## Kendall Hunt's Illustrative Mathematics 6-8 Math

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### Overall Summary

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectation for alignment to the CCSS. In Gateway 1, the instructional materials meet the expectations for focus by assessing grade-level content and spending at least 65 percent of class time on the major clusters of the grade, and they are coherent and consistent with the Standards. In Gateway 2, the instructional materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, and they connect the Standards for Mathematical Content and the Standards for Mathematical Practice.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for Gateway 1. These materials do not assess above-grade-level content and spend the majority of the time on the major clusters of each grade level. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are consistent with the mathematical progression in the standards, and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the materials meet the expectations for focusing on the major work of the grade, and the materials also meet the expectations for coherence.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectation for not assessing topics before the grade level in which the topic should be introduced. The materials do not include any assessment questions that were above grade level.

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The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for assessing grade-level content. The assessments are aligned to grade-level standards. Examples include:

• In Unit 5, Mid-Unit Assessment, Problem 5 provides a scenario in which bacteria is doubling over time. Students determine if the given population of bacteria is a function of the number of days. Students also determine if the function is linear and explain their reasoning (8.F.1). The problem reads, “Lin counts five bacteria under a microscope. She counts them again each day for four days, and finds that the number of bacteria doubled each day—from 5 to 10, then from 10 to 20, and so on. Is the population of bacteria a function of the number of days? If so, is it linear? Explain your reasoning.”
• In Unit 7, End-of-Unit Assessment, Problem 2, students select all expressions that equal  $$6^{-10}$$. The five choices are all variations of using the properties of exponents (8.EE.1).

Assessments are located on each Unit Page under the Assessments tab for each of the first eight units. Unit 9 is an optional unit and has no assessments. Assessments are limited to seven problems, but these are often broken into multiple prompts and assess numerous standards. Units 1 and 5 contain Mid-Unit Assessments for a total of 10 summative assessments.

#### Criterion 1.2: Coherence

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for having students and teachers using the materials as designed, devoting the large majority of class time to the major work of the grade. Overall, the materials devote at least 65 percent of class time to major work.

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Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for spending a majority of instructional time on major work of the grade.

• The number of units devoted to major work of the grade, including assessments and supporting work, is 7 out of 8, which is approximately 88 percent.
• The number of non-optional lessons devoted to major work of the grade, including assessments and supporting work, is 103 out of 121 total non-optional lessons, which is approximately 85 percent.
• The number of days devoted to major work, including assessments and supporting work, is 118 out of 149 days, which is approximately 79 percent.

A lesson-level analysis is most representative of the instructional materials because this calculation includes all lessons with connections to major work with no additional days factored in. As a result, approximately 85 percent of the instructional materials focus on major work of the grade. An analysis of days devoted to major work includes 18 days for review and assessment, but the materials do not indicate which items to use for the review.

#### Criterion 1.3: Coherence

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for being coherent and consistent with the standards. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and fosters coherence between the grades. Content from prior or future grades is clearly identified, and the materials explicitly relate grade-level concepts to prior knowledge from earlier grades. The objectives for the materials are shaped by the CCSSM cluster headings, and they also incorporate natural connections that will prepare a student for upcoming grades.

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Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Supporting standards/clusters are connected to the major standards/clusters of the grade. Multiple lessons in the Grade 8 curriculum incorporate supporting standards in ways that support and/or maintain the focus on major work standards. Examples of the connections between supporting work and major work include the following:

• In Unit 5, Lesson 17, students use volume formulas for three-dimensional figures to explore aspects of functions. Students compare functions presented as equations, graphs, and/or tables when exploring linear and nonlinear relationships. In the lesson Warm-Up, students determine if two quantities are functions by examining the distance a truck traveled in relation to the amount of gas used (8.F.1). Activities 3 and 4 address 8.F.1, 8.F.B, and 8.G.9 as: “Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height.” The lesson Cool-Down connects 8.G.C and 8.F.1 as students explore the height and volume of cylinders. In Lesson 18, students explore the effect on the volume of cylinders in functions where two variables are scaled. Students use volume formulas for cylinders and graphs of the functions in this exploration.
• In Unit 6, the first three lessons introduce students to associations in two variables and how to display and analyze those associations using scatter plots (8.SP.2). In the Lesson 4 Warm-Up, students estimate the slope of a given line that lies close to, but not directly on, two points. During lesson activities, students state the meaning of the slope in the given contexts and also whether outliers have an effect on the slope of the line of best fit. Lessons 5 through 8 continue to build students' understanding of lines and slope (8.EE.5) to build an understanding of scatter plots and bivariate data.
• In Unit 8, Lesson 3, Activities, students assess whether different rational numbers are solutions for square roots. In Lesson 4, students find solutions to functions that include square roots, and they estimate solutions to square roots in order to develop a deeper understanding of square roots as numbers and not just as solutions to the side length of a square when the area is known (8.NS.2, 8.EE.2).
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The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials for Kendall Hunt’s llustrative Mathematics 6-8 Math, Grade 8 meet expectations that the amount of content designated for one grade level is viable for one year.

The suggested amount of time and expectations of the materials for teachers and students are viable for one school year as written and would not require significant modifications. As designed, the instructional materials can be completed in a school year.

• The provided scope and sequence found in the Grade 8 Course Guide includes materials for 149 instructional days. There are 121 non-optional lessons, 18 assessment days (16 summative), and 10 optional lessons.
• 118 of these non-optional lessons are designed to address grade-level standards. 3 non-optional lessons provide problem contexts and activities that prepare students for the unit or connect work from prior grades.
• 4 of the optional lessons are present throughout the first eight units, and Unit 9 is an optional unit which includes 6 lessons.
• Units 1-8 are comprised of 11 to 22 lessons, and each lesson is designed for 45-50 minutes. Within each unit, lessons contain a Warm-Up, two or three Activities, a Lesson Synthesis, and a Cool-Down. Guidance regarding the number of minutes needed to complete each component of the lesson is provided in the Course Guide.
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Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for being consistent with the progressions in the standards. The instructional materials clearly identify content from prior and future grade levels and use it to support the progressions of the grade-level standards. The instructional materials also relate grade-level concepts explicitly to prior knowledge from earlier grades.

The materials are intentionally designed to address the standards the way they are laid out in the progressions, and the Grade 8 Narrative in the Course Guide describes how the standards and progressions are connected for educators. Units begin with lessons connected to the standards from prior grades that are relevant to the current topic. Standards from the grade level and prior grades, and standards that will be addressed later in the year are identified in the sections as “addressing,” “building on,” and “building towards,” respectively. For example:

• In Unit 3, Lesson 1, the Lesson Narrative states, “This lesson is the first of four where students work with proportional relationships from a Grade 8 perspective.” In the lesson, 7.RP.2 is identified as the standard that the lesson is “building on,” the standard the lesson is “addressing” is 8.EE.B, and the standard the lesson is “building toward” is 8.EE.5. According to the CCSSM progressions, the study of proportional relationships is a foundation for the study of functions. After Lesson 4, students move to linear relationships. The Lesson 5 Overview states, “After revisiting examples of proportional relationships in the previous lessons, this lesson is the first of four lessons that moves from proportional relationships to linear relationships with positive rates of change.”

The Warm-Ups in lessons frequently work with prior-grade standards in ways that support learning of grade-level problems and make connections to progressions from previous grades. For example:

• In Unit 3, Lessons 1 and 2 begin with looking at graphs and comparing proportional relationships (7.RP.2). The Lesson 1 Warm-Up has students compare graphs of proportional relationships with the same scale. The Lesson 2 Warm-Up includes different scales. Lessons 3 and 4 continue work with graphing proportional relationships with attention to graph features such as scale. Students pay attention to graph features in preparation for grade-level work with linear equations and graphs of linear equations.
• The Unit 8, Lesson 3 (6.EE.4, 5.NF.4) Warm-Up includes practice with multiplying fractions (5.NF.4) which is essential in the lesson for estimating square roots. This connection is explicit in the teacher materials that state, “The purpose of this warm-up is for students to review multiplication of fractions in preparation for the main problem of this lesson: estimating solutions to the equation $$x^2$$= 2. For example, 3⁄2 · 3⁄2 = 9⁄4, which is a value close to 2 so 3⁄2 is a value close to the $$\sqrt2$$.”

