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

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

Cluster 6.RP.A addresses understanding of ratio concepts and using ratio reasoning to solve problems. Units 2 and 3 explore a variety of real-world applications using multiple mathematical representations. Multiple opportunities exist for students to work with ratios that call specifically for conceptual understanding and include the use of visual representations, interactive examples, and different strategies, which then shifts to more abstract methods of finding equivalent ratios in later lessons in the unit. For example, in Unit 2:

• In Lessons 1 and 2, students use physical objects to develop ratio language to describe a relationship between two quantities (6.RP.1). Students sort and categorize concrete objects such as different color binder clips and analyze a picture of snap cubes to write a sentence to describe the ratio shown in their diagram.

• In Lessons 3 and 4, students develop a conceptual understanding of equivalent ratios (6.RP.3). Lesson 3 extends the concept of ratios as described in the lesson introduction: “Students see that scaling a recipe up (or down) requires multiplying the amount of each ingredient by the same factor, e.g., doubling a recipe means doubling the amount of each ingredient (MP7). They also gain more experience using a discrete diagram as a tool to represent a situation.”

• Lesson 6 introduces double number lines for students to use and interpret alongside the more familiar discrete diagrams and in the familiar context of recipes.

• Lesson 8 How Much For One? introduces students to the concept of unit price. They continue their work on ratios involving one unit “of something” in a real-world context (6.RP.2). For example, “Eight avocados cost $4. How much do 16 avocados cost? How much do 20 avocados cost? How much do 9 avocados cost?” Students also choose whether to draw double number lines or other representations to support their reasoning.

• In Lesson 10, a short video is used to show a person walking at a constant speed on a treadmill for a few seconds. Students then compare the length of time it takes two different people to run three miles and explain their reasoning.

• Lessons 11 through 15 explore tables, tape diagrams, and double number lines, as well as varying ratio problem types such as equivalent ratio problems and part-part-whole ratios.

• In Lesson 16, students use all the methods learned in Unit 2 “to solve ratio problems that involve the sum of the quantities in the ratio.”

Cluster 6.NS.A addresses applying and extending previous understandings of multiplication and division to divide fractions by fractions. A variety of additional applets are used within this unit by students to build understanding when dividing fractions. Unit 4 develops conceptual understanding of division of fractions. For example:

• In Lesson 1 Activities, students explore the size of quotients, based on divisors and dividends. Activity 1.2 includes an applet for students to model a variety of division problems and interpret the quotients. Students then examine the divisor and dividend (not perform the operation) and put them in order from least to greatest, group them as close to 0, close to 1, or much greater than 1. In Activity 1.3, students interpret division situations. The Cool-Down requires students to determine proximity to 1, based on the given division problems in order to demonstrate understanding of the concepts within the lesson.

• In Lessons 4 and 5, students manipulate pattern blocks to determine how many groups can be formed. Students begin by using pattern blocks to find how many times a fraction goes into a number starting with whole numbers, then mixed numbers, and finally, fractions. The Lesson 4 Cool-Down Student Facing Task states: “Answer the following questions. If you get stuck, use pattern blocks. a) How many ½ are in 3 ½? b) How many ⅙ are in ⅔?” c) How many ⅙ are in ⅔?” Students are encouraged to look at division of fractions from a multiplication perspective, and are encouraged to use a diagram to understand the connection between multiplication and division.

Cluster 6.EE addresses applying and extending previous understandings of arithmetic to algebraic expressions and developing reasoning to solve one-variable equations and inequalities. Unit 6 Expressions and Equations presents opportunities for students to develop their conceptual understanding. For example:

• Lesson 1 introduces students to tape diagrams to represent equations with and without variables, and then students match the equation with the related diagram and use the diagrams as needed throughout the unit to solve equations (6.EE.6).

• Lesson 2 introduces students to “hanger diagrams” (to represent balance scales) and students reason about concrete representations of equations. They identify what is true and/or false about the diagrams, as well as reason about how balanced hangers with two shapes are related when the shapes are not equally represented on each side, connecting the “hanger diagrams” to equations.

• In Lesson 3, students develop the concept of equivalency. In Activity 2, students use an applet to model equations and solve for the given variable (6.EE.7).

• In Lesson 6, students match equations to tape diagrams, match equations to situations, and solve equations.

• In Lesson 8, students draw diagrams of two separate expressions to show that they are equivalent for given values (6.EE.4).

• In Lesson 10 Activity 1, students calculate the area of partitioned rectangles as both a product of length and width and as the sum of the area of two smaller rectangles and write expressions to represent both calculations. In comparing their expressions students realize they are equivalent because of the distributive property. (6.EE.A.3 and 6.EE.A.4).

