit inside Figure 2. 2. Now arrange the shapes to fit inside Figure 3. 3. Check to see that Figure 3 is congruent to the large square in Figure 4. 4. Check to see that the 5 cut out shapes fit inside the two smaller squares in Figure 4. 5. If the right triangle in Figure 4 has legs a and b and hypotenuse c, what have you just demonstrated to be true?” The Cool-Down states, “Find the length of the hypotenuse in a right triangle if a is 5 cm and b is 8 cm.”

The materials reviewed for Imagine IM Grade 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 in Grade 6 devote at least 65 percent of instructional time to the major clusters of the grade:

• The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 6 out of 9, approximately 67%.

• The number of lessons devoted to major work of the grade (including assessments and related supporting work) is 96 out of 152, approximately 63%. 

• The number of days devoted to major work of the grade (including assessments and related supporting work) is 112 out of 173, approximately 65%. 

• The number of days devoted to major work of the grade (including assessments and supporting work, excluding optional lessons) is 96 out of 146, approximately 66%.

The instructional materials in Grade 7 devote at least 65 percent of instructional time to the major clusters of the grade:

• The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 6 out of 9, approximately 67%.

• The number of lessons devoted to major work of the grade (including assessments and related supporting work) is 87 out of 143, approximately 61%. 

• The number of days devoted to major work of the grade (including assessments and related supporting work) is 105 out of 161, approximately 65%. 

• The number of days devoted to major work of the grade (including assessments and supporting work, excluding optional lessons) is 84 out of 138, approximately 61%.

The instructional materials in Grade 8 devote at least 65 percent of instructional time to the major clusters of the grade:

• The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 8 out of 9, approximately 89%.

• The number of lessons devoted to major work of the grade (including assessments and related supporting work) is 108 out of 134, approximately 81%. 

• The number of days devoted to major work of the grade (including assessments and related supporting work) is 126 out of 152, approximately 83%. 

• The number of days devoted to major work of the grade (including assessments and supporting work, excluding optional lessons) is 122 out of 142, approximately 86%.

An instructional day analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 65% of materials in Grade 6, 65% of materials in Grade 7 and 83% of materials in Grade 8 focus on major work of the grade.

The materials reviewed for Imagine IM Grade 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 grades. These connections are listed for teachers within the Dependency Chart documents. 

An example of a connection in Grade 6 includes:

• Unit 7, Rational Numbers, Lesson 15, Activity 2 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 practice plotting coordinates in all four quadrants and find horizontal and vertical distances between coordinates in a puzzle. The Task Statement states, “The diagram shows Andre’s route through a maze. He started from the lower right entrance. 1. What are the coordinates of the first two and the last two points of his route? 2. How far did he walk from his starting point to his ending point? Explain or show your reasoning.”

An example of a connection in Grade 7 includes:

• Unit 1, Scale Drawings, Lesson 3, Cool-Down connects the supporting work of 7.G.1 (Solve problems involving scale drawings of geometric figures) and major work of 7.RP.2 (Recognize and represent proportional relationships between quantities.) Students draw scaled copies of simple shapes, applying what they have learned about corresponding parts and scale factors. The Student Task Statement reads, “1. Create a scaled copy of ABCD using a scale factor of 4. 2. Triangle Z is a scaled copy of Triangle M. Select all the sets of values that could be the side lengths of Triangle Z. A. 8, 11, and 14. B. 10, 17.5, and 25. C. 6, 9, and 11. D. 6, 10.5, and 15. E. 8, 14, and 20.” Triangle M is shown with sides of 4, 7, and 10.

An example of a connection in Grade 8 includes:

• Unit 8, Pythagorean Theorem and Irrational Number, Lesson 2, Activity 2 connects the supporting work 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.) to the major work of 8.EE.2 (Uses square root and cube root symbols to represent solutions to equations of the form x^{2}=p, and x^{3}=pwhere p is a positive rational number.). Students estimate the side length of a square using a geometric construction that relates the side length of the square to a point on the number line. The problem states, “1. Use the circle to estimate the area of the square shown here. Explain your reasoning. 2. Use the grid to check your answer to the first problem. Are you ready for more? One vertex of the equilateral triangle is in the center of the square, and one vertex of the square is in the center of the equilateral triangle. What is x?” Students are shown a coordinate plane with a circle. Covering part of the circle is a square with one of the vertices on the origin.

The instructional materials for Imagine IM Grade 6 through Grade 8 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Connections between major works are present throughout the grade-level materials where appropriate. These connections are listed for teachers in the Teacher Course Guides (Flipbook) in the Dependency Chart, and may appear in one or more phases of a typical lesson: Warm-Up, Instructional Activities, Lesson Synthesis, or Cool-Down. 

An example of a connection in Grade 6 includes:

• Unit 6, Expressions and Equations, Lesson 16, Activity 1 connects the major work of 6.EE.C (Represent and analyze quantitative relationships between dependent and independent variables.) to the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems.) Students write two equations relating the two quantities in the ratio and represent them with graphs. The Task Statement reads, “Lin needs to mix a specific color of paint for the set of the school play. The color is a shade of orange that uses 3 parts yellow for every 2 parts red. a. Lin needs to mix a specific shade of orange paint for the set of the school play. The color is a mixture of red and yellow paint. 1. Complete the table to show different combinations of red and yellow paint that will make the shade of orange Lin needs.” A chart is displayed with two columns: ‘cups of red paint’ and ‘cups of yellow paint.’ In the ‘cups of red paint’ column, the values shown are 2, 6, blank, 12, blank, 1, and blank. In the ‘cups of yellow paint’ column, the values are 3, blank, 12, blank, 21, blank, and 1. “2. Use the values in the table to create two graphs that can represent the relationship between cups of red paint and cups of yellow paint. 3. Describe the relationship between cups of red paint and cups of yellow paint in as many ways as you can. 4. Lin writes this equation to figure out the amount of yellow paint she will need if she knows the amount of red paint being used. In this equation, y represents cups of yellow paint and r represents cups of red paint. y=\frac{3}{2}r. Do you agree that the equation represents the quantities in the situation? Explain your reasoning. Are you ready for more? The owners of a fruit stand sell apples, peaches, and tomatoes. Today, they sold 4 apples for every 5 peaches. They sold 2 peaches for every 3 tomatoes. They sold 132 pieces of those four fruits in total. How many of each fruit did they sell?”