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems.

In the Course Guide under Lessons and Standards, there is a table which reflects the mathematics in the materials. All grade-level standards are represented across the 9 units. Tasks are aligned to grade-level work and are connected to prior-grade knowledge. For example:

• In the Course Guide, the Narrative for Unit 5 describes multiple ways that function understanding is developed by building upon prior-grade work, including input/output patterns in Grades 4 and 5, independent and dependent variables in Grade 6, work with ratios and proportions in Grades 6 through 8, and work with geometric measurement as early as Kindergarten. Students explore linear relationships built from work early in Units 1 and 2 with similarity of triangles. The unit develops from exploring input/output patterns in order to develop a formal understanding of the definition of a function in Lessons 1 and 2, to exploring different representations of functions in Lessons 3 through 7. This work includes verbal descriptions, equations, tables, and graphs. Multiple representations are used in each lesson. The lessons near the end of the unit work with volume formulas. Lesson 22, the final lesson in the unit, explicitly connects work with functions to volume and includes exposure to non-linear functions.
• In Unit 4, Lesson 13 consists of problems where students use graphs of systems of linear equations to reason about the solutions to equations. The first activity has students match the graph with the system of equations, then students work on a digital graph to find solutions to a system of equations. In the second activity, the students look at six different graphic representations of systems of equations and discuss the meaning of having no solution, one solution, or infinite solutions. In the Cool-Down, students look at the graphic representation of a system of equations and give possible solutions based on what the graph looks like. There are also five practice problems that are connected to the lesson.

A typical lesson has a Warm-Up, one or more Activities, Synthesis, and a Cool-Down. Additionally, every lesson provides practice problems that can be used as independent or group work. Some lessons also provide an “Are you ready for more?” problem. These problems are an opportunity for students to explore grade-level mathematics in more depth and often make connections between the topic in the lesson and other concepts at grade level. They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own.

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Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings, including:

8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.

• The Unit 4 overview describes how work in solving linear equations builds on work from Grades 6 and 7 with equations and expressions in one variable, then further builds to algebraic methods for solving equations. The Lesson 10 learning goals state: “Make connections between ordered pairs in tables and graphs,” and “Given descriptions of two linear relationships, interpret ordered pairs in contexts, focusing on when or where the same ordered pair makes each relationship true.” The Lesson 13 learning goals are also connected to the cluster heading, including: “Understand that a system of equations can have no solutions, one solution, or infinitely many solutions.” “Connect features of the graphs to the number of solutions the system has.” and “Start solving systems algebraically.”

8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

• The Unit 5 overview details the progression of geometric measurement since Kindergarten and lays out how the unit will extend concepts of volume to deepen students' understanding of the geometric features of three-dimensional shapes such as radii, bases, and heights. In addition, the unit integrates student work with functions as they develop a conceptual understanding of formulas for volume. All Unit 5 lessons include real-world or mathematical problems. Sample learning goals aligned to the cluster heading include: “Find volumes of cones in mathematical and real-world problems” in Lesson 15 and “Solve a variety of mathematical and real-world problems about the volume of a cylinder, cone, or sphere” in Lesson 21.

The materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Multiple examples of tasks connecting standards within and across clusters and domains are present.

• In Unit 5, Lesson 8, Activities 2 and 3, students model real-world problems with functions by interpreting the start value and the slope (8.F.B). This includes understanding linear equations as describing linear functions and involves comparing different representations of a function, with an emphasis on equations and graphs, but also includes verbal/written descriptions of the real-world scenario (8.F.A).
• In Unit 8, Lesson 2, 8.EE.A and 8.F.B are connected as students take measurements of area and side lengths, collect data, and determine y = $$x^2$$ as a function that models the relationship between the area of a square and a side length. Students make estimates of side lengths given the area of a square and begin to use square root symbols and terms.
• In Unit 3, Lesson 8, 8.G.A and 8.EE.B are connected. The Warm-Up task reviews 8.G.A.1 where students must identify lines that have been translated from a given line. The rest of the lesson addresses 8.EE.B with an emphasis on students using their understanding of translations to derive y = mx + b from a graph of y = mx. The series of tasks use contexts to help students make connections to the equation of a line and how translations can be helpful in interpretation in context.

### Rigor & Mathematical Practices

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectation for aligning with the CCSS expectations for rigor and mathematical practices. The instructional materials attend to each of the three aspects of rigor individually, and they also attend to the balance among the three aspects. The instructional materials emphasize mathematical reasoning, identify the Mathematical Practices (MPs), and attend to the full meaning of each practice standard.

##### Gateway 2
Meets Expectations

#### Criterion 2.1: Rigor

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance of all three aspects of rigor.

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Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level, and multiple opportunities exist for students to access concepts from different perspectives and independently demonstrate conceptual understanding throughout the grade.

In Unit 5, students define, evaluate, and compare functions (8.F.A), for example:

• Lesson 1 begins to develop the “idea of a function as a rule that assigns to each allowable input exactly one output.” In the first Activity of Lesson 1, students play "Guess My Rule" in an applet. Students enter any input value into Column A of the spreadsheet found in the applet, and the resulting output appears in Column B. Students generate a rule after completing as many iterations as needed. Functions' rules include additive, multiplicative, and exponential patterns.
• In Lesson 2, function language is introduced as students identify examples and nonexamples of functions in order to extend the idea that the output is dependent on the input. In the first Activity, students answer: “A number is 5. Do you know its square?” and the anticipated response is given as, “Yes, the square of 5 is 25.” Another prompt states, “The square of a number is 16. Do you know the number?” resulting in the anticipated response, “No, there are two different numbers whose square is 16, namely 4 and -4.” The term “function” is not introduced until the second Activity in which students use function language to express whether the given scenarios from the previous Activity are functions. The corresponding exemplar statements from the given examples are, “Yes, the square of a number depends on the number,” and “No, knowing the square of a number does not determine the number.”
• In Lesson 4, the second Activity, students examine three unlabeled continuous graphs in order to make connections between representations. They choose the matching equation and context (8.F.3), use the context to identify the dependent and independent variables, use the graph to identify the output when the input is 1, and interpret what that tells you about each situation (8.F.1). The graphs include a non-linear representation and two linear functions, one with positive slope and one with negative slope.

Unit 1, which begins with transformations, addresses congruence using physical models, transparencies, or geometry software (8.G.A).

• In Lessons 1 through 6, students spend most of the instructional time either physically moving shapes or imitating that movement in GeoGebra. In Lesson 1, students examine transformations as a way of moving objects in a plane; in Lesson 2, students define these movements in their own words.
• Lessons 11 through 13 explore congruence. In the Lesson 12 Warm-Up, students are given a variety of congruent triangles in different orientations and the following prompt: “All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle ABC under a translation.” Students develop the concept that a two-dimensional figure is congruent to another two-dimensional figure if the second can be obtained from the first by a sequence of transformations. The idea of “rotations and reflections usually (but not always) change the orientation of a figure” is discussed and further explored when students name a sequence of transformations to prove some of the non-shaded triangles congruent to triangle ABC.
• In Lessons 14 through 16, students establish informal arguments about angles. The second Activity in Lesson 14 states, "Lines ℓ and k are parallel, and t is a transversal. Point M is the midpoint of segment PQ.” Students use tracing paper to “Find a rigid transformation showing that angles MPA and MQB are congruent.”
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Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for attending to those standards that set an expectation of procedural skill. Materials attend to the Grade 8 expected procedural skills, particularly those related to solving linear equations and systems of equations.

