6th to 8th Grade - Gateway 3

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

The Course Guide contains sections titled What’s in an IM Lesson, Key Structures in This Course, Scope and Sequence, and Pacing Guides and Dependency Diagrams, which provide instructional guidance related to the use of student and ancillary materials. Examples include:

• Course Guide, Key Structures in This Course, Coherent Progression, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer the opportunity to observe students’ prior understandings. Each lesson starts with a Warm-up to activate students’ prior knowledge and to set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The Lesson Synthesis at the end consolidates understanding and makes the learning goals of the lesson explicit. In the Cool-down that follows, students apply what they learned. Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.” 

• Course Guide, Key Structures in This Course, Using the 5 Practices For Orchestrating Productive Discussions states, "Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. To facilitate these conversations, the IM curriculum incorporates the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). The 5 Practices are: anticipate, monitor, select, sequence, and make connections between students’ responses. All IM lessons support the practices of anticipating, monitoring, and selecting students’ work to share during whole-group discussions. In lessons in which students make connections between representations, strategies, concepts, and procedures, the Lesson Narrative and the Activity Narrative support the practices of sequencing and connecting as well, and the lesson is tagged so that these opportunities are easily identifiable. For additional opportunities to connect students’ work, look for activities tagged with MLR7 Compare and Connect. Similar to the 5 Practices routine, Compare and Connect supports the practices of monitoring, selecting, and making connections. In curriculum workshops and PLCs, rehearse and reflect on enacting the 5 Practices."

• Course Guide, What’s in an IM Lesson, Narratives Tell The Story, “The story of each grade is told across the units in the narratives. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each section within a unit has a narrative that describes the mathematical work in the section. Each lesson and each activity in a unit also have narratives. The Lesson Narrative explains: The mathematical content of the lesson and its place in the learning sequence. The meaning of any new terms introduced in the lesson. How the mathematical practices come into play, as appropriate. The Activity Narrative explains: The mathematical purpose of the activity and its place in the learning sequence. What students are doing during the activity. What to look for, while students are working on an activity, to orchestrate an effective Activity Synthesis. Connections to the mathematical practices, when appropriate.”

• The Course Guide, and Scope and Sequence list each of the nine units, explain connections to prior learning, describe the progression of learning throughout the unit, and integrate new terminology/vocabulary throughout each unit.

• The Course Guide, Scope and Sequence, Pacing Guides, and Dependency Diagrams show the interconnectedness between lessons and units within Grade 6 through Grade 8 and across all grades.

• The Glossary provides a visual glossary for teachers that includes both definitions and illustrations.

Materials include sufficient annotations and suggestions presented within the context of the specific learning objectives. Examples include:

• Grade 6, Unit 2, Introducing Ratios, Lesson 11, Activity 11.2, Narrative states, “In this activity, students are asked to find missing values for significantly scaled-up ratios. The work serves several purposes: To uncover a limitation of a double number line (namely, that it is not always practical to extend it to find significantly scaled-up equivalent ratios); To reinforce the multiplicative reasoning needed to find equivalent ratios (especially in cases when drawing diagrams or skip counting is inefficient); To introduce a table as a way to represent equivalent ratios. Monitor for the different ways students find equivalent ratios involving large values. Here are some strategies they may use, from less practical or less flexible to more accommodating: Extend the given double number line and then try to squeeze numbers on the extreme right end, ignoring the previously equal intervals. Draw a new double number line diagram that is longer or where the intervals represent multiples of 4 and 5 (rather than 4 and 5). Use multiplication (or division) and write expressions or equations to represent the given scenarios. Select students who use different strategies to share later, including those who find the given double number line inadequate—not long enough to accommodate large numbers, requiring more marking or writing, and so on—and are consequently motivated to find a more efficient strategy.”

• Grade 7, Unit 4, Proportional Relationships and Percentages, Lesson 10, Lesson Synthesis states, “Share with students, ‘Today we calculated sales tax and tips. These are two different examples of percent increase.’ To review the meanings of these terms, consider asking students: ‘What is sales tax?’ (Sales tax is an extra amount of money, added to the price of an item, that is paid to the government.) ‘If the tax rate is 7%, how can you calculate the price of an item including sales tax?’ (I multiply the price by 1.07.) ‘What is a tip?’ (A tip is an extra amount of money, added to the price of a meal, that is given to the server at a restaurant.) ‘If you want to leave a 15% tip, how can you calculate the total amount to pay for a meal?’ (I multiply the bill by 1.15.)”

• Grade 8, Unit 4, Linear Equations and Linear Systems, Unit Narrative, “In this unit, students work with writing equivalent equations and use reasoning to solve equations for a variable. Then students solve systems of linear equations using graphic and algebraic methods. The unit begins with a focus on moves that can be done to write equivalent equations. At first, students use hanger diagrams as an intuitive representation of equality and represent their reasoning by labeling arrows that connect equivalent representations. With the reintroduction of negative values, students move away from hanger diagrams to algebraic equations and writing equivalent equations with the intention of solving for a variable. Next, students examine the conditions under which equations could have 0, 1, or infinite solutions as a transition to thinking about similar situations involving systems of equations. Students finish the unit by examining systems of equations graphically and then finding solutions algebraically. They build on their understanding that the line representing an equation with 2 variables is made up of coordinate pairs that make the equation true. They find that the intersection of 2 lines is the point that makes both equations for the system true. Students also recognize when systems have no solution or infinite solutions based on the graphs and the slope and intercept.”

