The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that they assess grade-level content.

Each Topic contains diagnostic, formative, and summative assessments. Summative assessments provided by the program include: Topic Assessments Forms A and B, Topic Performance Tasks Forms A and B, and Cumulative/Benchmark Assessments. Assessments can be administered online or printed for paper/pencil format. No above grade-level assessment questions are present. Examples of grade-level assessment aligned to standards include: 

• Topic 1, Assessment Form A, Question 4, “Ron asked 18 classmates whether they prefer granola bars over muffins. He used a calculator to compare the number of classmates who said yes to the total number he surveyed. The calculator showed the result as 0.66666667. Part A: Write this number as a fraction. Part B: How many students prefer granola bars over muffins?” (8.NS.1)
• Topic 3, Performance Task Form A, Question 3, “Hector makes a graph to show the height of a shot put after it is thrown. Describe the behavior of the shot put based on the graph.” (8.F.5)
• Topics 1-4, Cumulative/Benchmark Assessment, Question 15, “Students at a community college were asked a survey question. The two-way frequency table shows the responses from full-time students and part-time students. Is there evidence that responding yes was related to attending the college full-time or part-time? Explain.” (8.SP.4)
• Topic 6, Assessment Form B, Question 5, “Consider the figures on the coordinate plane. Part A: Which two figures are congruent? Part B: Describe the sequence of transformations that maps the congruent figures.” (8.G.2)
• Topics 1-8, Cumulative/Benchmark Assessment, Question 9, “Jennie has 177 more songs downloaded on her mp3 player than Diamond. Together, they have 895 songs downloaded. Part A: What systems of equations could be used to determine how many songs each girl has downloaded? Part B: How many songs does each girl have?” (8.EE.8)

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations for spending a majority of instructional time on major work of the grade. 

• The approximate number of Topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6 out of 8, which is approximately 75%.
• The number of lessons (content-focused lessons, 3-Act Mathematical Modeling tasks, projects, Topic Reviews, and assessments) devoted to major work of the grade (including supporting work connected to the major work) is 68 out of 84, which is approximately 81%.
• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 149 out of 176, which is approximately 85%. 

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each Topic. As a result, approximately 81% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. Examples from the Teacher Resource include:

• Lesson 1-7, More Properties of Integer Exponents, Visual Learning, Example 3, students connect properties of exponents to irrational numbers, “Write the expression $$\frac{1}{7^{-3}}$$ with a positive exponent. Try It! Write each expression using positive exponents. a. $$\frac{1}{5^{-3}}$$ b. $$\frac{1}{2^{-6}}$$” This example connects the supporting work of 8.NS.1, know that numbers that are not rational are called irrational to the major work of 8.EE.1, know and apply the properties of integer exponents to generate equivalent numerical expressions.
• Lesson 4-1, Construct and Interpret Scatterplots, Visual Learning, Example 3, and Key Concept, students construct a scatter plot to model paired data and utilize a scatter plot to identify and interpret the relationship between paired data, “How do you know that the scatter plot does not show an association between the number of minutes played and the number of fouls committed? How do gaps and clusters help you understand how a scatter plot shows the relationship between paired data?” These questions connect the supporting work of 8.SP.1, construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between quantities to the major work of 8.F.4, use functions to model relationships between quantities.
• Lesson 4-3, Use Linear Models to Make Predictions, Visual Learning, Example 1, students write an equation for scatter plot trend lines and make predictions, “Michaela is a speed skater and hopes to compete in future Olympic games. She researched the winning times of the past 50 years. If the trend in faster speeds continues at the same rate, how can she use the information to predict what might be the time to beat in 2026?” This example connects the supporting work of 8.SP.3, use the equation of linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept to the major work of 8.F.4, construct a function to model a linear relationship between two quantities.
• Lesson 8-2, Find Volume of Cylinders, Visual Learning, Example 1, students apply their previous knowledge of solving linear equations when finding the volume of cylinders, “Jenna and Ricardo need to buy a tank that is large enough for 25 zebra fish. The tank needs to have a volume of 2,310 cubic inches. How can Jenna and Richardo determine whether the cylindrical tank can hold the zebrafish?” This example connects the supporting work of 8.G.9, know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems to the major work of 8.EE.2, use square root and cube root symbols to represent solutions to equations of the form x$$^2$$ = p and x$$^3$$ = p, where p is a positive rational number.
• Lesson 8-4, Find Volume of Spheres, Visual Learning, Do You Know How?, Item 4, students solve linear equations as they find the volume of cylinders, “Clarissa has a decorative bulb in the shape of a sphere. If it has a radius of 3 inches, what is its volume? Use 3.14 for $$\pi$$.” This question connects the supporting work of 8.G.9, know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems to the major work of 8.EE.7, solve linear equations in one variable.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the amount of content designated for one grade-level is viable for one year. As designed, the instructional materials can be completed in 152-176 days.

According to the Pacing Guide in the Teacher Resource, Program Overview, “enVision Mathematics 6-8 was designed to provide students rich opportunities to build understanding of important new mathematical concepts, develop fluency with key skills necessary for success in algebra, and to gain proficiency with the habits of mind and thinking dispositions of proficient mathematical students. To achieve these goals, the program includes content-focused lessons, 3-Act Mathematical Modeling lessons, STEM projects, and Pick a Project. All of these instructional activities are integral to helping students achieve success, and the pacing of the program reflects this. Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 day for the enVisions STEM project and/or Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessment.” For example:

• There are 8 Topics with 52 content-focused lessons for a total of 104 instructional days.
• Each of the 8 Topics contains a 3-Act Mathematical Modeling Lesson for a total of 8-16 instructional days.
• Each of the 8 Topics contains a STEM Project/Pick a Project for a total of 8 instruction days.
• Each of the 8 Topics contains a Topic Review and Topic Assessment for a total of 16 instructional days. 
• Materials allow 16-32 additional instructional days for remediation, fluency practice, differentiation, and other assessments.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations for the materials being consistent with the progressions in the Standards.

