6th Grade - Gateway 3

The materials reviewed for Math Nation 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 to guide their mathematical development. 

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Course Overview: A Course Overview (Unit 0) is found at the beginning of each course.  Within each Course Overview there is a Course Narrative, which contains a summary of the mathematical content contained in each course, and a Course Guide. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Resources, Corrections, and Cool-Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the About These Materials section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool-Down. The How to Use These Materials section contains guidance about the three phases of classroom activities (Launch-Work-Synthesize) and utilizing instructional routines. In the Scope and Sequence section, teachers will find a Pacing Guide which contains time estimates for coverage of each of the units.

• Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessments (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Student Task Statements Cool-downs, Practice Problems, Blackline Masters, and My Reflections all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Full Lesson Plan downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.

• Full Lesson Plan: Within each Teacher Edition lesson component, teachers can find a Full Lesson Plan that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Course Guide, Lesson Plans, Lesson Narratives, Overviews, and Warm-up provide useful annotations. Examples include:

• Course Guide, Assessments Overview, “Pre-Unit Diagnostic Assessments At the start of each unit is a pre-unit diagnostic assessment. These assessments vary in length. Most of the problems in the pre-unit diagnostic assessment address prerequisite concepts and skills for the unit. Teachers can use these problems to identify students with particular below-grade needs, or topics to carefully address during the unit. 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…What if a large number of students can’t do the same pre-unit assessment problem? Look for opportunities within the upcoming unit where the target skill could be addressed in context…What if all students do really well on the pre-unit diagnostic assessment? Great! That means they are ready for the work ahead, and special attention likely doesn’t need to be paid to below-grade skills.”

• Unit 1, Lesson 9, Full Lesson Plan, 1.9.3 Exploration Activity, “Anticipated Misconceptions The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide the measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the warm-up activity.”

• Unit 5, Lesson 8, Full Lesson Plan, “Lesson Narrative In this culminating lesson on multiplication, students continue to use the structure of base-ten numbers to make sense of calculations (MP7) and consolidate their understanding of the themes from the previous lessons. They see that multiplication of decimals can be accomplished by: thinking of the decimals as products of whole numbers and fractions; writing the non-zero digits of the factors as whole numbers, multiplying them, and moving the decimal point in the product; representing the multiplication with an area diagram and finding partial products; and using a multiplication algorithm, the steps of which can be explained with the reasonings above.”

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

The Course Guide, About These Materials sections, states the following note about standards alignment, “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.’” All lessons in the materials have this correlation information. An example:

• Unit 7, Lesson 4, Full Lesson Plan, Lesson Standards Alignment, Building on 4.NBT.2, 5.NBT.3b; Addressing 6.NS.C, 6.NS.6, 6.NS.6a, 6.NS.7; Building Towards 6.NS.7.

Explanations of the role of the specific grade-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative, and Full Lesson Plan. Examples include:

• Course Guide, Scope and Sequence, Unit 1: Area and Surface Area, “Work with area in grade 6 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students' work with geometric measurement began with length and continued with area…In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths…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…”

• Course Guide, Scope and Sequence, Unit 2: Introducing Ratios, “Work with ratios in grade 6 draws on earlier work with numbers and operations. In elementary school, students worked to understand, represent, and solve arithmetic problems involving quantities with the same units. In grade 4, students began to use two-column tables, e.g., to record conversions between measurements in inches and yards. In grade 5, they began to plot points on the coordinate plane, building on their work with length and area…Use of tables to represent equivalent ratios is an important stepping stone toward use of tables to represent linear and other functional relationships in grade 8 and beyond. Because of this, students should learn to use tables to solve all kinds of ratio problems, but they should always have the option of using discrete diagrams and double number line diagrams to support their thinking…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 ab as ‘ratios.’”

The materials reviewed for Math Nation Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials sections. 

The About These Materials section states the following about the instructional approach of the program, “What is a Problem Based Curriculum? In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student's education. Families, students, teachers, and administrators may need support making this shift. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.”

Examples of materials including and referencing research-based strategies include:

• “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…”

• “Supporting English Language Learners This curriculum builds on foundational principles for supporting language development for all students. This section aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional supports and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012).”

• “Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”

Within the Course Guide, How to Use These Materials, a Reference section is included.

