6th Grade - Gateway 3

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.