8th Grade - Gateway 3

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Within the Course Guide, several sections (Design Principles, A Typical Lesson, How to Use the Materials, and Key Structures in This Course) provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• Resources, Course Guide, About These Materials, The Five Practices, “Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem. The teacher circulates as students are working and notes groups using different approaches. Groups or individuals are selected in a specific, recommended sequence to share their approach with the class, and finally the teacher leads a whole-class discussion to make connections and highlight important ideas.”

• Resources, Course Guide, About These Materials, A Typical Lesson, “A note about optional activities: A relatively small number of activities throughout the course have been marked “optional.” Some common reasons an activity might be optional include: The activity addresses a concept or skill that is below grade level, but we know that it is common for students to need a chance to focus on it before encountering grade-level material. If the pre-unit diagnostic assessment (”Check Your Readiness”) indicates that students don’t need this review, an activity like this can be safely skipped. The activity addresses a concept or skill that goes beyond the requirements of a standard. The activity is nice to do if there is time, but students won’t miss anything important if the activity is skipped. The activity provides an opportunity for additional practice on a concept or skill that we know many students (but not necessarily all students) need. Teachers should use their judgment about whether class time is needed for such an activity. A typical lesson has four phases: 1. A Warm Up 2. One or more instructional activities 3. The lesson synthesis 4. A Cool Down.”

• Resources, Course Guide, How To Use These Materials, Each Lesson and Unit Tells a Story, “The story of each grade is told in nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson in the unit also has a narrative. Lesson Narratives explain: The mathematical content of the lesson and its place in the learning sequence. The meaning of any new terms introduced in the lesson. How the mathematical practices come into play, as appropriate. Activities within lessons also have narratives, which explain: The mathematical purpose of the activity and its place in the learning sequence. What students are doing during the activity. What teacher needs to look for while students are working on an activity to orchestrate an effective synthesis. Connections to the mathematical practices, when appropriate.”

• Resources, Course Guide, Scope and Sequence lists each of the nine units, a Pacing Guide to plan instruction, and Dependency Diagrams. These Dependency Diagrams show the interconnectedness between lessons and units within Grade 8 and across all grades.

• Resources, Glossary, provides a visual glossary for teachers that includes both definitions and illustrations. Some images use examples and nonexamples, and all have citations referencing what unit and lesson the definition is from.

Materials include sufficient annotations and suggestions presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section D: Angles in a Triangle, Lesson 15: Adding the Angles in a Triangle, Lesson Narrative, “In this lesson, the focus is on the interior angles of a triangle. What can we say about the three interior angles of a triangle? Do they have special properties? The lesson opens with an optional activity looking at different types of triangles with a particular focus on the angle combinations of specific acute, right, and obtuse triangles. After being given a triangle, students form groups of 3 by identifying two other students with a triangle congruent to their own. After collecting some class data on all the triangles and their angles, they find that the sum of the angle measures in all the triangles turns out to be 180 degrees. In the next activity, students observe that if a straight angle is decomposed into three angles, it appears that the three angles can be used to create a triangle. Together the activities provide evidence of a close connection between three positive numbers adding up to 180 and having a triangle with those three numbers as angle measures. A new argument is needed to justify this relationship between three angles making a line and three angles being the angles of a triangle. This is the topic of the following lesson.” 

• Unit 4: Linear Equations and Linear Systems, Unit Overview, “The second section focuses on linear equations in one variable. Students analyze ‘hanger diagrams’ that depict two collections of shapes that balance each other. Assuming that identical shapes have the same weight, they decide which actions of adding or removing weights preserve that balance. Given a hanger diagram that shows one type of shape with unknown weight, they use the diagram and their understanding of balance to find the unknown weight. Abstracting actions of adding or removing weights that preserve balance (MP7), students formulate the analogous actions for equations, using these along with their understanding of equivalent expressions to develop algebraic methods for solving linear equations in one variable. They analyze groups of linear equations in one unknown, noting that they fall into three categories: no solution, exactly one solution, and infinitely many solutions. They learn that any one such equation is false, true for one value of the variable, or (using properties of operations) true for all values of the variable. Given descriptions of real-world situations, students write and solve linear equations in one variable, interpreting solutions in the contexts from which the equations arose.”