The materials reviewed for Math Nation Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.  Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Resources), Teacher Editions, for each lesson, under Lesson Preparation (Required Material(s)), and in Teacher Guides for specific lessons. Examples include:

• Unit 1, Lesson 3, Lesson Preparation, Required Materials: “Blackline master for Activity 3.1, Cool-down, copies of blackline master, geometry toolkits (tracing paper, graph paper, colored pencils, scissors, and an index card)”

• Unit 3, Lesson 3, Lesson Preparation, Required Materials: “Blackline master for Activity 3.2, Cool-down, base-ten blocks, blank paper, cuisenaire rods, gallon-sized jug, graduated cylinders, household items, inch cubes, internet-enabled device, liter-sized bottle, materials assembled from the blackline master, metal paper fasteners, meter sticks, pre-assembled polyhedra, quart-sized bottle, rulers, salt, scale, straightedges, teaspoon, tray”

• Unit 7, Lesson 7, Lesson Preparation, Required Materials: “Cool-down, sticky notes”

The materials reviewed for Math Nation 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.

Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses. 

Suggestions to teachers for following up with students are provided throughout the materials via the Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.

Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:

• Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses. 

  • Restricted Constructed Response

    • Tier 1 response: Work is complete and correct.

    • Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.

    • Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.

  • Extended Response

    • Tier 1 response: Work is complete and correct, with complete explanation or justification. 

    • Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. 

    • Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. 

    • Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”

• Unit 5, End-of-Unit Assessment (B), Question 6, “A sign in a bakery gives these options: 12 muffins for $25; 24 muffins for $46; 50 muffins for $94 1. Find each unit price to the nearest cent, and show your reasoning. 2. Which option gives the lowest unit price? Solution 1. The unit prices are $2.08, $1.92, and $1.88. Reasoning varies, but long division is a reasonable approach. 2. 50 muffins for $94 has the lowest unit price.  Minimal Tier 1 response: Work is complete and correct. Sample: (Accompanied by work showing long division or other calculation methods.) The unit prices are $2.08, $1.92, and $1.88. 50 muffins for $94 is the best deal. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Acceptable errors: incorrect selection of lowest unit rate comes from errors in calculation of unit rates. Sample errors: one or two errors in long division; incorrect selection of the best deal despite having calculated all three unit rates correctly. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: calculation of unit rates involves a conceptual error, such as dividing the number of muffins by the price; three or more errors in long division; correct selection of the best deal without calculation of each unit rate.”

Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following up with students include:

• Course Guide, Cool-Down Guidance states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”

• Unit 1, Check-Your-Readiness (A), Question 5, “The content assessed in this problem is first encountered in Lesson 5: Bases and Heights of Parallelograms. In this unit, students will find the area of parallelograms and triangles by decomposing them into shapes with perpendicular sides and rearranging the pieces. Students will need to be familiar with perpendicular lines in order to make sense of the ‘height’ of a parallelogram or triangle. If most students struggle with this item, plan to start Lesson 5 Activity 2 by amplifying the term perpendicular for the students. Students may need some visual cues to support this concept.” 

• Unit 6, Lesson 15, Cool-down Guidance, “Support Level 2. Points to Emphasize. Notes If students struggle with using properties of exponents strategically in the cool-down, plan to focus on this idea when opportunities arise over the next several lessons. For example, in the practice problem set for Lesson 17, consider inviting students to reflect on the reasoning behind prompt 3.”

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

All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in-classroom. Examples Include:

• Unit 1, Lesson 2, Cool-down, demonstrates the full intent of 6.G.1. “The square in the middle has an area of 1 square unit. What is the area of the entire rectangle in square units? Explain your reasoning.” A figure is given that is composed of many different shapes with a square in the middle of it.

• Unit 3, Lesson 6, Cool-down, demonstrates the full intent of 6.RP.3 and MP2. “Two pounds of grapes cost $6. 1. Complete the table showing the price of different amounts of grapes at this rate. 2. Explain the meaning of each of the numbers you found.” A table is provided with two columns one labeled “grapes (pounds)” and the other labeled “price (dollars)”.

• Unit 7, End-of-Unit Assessment (A), Question 4, demonstrates the full intent of 6.NS.7. “Given x = -2, mark and place these expressions on the same number line. x, -x, |-1.5|, -4, |5|, |-6|”

The materials reviewed for Math Nation 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 mathematics.

