6th Grade - Gateway 3

The materials reviewed for Open Up Resources Grade 6 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 6 and across all grades.

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

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

• Unit 2: Introducing Ratios, Section D: Solving Ratio and Rate Problems, Lesson 11: Representing Ratios with Tables, Activity 1: A Huge Amount of Sparkling Orange Juice, Instructional Routines, “Here, students are asked to find missing values for significantly scaled-up ratios. The activity serves several purposes: To uncover a limitation of a double number line (e.g., that it is not always practical to extend it to find significantly scaled-up equivalent ratios), To reinforce the multiplicative reasoning needed to find equivalent ratios (especially in cases when drawing diagrams or skip counting is inefficient), and To introduce a table as a way to represent equivalent ratios. To find equivalent ratios involving large values, some students may simply try to squeeze numbers on the extreme right side of the paper, ignoring the previously equal intervals. Others may use multiplication (or division) and write expressions or equations to capture the given scenarios. Notice students’ reasoning processes, especially any struggles with the double number line (e.g., the lines not being long enough, requiring much marking and writing, the numbers being too large, etc.), as these can motivate a need for a more efficient strategy.”

• Unit 3: Unit Rates and Percentages, Unit Overview, “In this unit, tables and double number line diagrams are intended to help students connect percentages with equivalent ratios, and reinforce an understanding of percentages as rates per 100. Students should internalize the meaning of important benchmark percentages, for example, they should connect ‘75% of a number’ with ‘$$\frac{3}{4}$$ times a number’ and ‘0.75 times a number.’ Note that 75% (“seventy-five per hundred”) does not represent a fraction or decimal (which are numbers), but that ‘75% of a number’ is calculated as a fraction of or a decimal times the number.”

• Unit 7: Rational Numbers, Section A: Negative Numbers and Absolute Value, Lesson 7: Comparing Numbers and Distance from Zero, Lesson Narrative, “It is a common mistake for students to mix up ‘greater’ or ‘less’ with absolute value. A confused student might say that -18 is greater than 4 because they see 18 as being the ‘bigger’ number. What this student means to express is \lvert -18 \rvert > 4. The absolute value of -18 is greater than 4 because -18 is more than 4 units away from 0.” 

The materials reviewed for Open Up Resources Grade 6 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 4: Dividing Fractions, Section A: Making Sense of Division, Lesson 3: Interpreting Division Situations, Activity 1: Homemade Jams, Instructional Routines, “This activity allows students to draw diagrams and write equations to represent simple division situations. Some students may draw concrete diagrams; others may draw abstract ones. Any diagrammatic representation is fine as long as it enables students to make sense of the relationship between the number of groups, the size of a group, and a total amount. The last question is likely more challenging to represent with a diagram. Because the question asks for the number of jars, and because the amount per jar is a fraction, students will not initially know how many jars to draw (unless they know what 6\frac{3}{4}\div\frac{3}{4} is). Suggest that they start with an estimate, and as they reason about the problem, add jars to (or remove jars from) their diagram as needed.”

• Unit 6: Expressions and Equations, Section C: Expressions with Exponents, Lesson 14: Evaluating Expressions with Exponents, Activity 1: Calculating Surface Area, Activity Synthesis, “In finding the surface area, there is a clear reason to find 10^2 and then multiply by 6. Tell students that sometimes it is not so clear in which order to evaluate operations. There is an order that we all generally agree on, and when we want something done in a different order, brackets are used to communicate what to do first. When an exponent occurs in the same expression as multiplication or division, we evaluate the exponent first, unless brackets say otherwise. Examples: {(3\cdot4)}^2 = {12}^2 = 144, since the brackets tell us to multiply (3 $$\cdot$$ 4) first. But 3 \cdot 4^2 = 3 \cdot 16 = 48, because since there are no brackets, we evaluate the exponent before multiplying. If students bring up PEMDAS or another mnemonic for remembering the order of operations, point out that PEMDAS can be misleading in indicating multiplication before division, and addition before subtraction. Discuss the convention that brackets or parentheses indicate that something should be evaluated first, followed by exponents, multiplication or division (evaluated left to right), and last, addition or subtraction (evaluated left to right).” 

• Unit 7: Rational Numbers, Unit Overview, “Previously, when students worked only with non-negative numbers, magnitude and order were indistinguishable: if one number was greater than another, then on the number line it was always to the right of the other number and always farther from zero. In comparing two signed numbers, students distinguish between magnitude (the absolute value of a number) and order (relative position on the number line), distinguishing between ‘greater than’ and ‘greater absolute value,’ and ‘less than’ and ‘smaller absolute value’. Students examine opposites of numbers, noticing that the opposite of a negative number is positive.”

