6th Grade - Gateway 3

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that there is a clear distinction between problems and exercises in the materials.

There are eight Topics in each grade level. Each Topic presents lessons in a consistent structure. During the instructional sections, which include guided instruction, step-by step procedures, and problem solving, students work on examples and problems to learn new concepts.

At the end of the lesson, a variety of exercises allow students to independently show their understanding of the material. The exercises also include Higher Order Thinking, a Lesson Quiz, and Additional Practice.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that the design of assignments is intentional and not haphazard.

Lessons follow a consistent format that sequences assignments intentionally.

• In Solve and Discuss It, students are introduced to concepts and procedures through a problem-based situation.
• Visual Learning: This portion of instruction connects to the problem learned previously and is the substance of the lesson. There are three different examples explored to create student understanding.
• Do You Understand? These exercises generally promote conceptual understanding from the lesson.
• Do you Know How? These exercises generally promote procedural skill and fluency from the lesson.
• Practice and Problem Solving: These are a mix of rigorous exercises and include application problems.

Lessons are in a logical order that build coherence throughout the grade level. Exercises are intentional to encourage a progression of understanding and skills.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that the instructional materials prompt students to show their mathematical thinking in a variety of ways. For example:

• Lesson 1-5: Students explain the concept of dividing fractions using bar diagram and area models.
• Lesson 2-1: Students use a number line to compare integers.
• Lesson 3-5: Students defend or critique the work of others to show their understanding of Order of Operations.
• Lesson 8-2: Students explain their reasoning when using mean, median, mode, and range when describing data sets.
• Lesson 5-1: Students analyze double number lines and bar diagrams to understand ratios.

The instructional materials reviewed for enVision Florida 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, but the materials do not include physical manipulatives.

• Manipulatives and other mathematical representations are consistently aligned to the mathematical content in the standards, and virtual manipulatives are used for developing conceptual understanding, such as bar diagrams or geometric objects.
• The materials have manipulatives embedded within Visual Learning Animation Plus for many lessons and also within Independent Practice and Differentiated Intervention activities.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide teachers with quality questions for students.

In the Teacher Edition, facilitator notes for each activity include questions for the teacher to guide students' mathematical development and to elicit students' understanding. The materials indicate that questions provided are intended to provoke thinking and provide facilitation through the mathematical practices as well as getting the students to think through their work. For example:

• Lesson 1-1: “Why is it important to line up the digits by place value when you add decimals?”
• Topic 1 Introduction: “What picture or model could you draw to represent the problem?” and “What place value is to the right of the decimal point?”
• Lesson 2-6: “How can you find the perimeter of any polygon?” and “How are the methods used to convert between customary and metric units similar to those used to convert within one measurement system?”

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide teachers guidance for presenting the content and using embedded technology for student learning.

• The Topic Overview includes Math Background for the Topic. This provides a big picture of the learning throughout the lessons. The Math Background also looks at how Focus, Coherence, and Rigor, as well as the Mathematical Practices, are addressed within the Topic.
• Each STEM Activity provides teachers with background information on the math, science, and engineering and technology found in the activity. In addition, it gives information on how to present the topic, including discussion questions. It also gives direction on when to show the STEM video, and when the project can be launched.

In each lesson teachers are provided with the following guidance:

• Information about how to activate prior knowledge is given in the Review What You Know! section. A Vocabulary Review activity is also given, as well as instructions on preparing students for reading success.
• A Lesson Overview that includes the Objectives, Essential Understandings, descriptions of how the mathematical content builds over the grades and within the topic, and how rigor is addressed in the lesson. In addition, both the content and the practice standards addressed in the lesson are described.
• Tips on what to do before, during, and after problems are available for the teacher.
• Technology enhancement included in the online program is noted in the teacher materials, though specific games/lessons/videos/online tools must be accessed online for teachers to know what they include.  

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials explain the role of the grade-level mathematics in the context of the overall mathematics curriculum.

• Each topic opens with a Topic Overview that includes a Math Background for the Topic.
• The Coherence section has three parts: Look Back, Topic ____, and Look Ahead. Each section gives a clear, specific explanation of how the topic is connected across grades.
• Each topic includes a Review What You Know! section. The section includes a practice page for students, questions for teachers to activate prior student knowledge, and a vocabulary review.
• Teacher Edition Program Overview Materials contain an overview of mathematics for K-12.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

The Assessment Sourcebook and the Teacher Program Overview provide information about the use of assessments to gather information about students' prior knowledge. Every grade level includes a Grade Level Readiness test. The Topic Readiness Assessment in each Topic helps teachers gather information about students’ prior knowledge within and across grade levels. Topic Readiness assessments can also be taken online, where they are auto-scored, and interventions are auto-assigned.

There is a Review What You Know assignment at the beginning of each Topic that helps students activate prior knowledge and prepare for the skills needed in the Topic. Each of these assignments has questioning strategies for the teacher. Each lesson also provides information for the teacher about prior and current grade levels and future math that is used.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide strategies for teachers to identify and address common student errors and misconceptions.

Each lesson identifies common errors and misconceptions for the teacher to address in the independent practice. The misconception/error is followed with prompts that the teacher can ask to help students understand their mistakes.

• In Lesson 7-4, Error Intervention for Item 12 is “Students may have difficulty converting square feet to square yards.”
• In Lesson 2-1, Prevent Misconceptions for Item 3 includes “Explain that numbers that are not preceded by a sign are presumed to be positive.”

