8th Grade - Gateway 3

The materials reviewed for enVision Mathematics Grade 8 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 connect converting between terminating decimals and fractions to writing a repeating decimal as a fraction in the Visual Learning Bridge. Before Whole Class 1 Introduce the Problem Provide blank number lines, as needed. 2 Check for Understanding of the Problem Engage students with the problem by asking: What real-world values are often given in decimals? In fractions?”

• Topic 3, Lesson 3-3, Lesson 3-3, Convince Me!, “How can linear equations help you compare linear functions?” Teacher guidance: “Q: What is another way you can compare linear functions using an equation? [Sample answer: You can use an equation to identify the initial value or y-intercept.]”

• Topic 5, Lesson 5-1, Do You Understand?, Problem 1, “Essential Question How are slopes and y-intercepts related to the number of solutions of a system of linear equations?” Teacher guidance: “Essential Question Make sure students relate the number of intersections to the number of solutions.

The materials reviewed for enVision Mathematics Grade 8 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 5, Topic Overview, Advanced Concepts for the Teacher, “Operations on Systems of Equations Solving a system of linear equations is finding the set of all values for the variables that makes all equations in the system simultaneously true. When solving systems of equations using elimination, the goal is to use Properties of Equality to generate equivalent systems of equations that have the same solution set. Consider the system below. [an example is provided]...Strategy in Solving Systems of Equations Often, the strategy chosen for solving a system of equation is based solely on the form in which the equations are given. One example is always choosing to use graphing if both equations are in the form y = ax + b…”

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 2, Topic Overview, Math Background Coherence, Look Ahead, the materials state, “Grade 9… Functions In Grade 9, students will represent functions using graphs and algebraic expressions like (x) = a + bx. They will interpret functions in real-world contexts and build new functions from existing functions.”

The materials reviewed for enVision Mathematics Grade 8 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 8 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 3, Topic Overview, Math Background Coherence, the materials highlight three of the learnings within the topics: “Relations and Functions, Properties of Functions, and Qualitative Graphs” 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 7 and 8 connection, “Grade 7… Percents In Topic 3, students used the percent proportion and the percent equation to solve multistep problems involving simple interest, discounts, commissions, markups, and markdowns. Earlier in Grade 8 … Linear Equations Students studied slope by analyzing similar triangles. They derived the linear equation in the form y = mx +b, understanding that m represents the slope while b the y-intercept.”

• Topic 6, Topic Overview, Math Background Coherence, the materials highlight three of the learnings within the topic: “Transformations, Congruent and Similar Figures, and Angle Measurements” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Back” section asks the question, “How does Topic 6 connect to what students will learn earlier?” and provides a Grade 6 and 7 connection, “Grade 6 Geometry In Grade 6, students represented polygons on the coordinate plane. Grade 7 Geometry In Grade 7, students draw, construct, and describe geometrical figures and the relationships between them. They solve real-life and mathematical problems involving angle measure, area, surface area, and volume.” 

• Topic 7, Topic Overview, Math Background Coherence, the materials highlight one of the learnings within the topics: “Pythagorean Theorem” 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 and Algebra I connection, “Later in Grade 8 Apply the Pythagorean Theorem In Topic 8, students will compute the surface area and volume of figures. Students will use the Pythagorean Theorem to find the length of missing measurements such as the radius, height, or slant height of a cone. Algebra I Pythagorean Theorem In Algebra I, students use the Pythagorean Theorem to formally prove triangle similarity, and to solve application problems.”

The materials reviewed for enVision Mathematics Grade 8 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 8 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: “Graph paper, Graphing calculator/calculator, Index cards, Rulers...”

• The table indicates that Topic 4 will require the following materials: “Graph paper, Grip strength dynamometer (optional), Index cards, Scale...”

• The table indicates that Topic 8 will require the following materials: “Anglegs, Compass, Index cards, Scissors...”

The materials reviewed for enVision Mathematics Grade 8 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 2, Lesson 2-2,  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, “A red candle is 8 inches tall and burns at a rate of \frac{7}{10} inch per hour. A blue candle is 6 inches tall and burns at a rate of \frac{1}{5} inch per hour. After how many hours will both candles be the same height?”