Materials contain adult-level explanations and examples of concepts beyond grade 6 so that teachers can improve their knowledge of the subject. Examples include:

• Unit 1: Area and Surface Area, Unit 1 Overview, “In grade 8, students will understand “identical copy of” as “congruent to” and understand congruence in terms of rigid motions, that is, motions such as reflection, rotation, and translation. In grade 6, students do not have any way to check for congruence except by inspection, but it is not practical to cut out and stack every pair of figures one sees. Tracing paper is an excellent tool for verifying that figures ‘match up exactly” and students should have access to this and other tools at all times in this unit.”

• Unit 2: Introducing Ratios, Unit 2 Overview, “The terms proportion 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 \frac{a}{b} as “ratios.”

• Unit 5: Arithmetic in Base Ten, Section D: Dividing Decimals, Lesson 11: Dividing Numbers that Result in Decimals, Activity 2 Synthesis, "Problems like 1 ÷ 25 are challenging because the first step is 0: there are zero groups of 25 in 1. This means that we need to introduce a decimal and put a 0 to the right of the decimal. But one 0 is not enough. It is not until we add the second 0 to the right of the decimal that we can find 4 groups of 25 in 100. Because we moved two places to the right of the decimal, these 4 groups are really 0.04, which is the quotient of 1 by 25." "Problems like 1 ÷ 3 are not fully treated until grade 7. At this point, we can observe that the long division process will go on and on because there is always a remainder of 1."

The materials reviewed for Open Up Resources Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

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

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

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

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

• Unit 4: Dividing Fractions, Section B: Meanings of Fraction Division, Lesson 7: What Fraction of a Group?, “Addressing 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.”

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

• Unit 5: Arithmetic in Base Ten, Overview, “In this unit, students learn an efficient algorithm for division and extend their use of other base-ten algorithms to decimals of arbitrary length. Because these algorithms rely on the structure of the base-ten system, students build on the understanding of place value and the properties of operations developed during earlier grades (MP7). The unit begins with a lesson that revisits sums and differences of decimals to hundredths, and products of a decimal and whole number. The tasks are set in the context of shopping and budgeting, allowing students to be reminded of appropriate magnitudes for results of calculations with decimals. The next section focuses on extending algorithms for addition, subtraction, and multiplication, which students used with whole numbers in earlier grades, to decimals of arbitrary length.”

• Unit 7: Rational Numbers, Section C: The Coordinate Plane, Section Overview, “The third section of the unit focuses on the coordinate plane. In grade 5, students learned to plot points in the coordinate plane, but they worked only with non-negative numbers, thus plotted points only in the first quadrant. In a previous unit, students again worked in the first quadrant of the coordinate plane, plotting points to represent ratio and other relationships between two quantities with positive values. In this unit, students work in all four quadrants of the coordinate plane, plotting pairs of signed number coordinates in the plane. They understand that for a given data set, there are more and less strategic choices for the scale and extent of a set of axes. They understand the correspondence between the signs of a pair of coordinates and the quadrant of the corresponding point. They interpret the meanings of plotted points in given contexts (MP2), and use coordinates to calculate horizontal and vertical distances between two points.”

• Unit 8: Data Sets and Distributions, Section D: Median and IQR, Lesson 14: Comparing Mean and Median, Lesson Narrative, “In this lesson, students investigate whether the mean or the median is a more appropriate measure of the center of a distribution in a given context. They learn that when the distribution is symmetrical, the mean and median have similar values. When a distribution is not symmetrical, however, the mean is often greatly influenced by values that are far from the majority of the data points (even if there is only one unusual value). In this case, the median may be a better choice. At this point, students may not yet fully understand that the choice of measures of center is not entirely black and white, or that the choice should always be interpreted in the context of the problem (MP2) and should hinge on what insights we seek or questions we would like to answer. This is acceptable at this stage. In upcoming lessons, they will have more opportunities to include these considerations into their decisions about measures of center.”