The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. According to the Teacher Resource, Program Overview, “Connections to content in previous grades and in future grades are highlighted in the Coherence page of the Topic Overview in the Teacher’s Edition.” These sections are labeled Look Back and Look Ahead. Examples from the Teacher Resource include:

• Topic 1 Overview, Real Numbers, Math Background, Coherence, “In Grade 7, students learned about rational numbers and integers. They performed integer operations in Topic 1. Seventh graders learned that writing an expression in different forms can help when solving problems. The work students do in this Topic connects directly to Topic 8: Solve Problems Involving Surface Area and Volume: students will use what they learned about squares, square roots, and the irrational number $$\pi$$ to calculate the surface area of solids, volume of cones, cylinders, spheres or find the length of the radius. In Grade 9, students will explain why the sum or product of two rational numbers is rational. They will also justify the sum of a rational number and an irrational number is irrational. In addition, they will recognize that the product of a nonzero rational number and an irrational number is irrational. In Grade 9, students will connect their understanding of rational numbers and integer exponents to learn about rational exponents. They will write and evaluate expressions involving radical and rational exponents using the properties of exponents.” 
• Topic 2 Overview, Analyze and Solve Linear Equations, Math Background, Coherence, “In Grade 7 students learned to understand and write expressions by using variables to represent unknown quantities to solve problems. They used what they learned about order of operations to analyze and write equivalent expressions and solve multi-step equations using the Distributive Property. In Grade 9, students will rewrite an equation in an equivalent form. They will learn strategies to solve problems by manipulating complex equations into simpler equations. 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.”
• Topic 5 Overview, Analyze and Solve System of Linear Equations, Math Background, Coherence, “In Grade 7, students learned how to write expressions to represent situations, and to solve one-step and two-step equations. Earlier in Grade 8, In Topic 2, students reviewed how to solve one-step and 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 make given line graphs. In Algebra, students will write equations in two or more variables to represent relationships between quantities, and graph the equations on the coordinate plane. They will continue to work with systems of equations to solve simple systems of one linear equation, one quadratic equation in two variables, both graphically and algebraically.” 

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The Solve & Discuss It! presents students with high-interest problems that embed new math ideas, connect prior knowledge to new learning and provide multiple entry points. Example problems are highly visual, provide guided instruction and formalize the mathematics of the lesson. Try It! provides problems that can be used as formative assessment following Example problems and Convince Me! provides problems that connect back to the Essential Understanding of the lesson. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept and skill application. Examples from the Teacher Resource include:

• Lesson 1-1, Rational Numbers as Decimals, Visual Learning, Example 1, Try It!, students represent the decimal expansion of a number as a rational number, “In another baseball division, one team had a winning percentage of 0.444… What fraction of their games did this team win?” (8.NS.1)
• Lesson 1-2, Understanding Irrational Numbers, Visual Learning, Example 1, Convince Me!, students classify a number as rational or irrational, “Jen classifies the number 4.567 as irrational because it does not repeat. Is Jen correct?” (8.NS.1)
• Lesson 1-5, Solve Equations Using Square Roots and Cube Roots, Solve & Discuss It!, students solve equations and problems, in real-world context, involving square roots and cube roots, “Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model. What are the dimensions of the possible models that she can build? How many blocks would Janine use for each model? Explain.” (8.EE.2)
• Lesson 3-6, Sketch Functions From Verbal Descriptions, Example 2, students analyze and interpret the sketch of a graph of a function, “Danika sketched the relationship between altitude and time for one of her parasailing flights. Describe the behavior of the function in each interval based on her sketch.” (8.F.5)
• Lesson 4-5, Interpret Two-Way Relative Frequency Tables, Do You Know How?, Items 4-6, students construct two-way relative frequency tables and compare and make conjectures about the data displayed, “In 4-6, use the table. Round to the nearest percent. What percent of the people surveyed have artistic ability? What percent of left-handed people surveyed have artistic ability? What percent of the people who have artistic ability are left-handed?” (8.SP.4) 
• Lesson 6-4, Compose Transformations, Practice & Problem Solving, Question 11, students describe and perform a sequence of transformations and apply their knowledge of transformations to solve problems, “A student says that he was rearranging furniture at home and he used a glide reflection to move a table with legs from one side of the room to the other. Will a glide reflection result in a functioning table? Explain.” (8.G.1abc & 8.G.3)

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Each Lesson Overview contains a Coherence section that connects learning to prior grades. Examples include:

• Lesson 2-1, Combine Like Terms to Solve Equations, Lesson Overview, Coherence, “Students will be able to combine like terms, solve equations with like terms on one side of the equation, and make sense of scenarios and represent them with equations.” (8.EE.7b) “In Grade 7, students used variables to represent quantities and created simple equations to solve problems.”
• Lesson 6-1, Analyze Translations, Lesson Overview, Coherence, “Students will be able to use coordinates to describe the rules of a translation. Students will be able to translate a two-dimensional figure on a coordinate plane by mapping each of its vertices.” (6.G.1a,b,c & 6.G.3). “In Grade 6, students drew polygons on the coordinate plane given coordinates of the vertices.”
• Lesson 8-1, Find Surface Area of Three-Dimensional Figures, Lesson Overview, Coherence, “Students will be able to calculate the surface areas of cylinders, cones, and spheres.” (8.G.9) “Previously in Grade 7, students found the surface areas of cubes and right prisms and calculated the area of a circle.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Topics are divided into Lessons focused on domains. Grade 8 standards are clearly identified in each Topic Planner found in the Topic Overview. Additionally, each lesson identifies the Content Standards in the Mathematics Overview. Examples from the Teacher Resource include:

• Lesson 1-3, Compare and Order Real Numbers, Lesson Overview, Mathematics Objective, “Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g. $$\pi^2$$).” (8.NS.2)
• Lesson 3-5, Intervals of Increase and Decrease, Lesson Overview, Mathematics Objective, “Describe the behavior of a function in different intervals.” (8.F.5)
• Lesson 4-4 Interpret Two-Way Frequency Tables, Lesson Overview, Mathematics Objective, “Organize paired categorical data into a two-way frequency table. Compare and make conjectures about data displayed in a two-way frequency table.” (8.SP.4)
• Lesson 5-2, Solve Systems by Graphing, Lesson Overview, Mathematics Objective, “Create and examine graphs of linear systems of equations to determine the solution.” (8.EE.8a, 8.EE.8c)
• Lesson 6-6, Describe Dilations, Lesson Overview, Mathematics Objective, “Verify the properties of a dilation. Graph the image of a dilation given a fixed center and a common scale factor.” (8.G.3, 8.G.4) 

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples from the Teacher Resource include:

• Lesson 2-7, Analyze Linear Equations y=mx, Visual Learning, Example 1, students use the concept of similarity to define slope, “The students in Meg’s class are building a fence around the class garden. How can they use the pricing for the different lengths of fencing to determine the cost for 50 feet of fencing?” The steps to this problem are outlined for the students and include supporting text, “Drawing lines to find the rise and the run creates a right triangle.” and “Notice that the ratios of the $$\frac{rise}{run}$$ are equivalent, so the slope of the line is constant.” This example connects the work of 8.EE to the work of 8.G.
• Lesson 3-4, Construct Functions to Model Linear Relationships, Visual Learning, Example 2, students work with functions and proportional relationships, “The cost to manufacture 5 toys is $17.50: the cost to manufacture 10 toys is $30. Construct a linear function in the form of y = mx + b that represents the relationship between the number of toys produced and the cost of producing them.” This example connects the work of 8.F.B to the work of 8.EE.B
• Lesson 6-10, Angle-Angle Triangle Similarity, Visual Learning, Example 1, students work with congruent and similar figures and make connections to proportional relationships, “Justin designs another pair of flags for another model sailboat. The larger flag is 1.5 times the size of the smaller flag. How can Justin determine whether the triangles that represent the flags are similar?” This example connects the work of 8.G to the work of 8.EE. 
• Lesson 7-3, Apply the Pythagorean Theorem to Solve Problems, Visual Learning, Example 1, students determine which length of wood to use for a kite using the Pythagorean Theorem while solving equations. Three pieces of wood with measures of 28 in., 35 in., and 49 in. and an image of the wooden dowel with sides 28 in. and 21 in are shown. Example 1 states, “Kiana is using a kit to build the kite shown. The kit includes three different lengths of wooden dowels. How can Kiana decide which pieces of wood to use as the diagonal braces for the top or bottom of the kite?” This example connects the work of 8.G to the work of 8.EE. 
• Lesson 8-3, Find Volume of Cones, Visual Learning, Additional Example 2, students recognize that irrational numbers must be rounded because their decimal expansion does not terminate or repeat while finding volume of cones, “A small hanging planter is shaped like a cone. Its radius is 1.5 inches and its slant height is 5 inches. What is the greatest volume of the potting soil that will fit in the hanging planter?” This example connects the work of 8.NS to the work of 8.G.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples include:

• Lesson 1-2, Understand Irrational Numbers, Visual Learning, Example 1, students develop conceptual understanding of classifying rational and irrational numbers, “The Venn diagram shows the relationships among rational numbers. How would you classify the number 0.24758326… ?” Teachers then ask, “Is the given decimal a terminating decimal? Explain. Suppose the given decimal was written as 0.24758326. Would this decimal be a rational number? Explain.” (8.NS.1)
• Lesson 2-9, Analyze Linear Equations: y = mx + b, Visual Learning, Example 1, students develop conceptual understanding of recognizing the mathematical relationship between the equation of a line and the graph of that line, “The Middle School Student Council is organizing a dance that has $500 to pay for a DJ. DJ Dave will charge $200 for a set-up fee and the first hour, or $425 for a set-up fee and four hours. How can the Student Council determine whether they can afford to have DJ Dave play for 5 hours?” The teacher asks, “Why does 125 represent the y-intercept? Why does 75 represent the slope?” (8.EE.6)
• Lesson 3-2, Connect Representations of Functions, Visual Learning, Example 2, students deepen their understanding of functions as they explore graphs of linear and nonlinear functions, “How can you determine whether the relationship between side lengths and area is a function? How can you determine whether a relation in the table is a function? What do you notice about the shape of this graph? Is it a function?” (8.F.1)
• Lesson 4-4, Interpret Two-Way Frequency Tables, Visual Learning, Example 3, students develop conceptual understanding of representing, finding, and comparing sub-categories of data, “Two hundred people responded to a survey. Of those who had green eyes, 7 had blonde hair, 9 had brown hair, and 2 had red hair. Of those who had brown eyes, 76 had blonde hair, 89 had brown hair, and 17 had red hair. Construct a two-way table to display these data. Then identify the least common combination of eye and hair color. Explain.” (8.SP.4)
• Lesson 6-5, Understand Congruent Figures, Solve and Discuss It!, students develop conceptual understanding of congruent geometric figures produced through reflections, rotations, and translations, “Simone plays a video game in which she moves shapes into empty spaces. After several rounds, her next move must fit the blue piece into the dashed space. How can Simone move the blue piece to fit in the space?” (8.G.2, 8.G.3)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice and Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept. Examples include:

• Lesson 1-1, Rational Numbers as Decimals, Practice & Problem Solving, Item 10, students independently demonstrate conceptual understanding of writing repeating decimals as fractions, “Thomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students. His calculator showed the result as 0.9333…. A. Write this number as a fraction. B. How many students said that summer break should be longer?” (8.NS.1)
• Lesson 2-2, Solve Equations with Variables on Both Sides, Do You Know How?, Item 4, students independently demonstrate conceptual understanding of solving equations with like terms on both sides of the equation, “Maria and Liam work in a banquet hall. Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 puls a 10% commission on his food sales. What amount of food sales will result in Maria and LIam earning the same amount for the week?” (8.EE.7b)
• Lesson 3-1, Understand Relations and Functions, Do You Understand?, Item 3, students independently demonstrate their understanding of whether functions are relations, “Is a relation always a function? Is a function always a relation? Explain.” (8.F.1)
• Lesson 4-1, Construct and Interpret Scatter Plots, Visual Learning, Example 3, Try It!, students independently demonstrate understanding of scatter plots, “Avery also tracks the number of minutes a player plays and the number of points the player scored. Describe the association between the two data sets. Tell what the association suggests.” A graph of Basketball Scoring with minutes played and points scored is shown. (8.SP.1)
• Lesson 7-1, Understand the Pythagorean Theorem, Practice & Problem Solving, Item 9, students independently demonstrate conceptual understanding of the Pythagorean Theorem, “What is the length of the hypotenuse of the triangle when x = 15? Round your answer to the nearest tenth of a unit.” A picture of a triangle with legs 4x + 4 and 3x is shown. (8.G.6 and 8.G.7)

The instructional materials for enVision Mathematics Common Core Grade 8 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It!, And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples Include:

• Lesson 1-2, Understand Irrational Numbers, Do You Know How?, Item 4, students identify rational and irrational numbers, “Is the number 65.4349224... rational or irrational? Explain.” (8.NS.1)
• Lesson 2-2, Solve Equations with Variables on Both Sides, Visual Learning, Example 3, Try It!, students solve equations with variables on both sides, “Solve the equation. 96 - 4.5y - 3.2y = 5.6y + 42.80.” (8.EE.7)
• Lesson 3-1, Understand Relations and Functions, Do You Know How?, Item 5, students identify whether a relation is a function, “Is the relation shown below a function? Explain.” Students are shown a table with inputs 3, 4, 1, 5, 2 and corresponding outputs of 4, 6, 2, 8, 5. (8.F.1)
• Lesson 4-4, Interpret Two-Way Frequency Tables, Do You Know How?, Item 4, students explain how to display and interpret relationships between paired categorical data, “A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations.” A partially completed table of grade levels and basketball shots is provided. (8.SP.4)
• Lesson 6-4, Compose Transformations, Visual Learning, Example 1, Try It!, students describe a sequence of transformations involving floor plans using a coordinate plane, “Ava decided to move the cabinet to the opposite wall. What sequence of transformations moves the cabinet to its new position?” (8.G.3)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Fluency Practice page which engages students in fluency activities. Examples include:

• Topic 1 Review, Fluency Practice, students solve one-step equations, including those involving square roots and cube roots, “Crisscrossed: Solve each equation. Write your answers in the cross-number puzzle below. Each digit, negative sign, and decimal point of your answer goes in its own box. A. -377 = x - 1,000; C. x$$^3$$= -8; D. x + 7 = -209; F. x + 19 = -9.” (8.EE.7)
• Lesson 1-3, Compare and Order Real Numbers, Practice & Problem Solving, Item 10, students compare real numbers in various forms, “Does $$\frac{1}{6}$$, -3, $$\sqrt7$$, -6/5, or 4.5 come first when the numbers are listed from least to greatest? Explain.” (8.NS.2)
• Lesson 2-3, Solve Multistep Equations, Practice & Problem Solving, Item 12, students solve multi-step equations using the distributive property, “What is the solution of the equation 3(x + 2) = 2(x + 5)?” (8.EE.7).
• Lesson 3-4, Construct Functions to Model Linear Relationships, Practice & Problem Solving, Item 7, students calculate slope in order to write a linear equation, “A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.” (8.F.4)
• Lesson 4-3, Use Linear Models to Make Predictions, Practice & Problem Solving, Item 6, students calculate rates of change and initial values in order to make predictions in linear relationships, “If x represents the number of years since 2000 and y represents the gas price, predict what the difference between the gas price in 2013 and 2001 is? Round to the nearest hundredth.” (8.SP.3)
• Lesson 7-3, Apply the Pythagorean Theorem to Solve Problems, Practice & Problems Solving, Item 9, students solve problems using a$$^2$$ + b$$^2$$ = c$$^2$$, “A stainless steel patio heater is shaped like a square pyramid. The length of one side of the base is 19.8 inches. The slant height is 92.8 inches. What is the height of the heater? Round to the nearest tenth of an inch.” (8.G.7)

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied. 

The instructional materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues. For example:

• Topic 1, STEM Project, Going, Going, Gone, students use real numbers such as rational and irrational numbers to show the depletion rate of a natural resource, "Natural resources depletion is an important issue facing the world. Suppose a natural resource is being depleted at the rate of 1.333% per year. If there were 300 million tons of this resource in 2005, and there are no new discoveries, how much will be left in year 2045?” (8.NS.1, 8.NS.2, 8.EE.1,and 8.EE.2)
• Topic 3, STEM Project, Modeling Population Growth, students use population data to develop linear equations that model growth in urban areas. "Develop a linear model that presents the data on urbanization in India and the U.S. How does the model compare with those of population growth developed in the last topic?” (8.F.4)
• Topic 1, 3-Act Mathematical Modeling: Hard-Working Organs, Question 2, students compare large numbers, “How many times does your heartbeat in a decade? How does that number compare to the number of breaths you take in a decade?” (8.EE.1 and 8.EE.3)
• Topic 5, 3-Act Mathematical Modeling: Up and Downs, Questions 2, students develop a mathematical model to represent and propose a solution to a problem situation involving a system of equations. Students are presented with the main question, “Which route is faster?” Teachers ask, “Does your answer match the answer in the video? If not, what are some reasons that would explain the difference? Would you change your model now that you know the answer? Explain.” (8.EE.8)
• Topic 6, STEM Project, Forest Health, students use similar triangles and ratios to gather and interpret data, "Students use ratios and similar triangles to measure the health of various forest elements. Using what they know about similar triangles will allow students to measure tree heights. Students can use equivalent ratios to help generalize data to larger sections of forests.” (8.G.3 and 8.G.4)
• Topic 8, 3-Act Mathematical Modeling: Measure Up, students determine whether the liquid in one container will fit into a container with a different shape. Question 15, "Suppose you have a graduated cylinder half the height of the one in the video. How wide does the cylinder need to be to hold the liquid in the flask?" (8.G.9)