Procedural skills and fluencies develop with conceptual understanding and are built upon work students have accomplished with operations and equations from prior grades. Students practice developed procedures throughout practice problem sets that follow the units, and students use emerging fluencies in the context of solving problems. According to the How to Use the Materials, Design Principles, “Students are systematically introduced to representations, contexts, language, and notation. As their learning progresses, they see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency.” Number Talks included in many Warm-Ups often revisit fluencies developed in earlier grades and specifically relate to the Activities found in the lessons. Additionally, students demonstrate procedural skills throughout the year in a variety of practice problems. Examples of practice problems include:

8.EE.7 is addressed in Unit 4 as students develop procedural skill in solving linear equations in one variable, and students practice this skill in Unit 5.

• In Unit 4, Lesson 3 Warm-Up, students match hanger diagrams to equations and variables to their respective shapes within the diagram. In the first Activity, students begin to match the first “moves” in solving equations. In the additional activities in Lesson 3, students “[think] about strategically solving equations by paying attention to their structure” when they are presented two student work samples to evaluate and provide recommendations for solving.
• In Unit 4, Lesson 4, there is a mix of tasks that focuses on practicing solving equations such as matching, choosing solution steps, evaluating the work of sample student solution paths, as well as assessing similarities and looking for mistakes.
• In Unit 4, Lesson 5, students move toward a general method for solving linear equations using mental math to solve one-step equations for a variable on one side and then work with a partner to justify their steps with one another between each step.
• In Unit 4, Lesson 6, Strategic Solving states: “In this lesson, students learn to stop and think ahead strategically before plunging into a solution method. After a Warm-Up in which they construct their own equation to solve a problem, they look at equations with different structures and decide whether the solution will be positive, negative, or zero, without solving the equation. They judge which equations are likely to be easy to solve and which are likely to be difficult.” In the following lessons, students examine situations when an equation has many or no solutions.
• In Unit 5, Lesson 3, students solve and check the solutions to multi-step equations such as 4z + 5 = -3z − 8. In Lesson 7, students solve and check the solution to the multi-step equation -(-2x + 1) = 9 − 14x. In Lesson 9, Problem 3, students solve and explain the reasoning leading to the solutions of multi-step equations such as 4(2a + 2) = 8(2 − 3a).

Systems of equations (8.EE.8b) are formally introduced in the latter part of Unit 4 after students solve linear equations in one variable (8.EE.7), including writing, solving, and graphing equations as well as deciding what it means for an equation to be true. Students learn to interpret and solve systems of equations in Lessons 12 through 15 in preparation for applying their procedural knowledge in Lesson 16.

• In the Lesson 12 Warm-Up, students examine a situation and determine the features of the graph of the situation without actually graphing. All tasks involve graphing equations and discussing solutions in context of graphs. The term “system of equations” is introduced. Also in Lesson 12, students solve a system of two equations, both in slope-intercept form.
• In the Lesson 14 Warm-Up, students use substitution strategies to mentally solve systems of equations. In the first Activity, students analyze the structure of a system of equations before deciding on an efficient solution path. The equations lend themselves to suggesting substitution as the first step toward finding a solution and develop the procedure for substituting an expression in place of a variable.
• In Lesson 15, students write and interpret systems of equations from contexts with rational coefficients and continue to practice solving using various methods, including: “[Solving] the systems to find the number of solutions; [Using] the slope and y-intercept to determine the number of solutions; [Manipulating] the equations into another form, then compare the equations. [Noticing] that the left side of the second equation in system C is double the left side of the first equation, but the right side is not.”
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Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, both routine and non-routine, presented in a context in which the mathematics is applied.

Applications occur throughout the materials and are used throughout the curriculum to build conceptual understanding. The Grade 8 Course Guide states: “Students have opportunities to make connections to real-world contexts throughout the materials. Carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end.”

In Unit 5, Lessons 5 through 11, students model relationships with functions by identifying and creating tables, graphs, and equations (8.F.B).

• The Lesson 6 Warm-Up states: “The purpose of this Warm-Up is for students to realize there are different dependent variables that can be used when making a model of a context, and the choice of which we use affects how a graph of a function looks.” Students view five photographs of a dog taken at equal intervals of time and two graphs representing the scenario. Both graphs have the same independent variable but look dramatically different. Students determine how the dependent variable represents the perspective of the graph.
• In the Lesson 10 Warm-Up, students share what they notice on a graph of temperature data during different parts of the day. In the first Activity, they use piecewise linear graphs to find information about the real-life situation they represent. In the second Activity, students analyze a situation to calculate the rate of change.
• In Lesson 11, the second Activity, students investigate how the height of water in a graduated cylinder is a function of the volume of water in the graduated cylinder. Students make predictions about how the graph will look,  test their prediction by filling the graduated cylinder with different amounts of water, and gather and graph the data. Students use an applet to reason about the height of the water in a given cylinder. They graph the relationship and explain the meaning of specific points in their recorded data. Students use the context of the given cylinder in the graph to determine, “What would the endpoint of the graph be?” In the next phase of the Activity, students compare this relationship to ones in which the radius of the cylinder has been modified and explain how the slope is less steep in the given graphic representation.

In Unit 4, students solve real-world and mathematical problems leading to two linear equations in two variables (8.EE.8c). Lesson 16 includes opportunities for students to investigate applications of systems of equations.

• In the first Activity, students solve problems involving real-world contexts. For the first problem, students find the time at which two friends will meet if they are cycling toward one another. In the second problem, they determine how many grapefruits are sold if students are selling both grapefruits and nuts. The price of both items is given as well as the total number of items sold and the total money made in the fundraiser. In the third problem, students find the number of hours Andre and Jada must work to make the same amount of money when working different jobs and getting paid different rates. In each problem, students “explain or show [their] reasoning.” After engaging in these problems, students create their own situation and solve. These scenarios are then exchanged for other pairs of students to solve.
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Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The materials address aspects of rigor independently, and there are instances when multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized.

Examples of conceptual understanding include:

• In Unit 1, Lesson 3, the first Activity, students are given four figures on a grid and told: “In Figure 1, translate triangle ABC so that A goes to A′. In Figure 2, translate triangle ABC so that C goes to C′. In Figure 3, rotate triangle ABC $$90\degree$$ counterclockwise using center 0. In Figure 4, reflect triangle ABC using line l.” Activities similar to this are done repeatedly throughout the beginning lessons of Unit 1. Eventually, students use this information to draw conclusions about congruent figures, angles, and similar figures (8.G.A).
• In Unit 5, Lesson 6, students demonstrate conceptual understanding of graphs as they relate to context and determine the scale to represent the independent and dependent variables. Students determine the scale and create graphs for a given story/context. Discussion questions include: “Which quantity is a function of which? Explain your reasoning;" "Based on your graph, is his friend’s house or the park closer to Noah's home? Explain how you know;" and, "Read the story and all your responses again. Does everything make sense? If not, make changes to your work.”

Examples of procedural skills include:

• In Unit 4, Lesson 5, students develop procedural skill solving equations with one variable. During the Warm-Up, students solve the following equations mentally: 5 − x = 8, -1 = x − 2, -3x = 9, and -10 = -5x. In the first Activity, students are given a card with a more complex equation on it and told to work with a partner, taking turns after each step to solve the equation. They solve equations on four cards in all.
• In Unit 5, Lesson 16 Warm-Up, students develop skills using the formula for volume to solve: “27 = (1⁄3)h, 27 = (1⁄3)$$r^2$$, 12π= ( 1⁄3)πa, 12π = (1⁄3)π$$b^2$$.” In the first Activity, students practice finding relevant information and completing tables to apply the volume formula to find the value of unknown dimensions.

