7th Grade - Gateway 3

The materials reviewed for enVision Mathematics Grade 7 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 making sense of phrases to using integers to describe a real-world situation in the Visual Learning Bridge. Before Whole Class 1 Introduce the Problem Provide number lines, as needed. 2 Check for Understanding of the Problem Engage students with the problem by showing them footage of a rocket launch.”

• Topic 3, Lesson 3-3, Convince Me!, “How does the percent describe how the weights are related?” Teacher guidance: “Q: How does the definition of percent relate the proportional quantities? [Sample answer: The percent is the number of parts out of 100. So forever 100 pounds something weights on Earth, it would weigh about 17 pounds on the Moon.]”

• Topic 5, Lesson 5-2, Do You Understand?, Problem 1, “Essential Question How is solving a two-step equation similar to solving a one-step equation?” Teacher guidance: “Essential Question Students should understand that expressions on both sides of the equation must remain equal at all times. This is achieved by applying the properties of equality. Two-step equations require two properties of equality because they are solved using two operations.”

The materials reviewed for enVision Mathematics Grade 7 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 2, Topic Overview, Advanced Concepts for the Teacher, “Solving Proportions by Inspection Equivalent ratios are ratios that express the same multiplicative relationship between numbers. When two equivalent ratios have the same first or second term, then the other terms are equal. For example [an example is provided]…Understanding equivalency in ratios and solving proportional relationships is the foundation for slope of linear relationships, inverse relationships, and trigonometric ratios.”

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 3, Topic Overview, Math Background Coherence, Look Ahead, the materials state, “Grade 8 … Similarity In Grade 8, students will use similar triangles to investigate slopes, and develop the equations y = mx and y = mx +b, for lines.”

The materials reviewed for enVision Mathematics Grade 7 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 7 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 4, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “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 4 connect to what students will learn earlier?” and provides a Grade 6 connection, “Grade 6 Algebraic Expressions In Grade 6, students learned to read and interpret parts of an expression by using mathematical terms and viewing expressions as single entities. Students used elementary operations to write and evaluate expressions…” 

• Topic 6, Topic Overview, Math Background Coherence, the materials highlight three of the learnings within the topics: “Populations and Samples, Make Inferences” and “Compare Populations Informally” 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 8 and High School connection, “Grade 8 Display and Describe Numerical Data In Topic 4, students will extend their understanding of sample data sets and data displays to include bivariate data sets, and plot them on a scatter plot instead of a histogram, box plot, or dot plot. They will learn to apply the concepts of clusters, gaps, and outliers to bivariate data in a scatter plot…High School Understand and Evaluate Random Processes Students will extend the concepts of random representative samples as tools for making inferences about populations by using simulation models…”

• Topic 7, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Simple Probability” and “Compound Probability” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 7 connect to what students will learn later?” and provides a Grade 8 connection, “Two-Way Frequency Tables In Grade 8, students will continue to find probabilities of simple and compound events. They will extend this knowledge to finding probabilities and making inferences and predictions using a two-way frequency table.”

The materials reviewed for enVision Mathematics Grade 7 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 7 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: “Base-ten blocks, Colored pencils (optional), Different lengths of nails (optional), Fraction bars...”

• The table indicates that Topic 4 will require the following materials: “Algebra tiles, Base-ten blocks, Colored pencils (optional), Index cards...”

• The table indicates that Topic 8 will require the following materials: “Anglegs, Colored pencils (optional), Compass (optional), Graph paper...”

The materials reviewed for enVision Mathematics Grade 7 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 1-5, 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 1, “How would you subtract a negative fraction from a positive decimal? Explain.”

• Topic 4, Assessment Form A, Problem 7, “There are 11.5 ounces of cereal in a container. Additional cereal is poured into the container at a rate of 2.25 ounces per second. How many ounces are in the container after 4 seconds?” 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 K20, Standard 7.EE.A.1.