An example of a connection in Grade 7 includes:

• Unit 6, Expressions, Equations, and Inequalities, Lesson 18, Activity 2 connects the major work of 7.EE.A (Use properties of operations to generate equivalent expressions.) to the major work of 7.NS.A (Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.) Students use distributive property to write equivalent expressions with negative numbers. The problem states, “Write two expressions for the area of the big rectangle. 2. Use the distributive property to write an expression that is equivalent to \frac{1}{2}(8y+-x+-12). The boxes can help you organize your work.” Students are shown a rectangle with a width of \frac{1}{2} and segmented lengths of 8y, -x and -12.

An example of a connection in Grade 8 includes:

• Unit 3, Linear Relationships, Lesson 8, Activity 1 connects the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations.) to the major work of 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software.) Students find slopes and y-intercepts and write equations for lines using y=mx+b. The Task Statement reads, “1. Diego earns $10 per hour babysitting. He has no money saved before he starts babysitting and plans to save all of his earnings. Graph how much money, y, he has after x hours of babysitting. 2. Now imagine that Diego started with $30 saved before he starts babysitting. On the same set of axes, graph how much money, y, he would have after x hours of babysitting. 3. Compare the second line with the first line. How much more money does Diego have after 1 hour of babysitting? 2 hours? 5 hours? x hours?”

The instructional materials for Imagine IM 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. 

The Course Guide (Flipbook) contains a Scope and Sequence explaining content standard connections. Prior and Future connections are identified within materials in the Course Guide, Dependency Chart which states, “In the unit dependency chart, an arrow indicates that a particular unit is designed for students who already know the material in a previous unit. Reversing the order of the units would have a negative effect on mathematical or pedagogical coherence.” Some Unit Overviews, Lesson Narratives, and Activity Syntheses describe the progression of standards for the concept being taught. Each Lesson contains Preparation identifying learning standards (Building on, Addressing, or Building toward).

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

• Teacher Course Guide (Flipbook), Scope and Sequence for Grade 6, Unit 1, Area and Surface Area, connects 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents) and 6.G.2 (Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems) to the work of building a solid foundation for understanding and calculating volume and working with exponents in grades 7 and 8. The materials read, “Students will draw on the work here to further study exponents later in grade 6 and to find volumes of prisms and pyramids in grade 7. Their understanding of “two figures that match up exactly” will support their work on congruence and rigid motions in grade 8.”

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

• Teacher Course Guide (Flipbook), Scope and Sequence for Grade 7, Unit 2, Introducing Ratios, connects 7.RP.2 (Recognize and represent proportional relationships between quantities) to the work of linear functions in grade 8. “In this unit, students develop the idea of a proportional relationship. They work with proportional relationships that are represented in tables, as equations, and on graphs. This builds on grade 6 work with equivalent ratios and helps prepare students for the study of linear functions in grade 8.”

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

• Unit 7, Exponents and Scientific Notation, Lesson 5, connects 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions) to the work of reasoning and understanding of exponents in high school. About this lesson states, “In this lesson, students make sense of the rule that 10^{-n}=\frac{1}{10^{n}} and extend the rules they have developed for working with values with exponents to include situations with negative exponents. This type of reasoning appears again in a later course when students extend the rules of exponents to make sense of exponents that are not integers.”

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

• Teacher Course Guide (Flipbook), Scope and Sequence for Grade 6, Unit 2, Introducing Ratios connects 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations) to the foundational work on understanding measurements, relating two quantities, and making multiplicative comparisons in grades 3, 4, and 5. The materials state, “This unit introduces students to ratios and equivalent ratios. It builds on previous experiences students had with relating two quantities, such as converting measurements starting in grade 3, multiplicative comparison in grade 4, and interpreting multiplication as scaling in grade 5. The work prepares students to reason about unit rates and percentages in the next unit, proportional relationships in grade 7, and linear relationships in grade 8.”

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

• Teacher Course Guide (Flipbook), Scope and Sequence for Grade 7, Unit 6, Expressions, Equations, and Inequalities connects 7.EE.4 (Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities) to the work of solving linear equations and inequalities in grade 6. The materials state, “In this unit, students deepen their algebraic reasoning as they write and solve equations of the forms px + q = r and p(x + q) = r and inequalities of the forms px + q > r and \(p(x+q). Students also work with equivalent expressions that are more complex than what they have seen previously. This builds on grade 6 work with equations of the form p + x = q or px = q and with simpler equivalent expressions. Students will build on this work in grade 8 when they solve equations that have a variable on both sides of the equal sign and when they work with systems of equations.”

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

• Teacher Course Guide (Flipbook), Scope and Sequence for Grade 8, Unit 6, Associations in Data, connects 8.SP.1 (Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association) to foundational work on numerical data represented in plots on a number line, such as dot plots, histograms, and box plots, as students build on their ability to analyze data collected about a single variable. The materials state, “In prior grades, students analyzed data collected about one variable using dot plots, histograms, and box plots. This unit expands on that by considering the possible influence of a second variable on measurements about individuals.”