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 8: Analyzing Bivariate Data, Activity 2: Equal Body Dimensions, Instructional Routines, “In this activity students create another scatter plot to analyze the data they collected about their classmates in a previous lesson (MP4). A suggested linear model is compared to the data and a particular point is identified in both the scatter plot and data table. Although the scatter plots are left to students to organize, the only linear model considered is y = x which is symmetric when switching which variable is represented on each axis. If possible, identify any groups who have axes switched to bring up in the discussion. Note: Some students may be sensitive about their body measurements and providing alternate data allows the class to work with actual values without making students uncomfortable. Depending on your class, consider providing a similar data set to the one collected in the earlier lesson (measurements from the staff, a different class, or invented data that is similar to the data collected).”

The materials reviewed for Open Up Resources Grade 8 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their knowledge of the subject.

Unit Overviews, Instructional Routines, and Activity Synthesis sections within units and lessons include adult-level explanations and examples of the more complex grade-level concepts. Examples include:

• Unit 3: Linear Relationships, Unit Overview, “On using the terms ratio, rate, and proportion.In these materials, a quantity is a measurement that is or can be specified by a number and a unit, e.g., 4 oranges, 4 centimeters, ‘my height in feet’, or ‘my height’ (with the understanding that a unit of measurement will need to be chosen). The term ratio is used to mean an association between two or more quantities and the fractions \frac{a}{b} and \frac{b}{a} are never called ratios. The fractions \frac{a}{b} and \frac{b}{a} are identified as ‘unit rates’ for the ratio a : b. The word “per” is used with students in interpreting a unit rate in context, as in “$3 per ounce,” and “at the same rate” is used to signify a situation characterized by equivalent ratios.”

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Lesson 14: Solving More Systems, Activity 1: Challenge Yourself, Instructional Routines, “In this activity, students solve systems of linear equations that lend themselves to substitution. There are 4 kinds of systems presented: one kind has both equations given with the y value isolated on one side of the equation, another kind has one of the variables given as a constant, a third kind has one variable given as a multiple of the other, and the last kind has one equation given as a linear combination. This progression of systems nudges students towards the idea of substituting an expression in place of the variable it is equal to. Notice which kinds of systems students think are least and most difficult to solve. In future grades, students will manipulate equations to isolate one of the variables in a linear system of equations. For now, students do not need to solve a system like x + 2y = 7 and 2x - 2y = 2 using this substitution method.”

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 5: Negative Exponents with Powers of 10, Activity 1: Negative Exponent Table, Activity Synthesis, “One important idea is that multiplying by 10 increases the exponent, thus multiplying by \frac{1}{10} decreases the exponent. So negative exponents can be thought of as repeated multiplication by \frac{1}{10} , whereas positive exponents can be thought of as repeated multiplication by 10. Another key point is the effect that multiplying by 10 or \frac{1}{10} has on the placement of the decimal. Ask students to share how they converted between fractions, decimals, and exponents. Record their reasoning for all to see. Here are some possible questions to consider for whole-class discussion: Do you agree or disagree? Why? Did anyone think of this a different way? In your own words, what does 10^{-7} mean? How is it different from {10}^7? Introduce the visual display for {10}^{-n} = \frac{1}{10^n} and display it for all to see throughout the unit. For an example that illustrates the rule, consider displaying {10}^{-3} = \frac{1}{10} \cdot\frac{1}{10} \cdot\frac{1}{10} = \frac{1}{10^3}.”

Materials contain adult-level explanations and examples of concepts beyond grade 8 so that teachers can improve their knowledge of the subject. Examples include: 

• Unit 3: Linear Relationships, Unit 3 Overview, “In grades 6–8, students write rates without abbreviated units, for example as “3 miles per hour” or ‘3 miles in every 1 hour.’ Use of notation for derived units such as \frac{mi}{hr} waits for high school—except for the special cases of area and volume.”