The materials reviewed for Open Up Resources Grade 6 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 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

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

• Resources, Course Guide, Required Materials, “1/2-inch cubes, 1/4-inch graph paper. base-ten blocks, beakers, bingo chips, blank paper, colored pencils, Cuisenaire rods, decks of playing cards, demonstration nets with and without flaps, dot stickers, drink mix, empty containers, food coloring, four-function calculators, gallon-sized jug, geometry toolkits, glue or gluesticks, graduated cylinders, graph paper, graphing technology, grocery store circulars, household items, inch cubes, index cards, internet-enabled device, liter-sized bottle, markers, masking tape, measuring tapes, metal paper fasteners, meter sticks, nets of polyhedra, origami paper, paper cups, pattern blocks, pre-assembled or commercially produced tangrams, pre-assembled polyhedra, quart-sized bottle, rulers, rulers marked with centimeters, rulers marked with inches, salt, scale, scissors, snap cubes, sticky notes, stopwatches, straightedges, string, students’ collections of objects, tape, teacher’s collection of objects, teaspoon, tools for creating a visual display, tracing paper, tray, water, yardsticks.”

• Unit 2: Introducing Ratios, Section B: Equivalent Ratios, Lesson 5: Defining Equivalent Ratios, Required Materials, “tools for creating a visual display.”

• Unit 7: Rational Numbers, Section A: Negative Numbers and Absolute Value, Lesson 1: Positive and Negative Numbers, Required Materials, “rulers.”

The materials reviewed for Open Up Resources Grade 6 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: Arithmetic in Base Ten, Section D: Dividing Decimals, Lesson 10: Using Long Division, Cool Down: Dividing by 15, “6.NS.B.2, Use long division to find the value of 1,875 \div 15.”

• Unit 6: Expressions and Equations, Unit Assessments, Mid-Unit Assessment, Version A, Problem 4, “6.EE.B.6, 6.EE.B.7, “$$\frac{2}{9}$$ of the students in a school are in sixth grade. a. How many sixth graders are there if the school has 90 students? b. How many sixth graders are there if the school has 27 students? c. If the school has x students, write an expression for the number of sixth graders in terms of x. d. How many students are in the school if 42 of them are sixth graders?”

• Unit 7: Rational Numbers, Unit Assessments, End-of-Unit Assessment, Version B, Problem 2, “6.NS.B.4, Select ALL the numbers that are a common multiple of 8 and 12. A. 96 B. 80 C. 48 D. 32 E. 24 F. 20 G. 4.”

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, “MP2: I Can Reason Abstractly and Quantitatively: I can think about and show numbers in many ways. I can identify the things that can be counted in a problem. I can think about what the numbers in a problem mean and how to use them to solve the problem. I can make connections between real-world situations and objects, diagrams, numbers, expressions, or equations.”

The materials reviewed for Open Up Resources Grade 6 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 6: Expressions and Equations, Mid-Unit Assessment, Version A, Problem 6, students complete a table to represent the relationship between the number of raffle tickets sold and the amount of money earned then use this information to answer questions. “Diego is selling raffle tickets for $1.75 per ticket. a. Complete the table to show how much money he would earn if he sold each number of tickets. b. How many tickets would Diego need to sell to earn $140? Explain your reasoning.” Solution, “Minimal Tier 1 response: Work is complete and correct, with complete table and correct answer for part B. Sample: See table above1.75r = 140, r = 140 \div 1.75, r = 80. Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Acceptable errors: a reasonable response to part b is based on an incorrect expression in the last cell of the table. Sample errors: a substituted value for is recorded in the last column of the table, but keeps the multiplicative relationship of 1.75. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: The table reflects a lack of understanding of the multiplicative relationship, which affects the equation in part b. Work involves a misinterpretation of the situation that affects all or most problem parts, but work does show understanding of writing equations to represent situations and interpreting solutions to equations.” 

• Unit 8: Data Sets and Distributions, End-of-Unit Assessment, Version A, Problem 6, student find the median and interquartile range of a data set. “Ten students each attempted 10 free throws. This list shows how many free throws each student made. 8, 5, 6, 6, 4, 9, 7, 6, 5, 9 a. What is the median number of free throws made? b. What is the IQR (interquartile range)?” Solution, “a. 6 free throws. (The ordered list is 4, 5, 5, 6, 6, 6, 7, 8, 9, 9. The two middle terms in the ordered list are both b.) 3 free throws. (The first half of the data is 4, 5, 5, 6, 6; its median is 5. The second half of the data is 6, 7, 8, 9, 9; its median is 8. The IQR is 3, since 8 - 5 = 3.)”