In cluster 6.G.A, students solve real-world and mathematical problems involving area, surface area, and volume. Examples of supporting teachers and students on building upon conceptual understanding are present throughout Unit 1, Area and Surface Area. Students use concrete models to develop abstract representations using equations. For example:

• In Lesson 2 Warm-Up, the following guidance is provided for teachers: “Students may focus on how they have typically found the area of a rectangle—by multiplying its side lengths—instead of thinking about what ‘the area of any region’ means. Ask them to consider what the product of the side lengths of a rectangle actually tells us. (For example, if they say that the area of a 5-by-3 rectangle is 15, ask what the 15 means.)”

• In Lessons 2 and 3, students use physical tangrams to explore composing and decomposing two-dimensional figures to find area and reason about how the composed and decomposed shapes represent the area found.

• In Lesson 19 Activities 1 (Part 1) and 2 (Part 2) Tent Design, students work with partners and independently to design a tent given specific constraints. Both activities use models and visual representations from which students create expressions and equations to represent their tent design.

The instructional materials for Open Up Resources 6-8 Math, Grade 6 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. Materials attend to the Grade 6 expected fluencies, particularly fluency with multi-digit decimals and computing with them in expressions and equations.

Procedural skills and fluencies are intentionally built on conceptual understanding and the work students have accomplished with operations and equations from prior grades. Opportunities to formally practice developed procedures are found throughout practice problem sets that follow the units and include opportunities to use and practice emerging fluencies in the context of solving problems. According to the Design Principles within the Grade 6 Course Guide, “As the unit progresses, students are systematically introduced to representations, contexts, concepts, language, and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” Number Talks included in many lesson Warm-Ups often revisit fluencies developed in earlier grades and specifically relate to the Activities found in the lessons.

6.NS.2 and 6.NS.3 are found throughout Unit 5. The lessons in the unit address developing fluency in adding, subtracting, multiplying, and dividing, multi-digit decimals using the standard algorithm through visual models, word problems, and activities which build toward student understanding of the standard algorithm. Specific Unit 5 examples include:

• In Lesson 1, students review decimal work before they utilize the four operations to solve problems in real-world contexts such as using money or planning a party (6.NS.3), using strategies such as mental math to estimate with decimals.

• In Lesson 2, students use both a visual model and standard algorithm to calculate decimals.

• In Lesson 3, students add and subtract decimals, enabling them to work toward fluency. Students encounter decimals beyond thousandths, find missing addends, and work with decimals in the context of situations. Students are prompted to “evaluate mentally: 1.009 + 0.391.”

• In Lesson 5 Items 5 and 6, students solve procedural practice problems using addition and subtraction. Lesson 11 continues to use conceptual foundations from Lessons 9 and 10 (and from prior grades with Warm-Ups related to “unbundling”), but the second Activity uses the standard algorithm (6.NS.2).

• In Lesson 12, practice problems explicitly state to use “long division.”

6.EE.A is developed in Units 1 and 6 as students apply and extend previous understandings of arithmetic to algebraic expressions, beginning with using formulas for area in Unit 1 and computing with decimals and fractions embedded in expressions and equations in Unit 6.

• In Unit 1 Lessons 5 and 6, students find the area of parallelograms using the formula A=bh (6.EE.2c, 6.G.1). In Lesson 9, students complete a table, finding the area of triangles using the formula and substituting given quantities for the unknown variable. Students use computational skills and apply what they learned about the area formula as well as the base and height of a triangle with multiple given measurements. In Lessons 15 and 17, students continue to develop and use computational skills in order to evaluate expressions that arise from formulas used in real-world problems at specific values of their variables (6.EE.A). While 6.G.4 is identified as the standard being addressed in Lesson 18, students have opportunities to use computational skills involving whole number exponents as well as write, read, and evaluate expressions in which letters stand for numbers.

• In Unit 6, tape diagrams are used to represent equations in Lesson 1, and contexts are used in Lesson 2. Students see how an equation can represent a situation with an unknown amount. In Lesson 4, students solve a variety of equations with different structures and match equations to situations and solve them. In Lessons 9 through 11, students apply the distributive property to generate equivalent expressions, building upon what they know about rectangles, with variables to represent lengths of sides and areas of rectangles.

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

Work with applications of mathematics occurs throughout the materials in ways that enhance the focus on major work and when standards call for application in real-world or mathematical contexts. In addition, application contexts are used throughout the curriculum to build conceptual understanding. The Grade 6 Course Guide states: “Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on the mathematical contexts. The first unit on geometry is an example of this.” Connections between clusters and application extensions are also found in the multiple-day lessons found in optional Unit 9.