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

The Further Reading section, located within the Course Guide, connects research to pedagogy. It includes explanations and examples of both grade-level and above-grade-level content to support teacher understanding. Additionally, Unit Narrative, Lesson Narratives, and Activity Synthesis sections throughout the lessons provide similar support, offering explanations and examples of how grade-level concepts connect to content in other grade levels.

• Grade 6, Course Guide, Further Reading, “Unit 8, A Thread Through Early Algebra 1”, supports teachers with context for work beyond the grade. “In this blog post, Petersen and Black illustrate how the statistics work in middle school sets students up for success in Algebra 1.”

• Grade 6, Unit 2, Introducing Ratios, Unit Narrative, “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. First, students learn that a ratio is an association between two quantities, for instance, ‘There are 3 pencils for every 2 erasers.’ Students use sentences, drawings, or discrete diagrams to represent ratios that describe collections of objects and recipes. Next, students encounter equivalent ratios in terms of multiple batches of a recipe. ‘Equivalent’ is first used to describe a perceivable sameness of two ratios, such as two mixtures of drink mix and water that taste the same, or two mixtures of yellow and blue paint that make the same shade of green. Later, ‘equivalent’ acquires a more precise meaning: All ratios that are equivalent to a : b can be made by multiplying both a and b by the same non-zero number (non-negative, for now). Students then learn to use double number line diagrams and tables to represent and reason about equivalent ratios. These representations are more abstract than are discrete diagrams and offer greater flexibility. Use of tables here is a stepping stone toward use of tables to represent functional relationships in future courses. Students explore equivalent ratios in contexts such as constant speed and uniform pricing.” 

• Grade 6, Unit 6, Expressions and Equations, Lesson 14, Activity 14.2, Activity Synthesis states, “Point out that in finding the surface area, we need to find the area of one face of the cube, which is 10^{2}, before multiplying that number by 6. Tell students that sometimes it is not so clear in which order to perform the operations in an expression. However, there is an order that we all generally agree on, and when we want something done in a different order, we use parentheses, brackets, or other grouping symbols to communicate what to do first. In general: When an exponent occurs in the same expression as another operation, we evaluate the exponent first. For example, to find the value of 3 42 , we calculate 42 first, which is 16, and then multiply it by 3, which gives 48. When an expression involves grouping symbols, we perform the operation inside them first. For example, to find the value of 3\cdot 4^{2} , we calculate 4^{2} first, which is 16, and then multiply it by 3, which gives 48. When an expression involves grouping symbols, we perform the operation inside them first. For example, for (3\cdot 4)^{2}, we first multiply 3 and 4, which gives 12, and then square the 12 to get 144. If students bring up PEMDAS or another mnemonic for remembering the order of operations, point out that PEMDAS can be misleading in indicating multiplication before division, and addition before subtraction. Discuss the convention: First, find the value of any expression in brackets or parentheses. Next, find the value of any expression with an exponent. Then, perform multiplication or division, from left to right. Lastly, perform addition or subtraction, from left to right.”

• Grade 7, Unit 3, Measuring Circles, Lesson 3, Activity 3.2, Narrative states, “In this activity, students measure the diameter and circumference of different circular objects and graph their measurements. This echoes the structure of a previous activity in which they measured squares. Students notice that the two quantities appear to be proportional to each other. Based on the graph, they can estimate that the constant of proportionality is close to 3. From the table they may be able to estimate that it is a little bigger than 3. This activity provides evidence that there is a constant of proportionality between the circumference of a circle and its diameter. The best precision we can expect for the constant of proportionality in this activity is ‘around 3’ or possibly ‘a little bit bigger than 3.’ As students measure multiple circles and notice patterns in their measurements, they express regularity in repeated reasoning (MP8). Monitor for students who recognize that the relationship between diameter and circumference appears to be proportional, including: Notice that the points on the graph appear to lie on a line through the origin. Use the graph to estimate a constant of proportionality. Divide the values in the table to estimate a constant of proportionality. Plan to have students present in this order to support moving from less precise to more precise estimation. In the digital version of the activity, students use applets to measure the circumference of various circles and plot their measurements on the coordinate plane. The first applet allows students to see an animation of the circumference being unwrapped along a ruler. The second applet allows students to enter values into a table and see the points plotted on a coordinate plane. Use the digital version if physical circular objects are not available.” 

• Grade 7, Unit 5, Rational Number Arithmetic, Unit Narrative, “A note on using the terms ‘expression,’ ‘equation,’ and ‘signed number’: In these materials, an expression is built from numbers, variables, operation symbols (+,-,\cdot ,\div ), parentheses, and exponents. (Exponents—in particular, negative exponents—are not a focus of this unit. Students work with integer exponents in grade 8 and noninteger exponents in high school.) An equation is a statement that two expressions are equal, thus it always has an equal sign. Signed numbers include all rational numbers, written as decimals or in the form \frac{a}{b}.”

• Grade 8, Course Guide, Further Reading, “Unit 1, Proof in IM’s High School Geometry (A Sneak Preview)”, supports teachers with context for work beyond the grade. “In this blog post, Ray-Riek and Cardone describe the expectations for proof writing in high school geometry.”

• Grade 8, Unit 4, Linear Equations and Linear Systems, Lesson 14, Activity 14.2, Activity Narrative states, “In this activity, students solve systems of linear equations that lend themselves to substitution. There are four kinds of systems presented: One kind has both equations given with the y value isolated on one side of the equation, another kind has one of the variables given as a constant, a third kind has one variable given as a multiple of the other, and the last kind has one equation given as a linear combination. This progression of systems nudges students toward the idea of substituting an expression in place of the variable it is equal to. Notice which kinds of systems students think are the least and most difficult to solve. In future grades, students will manipulate equations to isolate one of the variables in a linear system of equations. For now, students do not need to solve a system like x+2y=7 and 2x-2y=2 using this substitution method.”