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 4: Dividing Fractions, End-of-Unit Assessment, Version B, Problem 7, students find volume of rectangular prisms. “Elena has two aquariums, each shaped like a rectangular prism. For each question, explain or show your reasoning. a. One aquarium has a length of \frac{7}{2} feet, a width of \frac{4}{3} feet, and a height of \frac{3}{2} feet. What is the volume of the aquarium? b. Elena paints the back of the second aquarium. It has a height of 1\frac{3}{4} feet. The painted area is 5\frac{5}{6} square feet. What is its length?” Guidance for teachers, “While most students should understand the context of the problem, some may still have difficulty understanding without a diagram. The second problem is about area, even though the aquarium is described as a rectangular prism. If students struggle to find the volume of a rectangular prism, provide additional instruction either in a small group or individually using OUR Math Grade 6 Unit 4 Lesson 15 Activity 2 and/or Practice Problems 1, 2, and 4.”

The materials reviewed for Open Up Resources Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level/ course-level standards and practices across the series.

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

• Unit 1: Area and Surface Area, Mid-Unit Assessment, Version B, Problem 5, students find area of parallelograms. “Draw two parallelograms, each with an area of 16 square units. The two parallelograms should not be identical copies of each other.” Students are given a grid to draw the parallelograms. (6.G.1)

• Unit 2, Introducing Ratios, End-of-Unit Assessment, Version B, Problem 2, students attend to precision as they, “Select all the ratios that are equivalent to 9:6. A. 6:9, B. 3:2, C. 13:10, D. 5:2, E. 18:12.” (MP6)

• Unit 4: Dividing Fractions, Section C: Algorithm for Fraction Division, Lesson 11: Using an Algorithm to Divide Fractions, Cool Down: Watering a Fraction of House Plants, Problem 2, students create and solve fraction division equations. “If \frac{4}{3} liters of water are enough to water \frac{2}{5} of the plants in the house, how much water is necessary to water all the plants in the house? Write an equation to represent the situation, and then find the answer.” (6.NS.1)

• Unit 6: Expressions and Equations, Section A: Equations in One Variable, Lesson 5: A New Way to Interpret a over b, Practice Problems, Problem 3, students use division using fractional notation to solve equations. “Solve each equation. A. 4a = 32 B. 4 = 32b C. 10c = 26 D. 26 = 100d.” (6.EE.7)

• Unit 6: Expressions and Equations, Mid-Unit Assessment, Version A, Problem 3, students model with mathematics as they choose expressions that represent an area model. “Select all the expressions that represent the total area of the rectangle. A. 4s B. \frac{1}{3}s + 12  C. \frac{1}{3}s + \frac{1}{3}s + 4 D. \frac{1}{3}s + 4 E. \frac{1}{3}(s + 12).” (MP4)

• Unit 8: Data Sets and Distributions, Mid-Unit Assessment, Version A, Problem 6, students construct and compare dot plots. “a. Draw two dot plots, each with 7 or fewer data points, so that: Both dot plots display data with approximately the same mean. The data displayed in Dot Plot A has a much larger MAD (mean absolute deviation) than the data displayed in Dot Plot B. b. How can you tell, visually, that one dot plot displays data with a larger MAD than another?” (6.SP.4 and 6.SP.5)

The materials reviewed for Open Up Resources Grade 6 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

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

• Unit 3: Unit Rates and Percentages, Section C: Rates, Lesson 9: Solving Rate Problems, Activity 1: Card Sort: Is It a Deal?, Supports for Students with Disabilities, “Representation: Comprehension, Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have identified which initial cards were good deals. Provides accessibility for: Conceptual Processing, Organization.”

• Unit 5: Arithmetic in Base Ten, Section B: Adding and Subtracting Decimals, Lesson 2: Using Diagrams to Represent Addition and Subtraction, Activity 1: Squares and Rectangles, Supports for Students with Disabilities, “Representation: Language and Symbols, Activate or supply background knowledge. Some students may benefit from continued access to physical base-ten blocks (if available), a paper version of the base-ten figures (from the Blackline Master), or the digital applet. Encourage students to begin with physical representations before drawing a diagram. Provides accessibility for: Conceptual Processing.”

• Unit 7: Rational Numbers, Section B: Inequalities, Lesson 9: Solutions of Inequalities, Activity 1: Amusement Park Rides, Supports for Students with Disabilities, “Representation: Language and Symbols, Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: inequality, solution to an inequality. Provides accessibility for: Conceptual Processing, Language, Memory.”

The materials reviewed for Open Up Resources Grade 6 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: Area and Surface Area, Section B: Parallelograms, Lesson 5: Bases and Heights of Parallelograms, Activity 1: The Right Height, Are You Ready for More?, “In the applet, the parallelogram is made of solid line segments, and the height and supporting lines are made of dashed line segments. A base (b) and corresponding height (h) are labeled. Experiment with dragging all of the movable points around the screen. Can you change the parallelogram so that … 1. its height is in a different location? 2. it has horizontal sides? 3. it is tall and skinny? 4. it is also a rectangle? 5. it is not a rectangle, and has b = 5 and h = 3?”

