7th Grade - Gateway 3

The materials reviewed for Open Up Resources Grade 7 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 7 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 and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Examples include:

• Unit 2: Introducing Proportional Relationships, Unit Overview, “Because this unit focuses on understanding what a proportional relationship is, how it is represented, and what types of contexts give rise to proportional relationships, the contexts have been carefully chosen. The first tasks in the unit employ contexts such as servings of food, recipes, constant speed, and measurement conversion, that should be familiar to students from the grade 6 course. These contexts are revisited throughout the unit as new aspects of proportional relationships are introduced. Associated with the contexts from the grade 6 course are derived units: miles per hour; meters per second; dollars per pound; or cents per minute. In this unit, students build on their grade 6 experiences in working with a wider variety of derived units, such as cups of flour per tablespoon of honey, hot dogs eaten per minute, and centimeters per millimeter. The tasks in this unit avoid discussion of measurement error and statistical variability, which will be addressed in later units. On using the terms quantity, ratio, proportional relationship, unit rate, and fraction. 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, MP6). The term ratio is used to mean a type of association between two or more quantities. A proportional relationship is a collection of equivalent ratios. A unit rate is the numerical part of a rate per 1 unit, e.g., the 6 in 6 miles per hour. 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 . 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 \frac{a}{b} as ‘ratios.’ 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. Students have worked with area since grade 3 and volume since grade 5. Before grade 6, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation in grade 6, they also use cm^2 and cm^3. A fraction is a point on the number line that can be located by partitioning the segment between 0 and 1 into equal parts, then finding a point that is a whole number of those parts away from 0. A fraction can be written in the form \frac{a}{b} or as a decimal.”

• Unit 4: Proportional Relationships and Percentages, Section C: Applying Percentages, Lesson 10: Tax and Tip, Lesson Narrative, “In this lesson students are introduced to contexts involving sales tax and tips. They can use tape diagrams and double number lines from their grade 6 work, but the lesson provides an opportunity to be more efficient by using an equation of the form y = kx. For example, if the tax rate is 6.2% they can calculate the tax, T, for any price, p, using the equation t = 0.062p. They do not necessarily write this equation out with variables, but rather repeatedly use it with specific values of p. By repeatedly calculating the tax for different prices and then generalizing the process they are engaging in expressing regularity in repeated reasoning (MP8). Questions about rounding naturally come up in this lesson. This lesson primarily involves dollar amounts, so it is sensible to round to the nearest cent (the nearest hundredth of a dollar). When students attend to precision and make decisions about what is the appropriate level of rounding, they engage in MP6.” 

• Unit 7: Angles, Triangles, and Prisms, Section C: Solid Geometry, Lesson 15: Distinguishing Volume and Surface Area, Activity 2: Card Sort: Surface Area or Volume, Instructional Routines, “The purpose of this activity is for students to sort cards with questions that have a context referring to either volume or surface area of a prism. In previous lessons, students focused on determining volume or surface area and the two concepts were never presented side by side. Here, students are asked to sort questions with a context to determine if it makes more sense to think about surface area or volume when answering the question. After sorting, students think about what information they need to answer a question and estimate reasonable measurements to calculate the answer to their question (MP2).”

The materials reviewed for Open Up Resources Grade 7 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 1: Scale Drawings, Unit Overview, “Students begin by looking at copies of a picture, some of which are to scale, and some of which are not. They use their own words to describe what differentiates scaled and non-scaled copies of a picture. As the unit progresses, students learn that all lengths in a scaled copy are multiplied by a scale factor and all angles stay the same. They draw scaled copies of figures. They learn that if the scale factor is greater than 1, the copy will be larger, and if the scale factor is less than 1, the copy will be smaller. They study how area changes in scaled copies of an image.”