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Lesson 1-3, Compare and Order Real Numbers, Practice & Practice Solving, Item 11, students compare irrational numbers, “A museum director wants to hang the painting on a wall. To the nearest foot, how tall does the wall need to be?” A painting is shown with a height of $$\sqrt90$$ ft. (8.NS.2)
• Topic 3, Pick a Project 3D, students design a video game element describing how perimeter and area dimensions change in regards to linear and non-linear relationships. “Make mock-ups of your designs and describe how the game will use the boxes. Calculate the interior and exterior perimeters and areas of each text box. Describe how perimeter and area relate to linear and nonlinear relationships.” (8.F.4)
• Lesson 3-3, Compare Linear and Nonlinear Functions, Practice & Problem Solving, Item 15, students apply their understanding of functions to solve real-world problems using equations and a table, “The students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time. Which snack did the students eat at a faster rate? Explain.” (8.F.2 and 8.F.3)
• Lesson 4-3, Use Linear Models to Make Predictions, Practice & Problem Solving, Item 10, students make predictions using slope of a trend line, “The graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows. What values could the actual temperature be after the freezer was on for five minutes? What is the equation that best represents the data in the graph? Use the equation for the trend line, how cold will the temperature be after 30 minutes? Do you think that will happen? Explain.” (8.SP.3)
• Lesson 5-3, Solve Systems by Substitution, Practice & Problem Solving, Item 15, students use substitution to solve real-world problems, “The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at $6.50. Let s represent the number of student tickets sold at $3.50 each. a. If all 500 seats are filled at the performance, how many of each type of ticket must have been sold for the member to raise exactly $2,050? b. At one performance there were 3 times as many student tickets sold as adult tickets. If there were 480 tickets sold at that performance, how much below the goal of $2,050 did ticket sales fall?” (8.EE.8)
• Topic 7, Pick a Project 7A, students find the distances in a coordinate plane they mapped of their community route, “On a coordinate grid, map out a metric bike route through your community, increase at least 5 stops. Use at least three diagonal line segments to represent different parts of your route. Calculate the distance between the stops.” (8.G.8)

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

All three aspects of rigor are present independently throughout the program materials. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, and application include:

• Topic 5 Review, Fluency Practice, students solve multi-step equations using the distributive property, “Pathfinder: Shade a path from START to FINISH. Follow the solutions to the equations from least to greatest. You can only move up, down, right, or left. 2(d + 1) = -38; 7x - 2(x - 11) = -103; 2w - 5(w + 4) = -14; -4(h - 2) = 8; 4(3 - 5k) = 92.” (8.EE.7)
• Lesson 6-1, Analyze Transitions, Solve & Discuss It!, students develop conceptual understanding of translating two dimensional figures, “Ashanti draws a trapezoid on the coordinate plane and labels it Figure 1. Then she draws Figure 2. How can she determine whether the figures have the same side lengths and the same angle measure?” (8.G.1 and 8.G.3)
• Lesson 7-4, Find Distance in the Coordinate Plane, Practice & Problem Solving, Item 11, students use application of the Pythagorean Theorem to calculate distance on the coordinate plane, “Suppose a park is located 3.6 miles east of your home. The library is 4.8 miles north of the park. What is the shortest distance between your home and the library?” (8.G.8)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Lesson 1-6, Use Properties of Integer Exponents, Do You Know How?, Item 7, students use properties of integer exponents, “A billboard has the given dimensions, 7$$^2$$ ft and 4$$^2$$ ft. Using exponents, write two equivalent expressions for the area of a rectangle.” This question develops conceptual understanding and procedural skill of 8.EE.1, know and apply the properties of integer exponents to generate equivalent numerical expressions.
• Lesson 2-1, Combine Like Terms to Solve Equations, Do You Know How?, Item 4, students solve equations in real-world context, “Henry is following the recipe card (shown) to make a cake. He has 95 cups of flour. How many cakes can Henry make?” This question develops procedural skill and application of 8.EE.7b, solve linear equations with rational number coefficients.
• Lesson 3-5, Intervals of Increase and Decrease, Practice & Problem Solving, Item 10, students describe the behavior of a function in different intervals, “The graph shows the speed of a car over time. What might the constant intervals in the function represent?” This question develops conceptual understanding and application of 8.F.5, describe qualitatively the functional relationship between two quantities by analyzing a graph.
• Lesson 4-1, Construct and Interpret Scatter Plots, Practice & Problem Solving, Item 9, students identify and interpret clusters, gaps, and outliers on a scatter plot, “The table shows the number of painters and sculptures enrolled in seven art schools. Jashar makes an incorrect scatter plot to represent the data. a. What error did Jashar likely make? b. Explain the relationship between the number of painters and sculptors enrolled in the art schools. c. Jashar’s scatter plot shows two possible outliers. Identify them and explain why they are outliers.” This question develops conceptual understanding and application of 8.SP.1, construct and interpret scatter plots for bivariate data to investigate patterns of association between two quantities.
• Lesson 4-4, Interpret Two-Way Frequency Tables, Do You Know How?, Item 4, students compare and make conjectures about data displayed in a two-way frequency table, “A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations” This question develops conceptual understanding and procedural skill of 8.SP.4, construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
• Lesson 6-3, Analyze Rotations, Practice & Problem Solving, Item12, students determine how a rotation affects a two-dimensional figure, “An architect is designing a new windmill with four sails. In her sketch, the sails’ center of rotation is the origin, (0,0), and the tip of one of the sails, Point Q, has coordinates (2,-3). She wants to make another sketch that shows the windmill after the sails have rotated 270° about the center of rotation. What would be the coordinates of ?” This question develops procedural skill and application of 8.G.3,describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All 8 MPs are clearly identified throughout the materials, with few or no exceptions. Math Practices identification in this program according to the Teacher Resource Program Overview include:

• Materials provide a Math Practices and Problem Solving Handbook for students, “A great resource to help students build on and enhance their mathematical thinking and habits of mind.” This handbook explains math practices in student-friendly language and digital animation videos for each math practice are also available.
• Opportunities to apply math practices are found in the Explore It, Explain It, and Solve & Discuss It portions of the lesson. “The Solve & Discuss It calls on students to draw on nearly all of the math practices, but especially sense-making and solution formulation as well as abstract and quantitative reasoning. The Explore It focuses students on mathematical modeling, generalizations, and structure of mathematical models. The Explain It emphasizes mathematical reasoning and argumentation. Students construct arguments to defend a claim or critique an argument defending a claim.”
• The Math Practices and Problem Solving Handbook Teacher Pages, “provide overviews of the math practices, offer instructional strategies to help students refine and enhance their thinking habits, and include student behaviors to listen and look for for each standard.”
• Each Topic Overview contains Math Practices Teacher Pages which include, “Two highlighted math practices with student behaviors to look for, and questions to help students become more proficient with these thinking habits.” For example, in Topic 8, Model with mathematics suggested questions state, “What patterns do you recognize in the formulas for the volume of a rectangular prism and the volume of a cylinder? What patterns do you recognize in the formulas for the volume of a cylinder and the volume of a cone?”
• Math Practices boxes found in the student text provide, “Reminders to be thinking about the application of the math practices as they solve problems.”
• Math Practices Run-in Heads found in the Practice & Problem Solving questions, “Remind students to apply the math practices as they solve problems.”

The majority of the time the MPs are used to enrich the mathematical content and are not treated separately. Examples include:

• MP1: Make sense of problems and persevere in solving them. Lesson 8-2, Find Volume of Cylinders, Practice & Problem Solving, Item 12, students make sense of cylinders as they determine the greatest volume a cylinder can hold, “A rectangular piece of cardboard with dimensions 6 inches by 8 inches is used to make the curved side of a cylinder-shaped container. Using this cardboard, what is the greatest volume the cylinder can hold? Explain.” 
• MP2: Reason abstractly and quantitatively. Lesson 1-1, Rational Numbers as Decimals, Practice & Problem Solving, Item 17, students expand their knowledge about rational numbers and the relationships with decimals and fractions when explaining their answer,” When writing a repeating decimal as a fraction, why does the fraction always have only 9s or 9s and 0s as digits in the denominator?” 
• MP4: Model with mathematics. Lesson 2-1, Combine Like Terms to Solve Equations, Visual Learning, Item 18, students combine like terms on one side of the equation to use inverse operations to solve, “Nathan bought one notebook and one binder for each of his college classes. The total cost of the notebooks and binders was $27.08. Draw a bar diagram to represent the situation. How many classes is Nathan taking?”
• MP5: Use appropriate tools strategically. Lesson 5-4, Solve Systems by Elimination, Practice & Problem Solving, Item 13b, students use elimination to solve for a linear system of equations, “A deli offers two platters of sandwiches. Platter A has 2 roast beef sandwiches and 3 turkey sandwiches. Platter B has 3 roast beef sandwiches and 2 turkey sandwiches. What is the cost of each sandwich?” A picture of Platters A and B with total cost are provided.
• MP6: Attend to precision. Lesson 2-8, Understand the y-intercept of a Line, Practice & Problem Solving, Item 11, students use the relationship between variables in a problem situation to explain the meaning of the y-intercept, “The line models the temperature on a certain winter day since sunrise. What is the y-intercept of the line? What does the y-intercept represent?” 
• MP7: Look for and make use of structure. Lesson 7-4, Find Distance in the Coordinate Plane, Practice & Problem Solving, Item 12, students use the structure of ordered pairs and the Pythagorean Theorem to find distance between two points, “Point B has coordinates (2, 1). The x-coordinate of coordinate A is -10. The distance between point A and point B is 15 units. What are the possible coordinates of point A?”
• MP8: Look for and express regularity in repeated reasoning. Lesson 5-1, Estimate Solutions by Inspection, Practice & Problem Solving, Item 13, students compare the equations in a linear system to determine the number of solutions of the system, “Describe a situation that can be represented by using this system of equations. Inspect the system to determine the number of solutions and interpret the solution within the context of your solution. y = 2x + 10 and y = x + 15.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

The materials do not attend to the full meaning of MP5: Use appropriate tools strategically. Examples include: 

• Lesson 1-3: Compare and Order Real Numbers, Visual Learning, Example 3, “Compare and order the numbers below. $$\pi^2$$, 9$$\frac{1}{2}$$, 9.8, 9.5, $$\sqrt{94}$$. Step 2, Plot each approximation on a number line.” Students are instructed to solve using a number line and one is provided in the text.
• Lesson 1-9, Understand Scientific Notation, Visual Learning, Example 3, “A. Kelly used a calculator to multiply large numbers. How can she write the number on her calculator screen in standard form? B. How can Charlie write the number on the calculator screen in standard form.” Use Appropriate Tools states, “Certain calculators may display scientific notation using the symbol EE or E. The number that follows is the power of 10.” Example 3a states, “Kelly used the calculator to multiply large numbers. How can she write that number on her calculator screen in standard form, 3.5 x 1015?” Students are not using appropriate tools or choosing appropriate tools because they are instructed on how to use calculators for scientific notation.
• Lesson 3-2, Connect Representations of Functions, Do You Understand? Item 2, “How can you use a graph to determine that a relationship is NOT a function?” Students are instructed to use a graph as a tool to identify a function. 
• Lesson 6-8, Angles, Lines, and Transversals, Solve & Discuss It!, Draw two parallel lines. Then draw a line that intersects both lines. Which angles have equal measures?” Use Appropriate Tools states, “What tools can you use to determine which angles have equal measures?” An image of a protractor, ruler, pencil, and straight edge are provided to students and they are instructed to draw a transversal to determine angle measures.