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Lesson 14: Solving More Systems, Activity 1: Challenge Yourself, “In future grades, students will manipulate equations to isolate one of the variables in a linear system of equations. For now, students do not need to solve a system like x + 2y = 7 and 2x - 2y = 2 using this substitution method.”

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section B: The Pythagorean Theorem, Lesson 7: A Proof of the Pythagorean Theorem, Warm Up Activity Synthesis, “Tell students that when you take a square and put a congruent right triangle on each side as shown on the left, they form a larger square (they will be able to prove this in high school). But it doesn’t work if the triangles are not right triangles.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information can be found within different sections of the Course Guide and within the Standards section of each lesson. Examples include:

• Resources, Course Guide, About These Materials, Task Purposes, “A note about standards alignments: There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as ‘building on’. When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as ‘building towards’. When a task is focused on the grade-level work, the alignment is indicated as ‘addressing’.”

• Resources, Course Guide, How To Use These Materials, Noticing and Assessing Student Progress in Mathematical Practices, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.”

• Resources, Course Guide, Scope and Sequence, “In the unit dependency chart, an arrow indicates that a particular unit is designed for students who already know the material in a previous unit. Reversing the order would have a negative effect on mathematical or pedagogical coherence.” Unit Dependency Diagrams identify connections between units and Section Dependency Diagrams identify specific connections within the grade level.

• Resources, Course Guide, Lesson and Standards, provides two tables: a Standards by Lesson table, and a Lessons by Standard table. Teachers can utilize these tables to identify standard/lesson alignment.

• Unit 4: Linear Equations and Linear Systems, Section B: Linear Equations in One Variable, Lesson 6: Strategic Solving, “Addressing 8.EE.C.7 Solve linear equations in one variable. 8.EE.C.7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.”

Explanations of the role of specific grade-level mathematics can be found within different sections of the Resources, Course Guide, Unit Overviews, Section Overviews, and Lesson Narratives. Examples include:

• Unit 2: Dilations, Similarities, and Introducing Slope, Overview, “Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts are sometimes contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have opportunities in the unit to tackle real-world applications. In the culminating activity of the unit, students examine shadows cast by objects in the sun. This is an opportunity for them to apply what they have learned about similar triangles (MP4).”

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Section Overview, “The third section focuses on systems of linear equations in two variables. It begins with activities intended to remind students that a point lies on the graph of a linear equation if and only if its coordinates make the equation true. Given descriptions of two linear relationships students interpret points on their graphs, including points on both graphs. Students categorize pairs of linear equations graphed on the same axes, noting that there are three categories: no intersection (lines distinct and parallel, no solution), exactly one intersection (lines not parallel, exactly one solution), and same line (infinitely many solutions).”

• Unit 7: Exponents and Scientific Notation, Section C: Scientific Notation, Lesson 9: Describing Large and Small Numbers Using Powers of 10, Lesson Narrative, “This lesson serves as a prelude to scientific notation and builds on work students have done in previous grades with numbers in base ten. Students use base-ten diagrams to represent different powers of 10 and review how multiplying and dividing by 10 affect the decimal representation of numbers. They use their understanding of base-ten structure as they express very large and very small numbers using exponents. Students also practice communicating—describing and writing—very large and small numbers in an activity, which requires attending to precision (MP6). This leads to a discussion about how powers of 10 can be used to more easily communicate such numbers.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials explain and provide examples of the program's instructional approaches and include and reference research-based strategies. Both the instructional approaches and the research-based strategies are included in the Course Guide. Examples include:

• Resources, Course Guide, About These Materials, Design Principles, The Five Practices, “Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem. The teacher circulates as students are working and notes groups using different approaches. Groups or individuals are selected in a specific, recommended sequence to share their approach with the class, and finally the teacher leads a whole-class discussion to make connections and highlight important ideas.”

