6th Grade - Gateway 3

The materials reviewed for enVision Mathematics 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 in order to guide their mathematical development. 

The Teacher’s Edition Program Overview provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: Overview of enVision Mathematics, User’s Guide, Correlation, and Content Guide.

• The Overview provides the table of contents for the course as well as a pacing guide. The authors provide the Program Goal and Organization, in addition to information about their attention to Focus, Coherence, Rigor, the Math Practices, and Assessment.

• The User’s Guide introduces the components of the program and then proceeds to illustrate how to use a ‘lesson’: Lesson Overview, Problem-Based Learning, Visual Learning, and Assess and Differentiate. In this section, there is additional information that addresses more specific areas such as STEM, Pick a Project, Building Literacy in Mathematics, and Supporting English Language Learners.

• The Correlation provides the correlation for the grade.

• The Content Guide portion directs teachers to resources such as the Scope and Sequence, Glossary, and Index.

Within the Teacher’s Edition, each Lesson is presented in a consistent format that opens with a  Lesson Overview, followed by probing questions to provide multiple entry points to the content, error intervention, support for English Language Learners, Response to Intervention, Enrichment and ends with multiple Differentiated Interventions.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Teacher’s Edition includes numerous brief annotations and suggestions at the topic and lesson level organized around multiple mathematics education strategies and initiatives, including the CCSSM Shifts in Instructional Practice (i.e., focus, coherence, rigor), CCSSM practices, STEM projects, and 3-ACT Math Tasks, and Problem-Based Learning. Examples of these annotations and suggestions from the Teacher’s Edition include:

• Topic 1, Lesson 1-1, Solve & Discuss It!, “Purpose Students engage in productive struggle to connect representing multiplying a whole number by a decimal on decimal grids to evaluating operations with decimal numbers during the Visual Learning Bridge. Before Whole Class 1 Introduce the Problem Provide additional decimal grids or decimal cube manipulatives as needed. 2 Check for Understanding of the Problem Activate prior knowledge by asking: What does each square of a decimal grid represent? What does an entire decimal grid represent?”

• Topic 3, Lesson 3-3, Do You Know How?, Problem 9, evaluate each expression “6 + 4 × 5 ÷ 3 - 8 × 1.5” Teacher guidance: “Prevent Misconceptions ITEM 9 Make sure students understand that multiplication and division together are done from left to right (not multiplication, then division.) Note: Although finding 8 × 1.5 before dividing 20 by 2 would not change the value of this particular expression, make sure students do not misinterpret the rule by saying that they must do it first thinking that multiplication comes before division. Q: Because there are no parentheses or exponents, what operation should you do first? [4 × 5]”

• Topic 5, Lesson 5-1, Practice & Problem Solving, Problem 19, “Reasoning Rita’s class has 14 girls and 16 boys. How does the ratio 14:30 describe Rita’s class?” Teacher guidance: “Error Intervention ITEM 19 Reasoning Students may think that the ratio 14:30 is incorrect because they may forget that ratios can compare a part to the whole. Q: What are the two ways that ratios can compare quantities? [Part to part and part to whole] Q: How can you find the total number of students in the class? [Add the number of girls, 14, to the number of boys, 16.]”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The materials provide professional development videos at two levels to help teachers improve their knowledge of the grade they are teaching.

• “Topic-level Professional Development videos available online. In each Topic Overview Video, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The video is a quick, focused ‘Watch me first’ experience as you start your planning for the topic.

• Lesson-level Professional Development videos available online. These Listen and Look For videos, available for some lessons in the topic, provide important information about the lesson.”

The Teacher’s Edition Program Overview, Professional Development section, states the “Advanced Concepts for the Teacher provides examples and adult-level explanations of more advanced mathematical concepts related to the topic. This professional development feature provides the teacher opportunities to improve his or her personal knowledge and build understanding of the mathematics in each topic. The explanations and examples in this section also support the teacher’s understanding of the underlying mathematical progressions.”

An example of an Advanced Concept for the Teacher:

• Topic 1, Topic Overview, Advanced Concepts for the Teacher, “Multiplying Decimals The standard algorithm for multiplying decimals extends the algorithm for multiplying multi-digit whole numbers by combining it with properties of multiplication and exponents. However, the manner in which decimal multiplication is typically represented presents some apparent contradictions. Consider the product of 4.32 and 1.8.[an example is provided]...This same principle applies to multiplying numbers in scientific notation involves explicit tracking of powers of 10, rather than transient adding and removing of ‘decimal places’, so no apparent contradictions arise in those standards algorithms.”