Course Guide, How to Use These Materials, Supporting Students with Disabilities sections states the following: “The philosophical stance that guided the creation of these materials is the belief that with proper structures, accommodations, and supports, all children can learn mathematics. Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students. While the suggested supports are designed for students with disabilities, they are also appropriate for many children who struggle to access rigorous, grade-level content. Teachers should use their professional judgment about which supports to use and when, based on their knowledge of the individual needs of students in their classroom.” Suggested supports are identified for teachers in the Full Lesson Plan to support learners of all levels. Lesson and activity-level supports, identified as “Support for Students with Disabilities,” are aligned to an area of cognitive functioning and are paired with a suggested strategy aimed to increase access and eliminate barriers. Supports are classified under the following categories: eliminate barriers, processing time, peer tutors, assistive technology, visual aids, graphic organizers, and brain breaks. Examples include:

• Eliminate Barriers: “Eliminate any barriers that students may encounter that prevent them from engaging with the important mathematical work of a lesson. This requires flexibility and attention to areas such as the physical environment of the classroom, access to tools, organization of lesson activities, and means of communication.” Unit 2, Lesson 15, Full Lesson Plan, 2.15.3 Exploration Activity, “Support for Students with Disabilities Executive Functioning: Eliminate Barriers. Chunk this task into more manageable parts (e.g., presenting one question at a time), which will aid students who benefit from support with organizational skills in problem solving”.

• Peer Tutors: “Develop peer tutors to help struggling students access content and solve problems. This support keeps all students engaged in the material by helping students who struggle and deepening the understanding of both the tutor and the tutee. For students with disabilities, peer tutor relationships with non-disabled peers can help them develop authentic, age-appropriate communication skills, and allow them to rely on a natural support while increasing independence.” Unit 6, Lesson 6, Full Lesson Plan, 6.6.3 Exploration Activity, “Support for Students with Disabilities… Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors…”

• Processing Time: “Increased time engaged in thinking and learning leads to mastery of grade level content for all students, including students with disabilities. Frequent switching between topics creates confusion and does not allow for content to deeply embed in the mind of the learner. Mathematical ideas and representations are carefully introduced in the materials in a gradual, purposeful way to establish a base of conceptual understanding. Some students may need additional time, which should be provided as required.” Unit 4, Lesson 12, Full Lesson Plan, 4.12.1 Warm Up, “Support for Students with Disabilities Memory: Processing Time. Provide sticky notes or mini whiteboards to aid students with working memory challenges. Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.”

There are several accessibility options (accessed via the wrench icon in the lower left-hand corner of the screen) available to help students navigate the materials. Examples include:

• Tools Menu allow students to change the language, and access a Demos Scientific and Graphing Calculator.

• Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, screen reader support. 

• UserWay, allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), Saturation (4 settings). 

Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.

The materials reviewed for Math Nation Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. 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 at grade level or that are outside of the standard K-12 curriculum. They are not routine or procedural, and intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems and it is not expected that any student works on all of them. Are You Ready for More? problems may also be good fodder for a Problem of the Week or similar structure.” If individual students would complete these optional activities, then they might be doing more assignments than their classmates.

Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:

• Unit 1, Lesson 15, 1.15.4 Exploration Extension: Are you Ready for More?, “1. Figure C shows a net of a cube. Draw a different net of a cube. Draw another one. And then another one. Show your work. 2. How many different nets can be drawn and assembled into a cube?” 

• Unit 2, Lesson 1, 2.1.4 Exploration Extension: Are you Ready for More?, “Use two colors to shade the rectangle so there are 2 square units of one color for every 1 square unit of the other color. 2. The rectangle you just colored has an area of 24 square units. Draw a different shape that does not have an area of 24 square units, but that can also be shaded with two colors in a 2:1 ratio. Shade your new shape using two colors.”

• Unit 8, Lesson 13, 8.13.3 Exploration Extension: Are you Ready for More?, “Invent a data set with a mean that is significantly lower than what you would consider a typical value for the data set.”

The materials reviewed for Math Nation 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.