• Resources, Course Guide, How to Use These Materials, Instructional Routines, “The kind of instruction appropriate in any particular lesson depends on the learning goals of that lesson. Some lessons may be devoted to developing a concept, others to mastering a procedural skill, yet others to applying mathematics to a real-world problem. These aspects of mathematical proficiency are interwoven. These lesson plans include a small set of activity structures and reference a small, high-leverage set of teacher moves that become more and more familiar to teachers and students as the year progresses. Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team. The purpose of each MLR is described here, but you can read more about supports for students with emerging English language proficiency in the Supports for English Language Learners section.”

• Resources, About These Materials, What is a “Problem-Based” Curriculum, Attitudes and Beliefs We Want to Cultivate, “Many people think that mathematical knowledge and skills exclusively belong to “math people.” Yet research shows that students who believe that hard work is more important than innate talent learn more mathematics. We want students to believe anyone can do mathematics and that persevering at mathematics will result in understanding and success. In the words of the NRC report Adding It Up, we want students to develop a “productive disposition—[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the Course Guide, Materials, there is a list of materials needed for each unit and each lesson. Lessons that do not have materials are indicated by none; lessons that need materials have a list of all the materials required. Examples include:

• Resources, Course Guide, Required Materials, “blank paper, colored pencils, compasses, dried linguine pasta, four-function calculators, geometry toolkits, graduated cylinders, graph paper, isometric graph paper, long straightedge, measuring tapes, protractors, rulers, rulers marked with centimeters, rulers marked with inches, scientific calculators, scissors, spherical objects, stopwatches, straightedges, string, tape, tools for creating a visual display, toothpicks, pencils, straws, or other objects, tracing paper.”

• Unit 3: Linear Relationships, Section B: Representing Linear Relationships, Lesson 5: Introduction to Linear Relationships, Required Materials, “graph paper, rulers.”

• Unit 5: Functions and Volume, Section C: Linear Functions and Rate of Change, Lesson 9: Linear Models, Required Materials, “straightedges.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

The materials consistently and accurately identify grade-level content standards for formal assessments in the Lesson Cool Down, Mid-Unit Assessments and End-of-Unit Assessments within each assessment answer key. Examples include:

• Resources, Course Guide, Assessments, Summative Assessments, “All summative assessment problems include a complete solution and standard alignment. Multiple-choice and multiple response problems often include a reason for each potential error a student might make. Restricted constructed response and extended response items include a rubric.”

• Unit 2: Dilations, Similarity, and Introducing Slope, Section C: Slope, Lesson 11: Writing Equations for Lines, Cool Down: Matching Relationships to Graphs, “8.EE.B.6, a. Explain why the slope of line a is \frac{2}{6}. b. Label the horizontal and vertical sides of the triangle with expressions representing their length. Explain why \frac{x-7}{x-5} = \frac{2}{6}.”

• Unit 5: Functions and Volume, Unit Assessments, Mid-Unit Assessment, Version A, Problem 2, “8.F.B.5, This graph shows the temperature in Diego’s house between noon and midnight one day. Select ALL the true statements. A. Time is a function of temperature. B. The lowest temperature occurred between 4:00 and 5:00. C. The temperature was increasing between 9:00 and 10:00. D. The temperature was 74 degrees twice during the 12-hour period. E. There was a four-hour period during which the temperature did not change.”

• Unit 8: Pythagorean Theorem and Irrational Numbers, Unit Assessments, End-of-Unit Assessment, Version B, Problem 1, “8.EE.A.2 Select all the numbers that are solutions to the equation x^2 = 15. A. 225 B. \sqrt{225} C. 7.5 D. \sqrt{15} E. -\sqrt{15}.” 