• Unit 3: Measuring Circles, Section A: Circumference of a Circle, Lesson 3: Exploring Circumference, Activity 1: Measuring Circumference and Diameter, Instructional Routines, “In this activity, students measure the diameter and circumference of different circular objects and plot the data on a coordinate plane, recalling the structure of the first activity in this unit where they measured different parts of squares. Students use a graph in order to conjecture an important relationship between the circumference of a circle and its diameter (MP5). They notice that the two quantities appear to be proportional to each other. Based on the graph, they estimate that the constant of proportionality is close to 3 (a table of values shows that it is a little bigger than 3). This is their first estimate of pi. This activity provides good, grade- appropriate evidence that there is a constant of proportionality between the circumference of a circle and its diameter. Students will investigate this relationship further in high school, using polygons inscribed in a circle for example. To measure the circumference, students can use a flexible measuring tape or a piece of string wrapped around the object and then measure with a ruler. As students measure, encourage them to be as precise as possible. Even so, the best precision we can expect for the proportionality constant in this activity is ‘around 3’ or possibly ‘a little bit bigger than 3’. This could be a good opportunity to talk about how many digits in the answer is reasonable. To get a good spread of points on the graph, it is important to use circles with a wide variety of diameters, from 3 cm to 25 cm. If there are points that deviate noticeably from the overall pattern (6.SP.B.5c), discuss how measurement error plays a factor.” 

• Unit 6: Expressions, Equations, and Inequalities, Section B: Solving Equations in the Form px + q = r and p(x + q) = r, Lesson 10: Different Options for Solving One Equation, Activity 2: Solution Pathways, Activity Synthesis, “Reveal the solution to each equation and give students a few minutes to resolve any discrepancies with their partner. Display the list of equations in the task, and ask students to help you label them with which solution method was easier, either “divide first” or “distribute first.” Discuss any disagreements and the reasons one method is easier than the other. (There is really no right or wrong answer here. Some people might prefer moves that eliminate fractions and decimals as early as possible. Some might want to minimize the number of computations.)”

Materials contain adult-level explanations and examples of concepts beyond grade 7 so that teachers can improve their knowledge of the subject. Examples include:

• Unit 1: Scaled Drawings, Unit 1 Overview, “Note that the study of scaled copies is limited to pairs of figures that have the same rotation and mirror orientation (i.e. that are not rotations or reflections of each other), because the unit focuses on scaling, scale factors, and scale drawings. In grade 8, students will extend their knowledge of scaled copies when they study translations, rotations, reflections, and dilations.”

• Unit 2: Introducing Proportional Relationships, Unit 2 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. Students have worked with area since grade 3 and volume since grade 5. Before grade 6, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation in grade 6, they also use {cm}^2 and {cm}^3.”

• Unit 3: Measuring Circles, Unit 3 Overview, “In the third and last section, students select and use formulas for the area and circumference of a circle to solve abstract and real-world problems that involve calculating lengths and areas. They express measurements in terms of \pi or using appropriate approximations of \pi to express them numerically. In grade 8, they will use and extend their knowledge of circles and radii at the beginning of a unit on dilations and similarity.”

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

The materials reviewed for Open Up Resources Grade 7 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 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: Proportional Relationships and Percentages, Section B: Percent Increase and Decrease, Lesson 7: One Hundred Percent, “Addressing 7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.”

The role of specific grade-level mathematics can be explained in Unit Overviews, Section Overviews, and Lesson Narratives. Examples include:

• Unit 1: Scale Drawings, Overview, “In this unit, students study scaled copies of pictures and plane figures, then apply what they have learned to scale drawings, e.g., maps and floor plans. This provides geometric preparation for grade 7 work on proportional relationships as well as grade 8 work on dilations and similarity. Students begin by looking at copies of a picture, some of which are to scale, and some of which are not. They use their own words to describe what differentiates scaled and non-scaled copies of a picture. As the unit progresses, students learn that all lengths in a scaled copy are multiplied by a scale factor and all angles stay the same. They draw scaled copies of figures. They learn that if the scale factor is greater than 1, the copy will be larger, and if the scale factor is less than 1, the copy will be smaller. They study how area changes in scaled copies of an image.”