The Topic Overview, Math Background Coherence, and Look Ahead sections, provide adult-level explanations and examples of concepts beyond the current grade as they relate what students are learning currently to future learning.

An example of how the materials support teachers to develop their own knowledge beyond the current grade:

• Topic 4, Topic Overview, Math Background Coherence, Look Ahead, the materials state, “Grade 7 Proportional Relationships Students will use tables and equations to represent proportional relationships and to graph proportional relationships on a coordinate plane. Solve Simple Equations and Inequalities Students will fluently solve algebraic word problems with relationships in the form px + q = r. Students will also represent and solve inequalities in the forms px + q < r and px + q > r.”

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

Standards correlation information is indicated in the Teacher’s Edition Program Overview, the Topic Planner, the Lesson Overview, and throughout each lesson. Examples include:

• The Teacher’s Edition Program Overview, Correlation to Grade 6 Common Core Standards, organizes standards by their Domain and Major Cluster and indicates those lessons and activities within the Student’s Edition and Teacher’s Edition that align with the standard. Lessons and activities with the most in-depth coverage of a standard are distinguished by boldface. The Correlation document also includes the Mathematical Practices. Although the application of the mathematical practices can be found throughout the program, the document indicates examples of lessons and activities within the Student’s Edition and Teacher’s Edition that align with each math practice.

• The Teacher’s Edition Program Overview, Scope & Sequence organizes standards by their Domain, Major Cluster, and specific component. The document indicates those topics that align with the specific component of the standard.

• The Teacher’s Edition, Topic Planner indicates the standards and Mathematical Practices that align to each lesson.

The Teacher’s Edition, Math Background Coherence provides information that summarizes the content connections across grades. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:

• Topic 2, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Rational Numbers” and “Coordinate Plane” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 2 connect to what students will learn later?” and provides a Grade 6 and 7 connection, “Later in Grade 6...Graph Equations In Topic 4, students will graph equations on the coordinate plane...Grade 7 Operations with Rational Numbers Students will add, subtract, multiply, and divide both positive and negative rational numbers. They will solve multistep problems involving operations on rational numbers, renaming numbers as appropriate.” 

• Topic 3, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Exponents and Expressions” and “Equivalent Expressions” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Back” section asks the question, “How does Topic 3 connect to what students will learn earlier?” and provides a Grade 5 connection, “Grade 5 Patterns with Exponents and Powers of 10 Students used whole-number exponents to express powers of 10. [an example is provided] Evaluate, Write, and Interpret Numerical Expressions Students used the order of operations to simplify numerical expressions that contain parentheses…” 

• Topic 6, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Concept of Percent” and “Find the Percent, the Whole, or a Part” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 6 connect to what students will learn later?” and provides a Grade 6 and 7 connection, “Later in Grade 6 Describe Data In Topic 8, students will use percentages when they summarize data distributions. Students will draw box plots to represent where 25%, 50%, and 75% of the data occur. Grade 7 Percent Problems Students will analyze and solve percent problems. They will solve percent increase and percent error problems. They will also solve mark-up and markdown problems as well as simple interest problems.”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Teacher’s Edition Program Overview provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted all examples are found in the Teacher’s Edition Program Overview.

Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

• The Program Goals section states the following: “The major goal in developing enVision Mathematics was to create a middle grades program that embodies the philosophy and pedagogy of the enVision series and was adapted for the middle school teacher and learner…enVision Mathematics embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.”