The materials consistently and accurately identify grade-level mathematical practice standards for formal assessments. Examples include:

• Resources, Course Guide, How to Use These Materials, Noticing and Assessing Student Progress in Mathematical Practices, How Can You Use the Mathematical Practices Chart, “No single task is sufficient for assessing student engagement with the Standards for Mathematical Practice. For teachers looking to assess their students, consider providing students the list of learning targets to self-assess their use of the practices, assigning students to create and maintain a portfolio of work that highlights their progress in using the Mathematical Practices throughout the course, monitoring collaborative work and noting student engagement with the Mathematical Practices. Because using the mathematical practices is part of a process for engaging with mathematical content, we suggest assessing the Mathematical Practices formatively. For example, if you notice that most students do not use appropriate tools strategically (MP5), plan in future lessons to select and highlight work from students who have chosen different tools. Since the Mathematical Practices in action can take many forms, a list of learning targets for each Mathematical Practice is provided to support teachers and students in recognizing when engagement with a particular Mathematical Practice is happening. The intent of the list is not that students check off every item on the list. Rather, the ‘I can’ statements are examples of the types of actions students could do if they are engaging with a particular Mathematical Practice.”

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress In Mathematical Practices, Standards for Mathematical Practice Student Facing Learning Targets, “MP7: I Can Look for and Make Use of Structure: I can identify connections between problems I have already solved and new problems. I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole. I can make connections between multiple mathematical representations. I can make use of patterns to help me solve a problem.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Materials provide opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance. Examples include:

• Resources, Course Guide, Assessments, Summative Assessments, End-of-Unit Assessments, “All summative assessment problems include a complete solution and standard alignment. Multiple-choice and multiple response problems often include a reason for each potential error a student might make. Restricted constructed response and extended response items include a rubric. Unlike formative assessments, problems on summative assessments generally do not prescribe a method of solution.”

• Unit 3: Linear Relationships, End-of-Unit Assessment, Version A, Problem 5, students compare proportional relationships between time and distance. “Three runners are training for a marathon. One day, they all run about ten miles, each at their own constant speed. This graph shows how long, in minutes, it takes Runner #1 to run d miles. The equation that relates Runner #2’s distance (in miles) with time (in minutes) is t=8.5t. Runner #3’s information is in the table. Which of the three runners has the fastest pace? Explain how you know.” Solution, “Minimal Tier 1 response: Work is complete and correct. Sample: Runner #1 goes at 10 minutes per mile, Runner #2 goes at 8.5 minutes per mile, and Runner #3 goes at 9 minutes per mile. Runner #2 is fastest, because they take the least time to run one mile. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Work contains correct unit rates for all three runners but concludes that runner #1 or #3 is the fastest or does not name a fastest runner; one unit rate is incorrect (possibly with an incorrect fastest runner identified as a consequence); insufficient explanation of work. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: two or more incorrect unit rates; the correct runner is identified but with no justification; response to the question is not based on unit rates or on similar methods such as calculating which runner has gone the farthest after 10 miles.” 

• Unit 6: Associations in Data, End-of-Unit Assessment, Version B, Problem 3, students look for associations between variables. “Select all the relationships that demonstrate a negative association between variables. A. Number of absences from school and final grades B. Outside temperature and ice cream sales C. Price of houses and house sales D. Number of rainy days and car accidents E. Number of hours playing video games and grades.” Solution, “A, C, and E.”

Materials provide opportunities to determine students' learning and general suggestions to teachers for following up with students. Examples include:

• Resources, Course Guide, Assessments, Pre-Unit Diagnostic Assessments, “What if a large number of students can’t do the same pre-unit assessment problem? Teachers are encouraged to address below-grade skills while continuing to work through the on-grade tasks and concepts of each unit, instead of abandoning the current work in favor of material that only addresses below-grade skills. Look for opportunities within the upcoming unit where the target skill could be addressed in context. For example, an upcoming activity might require solving an equation in one variable. Some strategies might include: ask a student who can do the skill to present their method, add additional questions to the Warm Up with the purpose of revisiting the skill, add to the activity launch a few related equations to solve, before students need to solve an equation while working on the activity, pause the class while working on the activity to focus on the portion that requires solving an equation. Then, attend carefully to students as they work through the activity. If difficulty persists, add more opportunities to practice the skill, by adapting tasks or practice problems.”