Examples of application include:

• In Unit 2, Lesson 13, the third Activity, students establish facts about similar triangles and angle relationships (8.G.5) to estimate the height of a tall object that cannot be measured directly. Students devise a method to estimate, justify, and test their estimate.
• In Unit 5, Lesson 21, students solve a variety of mathematical problems involving finding the volume of cones, cylinders, rectangular prisms, and spheres in the given figures (8.G.9).

Examples of lessons where two or three aspects of rigor are connected include:

• In Unit 5, Lessons 3 through 7, students develop their understanding of functions by comparing multiple representations. The majority of Activities use real-world contexts with frequent opportunities for students to interpret functions and their representations in specific contexts.
• The Practice Problems available for each lesson are arranged so that students practice and develop skills for a concept and engage with more complex applications. Typically, there are six problems included in the practice problems. Procedural practice, visual representations, contexts, and/or standard methods of solving said problems are present.

#### Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for practice–content connections. The materials identify and use the MPs to enrich the content, attend to the full meaning of each MP, support the Standards' emphasis on mathematical reasoning, and attend to the specialized language of mathematics.

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The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. All eight MPs are clearly identified throughout the materials. The MPs are initially identified in the narrative for each unit described within the course information, for example:

• In Unit 2, excerpts from the Unit Narrative include: “Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them (MP1), students use and extend their knowledge of geometry and geometric measurement.” Further, in the lesson narrative it states, “They use the definition of 'similar' and properties of similar figures to justify claims of similarity or non-similarity and to reason about similar figures (MP3)."
• In Unit 6, the Unit Narrative states: “[Students] return to the data on height and arm span gathered at the beginning of the unit, describe the association between the two, and fit a line to the data (MP4). The third section focuses on using two-way tables to analyze categorical data (MP4). Students use a two-way frequency table to create a relative frequency table to examine the percentages represented in each intersection of categories to look for any associations between the categories. Students also examine and create bar and segmented bar graphs to visualize any associations.”

Within a lesson, the MPs are identified within the Lesson Narratives or before each of the activities. Lesson Narratives often highlight when a Math Practice is particularly important for a concept or when a task may exemplify the identified Practice, for example:

• In Unit 1, Lesson 14, the narrative states, “One Mathematical Practice that is particularly relevant for this lesson is MP8. Students will notice as they calculate angles that they are repeatedly using vertical and adjacent angles and that often they have a choice which method to apply. They will also notice that the angles made by parallel lines cut by a transversal are the same and this observation is the key structure in this lesson.”
• In Unit 2, Lesson 2, the narrative discusses developing the idea of dilations by providing tools for students: “As with previous geometry lessons, students should have access to geometry toolkits so they can make strategic choices about which tools to use (MP5).”
• In Unit 4, Lesson 5, the narrative for the first Activity states, “The goal of this activity is for students to build fluency solving equations with variables on each side. Students describe each step in their solution process to a partner and justify how each of their changes maintains the equality of the two expressions (MP3).”
• In Unit 5, Lesson 3, the Warm-Up narrative states: “The purpose of this Warm-Up is for students to use repeated reasoning to write an algebraic expression to represent a rule of a function (MP8).”

The MPs are used to enrich the mathematical content and are not treated separately from the content in stand-alone lessons. MPs are highlighted and discussed throughout the lesson narratives to support a teacher’s understanding of the MP itself as the teacher is provided direction regarding how the content is connected to the MP, for example:

• In Unit 1, Lesson 13, the connection of MP6 to 8.G.2 is explained: “One of the mathematical practices that takes center stage in this lesson is MP6. For congruent figures built out of several different parts (for example, a collection of circles), the distances between all pairs of points must be the same. It is not enough that the constituent parts (circles, for example) be congruent: they must also be in the same configuration, the same distance apart. This follows from the definition of congruence: rigid motions do not change distances between points, so if Figure 1 is congruent to Figure 2 then the distance between any pair of points in Figure 1 is equal to the distance between the corresponding pair of points in Figure 2.”
• In Unit 9, Lesson 1, students extend their knowledge of transformations by creating a tessellation. The Lesson Narrative for the second Activity states, “Students look for and make use of structure (MP7), both when they try to put copies of the shape together to build a tessellation and when they examine whether or not it is possible to construct a different tessellation.”

The MPs are not identified in the student materials; however, there are questions posed with activities that engage students with MPs. For example, the student digital task accompanying Unit 9, Lesson 1 states,: “With your partner, choose one of the six shapes in the toolbar that you will both use. 1) Select the shape tool by clicking on it. Create copies of your shape by clicking in the work space. 2) When you have enough to work with, click on the Move tool (the arrow) to drag or turn them. 3) If you have trouble aligning the shapes, right click to turn on the grid.” After they have finished, the discussion prompts lead students to find structure in the patterns, “Compare your tessellation to your partner’s. How are they similar? How are they different?”

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Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for carefully attending to the full meaning of each practice standard. The materials attend to the full meaning of each of the 8 MPs. The MPs are discussed in both the unit and lesson narratives, as appropriate, when they relate to the overall work. They are also explained within individual activities, when necessary. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:

MP1 - Make sense of problems and persevere in solving them.

• In Unit 1, Lesson 6, the first Activity, students “make sense of problems by determining what information is necessary, and then ask for information they need to solve it. This may take several rounds of discussion if their first request does not yield the information they need.” The narrative for the second Activity includes: “Tell students they will continue to describe transformations using coordinates. Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of two. Provide access to graph paper. In each group, distribute a problem card to one student and a data card to the other student. They need to know which transformations were applied (i.e., translation, rotation, or reflection). They need to determine the order in which the transformations were applied. They need to remember what information is needed to describe a translation, rotation, or reflection.”
• In Unit 5, Lesson 21, the second activity, students engage in another Info Gap activity. One student is given a card that has a question related to volume equations of cylinders, cones, and spheres. The other student is given a card with all the information needed to answer the question. The student with the question asks the student with the data a series of questions that will give them the necessary information to solve the problem. It may take several rounds of discussion if their first request does not yield the information they need, creating a situation in which students have to persevere to solve a problem.

MP2 - Reason abstractly and quantitatively.

• In Unit 5, Lesson 7, the third activity, students compare “properties of functions represented in different ways. Students are given a verbal description and a table to compare and decide whose family traveled farther over the same time intervals. The purpose of this activity is for students to interpret and compare functions.”
• In Unit 7, students reason abstractly and quantitatively to solve problems involving operations with exponents. For example, in Lesson 2, the first Activity, students are shown a diagram of three base ten blocks: a hundred block, a ten block, and a one block. They must answer several questions: “If each small square represents 102, then what does the medium rectangle represent? The large square?” Additional questions change the chosen square and power of 10. The visual element provides both an abstract and quantitative entry point to the problem as students are introduced to the Laws of Exponents.

MP4 - Model with mathematics.

• In Unit 2, Lesson 13, students model a real-world context with similar triangles to find the height of an unknown object. Students examine the length of shadows of different objects to find that a proportional relationship exists between the height of the object and the length of its shadow. Students use their knowledge of similar triangles and the hypothesis that the rays of sunlight making the shadows are parallel to justify the proportional relationship between the object and its shadow. Students then go outside and make their own measurements of different objects and the lengths of their shadows and use this technique to estimate the height of these objects.
• In Unit 6, Lesson 8, Activity 2, students model and analyze data related to arm span and height measurements that were gathered in a previous lesson. The students create a scatter plot, identify and explain outliers, and explain whether the equation y = x is a good fit for the data. This activity is an opportunity for students to explore bivariate data that occurs in the real world and and is related to themselves.

MP5 - Use appropriate tools strategically.