Standard 6.RP.3 addresses students using ratio and rate reasoning to solve real-world and mathematical problems and is found in Units 2 and 3.

• Unit 2 introduces ratios, equivalence in ratios, and strategies for solving related problems. Reasoning using tables, double number lines, and tape diagrams are each applied in routine problems including a variety of contexts. Lesson 14 reads, “Lin read the first 54 pages from a 270-page book in the last three days. Diego read the first 100 pages from a 320-page book in the last four days. Elena read the first 160 pages from a 480-page book in the last five days. If they continue to read every day at these rates, who will finish first, second, and third? Explain or show your reasoning.”

• Students encounter non-routine word problems as they apply ratio and rate reasoning to problems with multiple solutions. In Lesson 15, students “invent another ratio problem that can be solved with a tape diagram and solve it. If you get stuck, consider looking back at the problems you solved in the earlier activity. Create a visual display that includes: The new problem that you wrote, without the solution, and enough work space for someone to show a solution. Trade your display with another group and solve each other’s problem. Include a tape diagram as part of your solution. Be prepared to share the solution with the class. When the solution to the problem you invented is being shared by another group, check their answer for accuracy.” In Lesson 16, a multiple-solution problem from openmiddle.com is included: “Use the digits 1 through 9 to create three equivalent ratios. Use each digit only one time. ____ : ____ is equivalent to ____ : ____ and ____ : ____.”

Standard 6.NS.1 addresses students solving word problems involving division of fractions by fractions and is found in Unit 4.

• In Lesson 3, students “analyze a division context and tell if it represents a “how many groups?” question, or a “how many in each group?" question.” Students use unit fractions, non-unit fractions with whole-number dividends, and mixed-number dividends with non-unit fraction divisors.

• Students encounter less routine word problems as they begin to divide with fractional dividends and divisors. In Lesson 11, the following division problem is included: “If 4/3 liters of water are enough to water 2/5 of the plants in the house, how much water is necessary to water all the plants in the house? Write a multiplication equation and a division equation for the situation, then answer the question. Show your reasoning.”

• The end of the unit includes opportunities to use division in multiplicative comparison word problems and problems involving length and area.

The instructional materials for Open Up Resources 6-8 Math, Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

There is evidence that the curriculum addresses standards, when called for, with specific and separate aspects of rigor and evidence of opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized.

Examples of conceptual understanding include:

• In Unit 1, concepts are built from the use of physical models and visual representations as students develop understanding of formulas. In Lessons 4 through 6, students work with parallelograms, and in Lessons 7 through 10 they work with triangles.

• In Unit 6 Lesson 1, conceptual understanding of the connections between multiplication and addition (6.EE.6) are reinforced. In the Warm-Up, visual models using tape diagrams are revisited. Students “draw a diagram that represents each equation, 4+3=7 and 4⋅3=12” in the first activity. Students then “use what they know about relationships between operations to identify multiple equations that match a given diagram.”

Examples of procedural fluency include:

• In Unit 5 Lessons 11 through 13, students divide decimals (6.NS.2) using the standard algorithm. First, students mentally solve four division problems using structure and patterns in the Warm-Up. In the second Activity, Using Long Division to Calculate Quotients, students evaluate the division algorithm as performed by a given student by answering questions such as, “Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?” Fluency is further developed over additional practice problems found in Lessons 11-13.

• In Unit 1 lesson 9, students look at examples and nonexamples to identify base and height. They then find a formula for the area of a triangle by looking at triangles on a grid and completing a table recording the base, height and area of the triangles.

Examples of application include:

• Unit 1 Lesson 19 intentionally addresses the language in standards 6.G.1 and 6.G.2: “Apply these techniques in the context of solving real-world and mathematical problems.” Students interpret a tent design problem and create a tent design that meets certain specifications. They also calculate surface area and estimate the amount of fabric they will need. In the second part of the lesson, students must present and justify their design to a peer and reflect on similarities and differences in the different designs of their group.

• Unit 2 Lesson 17 addresses the application found in 6.RP.A. Students are exposed to Fermi problems (e.g., “How many times does your heart beat in a year?”), clarify and narrow a problem, as well as apply what they’ve learned about rates and ratios to estimate a solution. Finally, students develop and create an estimated solution to their own Fermi problem.

• In Unit 3 Lesson 17, students work with application of 6.G.A and 6.RP.A to determine the area of bedroom walls, estimate the amount of paint needed, and determine the cost of materials. In this task, students make assumptions and decisions about what and how to model the situation as well as reflect upon and justify the decisions they make.