• Topic 8, Performance Task Form A, Problem 1, “Dave wants to build a rectangular screened-in porch that is 20 feet long, and it will extend 6 feet out from the back side of his house. 1. Dave wants to sketch a scale drawing of the porch. Choose a reasonable scale and make a scale drawing of the porch. Label the dimensions of the model.” The Scoring Rubric indicates 2: Correct response, 1: Partially correct response. The Item Analysis for Diagnosis and Intervention indicates for DOK 2, MDIS M36, Standards 7.G.A.1, 7.G.A.2 and MP.4. 

• Topics 1-8, Cumulative/Benchmark Assessment, Problem 11, “Wyatt uses 3.15 cups of flour in a recipe that makes 9 shortcakes. Cora uses 2.4 cups of flour in a recipe that makes 8 shortcakes. How much more flower per shortcake is needed for Cora’s recipe? (A) 0.05 cup (B) 0.20 cup (C) 0.25 cup (D) 0.50 cup”  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 the appropriate remediation activities available online. Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS M28 and M29, Standards 7.RP.A.1 and 7.RP.A.3.

The materials reviewed for enVision Mathematics Grade 7 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 7 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, “Error Intervention ITEM 13 Students may need to be reminded which value to use as the ‘whole’ in a percent error problem. Q: What is the whole in a percent error calculation? Which is the whole in this problem?”

• Topic 5, Lesson 5-2, RtI, “Use With Example 1, 2, & 3 There are three errors students need to avoid when solving two-step equations. They may add or subtract a value on one side of the equation but they forget to add or subtract that value on the other side of the equation. They also use the order or operations incorrectly.

  • Practice using the correct properties of equality to solve each equation: Q: 2x - 8  = 10 Q: 7x - 4 = 24 Q: 8x + 2 = 10”

The materials reviewed for enVision Mathematics Grade 7 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 2, Lesson 2-2, English Language Learners (Use with Example 2), “EXPANDING Ask students to review Example 2. Have students read the problem and then rewrite it, shortening it to include the necessary information and the question. Have students edit each other’s work to further shorten the summary. Q: Look at your problem again. Can you see a way to solve it using mental math? Have students share their suggestions.”

• Topic 5, Lesson 5-5, English Language Learners (Use with Example 1), “EMERGING Ask students to read Example 1. Q: What does ‘without going over’ mean? Q: Should Gina’s pot-bellied pig consume more than 18 ounces per day? Explain.”

• Topic 7, Lesson 7-3, English Language Learners (Use with Example 1), “BEGINING Before reading Example 1, have students work with a partner to explore the meaning of experimental probability, theoretical probability, and relative frequency using sentence frames. The relative frequency is the ___ of the number of times an event happens to the total number of trials. The experimental probability is the ___ as the relative frequency. You can find the theoretical probability of an event if you know all the possible outcomes and they are all ___.”

The materials for enVision Mathematics Grade 7 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 1-8, Explain It!, students are given number lines (or Teaching Tool 8) to connect the relationships between equations in the same fact family to dividing integers. “The shapes below are used to show the relationship between each of the four equations in the same fact family. A. Suppose the star represents -24. What values could the other shapes represent? B. What do you know about the square and circle if the star represents a negative number? C. What do you know about the star if the square and circle both represent a negative number?” An image is shown of four equations with shapes.

• Topic 4, Lesson 4-7, Explore It!, students are given algebra tiles (or Teaching Tool 17) to write expressions that represent scores of a football game. “The East Side Bulldogs and the West Side Bears are playing a football game. A fan is keeping score using T for a touchdown plus extra point, worth 7 points total, and F for a field goal, worth 3 points. A. How can you represent the score of each team using expressions? B. How can you represent the difference of the teams’ scores using an expression? C. How can you determine how many more points the winning team had than the losing team?”

• Topic 8, Lesson 8-9, Solve & Discuss It!, students use counters, rectangular prism blocks, or graph paper to determine how many rectangular prisms fill a larger rectangular prism. “Volunteers at a food pantry pack boxes of soup into crates. How many boxes of soup will fill each crate? Show your work.”