• In Unit 1, lesson plans suggest that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to select appropriate tools and use them strategically to solve problems. Lessons in this unit ensure the full depth of this MP by emphasizing choice. For example, the Lesson 1 Narrative states: “To make strategic choices about when to use which tools (MP5), students need to have opportunities to make those choices. Apps and simulations should supplement rather than replace physical tools.” In the Lesson 14 Warm-Up, the narrative includes the following guidance for teachers: “Some students may wish to use protractors, either to double check work or to investigate the different angle measures. This is an appropriate use of technology (MP5), but ask these students what other methods they could use instead.”
• In Unit 7, Lesson 10, students have opportunities to use digital tools and number lines, using them to think about how to rewrite expressions with exponents. In the second Activity, students are given a table showing how fast light waves or electricity can travel through different materials and a number line applet labeled 0, 1 x $$10^8$$, 2 x $$10^8$$, ...9 x $$10^8$$, 1 x $$10^9$$ with a magnifier that will expand the space between any two consecutive numbers. The students must convert the given speeds to a usable format and plot them on the given number line(s) as precisely as possible.

MP7 - Look for and make use of structure.

• In Unit 7, Lesson 3, the first activity, students look for patterns when 10 is raised to a power so that numerical expressions can be written with a single exponent, such as $$10\cdot10^3 = 10^4$$. In Lesson 6, students analyze the structure of exponents to make sense of expressions with multiple bases.
• In Unit 2, Lesson 11, the second Activity, students have already generated a rule to determine whether or not a point with coordinates (x,y) lies on a certain line when the line represents a proportional relationship. “In this activity, students find a rule to determine if a point (x,y) lies on a line that does not pass through (0,0).” Students use the structure of a line and properties of similar triangles to investigate rules relating pairs of coordinates on a line.

MP8 - Look for and express regularity in repeated reasoning.

• In Unit 3, Lesson 8, students write equations of lines using y = mx + b. In this lesson and the ones that lead up to this, students develop their understanding based on repeated reasoning about equations of lines. In a previous lesson, students wrote an equation of a line by making generalizations from repeated calculations using their understanding of similar triangles and slope. Additionally, they have written an equation of a linear relationship by reasoning about initial values and rates of change and have graphed the equation as a line in the plane. In this lesson, they develop the idea that any line in the plane can be considered a vertical translation of a line through the origin.
• In Unit 7, Lesson 6, students explore the idea that repeated division by two is equivalent to repeated multiplication by 1/2. This allows students to make sense of negative exponents later in the second activity, where students look for patterns when bases of 10 are raised to a power in the given chart. Finally, in Lesson 5, students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication of 1/10.
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Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
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Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for prompting students to construct viable arguments and/or analyze the arguments of others concerning key grade-level mathematics. The student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Students explain their reasoning and compare their strategies for solving in small group and whole class settings, and examples include:

• In Unit 2, Lesson 6, Cool-Down, students analyze and correct a series of transformations and a dilation intended to provide a similar figure. In Lesson 7, the first Activity, students analyze two figures that are reported to be similar. Students justify or deny the claim; they build reasoning as to what characteristics are found in polygons that are not similar.
• In Unit 5, Lesson 4, the first Activity, students answer, “For each function: What is the output when the input is 1?  What does this tell you about the situation? Label the corresponding point on the graph. Find two more input-output pairs. What do they tell you about the situation? Label the corresponding points on the graph.” Questions such as these are present throughout the lessons, providing students the opportunity to construct viable arguments in both verbal and written form.
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Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others throughout the program.

• The Unit 3, Lesson 5 Warm-Up provides guiding questions in the Activity Synthesis to help students practice MP3. This strategy is used repeatedly throughout the teacher materials. “To involve more students in the conversation, consider asking: Who can restate ____’s reasoning in a different way? Did anyone solve the problem the same way but would explain it differently? Did anyone solve the problem in a different way? Does anyone want to add on to ____’s strategy? Do you agree or disagree? Why?”
• The Unit 6, Lesson 10, second Activity provides guidance to the teacher during the observation of small groups using data displays to find a bivariate association. “As students work, identify groups that use the different segmented bar graphs to explain why there is an association between the color of the eraser and flaws…Select previously identified groups to share their explanation for noticing an association.”
• In Unit 3, Lesson 11, the first Activity, students explore vertical and horizontal lines in the coordinate plane. Teachers are prompted to: “...pause their work after question 2 and discuss which equation makes sense and why.”
• In Unit 1, Lesson 11, in the activity narrative, teachers facilitate a discussion to promote student debate so students can identify reasons and construct arguments as well as critique/analyze responses from others. The teacher guidance also provides ways to help students analyze/critique other’s arguments if it does not occur naturally by providing the teacher examples of what to say/suggest to promote more discourse: “For each pair of shapes, poll the class. Count how many students decided each pair was the same or not the same. Then for each pair of shapes, select at least one student to defend their reasoning.” Sample responses are provided for the teacher.
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Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for attending to the specialized language of mathematics. The materials provide explicit instruction on communicating mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In the teacher materials, the Grade 8 Glossary is located in the Course Guide. Lesson-specific vocabulary can be found in bold within the lesson, and is listed and defined at the end of the student lesson. In the student materials, the entire Glossary is accessible by a tab on the student home page.
• Both the unit and lesson narratives contain specific guidance for the teacher on methods to support students to communicate mathematically. Within the lesson narratives, new terms are in bold print and explained as related to the context of the material.
• Unit 1 develops the concept of Rigid Transformations. In the initial lessons, students use their own words to describe moving one figure to another. As the unit progresses, students build their understanding of transformations and name these movements with the proper terminology (translations, reflections, rotations).
• In Unit 6, Lessons 1 through 6 introduce scatter plots and related terminology. In Lessons 1 through 3, students are introduced to the definition of scatter plot, and they explore data represented by a scatter plot and the meaning of a specific point on a graph. In Lesson 4, the term outlier is introduced. In Lesson 5, students learn to explain trends in scatter plots, and in Lesson 6, they determine the slope of the “linear model”. Each lesson builds on the initial definition of scatter plot until students work with all aspects of them, understand the concept, and use related terminology.

No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.

### Usability

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for being well-designed and taking into account effective lesson structure and pacing. The instructional materials distinguish between problems and exercises, have exercises that are given in intentional sequences, have a variety in what students are asked to produce, and include manipulatives that are faithful representations of the mathematical objects they represent.

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The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson. It is clear when students solve problems to learn and when they apply skills.

Lessons include a Warm-Up, one or more Activities, Synthesis, and a Cool-Down. Practice Problems are in a separate section of the instructional materials, distinguishing between problems students complete and exercises in the lessons. Warm-Ups serve to either activate prior learning or as a hook to engage students in new material. Students learn and practice new mathematics in lesson Activities. In the Activity Synthesis, students build on their understanding of the new concept. Each activity lesson ends with a Cool-Down where students apply what they learned in the activities or are introduced to skills they may need in the next lesson.

Practice problems are consistently found in the “Practice Problem” sets that accompany each lesson. These sets of problems include questions that support students in developing mastery of the current lesson and unit concepts, in addition to review of material from previous units. When practice problems contain content from previous lessons, students apply their skills and understandings in different ways that enhance understanding or application (e.g., increased expectations for fluency, more abstract application, or a non-routine problem).

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Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for not being haphazard; exercises are given in intentional sequences.

Overall, clusters of lessons within units and activities within lessons, are intentionally sequenced so students develop understanding. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skill and fluency, and engage with multiple activities that are sequenced from concrete to abstract or increase in complexity. Lessons end with a Cool-Down which is aligned to the daily lesson objective. Unit sequences consistently follow the progressions outlined in the CCSSM Standards to support students' development of conceptual understanding and procedural skills.

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There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for having variety in what students are asked to produce.

The instructional materials prompt students to produce products in a variety of ways. Students produce solutions within Activities and Practice Problems, as well as participating in class, group, and partner discussions. Materials provide opportunities for students to construct viable arguments and critique the reasoning of their peers. Students use both a digital platform and paper-pencil to conduct and present their work. The materials consistently prompt students for solutions that represent the language and intent of the standards. Students use representations such as tables, number lines, double number lines, tape diagrams, and graphs, as well as strategically choose tools to complete their work (MP5). Lesson activities and tasks are varied within and across lessons.