• Grade 8, Unit 7, Exponents and Scientific Notation, Unit Narrative, “In this unit, students deepen their understanding of exponents, powers of 10, and place value before being introduced to scientific notation. They build on work done in a previous course where students focused on whole-number exponents with whole-number, fraction, decimal, or variable bases, but did not formulate rules regarding the use of exponents. Students begin this unit by identifying patterns that emerge when multiplying and dividing powers of 10, and when raising powers of 10 to another power. Students generalize these patterns to develop exponent rules. They extend these rules to see why 10^{0} must be equal to 1 and to understand what negative exponents mean. Next, students determine that the rules developed for powers of 10 also work with other bases, as long as the bases in both expressions are the same. They observe a new rule that applies when multiplying bases that are different if the exponents are the same. In the next section, students return to working with powers of 10 as they use multiples of powers of 10 to describe magnitudes of very large and very small quantities, such as the distance from Earth to the sun in kilometers or the mass of a proton in grams. Students plot these large and small values on number lines labeled using exponents and see how these numbers can be expressed in different ways — for example as 75\cdot 10^{5} or 7.5\cdot 10^{6}. After building a foundation connecting powers of 10 with place value, students are finally introduced to scientific notation as a specific and useful way of writing numbers as a power of 10. They compute sums, differences, products, and quotients of numbers written in scientific notation to make additive and multiplicative comparisons, estimate quantities, and make measurement conversions.”

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for including a year-long scope and sequence with standards correlation information.

The Course Guide includes multiple components that support planning and understanding of the program’s structure and standards alignment. 

Examples in Grade 6 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state: “IM Grade 6 begins with an exploration of area and surface area—an invitation for students to engage with novel ideas that they can represent concretely and visually, and reason about in intuitive ways. Starting with geometry also creates opportunities to elicit close observation, sense- and connection-making, and the exchange of ideas—elements of a healthy learning community. The next two units introduce ratios and rates, concepts that are also new. Students learn to represent, make sense of, and solve problems about equivalent ratios, rates, unit rates, and percentages. The mathematical reasoning here constitutes major work of the grade. In the two units that follow, students expand and deepen their prior knowledge of numbers and operations. In one unit, students explore division involving fractions, and work toward dividing a fraction by fraction. In the other, they learn to multiply and divide multi-digit, base-ten numbers, including decimals, using the standard algorithm for each operation. Building fluency with algorithms takes time and continues beyond the two units. Next, students further their understanding of equations and expressions, including those with variables. Students consider ways to represent, justify, and generate equivalent expressions. They also use expressions and equations to describe the relationship between quantities. From there, students are introduced to rational numbers. Students learn about negative numbers, and represent negative numbers on the number line and on the coordinate plane. They analyze and write inequalities that compare rational numbers. Toward the end of the course, students examine data sets and distributions. They learn about statistical questions, categorical data, and numerical data. They also explore ways to describe the center and the distribution of a data set. The final unit of the course is optional. The lessons provide students with additional opportunities to integrate and apply various ideas from the course to solve real-world and mathematical problems.”

• Lessons by Standard provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 6.EE.A.1 is addressed in Unit 1, Lessons 17 and 18, and Unit 6, Lessons 12, 13, 14, and 15.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 3, Lesson 5 includes 6.RP.A.2, 6.RP.A.3, and 6.RP.A.3.b.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 7, Lesson 3 integrates MP3, MP6, and MP7.

Examples in Grade 7 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state: “IM Grade 7 begins with students studying scale drawings, an engaging geometric topic that sets the stage for the subsequent work on proportional relationships in the following three units. Students also have opportunities to build fluency with IM Grade 6 arithmetic. They work with proportional relationships represented by tables, equations, and graphs. Geometry and proportional relationships are interwoven in the third unit, when the important proportional relationship between a circle's circumference and its diameter is studied. Then students work with percent increase and percent decrease. By the fifth unit, on operations with rational numbers, students have had time to brush up on and solidify their understanding of, and skill in, IM Grade 6 arithmetic. At this point, the emphasis becomes the role of the properties of operations in determining the rules for operating with negative numbers. This is a natural lead-in to the work on solving equations and simplifying expressions in the next unit. Students then put their arithmetical and algebraic skills to work in the last two units: on angles, triangles, and prisms, and on probability and sampling. The final unit of the course is optional. The lessons provide students with additional opportunities to integrate and apply various ideas from the course to solve real-world and mathematical problems.”

• Lessons by Standard provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 7.G.A.1 is addressed in Unit 1, Lessons 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13; Unit 2, Lesson 1; Unit 3, Lessons 6 and 11; and Unit 9, Lessons 3, 8, and 12.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 2, Lesson 5 includes 7.RP.A, 7.RP.A.2, 7.RP.A.2.b, and 7.RP.A.2.c.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 8, Lesson 7 integrates MP3, MP4, MP5, and MP8.

Examples in Grade 8 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state: “IM Grade 8 begins with transformational geometry. Students study rigid transformations and congruence, and then dilations and similarity. This provides background for understanding the slope of a line in the coordinate plane. Next, students build on their understanding of proportional relationships, from IM Grade 7, to study linear relationships. They use equations, tables, and graphs to represent linear relationships, and make connections across these representations. Students expand their ability to work with linear equations in one and two variables, extending their understanding of a solution to an equation in one or two variables to comprehend a solution to a system of equations in two variables. They learn that linear relationships are an example of a special kind of relationship called a function. Students apply their understanding of linear relationships and functions to contexts involving data with variability. The course ends the year with students extending their understanding of exponents to include all integers, and in the process codifying the properties of exponents. They learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities. They encounter irrational numbers for the first time and informally extend the rational-number system to the real-number system, motivated by their work with the Pythagorean Theorem. The final unit of the course is optional. The lessons provide students with additional opportunities to integrate and apply various ideas from the course to solve real-world and mathematical problems.”