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

• Unit 3 Lesson 1 provides students with facts about the Burj Khalifa (world’s tallest building) and then provides this information: “A window-washing crew can finish 15 windows in 18 minutes.” Students determine how long it would take the crew to wash all the windows of the Burj Khalifa. This task is designed to develop students' understanding of the utility of the unit rate in solving problems within this context. This lesson also extends students’ work in the last lesson of the previous unit (Unit 2 Lesson 17) with using rate and ratio reasoning to solve Fermi problems.

• On average there are six problems included in the practice problems. Procedural practice, visual representations, contexts, and/or standard methods of solving said problems are present. For example, in the Unit 6 Lesson 1 Practice Problems, students solve several problems involving tape diagrams to develop conceptual understanding and develop procedural skill and fluency with one-step equations. The last few problems spiral to Unit 3 material and require students to apply previous knowledge.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All eight MPs are clearly identified throughout the materials. The Math Practices are initially identified in the Teacher Guide under the narrative descriptions of each unit within the Course Information. For example:

• The Unit 1 Area and Surface Area narrative states, “[Students] learn strategies for finding areas of parallelograms and triangles and use regularity in repeated reasoning (MP8) to develop formulas for these areas, using geometric properties to justify the correctness of these formulas.”

• In Unit 4 Dividing Fractions, an excerpt from the Overview states, “The second section of the unit focuses on equal groups and comparison situations. It begins with partitive and quotitive situations that involve whole numbers, represented by tape diagrams and equations. Students interpret the numbers in the two situations (MP2) and consider analogous situations that involve one or more fractions, again accompanied by tape diagrams and equations.”

The MPs are identified within a lesson in the teacher narratives in the lesson overview in general and/or before each of the activities. Lesson narratives often highlight when a Math Practice is particularly important for a concept or when a task may exemplify the identified Practice. For example:

• MP6: The Unit 2 Lesson 1 narrative introduces ratios and ratio language, “Expressing associations of quantities in a context - as students will be doing in this lesson - requires students to use ratio language with care (MP6).”

• MP7: The Unit 6 Lesson 13 narrative accompanying Activity 1 states, “The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number.”

• MP6: The narrative associated with the Unit 2 Lesson 8 Warm-Up states, “Students choose whether to draw double number lines or other representations to support their reasoning. They continue to use precision in stating the units that go with the numbers in a ratio in both verbal statements and diagrams (MP6)."

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

• MP6: In the Unit 2 Lesson 2 introduction, an explanation is provided for ratio language and its connection to MP6, “Students used physical objects to learn about ratios in the previous lesson. Here they use diagrams to represent situations involving ratios and continue to develop ratio language. The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems. Both the visual and verbal descriptions of ratios demand careful interpretation and use of language (MP6).”

• MP2: In the first Activity in Unit 7 Lesson 1, understanding of positive and negative integers is enriched as “students reason abstractly and quantitatively when they represent the change in temperature on a number line (MP2).”

The MPs are not identified in the student materials; however, they are highlighted in the Teacher Guide in the narrative provided with each Activity. For example, Unit 7 Lesson 1 Activity 1 poses the following question in relation to MP2 (see previous bullet for teacher facing information): “Do numbers below 0 make sense outside of the context of temperature? If you think so, give some examples to show how they make sense. If you don’t think so, give some examples to show otherwise.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

Materials attend to the full meaning of each of the 8 MPs. The MPs are discussed in both the unit and lesson narratives, as appropriate, when they relate to the overall work. They are also explained within individual activities, when necessary. Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:

MP1 Make sense of problems and persevere in solving them.

• In Unit 2 Lesson 1, the first part of MP1 is captured as students make sense of ratios. Students sort shapes and various objects into categories with similar characteristics and then use their like traits to establish ratio relationships.

• In Unit 9 Lesson 2, student are given the following problem, “There are 7.4 billion people in the world. If the whole world were represented by a 30-person class: 14 people would eat rice as their main food, 12 people would be under the age of 20, 5 people would be from Africa. 1. How many people in the class would not eat rice as their main food? 2. What percentage of the people in the class would be under the age of 20? 3. Based on the number of people in the class representing people from Africa, how many people live in Africa?” In solving this problem, students have to look for entry points to the solution; analyze given information, constraints, relationships, and goals; and finally, make conjectures about the form and meaning of the solution and plan a solution pathway.

MP2 Reason abstractly and quantitatively.

• In Unit 5 Lesson 1, students are given time to think about solving problems in the context of money. For example, “Clare went to a concession stand that sells pretzels for $3.25, drinks for $1.85, and bags of popcorn for $0.99 each. She bought at least one of each item and spent no more than $10. Could Clare have purchased 2 pretzels, 2 drinks, and 2 bags of popcorn? Explain your reasoning.”

MP4 Model with mathematics.

