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.”

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.”

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.

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.

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?”

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.”

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.”

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’.”

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)

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.”

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.

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.

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.”

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.

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.

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.”

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.

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.

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.