• The Instructional Model section states the following: “Over the past twenty years, there have been numerous research studies measuring the effectiveness of problem-based learning, a key part of the core instructional approach used in enVision Mathematics. These studies have found that students taught partly or fully through problem-based learning showed greater gains in learning (Grant & Branch, 2005; Horton et al., 2006; Johnston, 2004; Jones & Kalinowski, 2007; Ljung & Blackwell, 1996; McMiller, Lee, Saroop, Green, & Johnson, 2006; Toolin, 2004). However, the interaction of problem-based learning, which fosters informal mathematical learning, and more explicit visual instruction that formalizes mathematical concepts with visual representations leads to the greatest gains for students (Barron et al., 1998; Boaler, 1997, 1998). The enVision Mathematics instructional model is built on the interaction between these two instructional approaches. STEP 1 PROBLEM-BASED LEARNING Introduce concepts and procedures with a problem-solving experience. Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014). Conceptual understanding results because the process of solving a problem that involves a new concept or procedure requires students to make connections of prior knowledge to the new concept or procedure. The process of making connections between ideas builds understanding. In enVision Mathematics, this problem-solving experience is called Solve & Discuss It. STEP 2 VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The important mathematics is the new concept or procedure students should understand. Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions for the Solve & Discuss It! task. Regardless of whether the important mathematics comes from discussing students’ thinking and work, understanding the important mathematics is further enhanced when teachers use an engaging and purposeful classroom conversation to explicitly present and discuss an additional problem related to the new concept or procedure…”

• Other research includes the following:

  • Resendez, M.; M. Azin; and A. Strobel. A study on the effects of Pearson’s 2009 enVisionMATH program. PRES Associates, 2009. 

  • What Works Clearinghouse. enVisionMATH, Institute of Education Sciences, January 2013.

• Throughout the Teacher’s Edition Program Overview references to research-based strategies are cited with some reference pages included at the end of some authors' work.

The materials reviewed for enVision Mathematics Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the online Teacher Resources for each grade, a Materials List is provided in table format identifying the required materials and the topic(s) where they will be used. Example includes:

• The table indicates that Topic 1 will require the following materials: “Blank paper, Decimal cubes, Fraction strips, Fraction tiles...”

• The table indicates that Topic 4 will require the following materials: “Algebra tiles, Base-ten blocks, Counters, Cubes/unit cubes, Different-sized boxes...”

• The table indicates that Topic 8 will require the following materials: “Graphing calculator/calculator, Index cards, Number cubes and other polyhedral game pieces (optional)...”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

The assessment system provides multiple opportunities to determine student’s learning throughout the lessons and topics. Answer keys and scoring guides are provided. In addition, teachers are given recommendations for Math Diagnosis and Intervention System (MDIS) lessons based on student scores. If assessments are given on the digital platform, students are automatically placed into intervention based on their responses.

Examples include:

• Topic 1, Lesson 1-2, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments.” I Intervention 0-3 points, O On-Level 4 points, A Advanced 5 points.” The materials provide follow-up activities—to be assigned at the teacher’s discretion—to students at each indicated level: Reteach to Build Understanding I, Additional Vocabulary Support I O, Build Mathematical Literacy I O, Enrichment O A, Math Tools and Games I O A, and Pick a Project and STEM Project I O A. For example, Problem 3, “The librarian purchased 22 copies of a best-selling book for $385.66. How much did each copy of the book cost?”

• Topic 3, Assessment Form A, Problem 4, “The same digits are used for the expression 3$$^4$$ and 4$$^3$$. Explain how to compare the values of the expressions.” The accompanying Scoring Guide gives the following recommendations based on the score: Greater than 85% /Assign the corresponding MDIS for items answered incorrectly. Use Enrichment activities with the student. 70% - 85% / Assign the corresponding MDIS for items answered incorrectly. You may also assign Reteach to Build Understanding and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Less Than 70% / Assign the corresponding MDIS for items answered incorrectly. Assign appropriate intervention lessons available online. You may also assign Reteach to Build Understanding, Additional Vocabulary Support, Build Mathematical Literacy, and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Item Analysis for Diagnosis and Intervention indicates Points 1, DOK 2, MDIS L2, Standard 6.EE.A.1.

• Topics 1-4, Cumulative/Benchmark Assessment, Problem 24, “Find the quotient. 1,107 ÷ 27”  The accompanying Scoring Guide gives the following recommendations based on the score: Greater than 85% /Assign the corresponding MDIS for items answered incorrectly. 70% - 85% / Assign the corresponding MDIS for items answered incorrectly. Monitor the student during Step 1 and Try It! parts of the lessons for personalized remediation needs. Less Than 70% / Assign the corresponding MDIS for items answered incorrectly. Assign appropriate remediation activities available online. Item Analysis for Diagnosis and Intervention indicates DOK 1, MDIS L31, Standard 6.NS.B.2.