• Unit 4: Proportional Relationships and Percentages, Section C: Applying Percentages, Section Overview, “In the third section of the unit, students begin by using their abilities to find percentages and percent rates to solve problems that involve sales tax, tip, discount, markup, markdown, and commission (MP2). The remaining lessons of the section continue the focus on situations that can be described in terms of percentages, but the situations involve error rather than change—describing an incorrect value as a percentage of the correct value rather than describing an initial amount as a percentage of a final amount (or vice versa).”

• Unit 6: Expressions, Equations, and Inequalities, Section A: Representing Situations of the Form px + q = r and p(x + q) = r, Lesson 6: Distinguishing Between Two Types of Situations, Lesson Narrative, “The purpose of this lesson is to distinguish equations of the form px + q = r and p(x + q) = r. Corresponding tape diagrams are used as tools in this work, along with situations that these equations can represent. First, students sort equations into categories of their choosing. The main categories to highlight distinguish between the two main types of equations being studied. Then, students consider two stories and corresponding diagrams and write equations to represent them. They use these representations to find an unknown value in the story.”

The materials reviewed for Open Up Resources Grade 7 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 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the Course Guide, Materials, there is a list of required materials 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 needed. Examples include:

• Resources, Course Guide, Required Materials, “blank paper, coins, colored pencils, compasses, cylindrical household items, drink mix, empty toilet paper roll, four-function calculators, fruits or vegetables, geometry toolkits, glue or gluesticks, graph paper, grocery store circulars, index cards, internet-enabled device, knife, maps or satellite images of the school grounds, markers, measuring cup, measuring spoons, measuring tapes, measuring tools, metal paper fasteners, meter sticks, metric and customary unit conversion charts, mixing containers, number cubes, paint, paper bags, paper clips, paper plates, pattern blocks, pre-assembled polyhedra, protractors, receipt tape, recipes, rulers, rulers marked with centimeters, rulers marked with inches, scissors, small disposable cups, snap cubes, sticky notes, stopwatches, straightedges, straws, string, tape, tools for creating a visual display, trundle wheels, water, yardsticks.”

• Unit 2: Introducing Proportional Relationships, Section D: Representing Proportional Relationships with Graphs, Lesson 12: Using Graphs to Compare Relationships, Required Materials, “colored pencils, rulers.”

• Unit 6: Expressions, Equations, and Inequalities, Section C: Inequalities, Lesson 16: Interpreting Inequalities, Required Materials, “tools for creating a visual display.”

The materials reviewed for Open Up Resources Grade 7 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 5: Rational Numbers Arithmetic, Unit Assessments, End-of-Unit Assessment, Version B, Problem 2, “7.NS.A.1.c, A sunken ship is resting at 3,000 feet below sea level. Directly above the ship, a whale is swimming 1,960 feet below sea level. Directly above the ship and the whale is a plane flying at 8,500 feet above sea level. Select the true statement. A. The distance between the heights of the whale and the plane is -10,460 feet. B. The difference in height between the whale and the plane is 10,460 feet. C. The difference in height between the whale and the ship is -1,040 feet. D. The distance between the heights of the whale and the ship is 1,040 feet.” 

• Unit 7: Angles, Triangles, and Prisms, Section B: Drawing Polygons with Given Conditions, Lesson 7: Building Polygons (Part 2), Cool Down: Finishing Elena’s Triangles, “7.G.A.2, a. Elena is trying to draw a triangle with side lengths 4 inches, 3 inches, and 5 inches. She uses her ruler to draw a 4 inch line segment AB. She uses her compass to draw a circle around point B with radius 3 inches. She draws another circle, around point A with radius 5 inches. What should Elena do next? Explain and show how she can finish drawing the triangle. B. Now Elena is trying to draw a triangle with side lengths 4 inches, 3 inches, and 8 inches. She uses her ruler to draw a 4 inch line segment AB. She uses her compass to draw a circle around point B with radius 3 inches. She draws another circle, around point A with radius 8 inches. Explain what Elena’s drawing means.” 

• Unit 8: Probability and Sampling, Unit Assessments, Mid-Unit Assessment, Version A, Problem 4, “7.SP.C.8b, “A movie theater sells 3 different sizes of popcorn: small, medium, and large. An order of popcorn comes with 3 different topping choices: butter, cheese, or caramel. How many unique ways can you order a bag of popcorn? (You can only select one topping.)”