• Lessons by Standard provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 8.EE.B.6 is addressed in Unit 2, Lessons 10, 11 and 12; and Unit 3, Lessons 7, 10, 11, 12, and 15.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 8, Lesson 11 includes 8.EE.A.2, 8.G.B.7, and 8.NS.A.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 5, Lesson 10 integrates MP1, MP2, and MP4.

In addition, the Pacing Guide and Dependency Diagram within the Course Guide outlines the number of lessons and suggested teaching days per unit, supporting year-long planning and implementation. Each lesson includes references to the standards addressed and often notes how the lesson builds on prior learning.

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for explaining the program’s instructional approaches, identifying research-based strategies, and explaining the role of the standards. Examples include:

• Course Guide, Problem-Based Teaching and Learning, “Illustrative Mathematics is a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, students’ thinking, and their own teaching practice. The curriculum and the professional-learning materials are designed to support students’ and teachers’ learning. This document defines the principles that guide IM’s approach to mathematics teaching and learning. It then outlines how each component of the curriculum supports teaching and learning, based on these principles.”

• Course Guide, Problem-Based Teaching and Learning, Learning Mathematics by Doing Mathematics, “Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving” (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the problem solvers learning the mathematics.”

• Course Guide, Key Structures in This Course, Coherent Progression, “The basic architecture of the materials supports all learners through a coherent progression of the mathematics, based both on the standards and on research-based learning trajectories. Activities and lessons are parts of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense.”

• Course Guide, What’s in an IM Lesson, Instructional Routines, Instructional Routines (IRs) are described in the materials as “designs for interaction that invite all students to engage in the mathematics of each lesson.” They are structured opportunities for students to connect personal experience and mathematical understanding through discussion, questioning, justification, and reasoning. The materials state that IRs “have a predictable structure and flow” and are intended to provide consistent support for both teachers and students. A finite set of routines is used across lessons to support pacing, reduce the need for repeated explanation of directions, and increase time spent on mathematical learning. Some IRs, identified as Mathematical Language Routines (MLRs), were developed by the Stanford University UL/SCALE team. MLRs are integrated into lessons either as embedded components or as optional supports for Multilingual Learners. The first instance of each routine includes detailed guidance for implementation, while subsequent instances offer abbreviated reminders. The materials also include Digital Routines (DRs), which identify required or suggested uses of technology. These routines are flagged within activities to support lesson planning and professional development.

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Guide includes a comprehensive Required Materials list detailing all materials needed for the grade-level content. At the lesson level, the Required Materials section specifies what is needed for that specific lesson. At the activity level, Materials To Gather specifies what is needed for that specific activity; if no materials are required, this section will indicate that by being left blank or stating "none."

Examples include: 

• Grade 6, Unit 7, Rational Numbers, Lesson 2, Activity 2.3, “Materials To Gather: Rulers marked with centimeters, Tracing paper.”

• Grade 7, Unit 8, Probability and Sampling, Lesson 14, Preparation, Required Materials, Activity 3, “Paper bags, Rulers marked with inches, Straws.” Required Preparation: “Prepare one paper bag containing straws cut to the specified lengths in the table for a demonstration. Length of straw in inches: \frac{1}{2}, 1, 2, 3, 4, 5. Number of straws: 6, 6, 8, 6, 5, 4. The demonstration will also require a ruler marked with inches to measure the straw pieces chosen in a sample.”

• Grade 8, Unit 3, Linear Relationships, Lesson 6, Activity 6.3, Materials To Copy (from Blackline Masters), “Slopes, Vertical Intercepts, and Graphs Cards.”

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for providing consistent opportunities to determine student learning throughout the school year. The assessment system provides sufficient teacher guidance for evaluating student performance and determining instructional next steps.

Each End-of-Unit Assessment and the Mid-Unit Assessment includes answer keys and standards alignment to support teachers in interpreting student understanding. According to the Course Guide and Assessment Guidance, “All summative assessment problems include a complete solution and standards alignment. Multiple-choice and multiple-response problems often include a reason for each potential error that a student might make. Restricted constructed-response and extended-response items include a rubric. Unlike formative assessments, problems on summative assessments generally do not prescribe a method of solution.” Examples include: 

• Grade 6, Unit 6, Expressions and Equations, Mid-Unit Assessment, Version A, Problem 6, students complete a table to represent the relationship between the number of raffle tickets sold and the amount of money earned, then use this information to answer problems. “Diego sells raffle tickets for a school fundraiser. He collects $1.75 for each ticket. 1. Complete the table to show how much money Diego would collect if he sold each number of tickets. 2. How many tickets would Diego need to sell to collect How many tickets would Diego need to sell to collect $140?” The Narrative for Question 6 states, “First, students complete a table to represent the relationship between the number of raffle tickets sold and the amount of money collected. Then they determine the number of tickets it would take to collect $140.” Solution states, “Minimal Tier 1 response: Work is complete and correct. Sample: 1. See table. 2. 1.75r=140, r=1401.75, r=80. Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Acceptable errors: Reasonable response to part b is based on an incorrect expression in the last cell of the table. Sample errors: Substituted value for r  is recorded in the last column of the table, but keeps the multiplicative relationship of 1.75. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Table reflects a lack of understanding of the multiplicative relationship, which affects the equation in part b; work involves a misinterpretation of the situation that affects all or most problem parts, but work does show understanding of writing equations to represent situations and interpreting solutions to equations.” The answer key aligns the item to standards 6.EE.6 and 6.EE.2.a, and includes sample representations to support teacher interpretation of student strategies.