• Resources, Course Guide, Assessments, Cool Downs, “What if the feedback from a Cool Down suggests students haven’t understood a key concept? Choose one or more of these strategies: Look at the next few lessons to see if students have more opportunities to engage with the same topic. If so, plan to focus on the topic in the context of the new activities. During the next lesson, display the work of a few students on that Cool Down. Anonymize their names, but show some correct and incorrect work. Ask the class to observe some things each student did well and could have done better. Give each student brief, written feedback on their Cool Down that asks a question that nudges them to re-examine their work. Ask students to revise and resubmit. Look for practice problems that are similar to, or involve the same topic as the Cool Down, then assign those problems over the next few lessons.”

• Unit 5: Functions and Volume, Mid-Unit Assessment, Version B, Problem 4, students calculate rate of change. “Mai hiked up a trail for 40 minutes. The graph shows the elevation in feet that she reached throughout her hike. Name the time period where Mai gained elevation at the fastest rate.” Guidance for teachers, “Check to see if students understand what “time period” means in this context. Some students may answer 10–22 minutes, looking at the longest section of the graph. Students answering 32–40 minutes may simply be choosing the section of the graph corresponding to the greatest height. If students struggle to calculate the different rates of change of a piecewise linear function using a graph, provide additional instruction either in a small group or individually using OUR Math Grade 8 Unit 5 Lesson 10 Activity 2. Ask students, ‘How would you describe a piecewise linear function to someone who has never seen one?’”

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level/ course-level standards and practices across the series.

Formative assessments include lesson activities, Cool Downs, and Practice Problems in each unit section. Summative assessments include Mid-Unit Assessments and End-of-Unit Assessments. Assessments regularly demonstrate the full intent of grade-level content and practice standards through various item types, including multiple-choice, multiple response, short answer, restricted constructed response, and extended response. Examples include:

• Unit 1: Rigid Transformations and Congruence, Mid-Unit Assessment, Version A, Problem 6, students apply rigid transformations to a single point. “Point A is located at coordinates (-4, 3). What are the coordinates of each point? a. Point B is the image of A after a rotation of 180° using (0, 0) as center. B. Point C is the image of A after a translation two units to the right, then a reflection using the x-axis. c. Point D is the image of A after a reflection using the y-axis, then a translation two units to the right.” (8.G.3)

• Unit 2: Dilations, Similarity, and Introducing Slope, Section B: Similarity, Lesson 7: Similar Polygons, Cool Down: How Do You Know?, students construct viable arguments and explain why two quadrilaterals are similar. “Explain how you know these two figures are similar.” (MP3)

• Unit 3: Linear Relationships, Section B: Representing Linear Relationships, Lesson 5: Introduction to Linear Relationships, Cool Down: Stackin More Cups, students analyze a graph of a non-proportional relationship. “A shorter style of cup is stacked tall. The graph displays the height of the stack in centimeters for different numbers of cups. How much does each cup after the first add to the height of the stack? Explain how you know.” Students are shown a graph of height in centimeters of stacked cups with the points (3, 5.5) and (8, 8) plotted. (8.EE.5)

• Unit 4: Linear Equations and Linear Systems, Lesson 12: System of Equations, Cool Down: Milkshakes Revisited, students graph and interpret systems of equations. “Determined to finish her milkshake before Diego, Lin now drinks her 12 ounce milkshake at a rate of \frac{1}{3} an ounce per second. Diego starts with his usual 20 ounce milkshake and drinks at the same rate as before, \frac{2}{3} an ounce per second. a. Graph this situation on the axes provided. b. What does the graph tell you about the situation and how many solutions there are?” (MP4)