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Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. The series includes a variety of virtual manipulatives and integrates hands-on activities that allow the use of physical manipulatives, for example:

• Manipulatives and other mathematical representations are consistently aligned to the expectations and concepts in the standards. The majority of manipulatives used are commonly accessible measurement and geometry tools.
• The materials provide digital applets for manipulating geometric shapes, such as GeoGebra applets, tailored to the lesson content and tasks. When physical, pictorial, or virtual manipulatives are used, they are aligned to the mathematical concepts they represent. For example, in Unit 4, Lesson 2, hanger diagrams are used to represent and support the conceptual development of balance as it relates to equality in the virtual applet practice.
• Examples of manipulatives for Grade 8 include: Tangram kits (or digital Tangram applets); Geometry toolkits containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles; and GeoGebra applets are used for both investigating the characteristics of shapes and area/perimeter as well as exploring coordinate and isometric grids.
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The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The visual design in Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 is not distracting or chaotic and supports students in engaging thoughtfully with the subject.

• The digital lesson materials for teachers follow a consistent format for each lesson. Teaching Notes with Supports for English Language Learners and Supports for Students with Disabilities are placed within the activity they support and are specific to the activity. Unit overviews follow a consistent format. The format of course overviews, units, and individual lessons are consistent across the Grade 8 materials.
• Student-facing printable materials follow a consistent format. Tasks within a lesson are numbered to match the teacher-facing guidance. The print and visuals on the materials are clear without any distracting visuals or overabundance of text features. Teachers can assign lessons and activities to students through the platform, enabling students to access digital manipulatives, practice problems, unit assessments, and lesson visuals.
• Printable student practice problem pages frequently include enough space for students to write their answers and demonstrate their thinking.

#### Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for supporting teacher learning and understanding of the standards. The instructional materials: support planning and providing learning experiences with quality questions; contain ample and useful notations and suggestions on how to present the content; contain full, adult-level explanations and examples of the more advanced mathematics concepts; and contain explanations of the grade-level mathematics in the context of the overall mathematics curriculum.

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Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each lesson consists of a detailed lesson plan accompanied by teaching notes. Included in these teaching notes are the objectives of the lesson, suggested questions for discussion, and guiding questions designed to increase classroom discourse and foster understanding of the concepts. For example, in Unit 1, Lesson 5, the following question is included, “What are some advantages to knowing the coordinates of points when you are doing transformations?” The teaching notes and questions for discussion support the teachers in planning and implementing lessons effectively.

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Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for providing teacher supports with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

• Each lesson includes the Learning Goals written for teachers and students, learning targets written for students, a list of Word/PDF documents that can be downloaded, CCSSM Standards that are “built upon” or “addressed” for the lesson, and any instructional routines to be implemented. Within the technology, there are expandable links to standards and instructional routines.
• Lessons include detailed guidance for teachers for the Warm-Up, Activities, and the Lesson Synthesis.
• Each lesson activity contains an overview and Launch narrative, guidance for teachers and student-facing materials, anticipated misconceptions, “Are you ready for more?”, and an Activity Synthesis. Included within these narratives are guiding questions and additional support for students.
• The teacher materials that correspond to the student lessons provide annotations and suggestions on how to present the content. “Launch” explains how to set up the activity and what to tell students. After the activity is complete, there is often “Anticipated Misconceptions” in the teaching notes, which describes how students may incorrectly interpret or misunderstand concepts and includes suggestions for addressing those misconceptions.
• The materials are available in both print and digital forms. The digital format has embedded GeoGebra applets. Guidance is provided to both the teacher and the student on how to use the Geometry Toolkit and applet. For example, in Unit 1, Lesson 3, teachers and students are provided with these directions on how to perform a translation: “Translate triangle ABC so that A goes to A′. a.) Select the Vector tool. b.) Click on the original point A and then the new point A′. You should see a vector. c.) Select the Translate by Vector tool. d.) Click on the figure to translate, and then click on the vector.”
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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials reviewed for the Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for the teacher edition containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge.

The narratives provided for each unit include information about the mathematical connections of concepts being taught. Previous and future grade levels are also referenced to show the progression of mathematics over time. Important vocabulary is included when it relates to the “big picture” of the unit.

Lesson narratives provide specific information about the mathematical content within the lesson and are presented in adult language. These narratives contextualize the mathematics of the lesson to build teacher understanding, and give guidance on what to expect from students and important vocabulary.

The Narrative for Unit 3 states, “A proportional relationship between two quantities represented by a and b is associated with two constants of proportionality: a/b and b/a. Throughout the unit, the convention is if a and b are represented by columns in a table and the column for a is to the left of the column for b, then b/a is the constant of proportionality for the relationship represented by the table.”

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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Course Guide and Narratives describe how mathematical concepts are built from previous grade-level and lesson material. For example, the Unit 4 narrative states, “In this unit, students build on their Grades 6 and 7 work with equivalent expressions and equations with one occurrence of one variable, learning algebraic methods to solve linear equations with multiple occurrences of one variable.”

For some units, there are explanations given for how the grade-level concepts fit into future grade-level work. For example, the Unit 3 Narrative for linear relationships states: “A proportional relationship is a collection of equivalent ratios. In high school—after their study of ratios, rates, and proportional relationships—students discard the term ‘unit rate’, referring to a to b, a:b, and a/b as ‘ratios’.”

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Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 provide a list of concepts in the Course Guide that cross-references the standards addressed and an estimated instructional time for each unit and lesson.

• The Course Guide includes a Scope and Sequence document that provides pacing information. A table, spanning 36 weeks of instruction, shows the unit that is taught each week, as well as the total number of days the unit should take to complete. In each lesson, the time an activity will take is included in the lesson's narrative. About These Materials in the Teacher Guide states, “Each lesson plan is designed to fit within a 45–50 minute period.”
• In the Course Guide under Lessons and Standards, there is a table that shows which standard each lesson addresses and another table to show where a standard is found in the materials.
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Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials reviewed for the Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Family Materials for each Unit include an explanation to family and caregivers on what their student will be learning over the course of the week. The Family Materials provide an overview of what the student will be learning in accessible language. For example, in Unit 8, the second week begins with: “This week your student will work with the Pythagorean Theorem, which describes the relationship between the sides of a right triangle. A right triangle is any triangle with a right angle.” In addition to the explanation of the current concepts and big ideas from the unit, there are diagrams and problems/tasks for families to discuss and solve.

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Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The instructional materials reviewed for the Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 contain explanations of the program's instructional approaches and identification of the research-based strategies.

The materials draw on research to explain and contextualize instructional routines and lesson activities. The Course Guide includes specific links to research, for example:

• “Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014).”
• How to Use These Materials: “Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team.”

In the Course Guide, all of the “Instructional Routines” are fully explained.

• Algebra Talks found in the Warm-Ups set a routine for collecting different strategies. “Algebra Talks build algebraic thinking by encouraging students to think about the numbers and variables in an expression and rely on what they know about structure, patterns, and properties of operations to mentally solve a problem. Algebra Talks promote seeing structure in expressions and thinking about how changing one number affects others in an equation. While participating in these activities, students need to be precise in their word choice and use of language (MP6).”
• Think-Pair-Share routines found in the Lesson Activities provide structure for engaging students in collaboration. “This is a teaching routine useful in many contexts whose purpose is to give all students enough time to think about a prompt and form a response before they are expected to try to verbalize their thinking. First they have an opportunity to share their thinking in a low-stakes way with one partner, so that when they share with the class they can feel calm and confident, as well as say something meaningful that might advance everyone’s understanding. Additionally, the teacher has an opportunity to eavesdrop on the partner conversations so that she can purposefully select students to share with the class.”

#### Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for offering teachers resources and tools to collect ongoing data about student progress on the standards. The instructional materials provide strategies for gathering information about students' prior knowledge, opportunities for identifying and addressing common student errors and misconceptions, ongoing review and practice with feedback, assessments with standards clearly denoted, and guidance to teachers for interpreting student performance and suggestions for follow-up.

##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels.

• Prior grade-level standards are indicated in the instructional materials. The lesson Warm-Up is designed to engage students' thinking about the upcoming lesson and/or to revisit previous grades' concepts or skills.
• Prior knowledge is gathered about students through the pre-unit assessments. In these assessments, prerequisite skills necessary for understanding the topics in the unit are assessed. Commentary for each question as to why the question is relevant to the topics in the unit and exactly which standards are assessed is provided for the teacher. For example, the Unit 4 Check Your Readiness Problem 1 states: “The distributive property will prove to be an important tool in solving linear equations.” (6.EE.A.3)
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Materials provide strategies for teachers to identify and address common student errors and misconceptions.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

Lesson Activities include teaching notes that identify where students may make a mistake or struggle. There is a rationale that explains why the mistake could have been made, suggestions for teachers to make instructional adjustments for students, and steps teachers can take to help clear up the misconceptions. For example, in Unit 1, Lesson 10, Anticipating Misconceptions give the following guidance: “Students may struggle to see the 180 rotation using center M. This may be because they do not understand that M is the center of rotation or because they struggle with visualizing a 180 rotation.”

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Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The lesson structure consisting of a Warm-up, Activities, Lesson Synthesis, and Cool-down provide students with opportunities to connect prior knowledge to new learning, engage with content, and synthesize their learning. Throughout the lesson, students have opportunities to work independently, with partners, and in groups where review, practice, and feedback are embedded into the instructional routine. Practice Problems for each lesson activity reinforce learning concepts and skills and enable students to engage with the content and receive timely feedback. In addition, discussion prompts provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

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Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for assessments clearly denoting which standards are being emphasized.

Assessments are located on the Assessment tab for each unit and are also available in print. For each unit, there is a Check Your Readiness and an End-Unit Assessment. Assessments begin with guidance for teachers on each problem, followed by the student-facing problem, solution(s), and the standard targeted.. Units 1 and 5 also include a Mid-Unit Assessment.

##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Assessments include an answer key, and when applicable, a rubric consisting of three to four tiers, ranging from Tier 1 (work is complete, acceptable errors) to Tiers 3 and 4 (significant errors, conceptual mistakes).

Assessments include multiple choice, multiple response, short answer, restricted constructed response, and extended response. Restricted constructed response and extended response items have rubrics that are provided to evaluate the level of student responses. The restricted constructed response items include a 3-tier rubric, and the extended constructed response items include a 4-tier rubric. For these types of questions, the teacher materials provide guidance as to what is expected for each tier as well as sample responses.

In the Assessment Teacher Guide for each Mid Unit and End of Unit Assessment, there are narratives about what may have caused students to choose an incorrect response before the problems are shown along with the correct responses and aligned standards. For example, in Unit 5, Mid Unit Assessment (B), Problem 2, the Assessment Teacher Guide states, “Students selecting A looked at the first two rows of the weight column and did not realize that the increase was over two weeks. Students selecting C may not understand how to find the constant rate of change. Students selecting one of D or E, but not both, may not realize that both can be true at once, or may think only D is true because age is the independent variable.

##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 include opportunities for students to monitor their own progress.

For every unit, there is a Lesson Synthesis that offers suggestions for self-monitoring such as, “... asking students to respond to prompts in a written journal, asking students to add on to a graphic organizer or concept map, or adding a new component to a persistent display like a word wall.” In the Lesson Synthesis, students have the opportunity to express their own thinking and understanding of the lesson content. For example, Unit 3, Lesson 12, Activity Synthesis has teachers invite students to answer questions such as, “According to the equation you wrote, if you bought 1/2 orange and 9 apples you would spend \$10. Do you think this situation is realistic when buying fruit in a store?”

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 8 meet the expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide a balanced portrayal of various demographic and personal characteristics. The instructional materials also consistently provide: strategies to help teachers sequence or scaffold lessons; strategies for meeting the needs of a range of learners; tasks with multiple entry-points; support, accommodations, and modifications for English Language Learners and other special populations; and opportunities for advanced students to investigate mathematics content at greater depth.

##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• Each lesson is designed with a Warm-Up that reviews prior knowledge and/or prepares all students for the activities that follow. The Cool-Down following lesson activities reviews the concepts of the lesson.
• Within a lesson, narratives provide explicit instructional supports for the teacher, including the Activity Launch, Anticipated Misconceptions, and Lesson Synthesis. This information assists teachers in making the content accessible to all learners.
• Lesson narratives often include guidance on where to focus questions in Activities or the Lesson Synthesis.
• Optional activities are often included that can be used for additional practice or support before moving to the next activity or lesson.
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

The lesson structure—Warm-Up, Activities, Lesson Synthesis, and Cool-Down—includes guidance for the teacher on the mathematics of the lesson, possible misconceptions, and specific strategies to address the needs of a range of learners. Embedded supports include:

• Mathematical Language Routines to support a range of learners to be successful are provided for the teacher throughout lessons to maximize output and cultivate conversation. For example:
• MLR1: Stronger and Clearer Each Time, in which “students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response.”
• MLR4: Information Gap, which “allows teachers to facilitate meaningful interactions by giving partners or team members different pieces of necessary information that must be used together to solve a problem or play a game...[S]tudents need to orally (and/or visually) share their ideas and information in order to bridge the gap.”
• MLR6: Three Reads, in order to "ensure that students know what they are being asked to do, and to create an opportunity for students to reflect on the ways mathematical questions are presented. This routine supports reading comprehension of problems and meta-awareness of mathematical language. It also supports negotiating information in a text with a partner in mathematical conversation.”
• Teaching notes appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These teaching notes state specific needs addressed in a recommended strategy that are relevant to the given task and include supports for Conceptual Processing, Expressive & Receptive Language, Visual-Spatial Processing, Executive Functioning, Memory, Social-Emotional Functioning, and Fine-motor Skills. For each support, there are multiple strategies teachers can employ, for example: Conceptual Processing includes strategies to Eliminate Barriers, Processing Time, Peer Tutors, Assistive Technology, Visual Aids, Graphic Organizers, and Brain Breaks.
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for embedding tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

The problem-based curriculum design engages students with complex tasks multiple times each lesson. The Warm-Up, Activities, Lesson Synthesis, and Cool-Down provide opportunities  for students to apply mathematics from multiple entry points.

Specific examples of strategies found in the materials include “Notice and Wonder” and “Which One Doesn’t Belong?” The lesson and task narratives provided for teachers offer possible solution paths and presentation strategies from various levels, for example:

• In Unit 3, Lesson 1, the second Activity, students write equations representing the speed of various insects on a number line. Teachers observe students using various strategies to write the equations, i.e., using unit rates or proportions, and choose some of these students to share during discussion.
• In Unit 3, Lesson 3, the first Activity, students create data tables and graphs for various situations. Teachers are given sample responses, as well as guidance in helping students scale the axes on the graphs. Teachers also “identify students using different scales for their graphs that show clearly the requested information to share during the discussion.”
##### Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for including support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The ELL Design is highlighted in the 6-8 Teacher Guide and embodies the Understanding Language/SCALE Framework from the Stanford University Graduate School of Education, which consists of four principles: Support Sense-Making, Optimize Outputs, Cultivate Conversation, and Maximize Meta-Awareness. In addition, there are eight Mathematical Language Routines (MLR) that were included “because they are the most effective and practical for simultaneously learning mathematical practices, content, and language.” "A Mathematical Language Routine refers to a structured but adaptable format for amplifying, accessing, and developing students’ language."