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

• Each Topic includes a Review what you know/Concept and Skills Review section that includes a Vocabulary review and Practice problems. This section includes review and practice on concepts that are related to the new Topic.
• The Cumulative/Benchmark Assessments found at the end of Topics 2, 4, 6, and 8 provide review of prior topics as an assessment. An Item Analysis is provided for diagnosis and intervention. Students can take the assessment online, with differentiated intervention automatically assigned to students based on their scores.
• The Math Diagnosis and Intervention System has practice pages which are specific to a skill or strategy (i.e., Markups and Markdowns and Mental Math).
• There are multiple pages of extra practice available at Pearson Realize online that give students extra opportunities to review skills assigned by the teacher. Each of these pages is able to be customized by the teacher or used as is.
• Different games online at Pearson Realize support students in practice and review of skills, as well procedural fluency.

Teacher materials in print and online complement each other in many instances; however, It is worth noting that teachers who only use the print materials will miss the extensive resources available online, as the print edition has some notes that point teachers to the digital materials but does not identify the full extent of the online resources. Digital assessments are auto-scored and generate reports that can help with grouping and differentiation.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that assessments clearly denote which standards are being emphasized.

The Item Analysis for Diagnosis and Remediation clearly denotes which standards are being assessed. This is found for all of the assessments, including the quizzes.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The Topic Overview in the Teacher Edition includes a section on Coherence which enhances the opportunity to scaffold instruction by identifying prerequisite skills that students should have.

All lessons include instructional notes and classroom strategies that provide teachers with sample questions, differentiation strategies, discussion questions, possible misconceptions, and what to look for from students, which provides structure for the teacher in making content accessible to all learners. Often, the Practice & Problem Solving section begins with scaffolded problems for students. In the Solve and Discuss It! section, teachers are given questions for before, during, and after the activity that provide scaffolding for students. For example:

• Topic 6, Lesson 6-6: Before: “What information are you given in the problem? What is the unknown in this problem situation?” During: “How does your model represent this problem?” After: “How does Adam use a double number line diagram to calculate the number of matches the soccer team played?"

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

• There are Response to Intervention strategies in each lesson. These sections give teachers “look fors” and suggestions to address the needs of students who are struggling. Questions for the teacher to ask are also included.
• Each lesson has at least one Additional Example that help students cement or extend their understanding of the concept being taught. It includes an extra problem for the teacher to use, as well as questions to help elicit meaningful responses.
• Each lesson has Differentiated Interventions for a wide-range of learners:
• Reteach to Build Understanding provides scaffolding to reteach.
• Additional Vocabulary Support helps students with key vocabulary.
• Build Mathematical Literacy provides support for struggling readers.
• Enrichment extends concepts from the lesson.
• Online Math Tools and Games build understanding and fluency.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials embed tasks with multiple entry-points.

Lessons begin with Problem-Based Learning, including Solve & Discuss It or Explore It, and can often be solved with a variety of solution strategies.  Also, 3-Act Mathematical Modeling and Performance Tasks include questions with multiple entry points that can be solved using a variety of representations.

• Lesson 7-6, Solve & Discuss It!: “Marianne orders boxes to pack gifts. When they arrive, she finds flat pieces of cardboard as shown below. Marianne needs to cover each face of the boxes with green paper. What is the least amount of paper needed to cover the box? Explain.” This is extended with “Suppose Melanie has only one large sheet of green paper that is 15 inches by 30 inches. Is the area of this sheet of paper great enough to cover all of the faces of one box? Explain.”

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations.

• Each lesson has support, accommodations and modifications for ELL students.
• During lessons, strategies are given for teachers to address different levels of English Language Learners such as:
• Emerging: “Ask students to identify which parts of the illustration show ‘four matches.’ Ask the students: 'What other meanings do you know for the word match? Explain. How can the illustration help you know which meaning to use?”
• Expanding: “English Language Learners may not understand the questions asked. Have students state the problem in their own words.”

An English Language Learners Toolkit suggests teaching strategies, assessment tips, vocabulary and reading strategies, and discussion and problem-solving grouping ideas.

• Two pages of multilingual thinking words include the following languages: English, Spanish, Chinese, Vietnamese, Korean, and Hmong.
• There is also a limited list of key vocabulary in the same 6 languages.
• Teaching Math to Culturally and Linguistically Diverse Students provides tips and strategies.

Each chapter has Prepare for Reading Success that gives teachers pre-reading strategies to help all students access the language of the topic. For example, one strategy is Making Predictions, where there are teacher questioning strategies, as well as work for students that involves making predictions regarding the content. It also includes an extension for all readers.

Another special population that is addressed is Early Finishers. Each lesson addresses these students' needs by asking an additional question for students to ponder that correlates with the essential question in the lesson.

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Each lesson offers differentiated instruction to extend the concepts in the lesson and provides opportunities to challenge advanced students. For example:

• Lesson 6-6 Enrichment example 3: “Challenge advanced students to create a game in which players match problems such as 4.5% of what number is 5.4? to the correct answer, which is 120 in this case.”
• Lesson 6-6 Challenge item 12: “Have students create a real-world problem using the numbers in the exercise.”

The instructional materials for enVision Florida Mathematics Grade 6 meet the expectations that materials provide a balanced portrayal of various demographic and personal characteristics.

For example:

• Different cultural names and situations are represented in the materials, ie., Sandra, Nadia, Nigel, Yoshi, Joaquin, Archie.
• The materials avoid pronouns, referencing a role instead, ie., the carpenter, the teacher, a plumber, the cross country team.
• There are few pictures of actual people in the Student Book, mostly objects or cartoonish drawings.