• Throughout the Activities in Unit 7 Lesson 1 students model with mathematics using number lines or a digital applet to represent thermometers and scenarios involving weather. The second Activity introduction states, “The purpose of this task is to present a second, natural context for negative numbers and to start comparing positive and negative numbers in preparation for ordering them.” Students again model a context using vertical number lines, but this time it is with elevation.

• In Unit 9 Lesson 1, students answer, “How long would it take an ant to run from New York City to Los Angeles?” The Fermi problem requires students to make a rough estimate for quantities that are difficult or impossible to measure directly. Often, they use rates and require several calculations with fractions and decimals, making them well-aligned to Grade 6 work. Fermi problems are examples of mathematical modeling because one must make simplifying assumptions, estimates, research, and decisions about which quantities are important and what mathematics to use.

MP5 Use appropriate tools strategically.

• Each lesson in Unit 1 Area and Surface Area lists a geometry toolkit containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles as Required Materials. The Unit 1 narrative explains, “Providing students with these toolkits gives opportunities for students to develop abilities to select appropriate tools and use them strategically to solve problems.” In addition, many lessons of Unit 1 include activities in which students use digital applets which allow for making simulations and exploring compositions and decompositions of figures. The unit narrative also explains, “Apps and simulations should be considered additions to their toolkits, not replacements for physical tools.”

• In the first activity of Unit 2 Lesson 2, students use beakers with blue and yellow water, one graduated cylinder, a permanent marker, a craft stick, and three opaque white cups to explore ratios. In Lesson 4, students use either the digital version or complete the activity with beakers and colored liquids. Students use appropriate tools to gather measurements and make sense of equivalent ratios through physical experiences. The narrative states, “Students mix different numbers of batches of a recipe for green water by combining blue and yellow water, students mix different numbers of batches of a color recipe to obtain a certain shade of green.”

MP7 Look for and make use of structure.

• In Unit 1 Lesson 7 Activity 2, students are given several quadrilaterals and directed to draw a line that would decompose them into two identical triangles. In order to make generalizations about quadrilaterals that can be decomposed into identical triangles, students first need to analyze the features of the given shapes and look for structure.

• In Unit 5 Lesson 4, students notice and use structure in the second Activity. The narrative states, “In this lesson, students practiced adding and subtracting numbers with many decimal places, both in and outside of the context of situations. They noticed the benefits of vertical calculations and used its structure to solve problems.”

MP8 Look for and express regularity in repeated reasoning.

• In the second Activity of Unit 1 Lesson 18, students are told that a cube has an edge length of x. These prompts follow: “1) Draw a net for the cube. 2) Write an expression for the area of each face. Label each face with its area. 3) Write an expression for the surface area. 4) Write an expression for the volume.” In doing this, students express regularity in repeated reasoning to write the formula for the surface area of a cube.

• In the optional activity in Lesson 8 of Unit 5, students have opportunities to solve problems with decimals and look at patterns in solving problems with decimals. First, students “write the following expressions as decimals (1−0.1, 1−0.1+10−0.01, 1−0.1+10−0.01+100−0.001). Describe the decimal that results as this process continues. What would happen to the decimal if all of the positive and negative signs became multiplication symbols? Explain your reasoning.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations that the instructional materials prompt students to construct viable arguments and/or analyze the arguments of others concerning key grade-level mathematics. Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others.

Students are consistently asked to explain their reasoning and compare their strategies for solving in small group and whole class settings. For example:

• In the Unit 1 Lesson 3 Warm-Up, students compare the areas of two shaded figures and explain their reasoning.

• In Unit 3 Lesson 6 Activity 2, students explore two unit rates related to a given ratio. They must decide which unit rate is correct and extend the unit rate into a related problem. In this scenario, both unit rates are correct, and the students could use either unit rate to solve the related problems.

• In the Unit 7 Lesson 6 Lesson Synthesis of the second Activity, students are asked, “What do you notice about the order of numbers after taking absolute value? Explain why this happens.” Questions such as these are present throughout the lessons, providing students the opportunity to construct viable arguments in both verbal and written form.

• In the Unit 6 Lesson 16 Warm-Up, students find the unit price to determine which price option is a better deal. Students engage in constructing arguments and critiquing the reasoning of their classmates. Students are asked: “Which one would you choose? Be prepared to explain your reasoning. A 5-pound jug of honey for $15.35 [or] three 1.5-pound jars of honey for $13.05?”

