6th Grade - Gateway 2

The materials reviewed in Grade 6 for this indicator meet the expectations by attending to conceptual understanding within the lesson.

• Module 1 assesses all of the specific content in 6.RP in various questions. Students were required to create and explain visual models (i.e., fraction models, ratio models, etc.) as part of their understanding on a regular basis.
• In Eureka-Grade 6, the development of division of fractions is structured in a way that includes several high-quality conceptual problems that allow students to work with several models, when applicable, and engage in discussion about the representations.
• The lessons are scaffolded in a way that gradually builds on and expands concepts from previous lessons.
• The lessons also structure the language of division in a way that helps transition students from the conceptual development to building fluency with the procedures.

The materials reviewed in Grade 6 for this indicator meet the expectations by attending to fluency and procedural work within the lessons.

• Procedural skill and fluency is most evident in modules 2, 3 and 4, which cover 6.NS and 6.EE. In addition to an abundance of examples throughout the lessons, there are also specific computation activities that provide practice with procedural skill designed to build fluency.
• Similarly, small-group and whole-class activities are sometimes included as opening or closing activities stressing the importance of procedural skill and fluency to the development of a concept, like simplifying ratios (e.g., module 1, lesson 12).
• Within the module on division of fractions, the review team identified the development of procedural fluency to be rushed. The unit spends a tremendous amount of time building conceptual understanding, but it is difficult to determine if there is sufficient time for students to gain fluency with the operations. Additionally, fluency in division of fractions is not assessed.
• However, the materials do continue to spiral back to previous concepts throughout the next lessons and modules.

The materials reviewed in Grade 6 for this indicator meet the expectations by attending to application within the lessons.

• Application of the mathematical concepts is abundant throughout each module. Overall, the introduction of new concepts is done through examples that involve applications, and lessons often follow that reinforce application skills.
• For example, module 1 uses application to develop 6.RP and the transition from ratios to tables to equations to graphs.
• Similarly, as students solve equations in module 4 (6.EE), they must develop the equation based on given information and then solve.
• In module 2, most of the problems are presented within a real-world context so that students can see the application of the math (division) to contextual situation.
• Students are also asked to apply mathematics to solve problems.
• There is a balance of introducing the concepts using real-world situation and also having students apply their understanding in order to solve application problems.

The materials reviewed in Grade 6 for this indicator meet the expectations by providing a balance of rigor. The three aspects are not always combined together nor are they always separate.

• Conceptual understanding, procedural skill and fluency, and application are integrated into each module as needed. When needed, separate fluency activities are included in the series. At other times, the procedural skill are part of application and conceptual understanding exercises.
• In module 1, lessons 1, 2, 10, 14, 15, 16 and 17 spend the majority of the time developing the concepts of ratios and unit rates.
• Lessons 8, 12, 13 and 18 continue to develop the concepts, but also require work finding equivalent ratios and rates using specific procedure to compute.
• The remaining lessons, as well as those already mentioned, give the students ample practice using ratios and unit rates in real-world application problems.
• There is a balance of the three aspects of rigor in included assessments.

The Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content.

• All modules at the Grade 6 level list the focus MPs in the module overview and expand on each practice's connection to the module.
• The summary in the overview is helpful because they specifically explain how the practice relates to the particular math concepts and lessons within the modules.
• For example, MP1, MP3, MP4 and MP6 are all called out in module 5 (page 7) and include some specific explanation of how they connect to the content in the module.
• The modules do not all find ways to address every practice standard, which is more realistic than forcing all of the practice standards into each module.
• Within the teacher lesson notes for each lesson, the mathematical practices are again called out when appropriate, sometimes several times throughout the lesson when practices are emphasized throughout the lesson. These notes assist teachers in understanding what students should be doing to engage with the practice standard.

Materials frequently prompt students to construct viable arguments concerning grade-level mathematics detailed in the content standards. Occasionally materials prompt students to analyze the arguments of others.

Throughout the discussion portion of each lesson, students are expected to explain the mathematics leading to understanding content and solving problems.

Students are also directed to explain responses in problem-set and exit ticket questions.

There are examples of students being asked to analyze the arguments of others in the lesson material or practice exercises.

Materials very explicitly attend to the specialized language of mathematics.

• Correct mathematical terminology is always used, enforced and reinforced.
• At the beginning of each module, terminology that is new or recent is specifically highlighted and defined (and examples provided in some cases) for teachers as are terms that should be more familiar.
• Explicit detail is always used in student-teacher discussion and explanation of process.
• The terminology that is used in the modules is consistent with the terms in the standards.
• Furthermore, relevant vocabulary is highlighted for students throughout the lessons and is reiterated at the end of each lesson (when relevant). This allows students to use their own resources in future lessons to review the relevant vocabulary and/or equations associated with such terms as "area" or "pi."