• Unit 5: Functions and Volume, Section B: Representing and Interpreting Functions, Lesson 7: Connecting Representations of Functions, Practice Problems, Problem 2, students compare input and output functions presented in different ways. “Elena and Lin are training for a race. Elena runs her mile at a constant speed of 7.5 miles per hour. Lin’s total distances are recorded every minute: a. Who finished their mile first? B. This is a graph of Lin’s progress. Draw a graph to represent Elena’s mile on the same axes. c. For these models, is distance a function of time? Is time a function of distance? Explain how you know.” A table and graph of Lin’s distances over time are shown. (8.F.2 and 8.F.3)

• Unit 8: Pythagorean Theorem and Irrational Numbers, End-of-Unit Assessment, Version B, Problem 4, students convert fractions to decimals. “What is the decimal expansion of \frac{16}{9}?” (8.NS.1)

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics, as suggested in each lesson. According to the Resources, Course Guide, Supports for Students with Disabilities, “Supplemental instructional strategies, labeled ‘Supports for Students with Disabilities,’ are included in each lesson. They are designed to help teachers meet the individual needs of a diverse group of learners. Each  is aligned to one of the three principles of Universal Design for Learning, to provide multiple means of engagement, representation, or action and expression, and includes a suggested strategy to increase access and eliminate barriers. These lesson specific supports can be used as needed to help students succeed with a specific activity, without reducing the mathematical demand of the task, and can be faded out as students gain understanding and fluency.” Examples of supports for special populations include: 

• Unit 3: Linear Relationships, Section D: Linear Equations, Lesson 13: More Solutions to Linear Equations, Activity 1: True or False: Solutions in the Coordinate Plane, Supports for Students with Disabilities, “Action and Expression: Executive Functions, Check for understanding by inviting students to rephrase directions in their own words. Provide the following sentence frame to support student explanations: ‘Statement ____ is true/false because …’ Provides accessibility for: Organization, Attention.”

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 4: Fitting a Line to Data, Activity 1: Shine Bright, Supports for Students with Disabilities, “Action and Expression: Expression and Communication, Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, ‘It looks like …’ and ‘We are trying to …’ Provides accessibility for: Language, Organization.”

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 3: Powers of Powers of 10, Activity 2: How Do the Rules Work?, Supports for Students with Disabilities, “Representation: Comprehension, Activate or supply background knowledge. Continue to display, or provide a physical copy of the visual display for the rule {({10}^n)}^m = {10}^{n \cdot m} from the previous activity. Provides accessibility for: Memory, Conceptual Processing.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found after activities and labeled “Are You Ready for More?” According to the Resources, Course Guide, How To Use The Materials, Are You Ready For More?, “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. We think of them as the ‘mathematical dessert’ to follow the ‘mathematical entrée’ of a classroom activity. Every extension problem is made available to all students with the heading “Are You Ready for More?” These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some of these problems extend the work of the associated activity, but some of them involve work from prior grades, prior units in the course, or reflect work that is related to the K–12 curriculum but a type of problem not required by the standards. They are not routine or procedural, and they are not just “the same thing again but with harder numbers.” Examples include:

• Unit 2: Dilations, Similarity, and Introducing Slope, Section A: Dilations, Lesson 3: Dilations with no Grid, Activity 2: Getting Perspective, Are You Ready for More? “Here is line segment DE and its image D’E’ under a dilation. a. Use a ruler to find and draw the center of dilation. Label it F. b. What is the scale factor of the dilation?”

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 5: Describing Trends in Scatter Plots, Activity 3: Practice Fitting Lines, Are You Ready for More? “These scatter plots were created by multiplying the x-coordinate by 3 then adding a random number between two values to get the y-coordinate. The first scatter plot added a random number between -0.5 and 0.5 to the y-coordinate. The second scatter plot added a random number between -2 and 2 to the y-coordinate. The third scatter plot added a random number between -10 and 10 to the y-coordinate. a. For each scatter plot, draw a line that fits the data. b. Explain why some were easier to do than others.” Three scatter plots are shown.