• The Unit 7 Lesson 1 Cool-Down includes the following prompts with which students must agree or disagree and explain their reasoning: “A temperature of 35 degrees Fahrenheit is as cold as a temperature of -35 degrees Fahrenheit. A city that has an elevation of 15 meters is closer to sea level than a city that has an elevation of -10 meters. A city that has an elevation of -17 meters is closer to sea level than a city that has an elevation of -40 meters.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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

• In Unit 1 Lesson 9, students study examples and non-examples of bases and heights in a triangle. Next, they select all the statements that are true about bases and heights in a triangle. The teacher is given the following direction: “As students discuss with their partners, listen for how they justify their decisions or how they know which statements are true.”

• The Unit 2 Lesson 4 Warm-Up provides guiding questions in the Activity Synthesis to engage students in MP3, such as: “Who can restate ___’s reasoning in a different way? Did anyone solve the problem the same way but would explain it differently? Did anyone solve the problem in a different way? Does anyone want to add on to _____’s strategy? Do you agree or disagree? Why?” This strategy is used repeatedly throughout the series.

• In Unit 4 Lesson 5, the first Activity provides guidance for the teacher as they observe student groups using pattern blocks to solve a task: “As students discuss in groups, listen for their explanations for the question ‘How many rhombuses are in a trapezoid?’ Select a couple of students to share later - one person to elaborate on Diego's argument, and another to support Jada's argument.”

• The Unit 7 Lesson 1 Warm-Up states: “The purpose of this task is to introduce students to temperatures measured in degrees Celsius.” This prompt assists teachers in engaging students in constructing viable arguments, precisely the types of questions teachers can ask to aid in the discussion and includes possible student responses.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations that the materials attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In the teacher materials, the Grade 6 Glossary is located in the Teacher Guide within the Course Information section. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples. In the student materials, the Grade 6 Glossary is accessible by a tab within each Unit or in the bottom margin of each lesson page. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples.

• Both the unit and the lesson narratives contain specific guidance for the teacher as to best methods to support students to communicate mathematically. Within the lesson narratives, new terms are in bold print and explained as related to the context of the material.

• Unit 2 Lesson 1 introduces ratios and ratio language to the students. Within the Warm-Up and the first Activity, students categorize items and verbally compare the sorted groups. The definition of ratio is developed and applied to the sorted groups using correct language. For example, “The ratio of purple to orange dinosaurs is 4 to 2.” or “There are 4 purple dinosaurs for every 2 orange dinosaurs.” Within the second Activity, students must write ratio sentences comparing two categories. The Lesson Synthesis provides further practice and discussion questions for the teacher that will solidify the concept of a ratio. “Consider posing some more general questions, such as: 'What things must you pay attention to when writing a ratio? What are some words and phrases that are used to write a ratio?'”

• In Unit 7, students interpret signed numbers in contexts (e.g., temperature above or below zero, elevation above or below sea level). Students use the context to build proper mathematical vocabulary. In Lesson 1, students explore the idea of a temperature that is less than zero. This activity is used to introduce the term negative as a way to represent a quantity less than zero.

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

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

Lessons include Warm-Up, Activities, and an Activity Synthesis. Practice Problems are in a separate section of the instructional materials, distinguishing between problems students complete and exercises in the lessons. Warm-Ups serve to either connect prior learning or prime students for learning new material in the lesson. Students learn and practice new mathematics in lesson Activities. In the Activity Synthesis, students have opportunities to build on their understanding of the new concept. Each activity lesson ends with a Cool-Down in which students have opportunities to apply what they have learned from the activities in the lesson and either provide preliminary practice or an introduction to skills they may need in the next lesson.

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

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation for not being haphazard; exercises are given in intentional sequences.

Overall, clusters of lessons within units and activities within lessons are intentionally sequenced so students develop understanding leading to content mastery. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skill and fluency, and engage with multiple activities that are sequenced from concrete to abstract or increase in complexity. Lessons close with a Cool-Down which is typically 1-2 activities aligned to the daily lesson objective. Unit sequences consistently follow the progressions outlined in the CCSSM Standards to support students' conceptual and skill development.

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

The instructional materials prompt students to produce products in a plethora of ways. Students not only produce answers and solutions within Activities and Practice Problems, but also in class, group and partner discussions. Students have opportunities to construct viable arguments and critique the reasoning of their peers in the instructional materials. Students use a digital platform (applets) and paper-pencil to conduct and present their work. Materials consistently call for student solutions that represent the language and intent of the standards. Students use representations such as tables, number lines, double number lines, tape diagrams, and graphs (MP4), as well as strategically choose tools to complete their work (MP5). Lesson activities and tasks are varied within and across lessons.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The series includes a variety of virtual manipulatives and integrates hands-on activities that allow the use of physical manipulatives. For example:

• Manipulatives and other mathematical representations are consistently aligned to the expectations and concepts in the standards. The majority of manipulatives used are commonly accessible measurement and geometry tools. In Unit 4 Lesson 2, students use graduated cylinders and beakers when measuring to create mixtures to explore ratio. A digital version of the task is also provided as an option.
• The materials also provide digital applets for manipulating geometric shapes, such as Tangram applets, tailored to the lesson content and tasks. When physical, pictorial, or virtual manipulatives are used, they are aligned to the mathematical concepts they represent. Unit 5 includes base 10 blocks (or a virtual applet) to support work with operations with decimals, ensure the use of mathematical vocabulary, and to bridge the concept of the place value to the procedural skill.
• Examples of manipulatives for Grade 6 include:
  • Tangram kits (or digital Tangram applet)
  • Geometry toolkits containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.
  • GeoGebra applets are used for both investigating the characteristics of shapes and area/perimeter as well as exploring coordinate and isometric grids.

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

Each section of each lesson contains an opening and closing narrative for the teacher. Included in these narratives are the objectives of the lesson, as well as suggested questions for discussion and guiding questions designed to increase classroom discourse and ensure understanding of the concepts. For example, in Unit 6 Lesson 6, the following questions are included: “What calculation did you do to arrive at that answer? Where are those measurements in the image?” The narratives, as well as the questions for discussion, support the teachers in planning and implementing lessons effectively.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectations for containing a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials also include teacher guidance on the use of embedded technology to support and enhance student learning.

• Each lesson opens with a table including Learning Goals written in both teacher and student language, learning targets written in student language, a list of Print-Formatted Word/PDF documents that can be downloaded, CCSSM Standards that are either being “built upon” or “addressed” for the lesson, and any instructional routines to be implemented. Within the technology, there are expandable links to standards and instructional routines.

• Lessons include detailed guidance for teachers for the Warm-Up, Activities and the Lesson Synthesis.

• Each lesson activity contains an overview and Launch narrative, guidance for teachers and student facing materials, anticipated misconceptions, “Are you Ready for More?” and an activity synthesis. Included within these narratives are guiding questions and additional supports for students.

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

• The materials are available in both print and digital forms. The digital format has an embed GeoGebra applet. Guidance is provided to both the teacher and the student on how to use the Geometry Toolkit and applet. For example, in Unit 1 Lesson 7 Activity 2, students use tracing paper to decompose parallelograms into triangles. The activity includes directions on how to decompose triangles to find the area of a figure: “Two polygons are identical if they match up exactly when placed one on top of the other. 1) Draw one segment to decompose each of the following polygons into two identical triangles, if possible. If you choose to, you can also draw the triangles. 2) Study the quadrilaterals that were, in fact, decomposable into two identical triangles. What do you notice about them? Write a couple of observations about what these quadrilaterals have in common.”

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

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

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

The Course Information and Scope and Sequence, Unit 4: Dividing Fractions states, “Multiplicative situations include three types: equal groups; comparisons of two quantities; dimensions of arrays or rectangles. In the equal groups and comparison situations, there are two subtypes, sometimes called the partitive (or measurement) and the quotitive interpretations of division. Students are not expected to identify the three types of situations or use the terms “partitive” or “quotitive.” However, they should recognize the associated interpretations of division in specific contexts (MP7)."

For example, in an equal groups situation when the group size is unknown, division can be used to answer the question, “How many in each group?” If the number of groups is unknown, division answers the question, “How many groups?” For example, if 12 pounds of almonds are equally shared among several bags: There are two bags. How many pounds in each bag? (partitive) There are six pounds in each bag. How many bags? (quotitive)”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Teacher Guide fully explains how mathematical concepts are built from previous grade-level and lesson material. For example, the Unit 4 Overview states the following regarding dividing fractions: “Work with fractions in Grade 6 draws on earlier work in operations and algebraic thinking, particularly the knowledge of multiplicative situations developed in Grades 3 to 5, and making use of the relationship between multiplication and division.”

There are limited explanations given for how the grade-level concepts fit into future grade-level work. For example, the Unit 2 Overview concludes, “The terms proportional and proportional relationship are not used anywhere in the Grade 6 materials. A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in Grade 7. In high school—after their study of ratios, rates, and proportional relationships—students discard the term ‘unit rate’, referring to a to b, a:b, and a/b as ‘ratios’.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels.

• Prior grade-level standards are indicated in the instructional materials. The lesson Warm-Up is designed to engage students thinking about the upcoming lesson and/or to revisit previous grades' concepts or skills.

• Prior knowledge is gathered about students through the pre-unit assessments. In these assessments, prerequisite skills necessary for understanding the topics in the unit are assessed. Commentary for each question as to why the question is relevant to the topics in the unit, and exactly which standards are assessed is provided for the teacher. For example, the Unit 3 Pre-Unit Assessment Problem 3 states: “Students will need to perform division when calculating unit rates and percentages. Keep an eye out for students who reverse the order of division or who misplace the decimal point. 5.NBT.B.7”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions.

