The instructional materials for Open Up Resources 6-8 Math, Grade 8 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding. Students access concepts from a number of perspectives and independently demonstrate conceptual understanding throughout the grade.

The conceptual understanding required in 8.F.A is addressed in Unit 5 as students define, evaluate, and compare functions. For example:

• Lesson 1 begins to develop the “idea of a function as a rule that assigns to each allowable input exactly one output.” This understanding is cultivated in the first Activity of Lesson 1 as student 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. Student pairs 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 after 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 are prompted: “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 asks, “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 Lessons 3 through 7, students use and compare verbal descriptions, tabular and graphic representations of functions, as well as equations. In the second Activity in Lesson 4, students are given 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, and finally, use the graph to identify the output when the input is 1 and interpret what that tells you about each situation in prompt 3 (8.F.1). The graphs include a non-linear representation and two linear functions, one with a positive slope and one with a negative slope.

Cluster 8.G.A addresses congruence and similarity using physical models, transparencies, or geometry software and is found in both Units 1 and 2. Unit 1 begins with transformations.

• In Lessons 1 through 6, students spend most of the instructional time either physically moving shapes or imitating that movement on a GeoGebra program. In Lesson 1, students first look at transformations as a way of moving objects in a plane. They define these movements in their own words in Lesson 2 and establish the actual definition of the movements in later lessons.

• 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 here 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.”

Cluster 8.G.6 builds understanding of the Pythagorean Theorem and is found in Unit 8.

• Lesson 6 Activity 1 prompts students to find the length of the sides of several triangles using a grid, some of which are right triangles and some are not. Through the investigation, students arrive at an understanding of the Pythagorean Theorem and that it only applies to right triangles. After the investigation the Pythagorean Theorem is officially introduced to the students.

The instructional materials for Open Up Resources 6-8 Math, Grade 8 meet expectations that they attend 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 are intentionally built on conceptual understanding and the work students have accomplished with operations and equations from prior grades. Opportunities to formally practice developed procedures are found throughout practice problem sets that follow the units, including opportunities to use and practice emerging fluencies in the context of solving problems. Units 1-3 include embedded review of rational number operations, and Units 5-9 include continued practice of content that is developed in Unit 4. According to the Design Principles within the Grade 8 Course Guide within each unit, “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 skill and fluency throughout the year in a variety of practice problems. Examples of lesson practice problems follow:

• 8.EE.7 is found in Unit 5.
  • In 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).
• 8.EE.8b is found in Unit 5.
  • In Lesson 1, students solve a system of two equations, both in slope-intercept form, using substitution.
  • In Lesson 3, students graph a system of equations with no solution and then write the equation of each graphed line.
  • In Lesson 4, students solve a system of two equations, both in slope-intercept form.
  • In Lesson 12, students solve a system of two equations, both in slope-intercept form.
  • In Lesson 20, students write and solve a system of equations representing the context given about a bicycle shop inventory in order to find the number of bicycles in the store.

8.EE.7 is found in Unit 4 where students use computational skills to solve linear equations in one variable.

• Lesson 1 through 4 connect to past grades with lessons and activities that incorporate the idea of balancing equations.

• In the Lesson 3 Warm-Up, students match hanger diagrams to equations and variables to their respective shapes within the diagram. Next, students begin to match the first “moves” in solving equations in the first Activity. 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 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 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 Lesson 6 Strategic Solving, “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 look at situations when an equation has many or no solutions. Once students are introduced to a general procedure for solving equations continuous practice is provided.

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 take a situation and features of the graphs without actually graphing, to develop fluency with the vocabulary and begin to visualize where systems of equations may have solutions. All tasks involve graphing equations and discussing solutions in context of graphs. The term “system of equations” is introduced.

• In the Lesson 14 Warm-Up, students use substitution strategies to mentally solve systems of equations. In Activity 1, 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 Open Up Resources 6-8 Math, Grade 8 meet expectations that the materials are 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.

Work with applications of mathematics occurs throughout the materials in ways that enhance the focus on major work and when standards call for application in real-world or mathematical contexts. 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.” Connections between clusters and application extensions are also found in the multiple-day lessons found in optional Unit 9.