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, “MP4: I Can Model with Mathematics: I can wonder about what mathematics is involved in a situation. I can come up with mathematical questions that can be asked about a situation. I can identify what questions can be answered based on data I have. I can identify information I need to know and don’t need to know to answer a question. I can collect data or explain how it could be collected. I can model a situation using a representation such as a drawing, equation, line plot, picture graph, bar graph, or a building made of blocks. I can think about the real-world implications of my model.”

The materials reviewed for Open Up Resources Grade 7 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 1: Scale Drawings, End-of-Unit Assessment, Version B, Problem 6, students reason about map scaling. “There are two different maps of California. The scale on the first map is 1 cm to 20 km. The distance from Fresno to San Francisco is 15 cm. The scale on the second map is 1 cm to 100 km. What is the distance from Fresno to San Francisco on the second map? Explain your reasoning.” Solution, “Minimal Tier 1 response: Work is complete and correct. Sample: Lengths on the second map are five times smaller because 1 cm represents 20 km instead of 100 km. Divide 15 cm by 5 to get 3 cm. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: multiplication or division errors in otherwise correct work; work involves a correct substantive intermediate step (such as the actual distance from Fresno to San Francisco) but goes wrong after that; one mistake involving an “upside down” scale factor (or multiplying when division is called for); a correct answer without explanation. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: work does not involve proportional reasoning; an incorrect answer without explanation, even if close; multiple mistakes that involve inversion of scale factors.” 

• Unit 4: Proportional Relationships and Percentages, End-of-Unit Assessment, Version B, Problem 2, students find percent error. “A graduated cylinder actually contains 7.5 milliliters of water. When Han measures the volume of the water inside the graduated cylinder, his measurement is 7 milliliters. Which of these is closest to the percent error for Han’s measurement? A. 107.1% B. 93.3% C. 7.1% D. 6.7%” Solution, “D.”

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 3: Measuring Circles, End-of-Unit Assessment, Version A, Problem 1, students calculate the area of a circle. “A circle has radius 50 cm. Which of these is closest to its area? A. 157 cm^2 B. 314 cm^2 C. 7,854 cm^2 D. 15,708 cm^2.” Guidance for teachers, “If students struggle to calculate the area of a circle, provide additional instruction either in a small group or individually using OUR Math Grade 7 Unit 3 Lesson 7 Activity 3 and Practice Problems 1-3.”

The materials reviewed for Open Up Resources Grade 7 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 3: Measuring Circles, End-Of-Unit Assessment, Version A, Problem 7, students analyze and make sense of a problem by working to understand the information in the problem and the question asked. “A groundskeeper needs grass seed to cover a circular field, 290 feet in diameter. A store sells 50-pound bags of grass seed. One pound of grass seed covers about 400 square feet of field. What is the smallest number of bags the groundskeeper must buy to cover the circular field. Explain or show your reasoning.” (MP1)

• Unit 4: Proportional Relationships and Percentages, Section A: Proportional Relationships with Fractions, Lesson 4: Half As Much Again, Practice Problems, Problem 3, students use distributive property to represent a situation. “Write a story that can be represented by the equation y = x + \frac{1}{4}x.” (7.RP.2)

• Unit 6: Expressions, Equations, and Inequalities, Section B: Solving Equations of the Form px + q = r and p(x + q) = r, Lesson 7: Reasoning About Solving Equations (Part I), Cool Down: Solve the Equation, students solve equations. “Solve the equation. If you get stuck, try using a diagram. 5x + \frac{1}{4} = \frac{61}{4}." (7.EE.4a)

