8th Grade - Gateway 3

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas, in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

Materials engage students in both problems and exercises through the grade level. Problems where students learn new mathematics are typically found in the Lesson’s Visual Learning Bridge. This portion of the lessons consists of visual examples that formalize the mathematics of the lesson by providing guided instruction of the math concepts with one example stepped-out. Examples from the Teacher Resource include:

• Lesson 7-2, Understand the Converse of the Pythagorean Theorem, Visual Learning, Example 1 students learn apply the Converse of the Pythagorean Theorem to identify right triangles, “$$\vartriangle$$ABC has side lengths a, b, and c such that a$$^2$$ + b$$^2$$ = c$$^2$$. Construct a logical argument to show that $$\vartriangle$$ABC is a right triangle.
• Lesson 3-3, Compare Linear and Nonlinear Functions, Visual Learning, Example 2, students learn how to determine relationships between two variables comparing two functions provided in various forms, “A square with side length s is shown. The table shows the relationship between the side length and perimeter as the side length increases. The graph shows the relationship between the side length and area. How do the two relationships compare?”
• Lesson 6-5, Understand Congruent Figures, Visual Learning, Example 1, students learn how a sequence of translations, reflections, and rotations result in congruent figures, “Ava wants to place a flame-resistant hearth rug in front of the sofa. How can she determine whether the rugs are the same size and shape?” A model of the room with visuals of how transformations are used is provided.

Exercises, where students apply learning to build mastery, are typically found in the Practice and Problem Solving section. These exercises build independent proficiency, challenge higher-order thinking, and simulate high-stakes testing questions. Examples from the Teacher Resource include:

• Lesson 1-7, More Properties of Integer Exponents, Practice & Problem Solving, Item 18, students solve an expression with negative exponents, then apply it to a given situation, “To win a math game, Lamar has to pick a card with an expression that has a volume greater than 1. The card Lamar chooses reads $$(\frac{1}{2})^{-4}$$. Does Lamar win the game? Explain.”
• Lesson 5-3, Solve Systems by Substitution, Practice & Problem Solving, Item 10, students write and solve a system of equations related to a given situation, “On a certain hot summer day, 481 people used the public swimming pool. The daily prices are $1.25 for children and $2.25 for adults. The receipts for admission totaled $865.25. How many children and how many adults swam at the public pool that day?”
• Lesson 7-2, Understand the Converse of the Pythagorean Theorem, Practice & Problem Solving, Item 13, students apply the Pythagorean Theorem to prove whether or not a right triangle exists under the given conditions, “Three students draw triangles with the side lengths shown. All three say their triangle is a right triangle. Which students are incorrect? What mistake might they have made? Student 1: 22, 33, 55; Student 2: 44, 33, 77; Student 3: 33, 44, 55.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the design of assignments is not haphazard: exercises are given in intentional sequences.

Lesson activities within each Topic are intentionally sequenced developing student understanding and leading towards mastery of the content. Students are introduced to concepts and procedures with a problem-solving experience, Solve & Discuss it. The Visual Learning Bridge provides direct instruction that makes the important mathematics explicit through class discussion of student thinking and solutions. Examples from the Teacher Resource include:

• Lesson 2-2, “Solve Equations with Variables on Both Sides,” Solve & Discuss It!, students are presented with the problem and work independently to solve using various strategies, after which discussion ensues to develop key concepts important in solving equations with variables on both sides. “Jaxson and Bryon collected cash and checks as shown below. If each check is written for the same amount, x, what is the total amount of money collected by both boys? Explain.” The picture shows Jaxson collected $15 and 14 checkers and Bryon collectd $50 and 7 checks.
• Lesson 4-3, “Use Linear Models to Make Predictions”, Solve & Discuss It!, students are presented with the problem and work independently to solve using various strategies, after which discussion ensues to develop key concepts, such as why solutions to the problem are not identical and the existing relationship between the two variables. “Bao has a new tracking device that he wears when he exercises. It sends data to his computer. How can Bao determine how long he should exercise each day if he wants to burn 5,000 Calories per week?” (data provided in the form of a scatter plot)
• Lesson 8-3, “Find Volume of Cones,” Solve & Discuss It!, students are presented with a situation allowing them to compare the volume of a pyramid and a prism and connect it to finding the volume of a cone. “A landscape architect uses molds for casting rectangular pyramids and rectangular prisms to make garden statues. He plans to place each finished pyramid on top of a prism. If one batch of concrete mix makes one prism or three pyramids, how does the volume of one pyramid compare to the volume of one prism? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that there is variety in what students are asked to produce. For example, students are asked to produce answers and solutions; but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc. Examples from the Teacher Resource include:

• Lesson 1-8, Use Powers of 10 to Estimate Quantities, Practice & Problem Solving, Item 14, students critique the work of another and justify their reasoning, “The diameter of one species of bacteria is shown. Bonnie approximates this measure as 3 x 10$$^{-11}$$ meter. Is she correct? Explain.” An illustration with 0.00000025691 m labeled is provided.
• Lesson 3-6, Sketch Functions from Verbal Descriptions, Practice & Problem Solving, Item 8, students visually represent the relationship between two variables, “Aaron’s mother drives to the gas station and fills up her tank. Then she drives to the market. Sketch the graph that shows the relationship between the amount of fuel in the gas tank of her car and time.”
• Lesson 6-4, Compose Transformations, Practice & Problem Solving, Item 10, students justify their understanding of transformations by graphing an image and its pre-image, “Map $$\vartriangle$$QRS to $$\vartriangle$$Q’R’P’ with a reflection across the y-axis followed by a translation 6 units down.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods. 

Students have access to Anytime Math Tools powered by Desmos to build understanding and are accessible from the Tools panel online. Desmos tools include a graphing calculator, a scientific calculator, and a geometry construction tool. In addition, students have access to digital math tools such as algebra tiles, integer chips, area models, and bar diagrams. Students see an icon with a wrench when tools are suggested for use during examples and questions. Examples from the Teacher Resource include:

• Lesson 1-4, Evaluate Square Roots and Cube Roots, Solve & Discuss It!, students use square tiles or graph paper to find square roots and cube roots of rational numbers, “Matt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet. What are all possible dimensions of the floor?”
• Lesson 2-1, Combine Like Terms to Solve Equations, Explore It!, students use algebra tiles to draw a representation and write an equation for a situation and combine like terms to solve the equation, “A superintendent orders the new laptops shown below (10 laptops are shown) for two schools in her district. She receives a bill for $7,500. Draw a representation that can show the relationship between the number of laptops and the total cost.”
• Lesson 6-9, Interior and Exterior Angles of Triangles, Solve & Discuss It!, students use graph paper, rulers, and protractors to find the measure of an angle inside a triangle and use it to find interior angle measures, “Nell cuts tile to make a decorative strip for a kitchen backsplash. She must cut the tiles precisely to be congruent triangles. She plans to place the tiles between two pieces of molding as shown. What is m∠1? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Effective Mathematics Teaching Practices (ETP) side notes provide quality questions that are designed to promote reasoning and problem solving, support productive struggle, and engage students in mathematical discourse. Establish the Mathematical Goal provides questions related to the Essential Question. Use and Connect Mathematical Representations and Pose Purposeful Questions provide probing questions to enrich the mathematics. Elicit Student Thinking is an opportunity to formatively assess students to determine their understanding of concepts learned. Examples from the Teacher Resource include:

• Lesson 3-4, Construct Functions to Model Linear Relationships, Visual Learning, Example 1, Try It!, Elicit and Use Evidence of Student Thinking, “How does the equation change now that the ramp has a 3:15 ratio? Will the height of the ramp be taller or shorter than the height of the ramp in Example 1?”
• Lesson 5-4, Solve Systems by Elimination, Visual Learning, Example 2, Pose Purposeful Questions, “What is the relationship in the equations between the terms with the variable x? Why is the second equation subtracted from the first equation? What would the solution be if you subtracted the first equation from the second equation? Explain.” 
• Lesson 6-7, Understand Similar Figures, Visual Learning, Example 1, Use and Connect Mathematical Representations, “How do you know that the two trapezoids are facing in opposite directions? Explain. How do you know the dilation of GHIK is a reflection of ABCD over the x-axis? Suppose the reflection of ABCD did not have the same coordinates as the dilation of GHJK. How could you use a transformation to map the reflection of ABCD onto the dilation of GHJK?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student materials and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning. 