Cluster 8.F.B addresses students using functions to model relationships between quantities and is found in Unit 5. Lessons 5 through 11 are specifically identified as addressing 8.F.B and use a variety of applications as students model relationships using functions. Students identify and create tables, graphs, and equations modeling relationships.

• 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 are asked to share what they notice on a graph of temperature data during different parts of the day. In Activity 1, they use piecewise linear graphs to find information about the real-life situation they represent. In Activity 2, students analyze a situation to calculate the rate of change.

• In Lesson 11 Activity 1, 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 and then test their prediction by filling the graduated cylinder with different amounts of water, gathering and graphing the data. Students use an applet to reason about the height of the water in a given cylinder. They graph the relationship and then explain the meaning of specific points in their recorded data. Students need to apply the context of the given cylinder to 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.

Standard 8.EE.8c addresses students solving real-world and mathematical problems leading to two linear equations in two variables and is found in Unit 4. Lesson 16 includes opportunities for students to investigate applications of systems of equations.

• In Activity 1, 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 Open Up Resources 6-8 Math, Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

There is evidence that the curriculum addresses standards, when called for, with specific and separate aspects of rigor and evidence of opportunities where 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:

• A conceptual understanding of transformations is developed in Unit 1. Students begin this unit by simply describing how one figure moves to another. Students use tracing paper or GeoGebra to make “the moves” and answer the questions with these visual, hands-on tools. In Lesson 3 Activity 1, 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 90o counterclockwise using center O. 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 create graphs for a given story/context determining the scale. 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 skill include:

• In Unit 5 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 four cards in all.

• In the Unit 5 Lesson 16 Warm-Up, students apply computational skills using the formula for volume to solve: “27=(1⁄3)h, 27=(1⁄3)r², 12π=(1⁄3)πa, 12π=(1⁄3)πb².” 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 5 Lesson 21, students practice 8.G.9 as they solve a variety of mathematical problems involving finding the volume cones, cylinders, rectangular prisms, and spheres in the given figures.

• In Unit 2 Lesson 13 Activity 3, students apply their understanding of both proportional relationships (7.RP.2) from the previous grade and arguments establishing facts about similar triangles and angle relationships (8.G.5) to estimate the height of a tall object that cannot be measured directly. Students use their learning from the unit and the previous activities to devise a method to estimate, justify, and test.

All three aspects of rigor are balanced throughout the course, including the unit assessments. There are multiple lessons where two or all three of the aspects are connected. For example:

• 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 strategically arranged so that procedural practice problems lead to opportunities for students to practice and develop proficiency and skills for a concept and engage with more complex application-level problems. On average 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 Open Up Resources 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, with few or no exceptions. The Math Practices are initially identified in the Teacher Guide under the narrative descriptions of each unit within the Course Information. For example:

• Unit 2 Dilations, Similarity and Introducing Slope excerpts 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 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 Associations in Data, the 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 teacher narratives accompanying the lesson in general 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:

• MP8: The Unit 1 Lesson 14 Lesson introduction 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.”

• MP5: The Unit 2 Lesson 2 introduction 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).”

• MP3: The Unit 4 Lesson 5 narrative accompanying Activity 1 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).”

• MP8: The Unit 5 Lesson 3 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 used to enrich the mathematical content and are discussed within narratives as pertaining to the learning target or specific task at hand. The narratives are used to support deepening a teacher’s understanding of the standard itself as the teacher is provided direction regarding how the content is connected to the MP. For example:

• MP6: In the Unit 1 Lesson 13 Overview, 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.”

• MP7: In Unit 9 Lesson 2, students extend their knowledge of transformations into the creation of a tessellation. The introduction to 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, they are highlighted in the Teacher Guide in the narrative provided with each Activity. For example, in the student task accompanying Unit 9 Lesson 2 Activity 2, the student facing directions state (see previous bullet for teacher facing information): “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 Open Up Resources 6-8 Math, Grade 8 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

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. Each practice is addressed multiple times throughout the year. 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 the Unit 1 Lesson 6 first Activity, students “make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need.” The following instructions are provided for the teacher: “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 Activity 3, 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 requests do 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 Activity 4, 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 continue building their skill interpreting and comparing functions.