“ELL Enhanced Lessons” are identified in the Unit Overview. These lessons highlight specific strategies for students who have a language barrier which affects their ability to participate in a given task. Throughout lessons, a variety of instructional routines are designed to assist students in developing full understanding of math concepts and terminology. These Mathematical Language Routines include:

• MLR2, Collect and Display, in which “The teacher listens for, and scribes, the student output using written words, diagrams and pictures; this collected output can be organized, revoiced, or explicitly connected to other language in a display for all students to use.”
• MLR5, Co-Craft Questions and Problems, which “[allows] students to get inside of a context before feeling pressure to produce answers, and to create space for students to produce the language of mathematical questions themselves.”
• MLR7, Compare and Connect, which “[fosters] students’ meta-awareness as they identify, compare, and contrast different mathematical approaches, representations, and language.”

Lesson narratives include strategies designed to assist other special populations of students in completing specific tasks. Examples of these supports for students with disabilities include:

• Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.
• Conceptual Processing: Eliminate Barriers. Assist students in seeing the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.
• Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.
• Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.
• Memory: Processing Time. Provide students with a number line that includes rational numbers.
• Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

All students complete the same lessons and activities; however, there are some optional lessons and activities that a teacher may choose to implement with students. In addition, Unit 9 Putting It All Together is an optional unit. Lessons in this unit tend to be multi-day, complex applications of the mathematics addressed over the year.

“Are you ready for more?” is included in some lessons to provide students additional interactions with the key concepts of the lesson. Some of these tasks would be considered investigations at greater depth, while others are additional practice.

There is no clear guidance for the teacher on how to specifically engage advanced students in investigating the mathematics content at greater depth.

##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for providing a balanced portrayal of various demographic and personal characteristics.

• The lessons contain a variety of tasks that interest students of various demographic and personal characteristics. All names and wording are chosen with diversity in mind, and the materials do not contain gender biases.
• The Grade 8 materials include a set number of names used throughout the problems and samples (e.g., Elena, Tyler, Lin, Noah, Diego, Kiran, Mia, Priya, Han, Jada, Andre, Clare). These names are presented repeatedly and in a way that does not appear to stereotype characters by gender, race, or ethnicity.
• Characters are often presented in pairs with different solution strategies. There does not appear to be a pattern in one character using more/less sophisticated strategies.
• When multiple characters are involved in a scenario they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias. For example, in Unit 3, Lesson 4, Activity 1, students analyze a variety of situations in which Elena babysits, Clare and Han have summer jobs, and Tyler starts a lemonade stand. The students’ tasks are not gender stereotypes.
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 provide opportunities for teachers to use a variety of grouping strategies.

The materials offer multiple opportunities to implement grouping strategies to complete the tasks of a daily lesson. Explicit instructions are found in the activity narratives. Grouping strategies range from partner to small group. For example, the narrative in Unit 8, Lesson 4, states, “Arrange students in groups of 2... Display the image for all to see. Ask students to signal when they have noticed or wondered about something.”

In addition, the Instructional Routines implemented in many lessons offer opportunities for students to interact with the mathematics with a partner or in a small group. These routines include: Take Turns Matching or Sorting, in which students engage in sorting and categories given sets of cards; Think-Pair-Share, where students think about and test ideas as well as exchange feedback before sharing their ideas with the class; and Gallery Walk and Group Presentations, in which students generate visual displays of a mathematical problem, and students from different groups interpret the work and find connections to their own work.

##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 encourage teachers to draw upon home language and culture to facilitate learning.

The 6-8 Math Course Guide includes Supporting English Language Learners from the Understanding Language/SCALE (UL/SCALE) at Stanford University’s Graduate School of Education. Promoting Language and Content Development explains the purpose of the document, the goal, and introduces the framework. The Supporting English-language Learners document in the Course Guide states: “The goal is to provide guidance to mathematics teachers for recognizing and supporting students’ language development processes in the context of mathematical sense making. UL/SCALE provides a framework for organizing strategies and special considerations to support students in learning mathematics practice, content, and language.” The section concludes with acknowledging the importance of the framework: “Therefore, while the framework can and should be used to support all students learning mathematics, it is particularly well-suited to meet the needs of linguistically and culturally diverse students who are simultaneously learning mathematics while acquiring English.”

#### Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math, Grade 6 integrate technology in ways that engage students in the Mathematical Practices. The digital materials are web-based and compatible with multiple internet browsers, but they do not include technological opportunities for assessing students' mathematical understandings and knowledge of procedural skills as students complete the assessments in printed formats. The instructional materials include opportunities for teachers to personalize learning for all students, and the materials offer opportunities for customized, local use. The instructional materials also include opportunities for teachers and/or students to collaborate.

##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The instructional materials reviewed for Kendall Hunt’s Illustrative MathematicsM 6-8 Math, Grade 8 are web-based and compatible with multiple internet browsers.

• The materials are platform-neutral and compatible with Chrome, ChromeOS, Safari, and Mozilla Firefox.
• Materials are compatible with various devices including iPads, laptops, Chromebooks, and other devices that connect to the internet with an applicable browser.
• Common Cartridge and LTI integration allows for materials to be integrated into all major Learning Management Systems.
##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 do not include opportunities to assess students' mathematical understanding and knowledge of procedural skills using technology.

Assessments are found under the Assessment tab. Assessments are available as PDFs and editable Word documents, and students complete the assessments in printed formats.

##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 include opportunities for teachers to personalize learning for all students.

• The online platform supports professional learning communities by being collaborative and allowing districts to customize the material.
• Lessons have been separated into components; warm-ups, activities, cool-downs, and practice problems can all be assigned to small groups and individual students, depending on the needs of a particular teacher.
• Common Cartridge and LTI integration allows for materials to be integrated into all major Learning Management Systems.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 can be adapted for local use.

Assessments are available as PDFs and editable Word documents.

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 incorporate technology that provides opportunities for teachers and/or students to collaborate with each other.

• Students and teachers have the opportunity to collaborate using the applets that are integrated into some of the lessons during activities.
• The Warm-Ups, Activities, Cool-Downs, and practice problems can be assigned to small groups to support student collaboration.
##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 integrate technology including interactive tools, virtual manipulatives/objects, and dynamic mathematics software in ways that engage students in the MPs.

Warm-Ups, Activities, Cool-Downs, and practice problems can be assigned to small groups or individuals. These sections consistently combine MPs and content.

Teachers and students have access to math tools and virtual manipulatives within a given activity or task, when appropriate. These applets are designed using GeoGebra, Desmos, and other independent designs, for example:

• In Unit 6, Lesson 4, students use a Desmos applet to investigate lines of best fit within a scatterplot. (MP1)
• In Unit 3, Lesson 7, students have opportunities to use the GeoGebra applet (in the event they are unable to conduct the experiment) to model and graph the results of water displacement to calculate volume. (MP4)

## Report Overview

### Summary of Alignment & Usability for Kendall Hunt's Illustrative Mathematics 6-8 Math | Math

#### Math 6-8

The instructional materials for Kendall Hunt's Illustrative Mathematics 6-8 Math meet the expectations for focus and coherence in Gateway 1. All grades meet the expectations for focus as they assess grade-level topics and spend the majority of class time on major work of the grade, and all grades meet the expectations for coherence as they have a sequence of topics that is consistent with the logical structure of mathematics. In Gateway 2, all grades meet the expectations for rigor and balance, and all grades meet the expectations for practice-content connections. In Gateway 3, all grades meet the expectations for instructional supports and usability. The instructional materials show strengths by being well designed and taking into account effective lesson structure and pacing, supporting teacher learning and understanding of the Standards, offering teachers resources and tools to collect ongoing data about student progress on the Standards, and supporting teachers in differentiating instruction for diverse learners within and across grades.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

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### Overall Summary

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###### Usability
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##### Gateway {{ gateway.number }}
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