6th Grade - Gateway 3

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson. It is clear when students solve problems to learn and when they apply skills.

Lessons include a Warm Up, Activities, Synthesis, and Cool Down. 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 activate prior learning or engage students for learning new material in the lesson. Students learn and practice new mathematics in lesson Activities. In the Activity Synthesis, students build on their understanding of the new concept. Each activity lesson ends with a Cool Down where students apply what they learned in the activities or are introduced 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 questions 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 enhance understanding or application (e.g., increased expectations for fluency, more abstract application, or a non-routine problem).

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations 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. 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 end with a Cool Down which is aligned to the daily lesson objective. Unit sequences consistently follow the progressions outlined in the CCSSM Standards to support students' development of conceptual understanding and procedural skills.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for having variety in what students are asked to produce.

The instructional materials prompt students to produce products in a variety of ways. Students produce solutions within Activities and Practice Problems, as well as participating in class, group, and partner discussions. Students construct viable arguments and critique the reasoning of their peers. Students use both a digital platform and paper-pencil to conduct and present their work. The materials consistently prompt students for 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, 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 McGraw-Hill Illustrative Mathematics Grade 6 meet expectations 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 2, Lesson 4, students use graduated cylinders and beakers to create mixtures when exploring ratio. A digital version of the task is also provided as an option.
• The materials 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. For example, 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 place value to the procedural skill.
• Examples of manipulatives for Grade 6 include: Tangram kits (or digital Tangram applets); Geometry toolkits containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles; and 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 McGraw-Hill Illustrative Mathematics 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 lesson consists of a detailed lesson plan accompanied by teaching notes. Included in these teaching notes are the objectives of the lesson, suggested questions for discussion, and guiding questions designed to increase classroom discourse and foster understanding of the concepts. For example, in Unit 6, Lesson 6, the following questions are included: “Did anyone solve the problem in a different way? Did anyone have the same strategy but would explain it differently?” The teaching notes and questions for discussion support the teachers in planning and implementing lessons effectively.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for providing teacher support with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

• Each lesson includes the Goals written for teachers, Student Learning Goals and Targets written for students, a list of Word/PDF documents that can be downloaded, CCSSM Standards that are “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 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 support for students.
• The teacher materials that correspond to the student lessons provide annotations and suggestions on how to present the content. “Launch” explains how to set up the activity and what to tell students. After the activity is complete, there is often “Anticipated Misconceptions” in the teaching notes, which describes how students may incorrectly interpret or misunderstand concepts and includes suggestions for addressing those misconceptions.
• The materials are available in both print and digital forms. The digital format has embedded applets. 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, the first Activity, students use tracing paper to decompose polygons 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 line segment to decompose each of the following polygons into two identical triangles, if possible. Use a straightedge to draw your line. If you choose to, you can also draw the triangles. 2) Which quadrilateral can be decomposed into two identical triangles? 3) Study the quadrilaterals that can, in fact, be decomposed 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 McGraw-Hill Illustrative Mathematics 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 include information about the mathematical connections of concepts being taught. Previous and future grade levels are also referenced to show the progression of 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, and give guidance on what to expect from students and important vocabulary.

The Narrative for Unit 4 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 McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Course Guide and Narratives describe how mathematical concepts are built from previous grade-level and lesson material. For example, the Unit 4 narrative states, “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.”

For some units, there are explanations given for how the grade-level concepts fit into future grade-level work. For example, the Unit 2 narrative concludes, “The terms proportional and proportional relationships 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 McGraw-Hill Illustrative Mathematics Grade 6 meet 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 Check Your Readiness Problem 4 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 McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

Lesson Activities include teaching notes 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, and steps teachers can take to help clear up the misconceptions. For example, in Unit 8, Lesson 5, Anticipating Misconceptions give 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 McGraw-Hill Illustrative Mathematics Grade 6 meet expectations 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. Practice Problems for each lesson activity reinforce learning concepts and skills and enable students to engage with the content and receive timely feedback. In addition, discussion prompts provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for assessments clearly denoting which standards are being emphasized.

Assessments are located on the Assessment tab for each unit and are also available in print. For each unit, there is a Check Your Readiness and an End-of-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, 6, and 8 also include a Mid-Unit Assessment.

The instructional materials reviewed McGraw-Hill Illustrative Mathematics Grade 6 

meet expectations for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Assessments include an answer key, and when applicable, a rubric consisting of three to four tiers, ranging from Tier 1 (work is complete, acceptable errors) to Tiers 3 and 4 (significant errors, conceptual mistakes).

Assessments include multiple choice, multiple response, short answer, restricted constructed response, and extended response. Restricted constructed response and extended response items have rubrics that are provided to evaluate the level of student responses. The restricted constructed response includes a 3-tier rubric, and the extended constructed response includes a 4-tier rubric. For these types of questions, the teacher materials provide guidance as to what is needed for each tier as well as some sample responses.

The End of Unit Assessment Teacher Guide includes Item Analysis to describe what may have caused an incorrect response. For example, in Unit 6, End of Unit Assessment, Problem 3, the End of Unit Assessment Teacher Guide states, “Students selecting A have made a sign error or did not notice the sign change in the expression. Students selecting B have not distributed the 4 to the 8d term. Students selecting D have not distributed the c to the 8d term.”

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations 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 activities that follow. The Cool Down following lesson activities reviews the concepts of the lesson.
• Within a lesson, narratives provide explicit instructional support for the teacher, including the Activity Launch, Anticipated Misconceptions, and Lesson Synthesis. This information assists teachers in making the content accessible to all learners.
• The Lesson Narrative often includes guidance on where to focus questions in Activities or the Lesson Synthesis.
• Optional activities are often included that can be used for additional practice or support before moving to the next activity or lesson.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations 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—includes 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. This routine supports reading comprehension of problems and meta-awareness of mathematical language. It also supports negotiating information in a text with a partner in mathematical conversation.”
• Teaching notes appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These teaching notes state 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 McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for embedding 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 complex tasks multiple times each lesson. The Warm Up, Activities, Lesson Synthesis, and Cool Down provide opportunities for students to apply mathematics from multiple entry points.

Specific examples of strategies found in the materials include “Notice and Wonder” and “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 Unit 1, Lesson 1, Warm Up, students 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.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for including 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 IM Implementation Guide, Courses 1-3, and embodies the Understanding Language/SCALE Framework from the Stanford University Graduate School of Education, which 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 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."

“ELL Enhanced Lessons” are identified in the Teacher Guide. These lessons highlight specific strategies for students who have a language barrier which affects their ability to participate in a given task. Throughout lessons, a variety of instructional routines are 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.”

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 McGraw-Hill Illustrative Mathematics Grade 6 meet expectations for providing 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 addressed 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.

There is no clear guidance for the teacher on how to specifically engage advanced students in investigating the mathematics content at greater depth.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics Grade 6 meet expectations 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.