• In Unit 7, students reason abstractly and quantitatively to solve problems involving operations with exponents. For example, in the first Activity of Lesson 2, students are given 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 data and practice several strategies for comparing bivariate data.

MP5 Use appropriate tools strategically.

• Throughout 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 Overview 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 108, 2 x 108, ...9 x 108, 1 x 109 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, students look for patterns when powers of 10 are raised to a power. The students write powers of 10 in expanded form and simplify expressions. The structure of the activity allows students to develop an understanding of powers raised to powers. In Lesson 6, students analyze the structure of exponents to make sense of expressions with multiple bases.

• Prior to Unit 2 Lesson 11 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 the Unit 7 Lesson 1 second Activity, students explore the idea that repeated division by two is equivalent to repeated multiplication by ½. This allows students to make sense of negative exponents later in Lesson 3, 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 Open Up Resources 6-8 Math, Grade 8 meet expectations that the instructional materials prompt students to construct viable arguments and/or analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Students are consistently asked to explain their reasoning and compare their strategies for solving in small group and whole class settings.

• In the Unit 2 Lesson 6 Cool-Down, students analyze and correct a series of transformations and a dilation intended to provide a similar figure. In the Unit 2 Lesson 7 first Activity, students analyze two figures that are claimed to be similar. Students justify or deny the claim; they are building reasoning as to what characteristics are found in polygons that are not similar.

• In the Unit 5 Lesson 4 first Activity, students are asked, “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 Open Up Resources 6-8 Math, Grade 8 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate, explaining their reasoning to each other.

• 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 Activity 2, 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 the Unit 1 Lesson 11 Overview of the first Activity, teachers are supported to facilitate a discussion to promote student debate so teachers can identify reasons and construct arguments as well as how they critique/analyze responses from others. The teacher guidance also provides ways to help students analyze/critique other’s arguments if it doesn’t 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 Open Up Resources 6-8 Math, Grade 8 meet expectations that the materials attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate 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 Teacher Guide within the Course Information section. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples. In the student materials, the Grade 8 Glossary is accessible by a tab within each unit or in the bottom margin of each lesson page. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples.

• Both the unit and the lesson narratives contain specific guidance for the teacher as to best 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, the students build their understanding of transformations and start naming these movements with the proper terminology (translations, reflections, rotations).

• Unit 6 Lessons 1 through 6 develop the concept of scatter plots and the related language. In Lessons 1 through 3, students are introduced to the definition of scatter plot, and then explore data represented by a scatter plot and the meaning of a specific point on a graph. In Lesson 4, the terms outlier and line of best fit are introduced, and in Lesson 5, students learn to explain trends in scatter plots followed by the slope of the line of best fit in Lesson 6. Each lesson builds on the initial definition of scatter plot until students work with all aspects of them, understand the concept, and use the related language effectively.

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

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson. It is clear when the students are solving problems to learn and when they are applying their skills to build mastery.

Lessons include Warm-Up, Activities, and an Activity Synthesis. 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 connect prior learning or prime students for learning new material in the lesson. Students learn and practice new mathematics in lesson Activities. In the Activity Synthesis, students have opportunities to build on their understanding of the new concept. Each activity lesson ends with a "Cool-Down" in which students have opportunities to apply what they have learned from the activities in the lesson and either provide preliminary practice or an introduction 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 problems 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 deepen understanding or application (e.g., increased expectations for fluency, more abstract application, or a non-routine problem).

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation 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 leading to content mastery. 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 close with a “Cool-Down” which is typically 1-2 activities aligned to the daily lesson objective. Unit sequences consistently follow the progressions outlined in the CCSSM Standards to support students' conceptual and skill development.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation for having variety in what students are asked to produce.