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section B: The Pythagorean Theorem, Lesson 6: Finding Side Lengths of a Triangle, Activity 2: Meet the Pythagorean Theorem, Are You Ready for More? “If the four shaded triangles in the figure are congruent right triangles, does the inner quadrilateral have to be a square? Explain how you know.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Teachers consistently provide guidance to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Resources, Course Guide, Supports for English Language Learners, Design, “Each lesson includes instructional strategies that teachers can use to facilitate access to the language demands of a lesson or activity. These support strategies, labeled ‘Supports for English Language Learners,’ stem from the design principles and are aligned to the language domains of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). They provide students with access to the mathematics by supporting them with the language demands of a specific activity without reducing the mathematical demand of the task. Using these supports will help maintain student engagement in mathematical discourse and ensure that the struggle remains productive. Teachers should use their professional judgment about which routines to use and when, based on their knowledge of the individual needs of students in their classroom.” Examples include:

• Unit 3: Linear Relationship, Section D: Linear Equations, Lesson 12: Solutions to Linear Equations, Activity 2: Solutions and Everything Else, Supports for English Learners, “Writing, Conversing: MLR1 Stronger and Clearer Each Time, Use this routine for students to respond in writing to the prompt: ‘What does a graph tell you about the solutions to an equation with two variables?’ Give students time to meet with 2–3 partners, to share and get feedback on their responses. Encourage the listener to press for supporting details and evidence by asking, ‘Could you give an example from your graph?’ or ‘Could you make a generalization about the solutions to an equation from the specific case you mentioned?’ Have the students write a second draft based on their peer feedback. This will help students articulate their understanding of the solution to an equation and clearly define it using a graph. Design Principles: Optimize output, Cultivate conversation”

• Unit 5: Functions and Volume, Section D: Cylinders and Cones, Lesson 12: How Much will Fit?, Activity 1: What’s Your Estimate?, Supports for English Language Learners, “Speaking: MLR8 Discussion Supports, Use this routine to support whole-class discussion. For each strategy that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to speak as they make sense of the reasoning of others. Design Principle: Support sense-making”

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 7: Observing More Patterns in Scatter Plots, Activity 1, Scatter Plot City, Supports for English Learners, “Speaking: MLR8 Discussion Supports, To support students’ explanations for sorting the scatter plots in the way they chose, display sentence frames for students to use when they are working with their partner. For example, ‘I think ____ because ____.’ or ‘I (agree/disagree) because ____.’ Design Principles: Support sense-making, Optimize output.”

The materials reviewed for Open Up Resources Grade 8 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Suggestions and/or links to manipulatives are consistently included within materials to support the understanding of grade-level math concepts. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section D: Angles in a Triangle, Lesson 14: Alternate Interior Angles, Warm Up: Angle Pairs, Instructional Routines, “This task is designed to prompt students to recall their prior work with supplementary angles. While they have seen this material in grade 7, this is the first time it has come up explicitly in grade 8. As students work on the task, listen to their conversations specifically for the use of vocabulary such as supplementary and vertical angles. If no students use this language, make those terms explicit in the discussion. 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.”

• Unit 5: Functions and Volume, Section B: Representing and Interpreting Functions, Lesson 6: Even More Graphs of Functions, Required Preparation, “Students are asked to make displays of their work in groups of 2–3. Prepare materials for creating a visual display in this way such as markers, chart paper, board space, etc.”

• Unit 7: Exponents and Scientific Notation, Section C: Scientific Notation, Lesson 10: Representing Large Numbers on the Number Line, Activity 1: Comparing Large Numbers with a Number Line, Instructional Routines, “This activity encourages students to use the number line to make sense of powers of 10 and think about how to rewrite expressions in the form b\cdot {10}^n, where b is between 1 and 10 (as in the case of scientific notation). It prompts students to use the structure of the number line to compare numbers, and to extend their use to estimate relative sizes of other numbers when no number lines are given. As students work, notice the ways in which they compare expressions that are not written as multiples of {10}^6. Highlight some of these methods in the discussion.”