• Grade 7, Unit 1, Scale Drawings, End-of-Unit Assessment, Version B, Problem 6, students reason about map scaling. “There are two different maps of California. On the first map, the scale is 1 cm to 20 km. The distance from Fresno to San Francisco is 15 cm. On the second map, the scale is 1 cm to 100 km. What is the distance from Fresno to San Francisco on the second map? Explain your reasoning.” The Narrative for Problem 6 states, “Students reason about maps drawn at two different scales. This problem can be solved using a variety of approaches, including tables and double number lines.” Solution states, “The distance is 3 cm. Sample explanations: On the 1 cm : 20 km scale map, each centimeter represents 5 times as much actual distance as on the 1 cm: 100 km map. That means that on the 1 cm : 5 km map the distance from Fresno to San Francisco will be one fifth as much, 3 cm. The actual distance from Fresno to California is 300 km, because 15\cdot 20=300. The distance on the second map that represents 300 km is 3 cm, because 300\div 100=3. Minimal Tier 1 response: Work is complete and correct. Sample: Lengths on the second map are five times smaller because 1 cm represents 20 km instead of 100 km. Divide 15 cm by 5 to get 3 cm. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Multiplication or division errors in otherwise correct work; work involves a correct substantive intermediate step (such as the actual distance from Fresno to San Francisco) but goes wrong after that; one mistake involving an ‘upside down’ scale factor (or multiplying when division is called for); a correct answer without explanation. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Work does not involve proportional reasoning; an incorrect answer without explanation, even if close; multiple mistakes that involve inversion of scale factors.” The answer key aligns the item to standard 7.G.1, and includes sample representations to support teacher interpretation of student strategies.

• Grade 8, Unit 3, Linear Relationships, End-of-Unit Assessment, Version A, Problem 5, students compare proportional relationships between time and distance. “Three runners are training for a race. One day, they all run a lap around a track, each at their own constant speed. The graph shows the distance in meters that Runner #1 runs with respect to the time in seconds. The equation that relates Runner #2’s distance (in meters) with time (in seconds) is d=6.5t. Runner #3’s information is in the table. Which of the three runners runs the fastest? Explain your reasoning.” The Narrative for Problem 5 states, “Students compare the pace of three different runners. The proportional relationship between time and distance is represented in three different ways. There is more than one way to do this problem correctly. For example, students could determine how long it takes each runner to run 5 miles to determine the fastest runner, or could determine each runner's speed in miles per minute or miles per hour.” Solution states, “Runner #2 runs the fastest. Sample reasoning: Using the points (0,0) and (50,200) from Runner #1’s graph, the slope is 4, showing that they run 4 meters every second. Runner #2’s equation shows that they run 6.5 meters every second. Within the table, the unit rate for Runner #3 is 5 meters per second because 45\div 9=5. Since Runner #2 travels farther every second, Runner #2 is the fastest. Minimal Tier 1 response: Work is complete and correct. Sample: Runner #1 goes at 4 meters per second, Runner #2 goes at 6.5 meters per second, and Runner #3 goes at 5 meters per second. Runner #2 is the fastest because they travel the greatest distance in 1 second. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Work contains correct unit rates for all three runners but concludes that runner #1 or #3 is the fastest or does not name a fastest runner; one unit rate is incorrect (possibly with an incorrect fastest runner identified as a consequence); insufficient explanation of work. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Two or more incorrect unit rates; the correct runner is identified but with no justification; response to the question is not based on unit rates or on similar methods, such as calculating which runner has gone the farthest after 10 miles.” The answer key aligns the item to standard 8.EE.5, and includes sample representations to support teacher interpretation of student strategies.

The materials also include guidance for determining next instructional steps, integrated into both formative and summative assessment opportunities. Most lessons conclude with a Cool-down task designed to assess student thinking in relation to the lesson’s learning objective. The Course Guide, Assessment Guidance, Cool-Downs includes the following description: 

• “If results from a Cool-down suggest most students are struggling with a key concept, consider implementing one or more of these strategies: During the next lesson, display the work of a few students on that Cool-down. Anonymize their names, and show both correct and incorrect work. Ask the class to observe what each student did well and what they could have done better. Consider using the MLR 3: Critique, Correct, Clarify structure. Give each student brief, written feedback on a Cool-down that nudges them to re-examine their work. Ask students to revise and resubmit. Identify a common error and create a sample response to the Cool-down, with the error, for students to identify and correct. Review activities and lessons, with related work, for optional content not yet used in class. Include this work in future lessons to give students additional time and different ways to make sense of key concepts.”