The instructional materials prompt students to produce products in a plethora of ways. Students not only produce answers and solutions within Activities and Practice Problems, but also in class, group and partner discussions. Students have opportunities to construct viable arguments and critique the reasoning of their peers in the instructional materials. Students use a digital platform (applets) and paper-pencil to conduct and present their work. Materials consistently call for student 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 (MP4), 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 Open Up Resources 6-8 Math, Grade 8 meet the 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 curriculum also provides 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. 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 applet)
  • Geometry toolkits containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.
  • 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 Open Up Resources 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 section of each lesson contains an opening and closing narrative for the teacher. Included in these narratives are the objectives of the lesson, as well as suggested questions for discussion and guiding questions designed to increase classroom discourse and ensure 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 narratives, as well as the questions for discussion, support the teachers in planning and implementing lessons effectively.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectations for containing a teacher edition 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 opens with a table including Learning Goals written in both teacher and student language, learning targets written in student language, a list of Print-Formatted Word/PDF documents that can be downloaded, CCSSM Standards that are either being “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 supports for students.

• The teacher materials that correspond to the student lessons provide annotations and suggestions on how to present the content. A “Launch” section follows which explains how to set up the activity and what to tell students. After the activity is complete there is often an “Anticipated Misconceptions” section, which describes how students may incorrectly interpret or misunderstand concepts and includes suggestions for addressing those misunderstandings.

• The materials are available in both print and digital forms. The digital format has an embed GeoGebra applet. 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 Open Up Resources 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 provide information about the mathematical connections of the concepts being taught. Previous and future grade levels are also referenced to show the progression of the 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, as well as guidance on what to expect from students and important vocabulary.

The Course Information and Scope and Sequence, Unit 3: Linear Relationships 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 Open Up Resources 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 Teacher Guide fully explains how mathematical concepts are built from previous grade-level and lesson material. For example, the Unit 4 Overview states the following regarding linear equations and linear systems: “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.”

There are limited 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 Open Up Resources 6-8 Math, Grade 8 meet the 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 Pre-Unit Assessment 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 Open Up Resources 6-8 Math, Grade 8 meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions.

• Lesson Activities include “Anticipated Misconceptions” 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, as well as steps teachers can take to help clear up the misconceptions. For example, in Unit 1 Lesson 10, the Anticipated Misconception section gives 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 Open Up Resources 6-8 Math, Grade 8 meet the expectation 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. In addition, Practice problems for each lesson activity reinforce learning concepts and skills and enable them to engage with the content and receive timely feedback. In addition, discussion prompts in the Teacher Guide provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation for assessments clearly denoting which standards are being emphasized.

Assessments are located on a separate tab at the top of the grade-level page and can be accessed at any time. For each unit there is a Pre-Unit Assessment 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, 3, and 5 also include a Mid-Unit Assessment.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation 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 following activities. The Cool-Down following lesson activities solidifies 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 sections. This information assists a teacher in making the content accessible to all learners.

• Lesson narratives often include guidance on where to focus questions in Activities or in the Lesson Synthesis portions.

• Optional activities are often included that can be used for additional practice or support before moving on to the next activity or lesson.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation for providing teachers with strategies for meeting the needs of a range of learners.

• Lessons are designed to introduce concepts from concrete to abstract, thus building understanding for all students. Progression of the lessons also builds off existing knowledge, making the transition to new material more effective. Time is also provided for students to think through problems and situations independently before engaging in conversation with a partner or small group.

• 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,… and [support] negotiating information in a text with a partner in mathematical conversation.”
• Sidebar text features appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These text-boxes call out 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 Open Up Resources 6-8 Math, Grade 8 meet the expectation that materials embed 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 rigorous tasks multiple times each lesson. The Warm-Up, Activities, and Cool-Down all provide opportunity for students to apply mathematics from multiple entry points.

Specific examples of strategies found in the materials include “Notice and Wonder” sections as well as “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 Activity 2, students are asked to write equations representing the speed of various insects on a number line. Teachers are encouraged to monitor for 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 Activity 1 students have the task of creating 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 are also encouraged to “Identify students using different scales for their graphs that show clearly the requested information to share during the discussion.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation that the materials include 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 Teacher Guide and embodies the Understanding Language/SCALE Framework from the Stanford University Graduate School of Education and 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.”

In addition, “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, the use of one of a variety of instructional routines 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.”

In addition, 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 Open Up Resources 6-8 Math, Grade 8 meet the expectation that the materials provide 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 covered 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.

It should be noted that there is no clear guidance for the teacher on how to specifically engage advanced students in going deeper.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet the expectation 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 gendered stereotypes.