• Topic 6, Performance Task Form A, Problem 2, “Ed wants to borrow $20,000 from a bank to open a small gym. Three banks charge different interest rates. 2. To help decide the best loan option, Ed wants to know the percent profit he will make each month. Part A The cost to run the gym each month is $5,000. Find Ed’s total monthly expenses for each loan option. Enter this in Table B. Part B Ed expects to average $7,500 in sales per month the first year.  Find Ed’s average monthly profit. Explain how to find the monthly profit. Then complete Table B. Part C Complete Table B for the estimated percent profit. Explain how you found the estimated percent profit.” The Scoring Rubric indicates 1 possible point for part A (correct answers), 2 possible points for parts B and C (2 points: Correct answers and explanation, 1 point: Correct answers or explanation). The Item Analysis for Diagnosis and Remediation indicates for 2A, DOK 2, MDIS M16, Standards 6.RP.A.3c, MP.2, for 2B, DOK 2, MDIS M16, Standards 6.RP.A.3c, MP.2, and for 2C, DOK 2, MDIS M41, Standards 6.RP.A.3c, MP.3.

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

The materials provide formative and summative assessments throughout the grade as print and digital resources. As detailed in the Assessment Sourcebook, the formative assessments—Try It! and Convince Me!, Do You Understand? and Do You Know How?, and Lesson Quiz—occur during and/or at the end of a lesson. The summative assessments—Topic Assessment (Form A and Form B), Topic Performance Task (Form A and Form B), and Cumulative/Benchmark Assessments—occur at the end of a topic, group of topics, and at the end of the year. The four Cumulative/Benchmark Assessments address Topics 1-2, 1-4, 1-6, and 1-8. 

• Try It! and Convince Me! “Assess students’ understanding of concepts and skills presented in each example; results can be used to modify instruction as needed.”

• Do You Understand? and Do You Know How? “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.”

• Lesson Quiz “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to prescribe differentiated instruction.”

• Topic Assessment, Form A and Form B “Assess students’ conceptual understanding and procedural fluency with topic content. Additional Topic Assessments are available with ExamView CD-ROM.”

• Topic Performance Task, Form A and Form B “Assess students’ ability to apply concepts learned and proficiency with math practices.”

• Cumulative/Benchmark Assessments “Assess students’ understanding of and proficiency with concepts and skills taught throughout the school year.”

The formative and summative assessments allow students to demonstrate their conceptual understanding, procedural fluency, and ability to make applications through a variety of item types. Examples include: 

• Order; Categorize

• Graphing

• Multiple choice

• Fill-in-the-blank

• Multi-part items

• Selected response (e.g., single-response and multiple-response)

• Constructed response (i.e., short or extended responses)

• Technology-enhanced items (e.g., drag and drop, drop-down menus, matching)

The materials reviewed for enVision Mathematics Grade 6 meet expectations for providing strategies and support for students in special populations to support their regular and active participation in learning grade-level mathematics. 

At the end of each lesson, there is a differentiated intervention section, these resources are assigned based on how students score on the lesson quiz taken on or offline. If taken online the resources are automatically assigned as the quiz is automatically scored. Resources are assigned based on the following scale based on the following scale: I = Intervention 0-3 points, O = On-Level 4 points, and A = Advanced 5. The types of resources include the following:

• Reteach to Build Understanding (I) - Provides scaffolded reteaching for the key lesson concepts.

• Additional Vocabulary Support (I, O) - Helps students develop and reinforce understanding of key terms and concepts.

• Build Mathematical Literacy (I, O) - Provides support for struggling readers to build mathematical literacy.

• Enrichment (O, A) - Presents engaging problems and activities that extend the lesson concepts.

• Math Tools and Games (I, O, A) - Offers additional activities and games to build understanding and fluency.

• Pick a Project and STEM Project (I, O, A) - Provides an additional opportunity for students to demonstrate understanding of key mathematical concepts.

Other resources offered are personalized study plans to provide targeted remediation for students, as well as support for English Language Learners and Enrichment. Additionally, Virtual Nerd tutorials are available for every lesson and can be accessed online.

Examples of the materials providing strategies and support for students in special populations include:

• Topic 3, Lesson 3-4, RtI, “Use With Example 3 Some students may need additional help identifying parts of an expression. Provide students with strips of paper and scissors. Have them write expressions with at least three terms on the strips and then trade the strips with a partner. Q: How many terms does the expression on your strip have? Q: Make a cut in your strip after each operation to separate the terms. How many pieces do you have? Students should check this answer against their answer to the first question, Discuss any discrepancies.”