• Topic 5, Performance Task Form A, Problem 1, “Jayden and Carson are selling T-shirts and sweatshirts with the school logo. Jayden sells 9 T-shirts and 3 sweatshirts for $288. Carson sells 1 T-shirt and 6 sweatshirts for $270. Find the selling prices of each item. 1. Write a system of equations to represent the situation.” The Scoring Rubric indicates 2: Two correct equations, 1: One correct equation. The Item Analysis for Diagnosis and Intervention indicates for DOK 3, MDIS K27 and K28, Standard 8.EE.C.8c. 

• Topics 1-6, Cumulative/Benchmark Assessment, Problem 5, “A truck rental company charges $27 per day plus $0.79 per mile. What is the equation of the line in slope-intercept form?”  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 Topics 1-6 Benchmark Assessment indicates Points 1, DOK 3, MDIS K52, Standard 8.F.B.4.

• Topic 8, Assessment Form A, Problem 11, “A laser pointer in the shape of a cylinder is 13 centimeters long with a radius of 0.75 centimeter. What is the volume of the laser pointer? Express your answer in terms of \pi and round to the nearest cubic centimeter.” 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 N53, Standard 8.G.C.9.

The materials reviewed for enVision Mathematics Grade 8 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 8 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 5, Lesson 5-2, RtI, “USE WITH EXAMPLE 1 Some students may have difficulty graphing equations with decimal slopes. 

  • Convert decimal slopes to fractions. Q: How do you write 0.20 as a fraction in lowest terms? How did you find this fraction? Q: What does a slope of \frac{1}{5} mean? Q: How do you write 0.25 as a fraction in lowest terms? How did you find this fraction? Q: What does a slope of \frac{1}{4} mean?”

• Topic 8, Lesson 8-1, RtI, “Error Intervention ITEM 10 Students may have difficulty using the circumference of a sphere to find the surface area of the sphere. Q: The circumference formula is C = 2$$\pi$$r. If the circumference is 514.96 yards, what is the radius?”

The materials reviewed for enVision Mathematics Grade 8 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 the Examples 1-3), “Entering As students work through Examples 1-3, be sure to provide students with the key vocabulary terms, relation and function. Present additional relations as ordered pairs and do a think-aloud for students to follow: This relation is a function because each input has exactly one output. OR This relation is not a function because at least one input has more than one output.”

• Topic 6, Lesson 6-1, English Language Learners (Use with Example 2), “INTERMEDIATE, Use with Example 2. Writing the terms in sentences can reinforce skills and solidify vocabulary. Have students complete the sentences. Q: A ____ is a change in position, shape, or size of a figure. Q: A ____ moves every point of a figure the same distance and the same direction. Q: In a transformation, the original figure is the ____, and the resulting figure is the ____.”

• Topic 7, Lesson 7-4, English Language Learners (Use with Example 3), “EXPANDING Solve Example 3. Have students restate the problem in their own words. Then have students work with a partner and take turns explaining how to solve for the third vertex of Li’s triangle. Listen for students who use academic vocabulary and develop fluency.”

The materials for enVision Mathematics Grade 8 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, Solve & Discuss It!, students are provided with square titles or graph paper to find the possible dimensions of a given area. “Matt and his dad are building a tree house. They buy enough flooring materials to cover an area of 36 square feet. What are all possible dimensions of the floor?”

• Topic 2, Lesson 2-1, Explore It!, students are provided with algebra titles (or Teaching Tool 11) to aid in drawing a representation of a relationship and writing an equation. “A superintendent orders the new laptops shown below for two schools in her district. She receives a bill for $7,500. A. Draw a representation to show the relationship between the number of laptops and the total cost. B. Use the representation to write an equation that can be used to determine the cost of one laptop.”

• Topic 6, Lesson 6-3, Explain It!, students are provided with grid paper, a compass, and a protractor to explore the position of a point on a Ferris wheel after a rotation. “Maria boards a car at the bottom of the Ferris wheel. She rides to the top, where the car stops. Maria tells her friend that she completed \frac{1}{4} turn before the car stopped. A. Do you agree with Maria? Explain. B. How could you use angle measures to describe the change in position of the car?”