The materials do attend to the full meaning of the following MPs. For example:

• MP1: Make sense of problems and persevere in solving them. Lesson 2-5, Compare Proportional Relationships, Practice & Problem, Item 9, “Beth, Manuel, and Petra are collecting sponsors for a walk-a-thon. The equation y = 20x represents the amount of money Beth raises for walking x miles. The table shows the relationship between the number of miles Manuel walks and the amount of money will raise. Petra will earn $15 for each mile that she walks. a. In order to compare the proportional relationships, what quantities should you use to find the unit rate? b. Compare the amount of money raised per mile by the three people.” Students interpret and make sense of the quantities presented in real-world situations and identify the relationships between them using unit rates from different types of representations.
• MP2: Reason abstractly and quantitatively. Lesson 5-2, Solve Systems by Graphing, Teacher’s Edition, Explore It!, students reason abstractly and quantitatively when they interpret graphs of linear systems of equations and make meaning by understanding that the solution is the intersection point(s). “Beth and Dante pass by the library as they walk home using separate straight paths. A. The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library. B. Write a system of equations that represents the paths taken by Beth and Dante. What does the point of intersection of the lines represent in this situation?”
• MP4: Model with mathematics. Lesson 6-4, Compose Transformations, Practice & Problem Solving, Item 8, “A family moves a table, shown as rectangle EFGH, by translating it 3 units left and 3 units down followed by a 90° rotation about the origin. Graph E’ F’ G’ H’ to show the new location of the table.” Students apply what they know about transformations and model how to move a table. 
• MP6: Attend to precision, Lesson 6-7, Understand Similar Figures, Do You Understand?, Item 2, “How do the angle measures and side lengths compare in similar figures?” Students perform a sequence of transformations to identify similar figures.
• MP7: Look for and make use of structure. Lesson 8-2, Find Volume of Cylinders, Do You Understand?, Item 2, “What two measurements do you need to know to find the volume of a cylinder?” Students recognize the relationship between the formulas for the volume of a rectangular prism and the volume of a cylinder.
• MP8: Look for and express regularity in repeated reasoning. Lesson 5-2, Solve Systems by Graphing, Do You Understand?, Item 2, “If a system has no solution what do you know about the lines being graphed?” Students create and examine graphs of linear systems of equations to determine the solution.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Lesson 1-5, Solve Equations Using Square Roots and Cube Roots, Do You Understand?, Item 3, students construct arguments related to the solutions of square and cube root equations. “There is an error in the work shown below. Explain the error and provide a correct solution. x$$^3$$ = 125.  $$\sqrt[3]{x^3}$$ = $$\sqrt[3]{125}$$. x = 5 and x = -5.”
• Lesson 2-5, Compare Proportional Relationships, Solve & Discuss It!, students use their understanding of proportional relationships to construct arguments and support their response, “Mei Li is going apple picking. She is choosing between two places. The cost of a crate of apples at each place is shown. Where should Mei Li go to pick her apples? Explain.”
• Lesson 3-1, Understand Relations and Functions, Practice & Problem Solving, Item 10, students use their understanding of relations and functions to justify their  arguments. “During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in). (0, 15), (5, 20), (10, 50), (15, 80), (20, 100), (25, 100). Is the relation a function? Explain.”
• Lesson 4-2, Analyze Linear Associations, Solve & Discuss It!, students construct arguments as they analyze bivariate data and connect it to linear associations. “What other factors should Angus also take into consideration to make a decision? Defend your response.”
• Lesson 6-2, Analyze Reflections, Practice & Problem Solving, Item 10, students use their understanding of reflections to construct arguments, “Your friend incorrectly says that the reflection of $$\vartriangle$$EFG to its image $$\vartriangle$$E’F’G’ is a reflection across the x-axis. a. What is your friend’s mistake? b. What is the correct description of the reflection?” 

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Lesson 1-4, Evaluate Square Roots and Cube Roots, Practice & Problem Solving, Item 15, students analyze the argument of others as they evaluate cubes and cube roots, “Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8. Is Diego correct? Explain.”
• Lesson 4-5, Interpret Two-Way Relative Frequency Tables, Do You Understand?, Item 3, students analyze the arguments of others as they find the relative frequency from a two-way table, “Maryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table. Do you agree? Explain why or why not.” 
• Lesson 5-3, Solve Systems by Substitution, Explain It!, students analyze the arguments of others as they graph a system of equations to determine the most cost effective cab company, “Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On Time Cabs will cost less. Do you agree with Jackson? Explain. Which taxi service company should Jackson call? Explain your reasoning.”
• Lesson 6-3, Analyze Rotations, Explore It!, students analyze the arguments of others as they explore a point on a circle 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 friends 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?”
• Lesson 8-1, Find Surface Area of Three-Dimensional Figures, Do You Understand?, Item 3, students analyze the arguments of others as they use formulas for polygons, “Aaron says that all cones with a base circumference of 8x inches will have the same surface area. Is Aaron correct? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in constructing viable arguments frequently throughout the program. Examples include:

• Lesson 1-6, Use Properties of Integer Exponents, Visual Learning, Example 4, Try It! Question 4d, “Write equivalent expressions using the properties of exponents. 8$$^9$$ ÷ 8$$^3$$” ETP (Effective Teaching Practices) Elicit and Use Evidence of Student Thinking teacher prompt states, “In part d, why not divide the exponent to get 8$$^3$$?”
• Lesson 2-2, Solve Equations with Variables on Both Sides, Visual Learning, Example 2, “Teresa earns a weekly salary of $925 and a 5% commission on her total sales. Ramon earns a weekly salary of $1,250 and a 3% commission on sales. What amount of sales, x, will result in them earning the same amount for the week?” ETP Pose Purposeful Questions teacher prompt states, “Could you solve this equation by subtracting 0.5x from both sides? Explain.” Within the example, the equation 0.5x + 925 = 0.3x + 1,250 is solved by first subtracting 0.3x from both sides.
• Lesson 6-1, Analyze Translations, Visual Learning, Example 3, “What is the rule that describes the translation that maps trapezoid PQRS onto trapezoid P’Q’R’S’?” ETP Pose Purposeful Questions teacher prompt states, “Suppose points P, Q, and R were translated 3 units right and 4 units up, but point S was not. Is this still a translation? Explain.” 