• Topic 5, Lesson 5-2, RtI, “Error Intervention ITEM 19 Use Appropriate Tools Students may not understand how they can use a table to find equivalent ratios. Q: As you go across each row, what does each of the numbers represent? Q: If you look at the column with 3 at the top, what do 6 and 15 in that column represent?”

The materials reviewed for enVision Mathematics 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 Teacher’s Edition Program Overview, Supporting English Language Learners section, list the following strategies and supports: 

• “Daily ELL instruction is provided in the Teacher’s Edition. 

• Levels of English language proficiency are indicated, and they align with the following levels identified in WIDA (World-Class Instructional Design and Assessment): Entering, Emerging, Developing, Expanding, Bridging.

• ELL Principles are based on Jim Cummins’ work frame.

• Visual Learning Animation Plus provides stepped-out animation to help lower language barriers to learning. Questions that are read aloud also appear on screen to help English language learners connect oral and written language.

• Visual Learning Example often has visual models to help give meaning to math language. Instruction is stepped out to organize important ideas visually.

• Animated Glossary is always available to students and teachers while using digital resources. The glossary is in English and Spanish to help students connect Spanish math terms they may know to English equivalents.

• Pictures with a purpose appear in lesson practice to help communicate information related to math concepts or to real-world problems.”

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

• Topic 3, Lesson 3-1, English Language Learners (Use with Example 1), “ENTERING Ask students to review Example 1. Q: What words are new to you? Circle them. Write the words with simple definitions. Refer to them as you discuss the examples. Q: The word base is a homonym. It has more than one meaning. Where have you heard the word base before? Students may suggest bases in baseball or base, the safe place when playing tag, or the bottom of something like a statue.”

• Topic 4, Lesson 4-7, English Language Learners (Use with Example 1), “DEVELOPING Draw a number line from 0 to 10. Write the inequality x < 6. Point to 6 in the inequality, and draw an open circle at 6 on the number line. Ask students to share with partners the direction indicated by < 6. Q: What conclusions can be made about the solutions? Have students respond with sentences using inequality vocabulary. Repeat in a similar manner for x > 6.”

• Topic 6, Lesson 6-1, English Language Learners (Use with Example 2), “ENTERING Help students read the labels on the line segments in Example 2. Q: What word does the symbol % stand for? Q: What does the abbreviation in. represent? Q: What do the letters above the line segments represent on the line segments?”

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

The Teacher’s Edition Program Overview, Concrete, Representational, Abstract section states the following: “Digital interactivities Digital interactivities can simulate work with concrete models and can let students interact with pictorial representations. Using the digital Math Tools, students can move counters around on the screen, arrange fraction strips, manipulate geometric figures, and more. Many of the interactivities in the Visual Learning Animation Plus provide those same opportunities. Physical Manipulatives Physical manipulatives, including algebra tiles, counters, cubes, geoboards, and anglegs, provide opportunities for students to engage in concrete modeling when developing abstract thinking with mathematical concepts. A recommended set of manipulatives is available for each grade…Digital versions of the manipulatives are also available online.”

Examples of how manipulatives, both virtual and physical, are representations of the mathematical objects they represent and, when appropriate are connected to written methods, include:

• Topic 1, Lesson 1-4, Explore It!, students are given number lines (or Teaching Tool 6) to represent a problem based on a diagram. “Students are competing in a 4-kilometer relay race. There are 10 runners. A. Use the number line to represent the data for the race. B. Use multiplication or division to describe your work on the number line.”

• Topic 4, Lesson 4-6, Solve & Discuss It!, students are given number lines (or Teaching Tool 6) to assist in solving a problem that has to do with breaking records. “The record time for the girls’ 50-meter freestyle swimming competition is 24.49 seconds. Camilla has been training and wants to swim the 50-meter freestyle in less time. What are some possible times Camilla would have to swim to break the current record? Solve this problem any way you choose.”

• Topic 7, Student Interactive Lesson 7-2, Solve & Discuss It!, students use an applet to form a triangle and find the area. “Connect point A to B, B to C, C to D, and D to A. Then draw a diagonal line connecting opposite vertices in the figure and find the area of each triangle formed.”