In addition to this formative support, the materials provide teachers with structured guidance following summative assessments. The Course Guide, Assessment Guidance also describes how teachers might observe patterns of student understanding and offers suggestions for addressing unfinished learning alongside upcoming grade-level instruction. For example, “Each lesson ends with a Cool-down to formatively assess students’ thinking in relation to an important math concept from the day’s learning. Each Cool-down is accompanied by guidance on how to continue teaching grade-level content, with appropriate and aligned practice and support for students.” This guidance is categorized as “More Chances,” which signals students will revisit the concept without altering instruction; “Points to Emphasize,” which recommends minor instructional adjustments; and “Press Pause,” which suggests more time is needed before advancing. Teachers are encouraged to “display the work of a few students on that Cool-down,” “give each student brief, written feedback,” and “ask students to revise and resubmit.” Similarly, “each section ends with a Checkpoint of one to three problems that assess the section learning goals,” and this guidance “falls into the same three categories as guidance for the Cool-down, with suggestions on next steps if most students struggle.” For example: 

• Grade 6, Unit 4, Dividing Fractions, Lesson 16, Cool-Down states, “A box of pencils is 514 inches wide. Seven pencils, laid side by side, take up 258 inches of the width. Problem 1. How many inches of the width of the box is not taken up by the pencils? Problem 2. All 7 pencils have the same width. How wide is each pencil? Explain or show your reasoning. Student Response: 1. 2\frac{5}{8} inches, because 5\frac{1}{4}-2\frac{5}{8}=2\frac{5}{8}. 2. \frac{3}{8} inches, because 2\frac{5}{8}\div 7=\frac{21}{8}\cdot \frac{1}{7}=\frac{1}{3}. Responding To Student Thinking, Press Pause, If students struggle with interpreting the division problem or finding the quotient, make time to revisit related work in earlier lessons. For example, ask students to interpret each situation in the activity referred to here, and discuss ways to reason about the answer to each question. Grade 6, Unit 4, Lesson 9, Activity 3: Amount in One Group.”

• Grade 6, Unit 7, Rational Numbers, Section A Checkpoint, Problem 2, Responding To Student Thinking, “Points To Emphasize: If most students struggle with using inequality signs to compare values, especially absolute values, reinforce the idea in the next section. For example, when finding the distances between points in the coordinate plane in the activity referred to here, relate the idea of finding the distance of each point to the x- or y-axis with absolute value. Grade 6, Unit 7, Lesson 14, Activity 2: Signs of Numbers in Coordinates.”

• Grade 7, Unit 3, Measuring Circles, Lesson 2, Cool-Down Student Task Statement states, “Here are two circles. Their centers are A and F. 1. What is the same about the two circles? What is different? 2. What is the length of segment AD? How do you know? 3. On the first circle, what segment is a diameter? How long is it? Student Response: 1. Because they are both circles, they are both round figures, without corners or straight sides, enclosing a two-dimensional region, that are the same distance across (through the center) in every direction. Both circles are the same size. They have the same diameter, radius, and circumference. The only difference is which additional segments (radii) are drawn. 2. Segment AD is 4 cm long because it is also a radius of the circle. 3. The diameter, segment EB, is 8 cm long. Building on Student Thinking, Students might not realize that the diameter or radius of a circle can be drawn at any endpoint that connects to the center. Responding To Student Thinking, Points To Emphasize: If students struggle with identifying radius and diameter, revisit the vocabulary when opportunities arise over the next several lessons. For example, make sure to invite multiple students to share their thinking about how they solved the problem in this activity: Grade 7, Unit 3, Lesson 4, Activity 3: Hopi Basket Weaving.” 

• Grade 7, Unit 7, Angles, Triangles, and Prisms, Section C Checkpoint, Problem 2, Responding To Student Thinking, “Press Pause: By this point in the unit, there should be some student mastery of finding surface area and volume of prisms. If most students struggle, make time to revisit related work in the lesson referred to here. See the Course Guide for ideas to help students re-engage with earlier work. Grade 7, Unit 7, Lesson 15: Distinguishing Volume and Surface Area.”

• Grade 8, Unit 5, Functions and Volume, Lesson 9, Cool-Down, Student Task Statement states, “A small company is selling a new board game, and they need to know how many to produce in the future. After 12 months, they sold 4 thousand games. After 18 months, they sold 7 thousand games. And after 36 months, they sold 15 thousand games. Could this information be reasonably estimated using a single linear model? If so, use the model to estimate the number of games sold after 48 months. If not, explain your reasoning. Student Response: Predictions between 20 and 22 thousand sales, depending on the data points used for the model, are reasonable. Sample response: Yes. After 48 months, they sold about 20.5 thousand games. From Month 12 to Month 38, the rate of games sold was about \frac{11}{24} thousand games per month. This means the amount sold during the 12 months from Month 36 to Month 48 was 5.5 thousand, since \frac{11}{24}\cdot 12=5.5, and 5.5 thousand added to 15 thousand is 20.5 thousand. Responding To Student Thinking, Points To Emphasize: If most students struggle with identifying a single linear model, revisit linear models as opportunities arise over the next several lessons. For example, in the activity referred to here, invite multiple students to share their thinking about the lines they drew and the meaning of the slopes for each piece. Also highlight the question in the Activity Synthesis asking about using the information to make predictions. Grade 8, Unit 5, Lesson 10, Activity 2: Modeling Recycling.” 

• Grade 8, Unit 8, Pythagorean Theorem and Irrational Numbers, Section D Checkpoint, Problem 1, Responding To Student Thinking, “Press Pause: If most students struggle with representing a fraction as a decimal, make time to review long division when reviewing the practice problem referred to here. The Course Guide provides additional ideas for revisiting earlier work. Grade 8, Unit 8, Lesson 17, Problem 2.”

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for providing strategies and support for students in special populations to work with grade-level content and meet or exceed grade-level standards, which support their regular and active participation in learning. 

Examples include:

• Course Guide, Advancing Mathematical Language and Access For Multilingual Learners, Mathematical Language Routines states, “The Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. The MLRs emphasize the use of language that is meaningful and purposeful, and isn’t only about getting answers. The routines included in this curriculum were selected because they simultaneously support students’ learning of mathematical practices, content, and language. They are particularly well-suited to meet the needs of linguistically and culturally diverse students, who are learning mathematics while concurrently acquiring English. Adapt and incorporate these flexible MLRs across the lessons in each unit to support students at all stages of language development in improving their use of English and disciplinary language.” MLRs are included in select activities of each unit, and are described in the Teacher Guide for the lessons in which they appear within the Instructional Routines and Supporting Multilingual Learners.