• Unit 4: Dividing Fractions, Section B: Meanings of Fraction Division, Lesson 5: How Many Groups? (Part II), Activity 2: Drawing Diagrams to Show Equal-sized Groups, Are You Ready for More?, “How many heaping teaspoons are in a heaping tablespoon? How would the answer depend on the shape of the spoons?”

• Unit 5: Arithmetic in Base Ten, Section C: Multiplying Decimals, Lesson 5: Decimal Points in Products, Activity 2: Fractionally Speaking: Multiples of Powers of Ten, Are You Ready for More?, “Ancient Romans used the letter I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1,000. Write a problem involving merchants at an agora, an open-air market, that uses multiplication of numbers written with Roman numerals.”

The materials reviewed for Open Up Resources Grade 6 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

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

• Unit 3: Unit Rates and Percentages, Section B: Unit Conversion, Lesson 4: Converting Units, Warm Up: Number Talk: Fractions of a Number, Supports for English Language Learners, “Speaking: MLR8 Discussion Supports, Display sentence frames to support students when they explain their strategy. For example, ‘First, I ____ because …’ or ‘I noticed ____ so I …’ Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Design Principle: Optimize output.”

• Unit 6: Expressions and Equations, Section C: Expressions with Exponents, Lesson 13: Expressions with Exponents, Activity 1: Is the Equation True? Supports for English Language Learners, “Speaking, Writing: MLR8 Discussion Supports, Revoice language and push for clarity in reasoning when students discuss their strategies for determining whether the equations are true or false. Provide a sentence frame such as ‘The equation is true (or false) because ____.’ This will strengthen students’ mathematical language use and reasoning when discussing the meaning of exponents and operations that can make the equivalence of expressions true or false. Design Principle: Maximize linguistic & cognitive meta-awareness.”

• Unit 7: Rational Numbers, Section A: Negative Numbers and Absolute Value, Lesson 3: Comparing Positive and Negative Numbers, Activity 2: Rational Numbers on a Number Line, Supports for English Language Learners, “Speaking: MLR2 Collect and Display, During the class discussion, record and display words and phrases that students use to explain why they decided certain inequality statements are true or false. Highlight phrases that include a reference to ‘to the right of,’ ‘to the left of,’ and a distance from zero. If students use gestures to support their reasoning, do your best to connect words to the gestures. Design Principles: Optimize output, Support sense-making.”

The materials reviewed for Open Up Resources Grade 6 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 2: Introducing Ratios, Section B: Equivalent Ratios, Lesson 4: Color Mixtures, Required Preparation, “Mix blue water and yellow water; each group of 2 students will need 1 cup each. To make colored water, add 1 teaspoon of food coloring to 1 cup of water. It would be best to give each mixture to students in a beaker or another containing a pour spout. If possible, conduct this lesson in a room with a sink. Note that a digital version of this activity is available. It is embedded in the digital version of the student materials, but if classrooms using the print version of materials have access to enough student devices, it could be used in place of mixing actual colored water.” 

• Unit 3: Unit Rates and Percentages, Section B: Unit Conversion, Lesson 3: Measuring with Different-Sized Units, Warm Up: Width of Paper, Launch, “This activity is written to use 9-cm and 6-cm Cuisenaire rods, which are often blue and dark green, respectively. If your set of Cuisennaire rods has different colors, or if using small and large paper clips as substitutes, instruct students to modify the task accordingly. Hold up two sizes of rods or paper clips for students to see. Give them quiet think time but not the manipulatives. Later, allow students to use the rods or paper clips to measure the paper if they need or wish to do so.” 

• Unit 7: Rational Numbers, Section A: Negative Numbers and Absolute Value, Lesson 2: Points on a Number Line, Required Preparation, “Each student needs access to a ruler marked with centimeters and at least 1 sheet of tracing paper. If the tracing paper is less than 20 cm wide, then students will need to construct their number lines in the “Folded Number Lines” activity to go from -7 to 7, or otherwise construct their number line on the diagonal of the tracing paper.” Warm up, “The purpose of this activity is to prime students for locating negative fractions on a number line. Students discern the value of a number by analyzing its position relative to landmarks on the number line. In these cases, students estimate that the point is halfway between 2 and 3 and use their understanding about fractions and decimals to identify numbers equal or close to 2.5. In later activities, students do the same process when describing negative rational numbers, except with those numbers increasing in magnitude going from left to right. Notice students who argue that 2.49 is correct or incorrect.”