• Lesson Activities include “Anticipated Misconceptions” that identify where students may make a mistake or struggle. There is a rationale that explains why the mistake could have been made, suggestions for teachers to make instructional adjustments for students, as well as steps teachers can take to help clear up the misconceptions. For example, in Unit 8 Lesson 5, the Anticipated Misconception gives the following guidance: “Students may neglect to change the rate given (from minutes per week to hours per week or to minutes per day) and may draw incorrect conclusions as a result. Ask them to think about the unit they are using in their responses.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

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

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

Assessments are located on a separate tab at the top of the grade-level page and can be accessed at any time. For each unit there is a Pre-Unit Assessment and an End-Unit Assessment. Assessments begin with guidance for teachers on each problem followed by the student facing problem, solution(s), and the standard targeted. Units 1, 4, 5, and 8 also include a Mid-Unit Assessment.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• Each lesson is designed with a Warm-Up that reviews prior knowledge and/or prepares all students for the following activities. The Cool-Down following lesson activities solidifies the concepts of the lesson.

• Within a lesson, narratives provide explicit instructional supports for the teacher, including the Activity Launch, Anticipated Misconceptions, and Lesson Synthesis sections. This information assists a teacher in making the content accessible to all learners.

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

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

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

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

• Mathematical Language Routines to support a range of learners to be successful are provided for the teacher throughout lessons to maximize output and cultivate conversation. For example:
  • MLR1: Stronger and Clearer Each Time, in which “students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response.”
  • MLR4, Information Gap, which “allows teachers to facilitate meaningful interactions by giving partners or team members different pieces of necessary information that must be used together to solve a problem or play a game...[S]tudents need to orally (and/or visually) share their ideas and information in order to bridge the gap.”
  • MLR6, Three Reads, in order to ensure that students know what they are being asked to do, and to create an opportunity for students to reflect on the ways mathematical questions are presented,… and [support] negotiating information in a text with a partner in mathematical conversation.”

• Sidebar text features appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These text-boxes call out specific needs addressed in a recommended strategy that are relevant to the given task and include supports for Conceptual Processing, Expressive & Receptive Language, Visual-Spatial Processing, Executive Functioning, Memory, Social-Emotional Functioning, and Fine-motor Skills. For each support there are multiple strategies teachers can employ; for example, Conceptual Processing includes strategies to Eliminate Barriers, Processing Time, Peer Tutors, Assistive Technology, Visual Aids, Graphic Organizers, and Brain Breaks.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

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

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

• In the Unit 1 Lesson 1 Warm-Up, students are asked to identify which tile pattern from a set of four does not belong. All tile patterns have a reason they may not belong (colors, shapes used, etc.), allowing for all students to participate in the task while also focusing instruction on the mathematical concept of area.

• In Unit 3 Lesson 9 (“Card Sort: Is it a Deal?”), students are given cards with an original price per unit listed and a new price. Students must determine if they should take the deal or not. The teacher is encouraged to look for multiple solution paths, and examples of different solution paths or student explanations are provided to help the teacher anticipate student solution strategies.

• In Unit 4 Lesson 10 Activity 10.2, students must complete tables to determine the tax rate and formula for the tax rate for two different cities. Multiple strategies are encouraged, and the teacher is prompted to “monitor for different strategies, especially students who note that they can always multiply by the same factor and students who set up and use an equation.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation that the materials include support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

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

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

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

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

• Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.
• Conceptual Processing: Eliminate Barriers. Assist students in seeing the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.
• Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.
• Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.
• Memory: Processing Time. Provide students with a number line that includes rational numbers.
• Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth.

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

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

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

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics.

• The lessons contain a variety of tasks that interest students of various demographic and personal characteristics. All names and wording are chosen with diversity in mind, and the materials do not contain gender biases.

• The Grade 6 materials include a set number of names used throughout the problems and examples (e.g., Elena, Tyler, Lin, Noah, Diego, Kiran, Mia, Priya, Han, Jada, Andre, Clare). These names are presented repeatedly and in a way that does not appear to stereotype characters by gender, race, or ethnicity.

• Characters are often presented in pairs with different solution strategies. There does not appear to be a pattern in one character using more/less sophisticated strategies.

• When multiple characters are involved in a scenario they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias. For example, in Unit 3 Lesson 6 Activity 6.2, Priya, Han, Lin, and Diego are on a camping trip together. Priya and Han cook together one week. Lin and Diego cook together the next week. There is no differentiation of what roles the characters take when cooking together that might suggest a gender, racial, or ethnic bias.