• Course Guide, Universal Design for Learning and Access for Students with Disabilities, Access for Students with Disabilities states, “Supplemental instructional strategies, labeled Access for Students with Disabilities, are included in each lesson. They are designed to help meet the individual needs of a diverse group of learners. Each support is aligned to one of the three principles of Universal Design for Learning, to provide multiple means of engagement, representation, or action and expression, and includes a suggested strategy to increase access and eliminate barriers. Use these lesson-specific supports, as needed, to help students succeed with a specific activity, without reducing the mathematical demands of the task. Phase them out as students gain understanding and fluency.”

• Course Guide, Universal Design for Learning and Access for Students with Disabilities, Accessibility For Students With Visual Impairments states, “For students with visual impairments, accessibility features are built into the materials: 1. A palette of colors distinguishable to people with the most common types of color blindness. 2. Tasks and problems designed so that success does not depend on the ability to distinguish between colors. 3. Mathematical diagrams, presented in scalable vector graphic (SVG) format, can be magnified, without loss of resolution, and rendered in Braille. 4. Where possible, text associated with images is not part of the image file, but rather included as an image caption accessible to screen readers. 5. Alt text on all images makes interpretation easier for users accessing the materials with a screen reader. If students with visual impairments are accessing the materials, using a screen reader, it is important to understand: All images in the curriculum have alt text: a very short indication of the image’s contents, so that the screen reader doesn’t skip over as if nothing is there. Some images have a longer description to help a student with a visual impairment recreate the image in their mind. Understand that students with visual impairments likely will need help accessing images in lesson activities and assessments. Prepare appropriate accommodations. Accessibility experts, who reviewed this curriculum, recommended that eligible students have access to a Braille version of the curriculum materials, because a verbal description of many of the complex mathematical diagrams is inadequate for supporting their learning.”

• Grade 6, Unit 7, Rational Numbers, Lesson 11, Activity 11.2, Access for Students with Disabilities, “Representation: Develop Language and Symbols. Provide students with access to a reference that shows the locations of the x- and y-axes and explains that the first number in a coordinate pair represents the x-coordinate value, while the second number in the coordinate pair represents the y-coordinate value. Supports accessibility for: Language, Memory.”

• Grade 7, Unit 2, Introducing Proportional Relationships, Lesson 7, Activity 7.2, Access for Students with Disabilities, “Representation: Develop Language and Symbols. Represent the problem in multiple ways to support understanding of the situation. For example, by acting out or having students draw quick sketches of the different cases. In either case, be sure students have accounted for the cost of the vehicle in their calculations of the total entrance cost. Supports accessibility for: Fine Motor Skills, Organization.”

• Grade 8, Unit 7, Exponents and Scientific Notation, Lesson 3, Activity 3.2, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Have students complete one problem at a time. Check in with students to provide feedback and encouragement after each chunk. Supports accessibility for: Attention, Social-Emotional Functioning.”

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for regularly providing extensions and/or opportunities for advanced students to engage with grade-level mathematics at greater depth. Examples include:

• Course Guide, Key Structures in This Course, Are You Ready For More?, states, “Select classroom activities offer differentiation for students ready for a greater challenge. These opportunities are the ‘mathematical dessert’ that follows the ‘mathematical entrée’ of a classroom activity. Every extension problem is made available to all students, with the heading Are You Ready for More? These problems go deeper into grade-level mathematics, and often make connections between the topic at hand and other concepts. Some problems extend the work of the associated activity, while others involve work from prior grades, prior units in the course, or reflect work that is related to the K–12 curriculum but are a type of problem not required by the standards. The problems are not routine or procedural, and they are not just ‘the same thing again but with harder numbers.’ They are intended for use on an opt-in basis by students—if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems, and it is not expected that any single student works on all of them. Are You Ready for More? problems also are good fodder for a Problem of the Week or similar structure.” While there are no instances where advanced students do more assignments than their classmates, materials do provide multiple opportunities for students to investigate grade-level content at greater depth.

• Course Guide, Key Structures in This Course, Teacher Learning through Curriculum, Building on Student Thinking states, “Certain activities within each lesson plan include ways to provide guidance, based on students’ understandings and ideas. Building on Student Thinking offers look-fors and questions to support students as they engage in an activity. Effective teaching requires supporting students as they work on challenging tasks, without taking over the process of thinking for them (Stein, Smith, Henningsen, & Silver, 2000). Monitor during the course of an activity to gain insight into what students know and are able to do. Based on these insights, the Building on Student Thinking section provides questions that advance students’ understanding of mathematical concepts, strategies, or connections between representations.”

• Grade 6, Unit 1, Area and Surface Area, Lesson 5, Activity 5.3, Are You Ready for More? states, “1. What happens to the area of a parallelogram if the height doubles but the base is unchanged? If the height triples? If the height is 100 times the original? 2. What happens to the area if both the base and the height double? Both triple? Both are 100 times their original lengths?” Building On Student Thinking states, “Finding a height segment outside of the parallelogram may still be unfamiliar to students. Have examples from the ‘The Right Height?’ activity visible so they can serve as a reference in finding heights. Students may say that the base of Parallelogram D cannot be determined because, as displayed, it does not have a horizontal side. Remind students that in an earlier activity we learned that any side of a parallelogram could be a base and that rotating our paper can help us see this. Ask students to see if there is a side whose length can be determined.”