• Unit 6: Expressions, Equations, and Inequalities, Mid-Unit Assessment, Version A, Problem 2, students recognize both the insight to be gained from a tape diagram, and its limitations. “Select all the situations that can be represented by the tape diagram. A. Clare buys 4 bouquets, each with the same number of flowers. The florist puts an extra flower in each bouquet before she leaves. She leaves with a total of 99 flowers. B. Andre babysat 5 times this past month and earned the same amount each time. To thank him, the family gave him an extra $4 at the end of the month. Andre earned $99 from babysitting. C. A family of 5 drove to a concert. They paid $4 for parking, and all of their tickets were the same price. They paid $99 in total. D. 5 bags of marbles each contain 4 large marbles and the same number of small marbles. Altogether, the bags contain 99 marbles. E. Han is baking five batches of muffins. Each batch needs the same amount of sugar in the muffins, and each batch needs four extra teaspoons of sugar for the topping. Han uses 99 total teaspoons of sugar.” (MP5)

• Unit 7: Angles, Triangles, and Prisms, End-of-Unit Assessment, Version A, Problem 2, students identify cross sections of a square pyramid. “A square pyramid is sliced parallel to the base and halfway up the pyramid. Which of these describes the cross section? A. A square smaller than the base B. A quadrilateral that is not a square C. A square the same size as the base D. A triangle with a height the same as the pyramid.” (7.G.3)

• Unit 8: Probability and Sampling, End-of-Unit Assessment, Version B, Problem 7, students use statistical measures to compare data sets. “Students are conducting a class experiment to see if there is a meaningful difference between two groups of plants that have begun to sprout leaves. The teacher randomly selects 8 plants from each group and counts the number of leaves on each plant. Group A: 2, 2, 3, 3, 5, 5, 7 Group B: 10, 9, 7, 8, 10, 12, 14, 10 Is there a meaningful difference between the two groups? Show all calculations that lead to your answer.” (7.SP.3 and 7.SP.4)

The materials reviewed for Open Up Resources Grade 7 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 2: Introducing Proportional Relationships, Section C: Comparing Proportional and Nonproportional Relationships, Lesson 7: Comparing Relationships with Tables, Activity 1: Visiting the Skate Park, Supports for Students with Disabilities, “Action and Expression: Executive Functions, To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have accounted for the cost of the vehicle in their calculations of the total entrance cost for 4 people and 10 people. Provides accessibility for: Memory, Organization.”

• Unit 6: Expressions, Equations, and Inequalities, Section A: Representing Situations of the Form px + q = r and p(x + q) = r, Lesson 5: Reasoning About Equations and Tape Diagrams (Part 2), Warm Up: Algebra Talk: Seeing Structure, Supports for Students with Disabilities, “Representation: Comprehension, To support working memory, provide students with sticky notes or mini whiteboards. Provides accessibility for: Memory, Organization.”

• Unit 8: Probability and Sampling, Section B: Probability of Multi-step Events, Lesson 9: Multi- Step Experiments, Activity 1: Spinning a Color and Number, Supports for Students with Disabilities, “Representation: Perception, Provide access to concrete manipulatives. Provide spinners for students to view or manipulate. These hands-on models will help students identify characteristics or features, and support finding outcomes for calculating probabilities. Provides accessibility for: Visual-Spatial Processing, Conceptual Processing.”

The materials reviewed for Open Up Resources Grade 7 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 1: Scaled Drawings, Section A: Scaled Copies, 2: Corresponding Parts and Scale Factors, Activity 2: Scaled Triangles, Are You Ready For More? “Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.”

• Unit 4: Proportional Relationships and Percentages, Section A: Proportional Relationships with Fractions, Lesson 2: Ratio and Rates with Fractions, Activity 2: Comparing Running Speeds, Are You Ready for More? “Nothing can go faster than the speed of light, which is 299,792,458 meters per second. Which of these are possible? a. Traveling a billion meters in 5 seconds. b. Traveling a meter in 2.5 nanoseconds. (A nanosecond is a billionth of a second.) c. Traveling a parsec in a year. (A parsec is about 3.26 light years and a light year is the distance light can travel in a year.)”

• Unit 7: Angles, Triangles, and Prisms, Section A: Angle Relationships, Lesson 4: Solving for Unknown Angles, Activity 2: What’s the Match? Are You Ready for More? “What is the angle between the hour and minute hands of a clock at 3:00?”