The Course Guide, How to Use These Materials section states the following: “The framework for supporting English language learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” The four design principles are, support sense-making, optimize output, cultivate conversation, and maximize meta-awareness. Each design principle has an explanation that goes into more detail about how teachers can use it to support students. The routines are the Mathematical Language Routines (MLRs), the materials state, “The mathematical language routines (MLRs) were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize uses of language that is meaningful and purposeful, rather than about just getting answers. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. Each MLR facilitates attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessment for all learners. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others' ideas.” These design principles and routines are referenced under Instructional Routines, in the Full Lesson Plan for lesson, to assist teachers with lesson planning. The “Supports for English Language Learners” section within the Full Lesson Plan contains explanations of how to implement the MLRs. 

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Unit 1, Lesson 18, Full Lesson Plan, 1.18.3 Exploration Activity, “Support for English Language Learners Conversing: MLR 3 Clarify, Critique, Correct. Present an incorrect response such as, ‘If the cube has edge length s, then the area of each face is 2s because ‘$$s×s=2s$$.’ Ask students to identify the error and to offer a correct argument to write an expression for the area of each face. This will help students to use symbolic representations while generalizing calculations related to surface area. Design Principle(s): Optimize output (for generalization); Maximize meta-awareness”

• Unit 2, Lesson 16, Full Lesson Plan, 2.16.2 Exploration Activity, “Support for English Language Learners Speaking: MLR 8 Discussion Supports. Provide sentence frames for students to state their reasoning (e.g., ‘I liked this method of solving the problem because _______’; ‘This way worked best because ________’; ‘The _____ strategy is the same as / different from the _____ strategy because _____ ‘). The helps students place extra attention on the language used to engage in mathematical communication and reasoning. Design Principle(s): Maximize meta-awareness; Optimize output (for generalization)”

• Unit 6, Lesson 4, Full Lesson Plan, 6.4.3 Exploration Activity, “Support for English Language Learners Reading: MLR 6 Three Reads. Demonstrate this routine with the first situation to support reading comprehension. Use the first read to help students understand context. Ask, ‘What is this situation about?’ (e.g., Clare and Mai each have a different number of books). After the second read, ask students ‘What are the quantities in the situation?’ (e.g., the number of books Mai has, the number of books Clare has). After the third read, ask students to brainstorm possible strategies to connect the situation with the appropriate equation(s). Encourage students to repeat this routine themselves for each situation. This helps students connect the language in the word problem with the equation(s) while keeping the intended level of cognitive demand in the task. Design Principle(s): Support sense-making”

The materials reviewed for Math Nation Grade 6 meet expectations for providing manipulatives, both virtual and physical, that are representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Unit 1, Lesson 4, 1.4.2 Exploration Activity, students use an applet to find the area of parallelograms by decomposing a parallelogram into pieces to find the area. “1. Find the area of each parallelogram. Show your reasoning. 2. Change the parallelogram by dragging the green points at its vertices. Find its area and explain your reasoning. 3. If you used the polygons on the side, how were they helpful? If you did not, could you use one or more of the polygons to show another way to find the area of the parallelogram?” Two Geogebra applets are provided with seven parallelograms on each, which students can manipulate to find the area.

• Unit 4, Lesson 14, 4.14.3 Exploration Activity, Questions 1 and 2, students use physical cubes or an applet of snap cubes to model the volume of a rectangular prism. “Use cubes or the applet to help you answer the following questions. 1. Here is a drawing of a cube with an edge length of 1 inch. a. How many cubes with an edge length of  \frac{1}{2}inch are needed to fill this cube? b. What is the volume, in cubic inches, of a cube with an edge length of \frac{1}{2}inch? Explain or show your reasoning. 2. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of \frac{1}{2}inch. Sketch the prism, and find its volume in cubic inches.” A GeoGebra applet is available that allows students to make prisms by manipulating the block length, width, and height. 

• Unit 6, Lesson 8, 6.8.6 Practice Problems, Question 1, students use an applet to show and explain when two equations are equivalent. A. Draw a diagram of x + 3 and a diagram of 2x when x is 1. B. Draw a diagram of x + 3 and 2x when x is 2. C. Draw a diagram of x + 3 and 2x when x is 3. D. Draw a diagram of x + 3 and 2x when x is 4…” Students have access to an applet with a grid where they can draw the diagrams.