• Grade 7, Unit 1, Scale Drawings, Lesson 2, Activity 2.3, Are You Ready For More? states, “Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.” Building On Student Thinking states, “Students may think that Triangle F is a scaled copy because just like the 3-4-5 triangle, the sides are also three consecutive whole numbers. Point out that corresponding angles are not equal.”

• Grade 8, Unit 6, Associations in Data, Lesson 5, Activity 5.4, Are You Ready for More? states, “These scatter plots were created by multiplying the x-coordinate by 3 then adding a random number between two values to get the y-coordinate. The first scatter plot added a random number between -0.5 and 0.5 to the y-coordinate. The second scatter plot added a random number between -8 and 8 to the y-coordinate. The third scatter plot added a random number between -20 and 20 to the y-coordinate. 1. For each scatter plot, draw a line that fits the data. 2. Explain why some were easier to do than others.” Three scatter plots are shown.

The materials reviewed for Kendall Hunt IM v.360 Grade 6 through Grade 8 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they support and, when appropriate, are connected to written methods.

Course Guide, Key Structures in This Course, Purposeful Representations states, Purposeful Representations states, “Several considerations guide the choice of representations, the timing of their introduction, and the sequence in which they are presented, such as the extent to which they: Serve the mathematical learning goals. Help to build conceptual understanding. Support a coherent progression of mathematical ideas. Are relevant across cases and contexts. The principle of ‘concrete before abstract’ also guides the use of representations. This principle comes into play in two ways: First, more concrete representations are introduced before those that are more abstract. For example, to develop an understanding of ratios, students first use physical objects to represent quantities related by a ratio. Next, they create discrete diagrams to represent such quantities, followed by double-number-line diagrams and tables of equivalent ratios. This thinking eventually bridges into representing non-proportional linear relationships and the multiple ways to represent functions that starts in IM Grade 8 and continues throughout high school. Each representation is less concrete, more abstract, and offers greater flexibility and efficiency than the one before it. Second, a representation can show concrete concepts and numbers in earlier courses, and abstract ideas and quantities in later courses. For example, starting in IM Grade 3, students use rectangular diagrams to represent the multiplication of whole numbers. In IM Grade 6, they use such diagrams to represent the product of multi-digit decimals. In these early uses, students relate the discrete factors and product to concrete attributes of the rectangle: its side lengths and area. Later, students use rectangles to represent the multiplication of expressions that include variables, such as 5\cdot (3x+8) and a(2b + c). The same diagram is now used in a more abstract sense: to represent and organize decomposable factors and their partial products. In high school, students use the same diagram to reason about the multiplication and division of binomials and higher-degree polynomials.” Manipulatives are referenced within lessons as appropriate to support concept development. Examples include:

• Grade 6, Unit 2, Introducing Ratios, Lesson 4, Activity 4.2, students experiment with mixing different batches of a color recipe using physical tools or a digital applet to explore equivalent and non-equivalent ratios of blue to yellow that affect the resulting shade of green. Launch states, “Read aloud the first sentence of the activity statement. Demonstrate how to pour liquid from the beakers into the graduated cylinder to measure 5 ml of blue water and 15 ml of yellow water. Show students how to get an accurate reading on the graduated cylinder by working on a level surface and by reading the measurement at eye level. Next, label an empty cup with the ratio of blue water and yellow water. Then, pour the colored water from the cylinders into the cup and mix. Tell students this is a single batch of the recipe, and that they will experiment with different mixtures of green water—doubling the recipe, tripling the recipe, and inventing a new recipe—and observe the resulting shades. Arrange students in groups of 2–4. (Smaller groups are preferable, but group size might depend on available equipment.) Each group needs a beaker of blue water and one of yellow water, one graduated cylinder, a permanent marker, a craft stick, and 3 opaque cups (styrofoam, paper, or plastic) with a white interior. Each group also needs a cup with a single batch of the original recipe (20 ml of green water) for comparison purposes. Before students begin, point out that for each experiment, there are several steps to take before mixing the blue and yellow water, and one step to complete afterward. Briefly review the directions for doubling the original recipe. When discussing how to draw a diagram, ask students how they might represent the amounts of blue water and yellow water in the single batch that was just mixed.”

• Grade 7, Unit 3, Measuring Circles, Lesson 2, Activity 2.4, students practice identifying and using the terms radius, diameter, and center by drawing circles. Launch states, “Distribute rulers. Give students a few minutes of quiet work time for the first two questions. If a student asks for a circular object to trace, graph paper, a protractor, or a compass, make that available. After drawing Circles A and B, but before drawing Circles C and D, ask students: ‘What was difficult about drawing the circles?’ ‘How could you make your drawings more precise?’ ‘What tools might be helpful?’ Once students realize that a compass would be a good tool for this task, distribute compasses to all students. Highlight the connection between the usefulness of a compass and the fact that any point on a circle is the same distance from the center.”

• Grade 8, Unit 7, Exponents and Scientific Notation, Lesson 10, Activity 10.2, students use a number line, either physical or digital, to explore powers of 10, compare and estimate relative sizes of numbers, and begin reasoning about rewriting expressions in scientific notation by identifying patterns in the structure of the number line. Digital Launch states, “A device that can run the applet is needed for every student. Arrange students in groups of 2. Give students 5 minutes of quiet time to work on the first two problems, followed by 1–2 minutes to discuss their work with their partner for the last problem. Follow with a brief whole-class discussion. Tell students to drag the points to their location on the number line. When all five points are on the line, feedback is available by clicking the ‘Check Points’ button, though feedback will not appear if none of the points are in the correct location. Note that labels are placed above or below the points only to avoid crowding on the number line.”