The materials reviewed for Open Up Resources Grade 7 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 1: Scare Drawings, Section A: Scaled Copies, Lesson 4: Scaled Relationships, Activity 1: Three Quadrilaterials (Part 2), Supports for English Language Learners, “Speaking: MLR7 Compare and Connect, Use this routine to call attention to the different ways students may identify scale factors. Display the following statements: “The scale factor from EFGH to IJKL is 3,” and “The scale factor from EFGH to IJKL is \frac{1}{3}.” Give students 2 minutes of quiet think time to read and consider whether either or both of the statements are correct. Invite students to share their initial thinking with a partner before selecting 2–3 students to share with the class. In this discussion, listen for and amplify any comments that refer to the order of the original figure and its scaled copy, as well as those who identify corresponding vertices and distances. Draw students’ attention to the different ways to describe the relationships between scaled copies and the original figure. Design Principle: Maximize linguistic & cognitive meta-awareness.”

• Unit 4: Proportional Relationships and Percentages, Section A: Proportional Relationships with Fractions, Lesson 3: Revisiting Proportional Relationships, Activity 2: Swimming, Manufacturing, and Painting, Supports for English Learners, “Writing: MLR3 Critique, Correct, and Clarify, Present an incorrect response to the question about mixing 4 quarts of blue paint. For example, ‘Since there are 0.3 quarts of blue paint to white paint, you need 1.2 quarts of white paint.’ Prompt students to identify the error (e.g., ask students, ‘Do you agree with the statement? Why or why not?’), and then write a correct version. This helps students evaluate, and improve on, the written mathematical arguments of others. Design Principle: Maximize linguistic & cognitive meta-awareness.”

• Unit 6: Expressions, Equations and Inequalities, Section C: Inequalities, Lesson 13: Reintroducing Inequalities, Activity 1: The Roller Coaster, Supports for English Learners, “Conversing, Writing: MLR5 Co-Craft Questions and Problems, Use this routine to help students consider the context of the first problem and to increase awareness about language used to describe situations involving inequalities. Begin by displaying only the initial text and photo of the roller coaster, without revealing the follow-up questions. In groups of 2, invite students to write down mathematical questions they have about this situation. Ask pairs to share their questions with the whole class. Amplify questions that highlight the mathematical language of ‘at least’. Design Principles: Cultivate conversation, Maximize linguistic & cognitive meta-awareness.”

The materials reviewed for Open Up Resources Grade 7 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 3: Measuring Circles, Section A: Circumference of a Circle, Lesson 2: Exploring Circles, Activity 3: Drawing Circles, Instructional Routines, “The purpose of this activity is to reinforce students’ understanding of the terms diameter, center, and radius and also for students to see what a compass is good for (MP5). Before using the compass, students first attempt to draw a circle freehand. Then, they recognize the compass as a strategic tool for drawing circles. However, the compass is useful not just for drawing circles but also for transferring lengths from one location to another for many different purposes. Students will apply this understanding in later units, for example, when they construct a triangle given the lengths of its three sides. This activity prepares students for that application by asking them to make the radius of the circle match another length they have already drawn. If this is a student’s first time using a compass, direct instruction may be needed on how to use one. The circles students draw may not be perfect, but as they gain more experience with a compass, they will improve. A digital version of the activity is provided for classrooms that do not have access to compasses but do have access to appropriate electronic devices.” 

• Unit 7: Angles, Triangles, and Prisms, Section A: Angle Relationships, Lesson 2: Adjacent Angles, Required Materials, “Cut blank paper in half so that each student can have 2 half sheets of paper. It is very important that these cuts are completely straight and exactly perpendicular to the sides being cut for this activity to work. Prepare to distribute scissors, straightedges, and protractors.”

• Unit 8: Probability and Sampling, Section B: Probabilities of Multi-Step Events, Lesson 10: Designing Simulations, Required Preparation, “Every 3 students need 2 coins for the Breeding Mice activity. Print and cut up questions from the Designing Simulations Blackline Master. Use one question for every 3 students. Groups will need access to number cubes, protractors, rulers, compasses, paper clips, bags, snap cubes, and scissors to simulate their scenarios.”