Effective Mathematics Teaching Practices (ETP) side notes provide Before, During, and After suggestions regarding lesson implementation. Examples from the Teacher Resource include:

• Lesson 1-1, Rational Numbers as Decimals, Solve & Discuss It!, ETP: Before, “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?’”
• Lesson 4-4, Interpret Two-Way Frequency Tables, Explore It!, ETP: During, “3. Observe Student Work: Students might determine the preferred activity of people in each age group, or they might determine the most common age group for each activity. If needed, ask: ‘How can you compare the data in the table?’ Early Finishers: ‘How would the problem change if 38 people aged 35 and under preferred skiing?’ [Sample answer: The total number of people surveyed increases, and the majority of both people 35 and under and people over 35 prefer skiing over snowboarding.]”
• Lesson 8-2, Find Volume of Cylinders, Explain It!, ETP: After, “4. Discuss Solution Strategies and Key Ideas: Consider having groups share their solutions to part (A) first, followed by part (B). Have students discuss what mathematical quantity is being compared in this problem; volume. Have them discuss how to compute the volume of the rectangular tank; they can find the area of the circular base and multiply it by the height of the prism. Have students consider and discuss how they could find the volume of the cylindrical tank; they can find the area of the circular base and multiply it by the height of the cylinder. 5. Consider Instructional Implications: After presenting Example 1, have students refer back to the problem in the Explain It and have them calculate how much larger the volume of rectangular tank is than the cylindrical tank; the rectangular tank has a volume of 27,648 in$$^3$$ and the cylindrical tank has a volume of about 21,704 in$$^3$$. The difference is about 5,944 in$$^3$$.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Each Topic contains a Topic Opener, Math Background: Focus section that provides a discussion of the math content in the topic along with sample work and strategies that illustrate the underlying concepts to help teachers anticipate the works students will do. The Topic Opener also contains Advanced Concepts for the Teacher that provides examples and adult-level explanations of more advanced mathematical concepts related to the topic with explanations and examples to support teacher understanding of the underlying mathematical progressions. Examples from the Teacher Resource include:

• Topic 2, Analyze and Solve Linear Equations, Math Background, “Solution Sets of Equations: The solution to an equation is the set of values for the variable(s) that make the equation true. For example, in the equation y = 2x, all value pairs (x, y) where y is twice x are in the solution set for the equation. This solution set is infinite. When an equation is represented by a graph in the coordinate plane, the graph represents the elements in the solution set for that equation. Solution Sets of Systems of Equations: The solution to a system of equations is the set of points that simultaneously satisfy each equation. The point(s) of intersection of the graphs of a system represent the solution set. Consider the equation mx + b = nx + c, where m, b, n, and c are real numbers. This could be solved algebraically or graphically. To solve by graphing, graph each side of the equation as its own linear equation: y = mx + c and y = mx + c. Where these graphs intersect (or overlap) are the values of x where each side of the original equation is equivalent for the same value of x.” Visual representations are provided.
• Topic 4, Investigate Bivariate Data, Math Background, “Scatter Plots: In Lesson 4-1, students will construct and interpret scatter plots that show the relationship between two sets of data. Scatter plots give a visual representation of the relationship between the two quantities being measured. Students decide whether the data have a positive or negative association, or no association at all. Linear Associations: In Lesson 4-2, students extend their work with scatter plots to begin drawing trend lines to represent the relationship that exists between the quantities. The students will use trend lines to decide whether the paired data show a linear association, a nonlinear association, or no association.” Visual illustration with examples of vocabulary terms cluster, gap, and outlier are provided.
• Topic 6, Congruence and Similarity, Math Background, “Compose Transformations: In Lesson 6-4, students apply their knowledge to perform a sequence of transformations. They will describe and perform two different forms of transformation, one at a time, to map a preimage to its image. In Lesson 6-5, students develop a deeper understanding of reflections, rotations, and translations by performing a sequence of transformations to identify congruent geometric objects. Similarity: In Lesson 6-7, students expand on previous lessons to recognize similar figures and will be able to prove the similarity by performing a sequence of transformations using rotations, reflections, translations, and dilations. In Lesson 6-10, they determine whether triangles are similar and solve triangle problems.” An example similarity problem is provided.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Each Topic Opener contains a section Math Background: Coherence that summarizes the content connections through the materials to prior and future grades. Look Back illustrates connections to previously taught concepts and skills include those within the grade, across content, or across grades. Look Ahead illustrates connections within or across grades. Examples from the Teacher Resource include:

• Topic 3, Use Functions to Model Relationships, Math Background, Look Back, “Grade 7: Proportional Reasoning - In Topic 2, students learned to reason about proportional relationships. They identified the constant of proportionality and used the equation y = kx, where k represents the constant of proportionality. Percents - In Topic 3, students used the percent proportion and the percent equation to solve multi-step problems involving simple interest, discounts, commissions, markups, and markdowns.”
• Topic 5, Analyze and Solve Systems of Linear Equations, Math Background, Look Back, “Earlier in Grade 8: Solve Equations - In Topic 2, students reviewed how to solve one-step, two-step, and multi-step equations, and extended their understanding to include equations with real number coefficients. Students gained experience with equations that had zero, one, or infinitely many solutions. They also graphed linear equations, and found equations to match given line graphs.”
• Topic 7, Understand and Apply the Pythagorean Theorem, Math Background, Look Ahead, “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 instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Materials provide strategies for gathering students’ prior knowledge. Examples include:

• Grade Level Readiness Test diagnoses students’ readiness for learning by assessing prerequisite content. This assessment is also available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource. 
• Topic Readiness Assessment diagnoses students’ proficiency with Topic prerequisite concepts and skills. This assessment is available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource. 
• Review What You Know, found at the beginning of each Topic, checks for understanding of key math concepts previously learned. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials provide strategies for teachers to identify common student errors and misconceptions. 

Materials provide strategies to identity student errors. Prevent Misconceptions are found in the Teacher Resource sidenotes for the Visual Learning portion of the lesson and Error Interventions are found in the Practice & Problem Solving Section. Examples from the Teacher Resource include:

• Lesson 2-1, Combine Like Terms to Solve Equations, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 5, “Some students may not set up the problem correctly so that it demonstrates a decrease in population. Q: How can you check whether your answer is reasonable? [Sample answer: If the population has decreased to 350,000, then the answer must have been greater than 350,000.]”
• Lesson 3-1, Understand Relations and Functions, Practice & Problem Solving, Error Intervention, Item 11, “Confirm students’ thinking around whether a relation is a function. Q: How do you determine whether the relation between the grade and number of students is a function? [Sample answer: If each input value (grade) in the table has a unique output value (number of students), the relation is a function.]”
• Lesson 8-3, Find Volume of Cylinders, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 5, “If students find this difficult suggest that they solve the problem using these steps. Q: What is the area of the base? [3.14 square feet]. Q: How can you use the area of the base and the volume to find the height? [Sample answer: Divide the volume by the area of the base.]”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review of concepts and skills. Examples Include:

• Each Topic includes Review What You Know to activate prior knowledge and and review prerequisite skills needed for the Topic. Both vocabulary and practice problems are provided.
• The Cumulative/Benchmark Assessments are found at the end of Topics 2, 4, 6 and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the year. 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.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials offer ongoing formative and summative assessments, clearly denoting which standards are being emphasized. 

Formative and summative assessments clearly denote standards being assessed. Examples include:

• Try It! and Convince Me! are found following the Visual Learning Examples and assess students’s understanding of concepts and skills presented in each Example and results can be used to modify instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Do You Understand? And Do You Know How? are found after the Visual Learning instruction and assess students’ conceptual understanding and procedural fluency and results can be used to review content. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Following each lesson is a Lesson Quiz that assesses students’ conceptual understanding and procedural fluency with the lesson content. Results can be used to determine differentiated instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• At the end of each Topic there is a Topic Assessment with 2 forms, Form A and Form B, that assesses students’ conceptual understanding and procedural fluency with the topic content. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• At the end of each Topic there is a Performance Task with 2 forms, Form A and Form B, that assess students’ ability to apply concepts learned and proficiency with math practices. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• Cumulative/Benchmark Assessments found at the end of Topics 2, 4, 5, and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the school year; results can be used to determine intervention. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials offer ongoing formative and summative assessments, which include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Following Lesson Quizzes, Topic Assessments, Topic Performance Task and Cumulative/Benchmark Assessments Scoring Guides are provided. Teachers can also assign these assessments online where they are auto-scored and differentiated intervention is automatically assigned to students based on their scores. Examples from the Teacher Resource include:

• Lesson 1-9, Understand Scientific Notation, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments. Intervention 0-3 Points. On-Level 4 Points. Advanced 5 Points. You may opt to have students take the Lesson Quiz online. The Lesson Quiz will be automatically scored and appropriate remediation, practice, or enrichment will be assigned based on student performance.” 
• Topic 3, Use Functions to Model Relationships, Topic Assessment, Form A, “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 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.”
• Topic 7, Understand and Apply the Pythagorean Theorem, Performance Task, Form A, students solve Item 2, “A triangular bookshelf has a base of 10 inches, 12 inches, and 18 inches. Will the bookshelf fit in the corner of a square living room? Explain.” Two charts are provided for the teacher, Item Analysis for Diagnosis and Intervention and Scoring Rubric for forms A and B. The Item Analysis for Diagnosis and Intervention Chart contains information to help the teacher with RTI such as DOK, MDIS, and standard. The scoring rubric provides the teacher with solutions and scoring explanations. “Item 2, Form A 2 Points: Correct answer and explanation. 1 Point: Correct answer or explanation.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet 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 Resource provides a coherence section which enhances the opportunity to scaffold instruction by identifying prerequisite skills needed. All lessons include instructional notes and classroom strategies in the side matter labeled ETP, Effective Teaching Practices. ETP notes provide teachers with sample questions, differentiation strategies, discussion questions, possible misconceptions, and student “look fors” to assist in making content accessible to all learners. Additionally, the Solve and Discuss It! Section provides teachers with Before, During, and After instruction notes to help scaffold learning for students. Examples from the Teacher Resource include:

• Lesson 2-3, Solve Multistep Equations, Example 1, ETP: Use and Connect Mathematical Representations, “Q: What does x represent in this problem? [Sample answer: The variable, x, is the number of miles that the teacher rode on Thursday.] Q: What does the expression 4x + 3 represent? [Sample answer: The expression 4x + 3 is the total number of miles that the teacher rode Monday through Wednesday.] Q: Why would you use 2(x + 7) instead of x + 7 + x + 7 in the equation to represent the distance ridden on Friday and Saturday? Explain? [Sample answer: It is easier to read the right side of the equation as x + 2(x + 7).]”
• Lesson 4-4, Interpret Two-Way Frequency Tables, Example 3, ETP: Pose Purposeful Questions, “Q: How is this table different from the tables in Examples 1 and 2? Is it still a two-way frequency table? Explain. [Yes: Sample answer: This table shows three categories under Hair Color but it is still a two-way relative frequency table because it compares two types of related data- Eye Color and Hair Color.]”
• Lesson 8-1, Find Surface Area of Three Dimensional Figures, Explore It!, ETP: Before, “1. Introduce the Problem. Provide scrap paper and scissors, as needed. 2. Check for Understanding of the Problem. Engage students with the problem by asking: What objects in the real world are shaped like tubes or cylinders?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Each lesson contains Response to Intervention and Enrichment strategies in each lesson. Additional Examples and Additional Practice are provided if students need more support. At the end of each lesson Differentiated Intervention is provided for Intervention, On-Level, and Advanced learners. Examples from the Teacher Resource include:

• Lesson 3-3, Compare Linear and Nonlinear Functions, Response to Intervention, “Use with Example 1: Some students may need to review how to identify the constant of proportionality and represent proportional relationships using equations. Q: What is the unit rate for this situation: Sidney purchased three video games for $45.75. Each game cost the same. What was the cost for one video game? Q: Write an equation to represent the total cost for the number of movie tickets purchased, where c represents the total cost, t represents the number of movie tickets purchased, and each movie ticket costs $9.75. What is the constant of proportionality?”
• Lesson 5-2 Solve Systems by Graphing, Enrichment, “Use with Example 1, Challenge students to further explore graphing linear systems to find a solution. Analyze Example 1. Q: How could you change the equations so that they would not be a solution? Q: Give two possible equations that would have infinitely many solutions. Q: Give an equation for a third plan that also costs $95 for 100 minutes. Then, explain what the equation means in context.”
• Lesson 7-2 Understand the Converse of thePythagorean Theorem, Differentiate Intervention, Reteach to Build Understanding, Problem 1, “The side lengths of a triangle are 8, 11, and $$\sqrt{185}$$ inches. Is the triangle a right triangle? Explain. Which side lengths are a,b, and c? a = _, b = _, c = __.”

The instructional materials reviewed for enVision Mathematics 2021 Common Core Grade 8 meet expectations that materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Each lesson begins with a Problem-Based Learning activity, Solve & Discuss It, Explore It or Explain it! that offer multiple entry-points. 3-Act Mathematical Modeling tasks and Performance Tasks also include questions with multiple entry points that can be solved using a variety of representations. Examples from the Teacher Resource include:

• Topic 1, Real Numbers, 3-Act Mathematical Modeling, Hard-Working Organs, students are shown a video and then encouraged to consider the situation and ask any questions that come to mind. Teachers pose the Main Question, “How many times does your heart beat in a decade? How does that number compare to the number of breaths you take in a decade?” Teachers are given questions and tips to facilitate discussion about the 3-Act Mathematical Modeling activities, “Why do you think your prediction is the answer to the Main Question? Who had a similar prediction? How many agree with that prediction? Who has a different prediction?” 
• Lesson  3-5, Intervals of Increase and Decrease, Solve & Discuss It!, “Martin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill. How do you think the routes will differ? What do you think about the relationship of speed and time? How do the characteristics of each route affect Martin’s travel time and speed?” 
• Topic 7, Understand and Apply the Pythagorean Theorem, Performance Task Form A, Item 4, “Once he has completed the table, Cameron decides to make corner shelves from left-over triangular boards. The side lengths of the boards are 18 inches, 18 inches, and 24 inches. Part A. Can Cameron use the boards as they are for his corner shelves? Explain. Part B. Cameron decides to cut down the left-over boards. He wants two sides of each shelf, which will fit in the corner, to have the same side length. To the nearest whole inch, what is the length of each side of the largest corner shelf Cameron can make using the boards?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Each lesson contains instructional strategies for Emerging, Developing, and Expanding English Language Learners. Additionally, the Language Support Handbook provides Topic and Lesson instructional support and online academic vocabulary activities. Examples from the Teacher Resource include:

• Lesson 2-3, Solve Multistep Equations, English Language Learners, “Developing: See Example 1. Have students reread the problem and summarize the known information and the question aloud. Ask students to share their summaries. Q: What do the expressions on each side of the equation represent? Q: Why can the Monday-Wednesday expression be set equal to the sum of Thursday, Friday, and Saturday?”
• Lesson 5-4, Solve Systems by Elimination, English Language Learners, “Expanding: Example 1. Students will benefit from articulating how the elimination method compares with the substitution method. Write this system on the board. 3x + 2y = 3. x + y = 2. Divide students into two groups. Have one group solve the system by substitution and the other group by elimination. Then, have the groups share their work. Have students describe their solution processes and compare their descriptions.”
• Lesson 7-1, Understand the Pythagorean Theorem, English Language Learners, “Entering: Ask students to review Example 1. Q: What is a right triangle? [A right triangle is a triangle that has an angle that measures 90°.] Q: What word(s) describe the sides that form the right angle? [Sample answer: Perpendicular, adjacent] Write c2on the board. Then erase the c and write 5. Say: Substitute 5 for c. Q: What does substitute mean? [Sample answers: Change or replace].”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Each lesson provides an Enrichment side note with instructional strategies for advanced learners. The Problem-Based Learning activity provides instructional strategies During the lesson for Early Finishers. A Challenge question is presented in the teacher side notes for Practice & Problem Solving. Examples from the Teacher Resource include:

• Lesson 2-7, Analyze Linear Equations: y = mx, Enrichment, “Use with Example 3, Challenge advanced students to compute slope using decimal values and then to write and graph a linear equation. Q: What is the slope of the line through the points (1.6, 6.4) and (2.3, 9.2)? Q: What is the equation of the line?”
• Lesson 4-3, Use Linear Models to Make Predictions, Solve & Discuss It!, ETP: During, “Early Finishers, How would the problem change if Bao wants to burn 8,000 calories per week?”
• Lesson 8-2, Find Volume of Cylinders, Practice & Problem Solving, Item 13, “Challenge: Reason Quantitatively, Interested students might want to explore how a change in the radius can impact the height of a cylinder when the volume is the same. Q: Two cylinders have the same volume. Cylinder X has a height of 10 inches. The radius of Cylinder Y is twice as long as the radius of Cylinder X. What is the height of Cylinder Y? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics.

Different cultural names and situations are represented. Role names are used instead of pronouns referencing gender. Objects, animals, and cartoon drawings are used in place of actual people. Examples from the Teacher Resource include:

• Lesson 1-7, More Properties of Integer Exponents, Explore It!, two males are shown competing in a work-out: one caucasion and one African American. Students determine a representation between the set number and the number of sit-ups.
• Lesson 4-4, Interpret Two-Way Frequency Tables, Practice & Problem Solving, Item 7, students complete two-way frequency tables, “A company surveyed 200 people and asked which car model they preferred. Complete the two-way frequency table to show the results of the survey.”
• Lesson 8-2, Find Volume of Cylinders, Explain It!, students compare the size of two fish tanks, “Jenna and Ricardo are buying a new fish tank for the growing population of zebrafish in their science lab. Jennna says the tanks hold the same amount of water because they have the same dimensions. Ricardo says that he can fill the bottom of the rectangular tank with more cubes, so it can hold more water.”