Teacher materials assist teachers in engaging students in analyzing the arguments of others frequently throughout the program. Examples include:

• Lesson 5-2, Solve Systems by Graphing, Explore It!, “Beth and Dante pass by the library as they walk home using separate straight paths. The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library.” ETP After teacher prompt states, “Have students present their graphs and discuss how they chose and drew their lines. Ask students how they could have drawn infinitely many different pairs of lines that intersect at the same point; a line can be defined by two points. You can choose one of those points to always be the point of intersection. Then have them explain how they found the slope and y-intercepts of their lines. Have students discuss what a system of equations is and how they can use the equations of their two lines to write a system of equations to represent the situation; a system of equations is a set of two or more equations that have the same unknowns.”
• Lesson 6-8, Angles, Lines, and Transversals, Visual Learning, Example 1, Try It!, “Which angles are congruent to 8? Which angles are supplementary to ∠8?” A diagram of parallel lines intersected by the transversal with angles labeled is shown. ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “Janine says that ∠8 and ∠2 are congruent because they are alternate interior angles. Do you agree? Explain.”
• Lesson 7-1, Understand the Pythagorean Theorem, Visual Learning, Example 1, an image of a triangle is provided. “is a right triangle with side lengths a, b, c. Construct a logical argument to show a$$^2$$+ b$$^2$$= c$$^2$$.” ETP Pose Purposeful Questions teacher prompt states, “How can you be certain that both larger squares are the same size? How can you justify that the areas of the smaller white squares inside the larger square are equal? What does the equation in the last section represent? Explain.” 

Teacher materials assist teachers in engaging students in both the construction of viable arguments and analyzing the arguments or reasoning of others frequently throughout the program. Each Topic Overview highlights specific Math Practices and suggests look fors in student behavior and provides questioning strategies. Examples include:

• Topic 6, Congruence and Similarity, Math Practices, look fors, “Mathematically proficient students: Justify their conclusions about reflections with mathematical ideas. Consider their reasoning about why they rotate figures a certain way or apply certain rules to the rotation of a figure. Make sense of accuracy in transformations and understand angles and congruence relationships. Use their understanding of the definition of the dilation to verify images that are the product of dilation.”
• Topic 6, Congruence and Similarity, Math Practices, questioning strategies, “What evidence did you use to support your solution? Are there similarities between reflections and rotations? Explain. How could you prove that figures are similar? How did you check whether your approach worked?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. Each Topic Overview provides a chart in the Topic Planner that lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow. Lesson practice includes questions to reinforce vocabulary comprehension and students write using math language to explain their thinking. Each Topic Review contains a Vocabulary Review section for students to review vocabulary taught in the Topic. Students have access to an Animated Glossary online in both English and Spanish. Examples include:

• Lesson 2-8, Understand the y-intercept of a Line, Visual Learning, Example 1, “The y-coordinate of the point where the line crosses the y-axis is the y-intercept.”
• Topic 4, Investigate Bivariate Data, Use Vocabulary in Writing, “Describe the scatter plot at the right. Use vocabulary terms in your description.” Students are provided a word bank containing, “categorical data, outlier(s), cluster(s), relative frequency, measurement data, and trend line.”
• Lesson 5-1, Estimate Solutions by Inspection, Visual Learning, Example 1, “A system of linear equations is formed by two or more linear equations that use the same variables. A solution to a system of linear equations is any ordered pair that makes all equations in the system true.”
• Lesson 6-3, Analyze Rotations, Visual Learning, Example 1, “A rotation is a transformation that turns a figure around a fixed point, called the center of rotation. The angle of rotation is the number of degrees the figure rotates. A positive angle of rotation turns a figure counterclockwise.”
• Topic 8, Solve Problems Involving Surface Area and Volume, Mid-Topic Checkpoint, Question 1, “Select all the statements that describe surface area and volume. Surface area is the sum of the areas of all the surfaces of a figure. Volume is the distance around a figure. Surface area is a three-dimensional measure. Volume is the amount of space a figure occupies. Volume is a three-dimensional measure.” 

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. A Vocabulary Glossary is provided in the back of Volume 1 and lists all the vocabulary terms and examples. Teacher side notes, Elicit and Use Evidence of Student Thinking and Pose Purposeful Questions, provide specific information about the use of vocabulary and math language. Examples include:

• Lesson 3-4, Construct Functions to Model Linear Relationships, Visual Learning, Example 3, Pose Purposeful Questions, “How is the initial value of a function represented in the equation y = mx + b? How is the constant rate of change represented in the equation y = mx + b?”
• Lesson 4-2, Analyze Linear Associations, Visual Learning, Example 1, Try It!, Elicit and Use Evidence of Student Thinking, “How did you decide where to draw your trend line? What type of relationship is it? Explain.”
• Lesson 6-7, Understand Similar Figures, Visual Learning, Example 1, Pose Purposeful Questions, “How do you know that the two trapezoids are facing in the opposite directions? Explain. How do you know the dilation of GHJK is a reflection of ABCD over the x-axis?”
• Lesson 7-1, Understand the Pythagorean Theorem, Visual Learning, Example 2, Pose Purposeful Questions, “Why is it possible to solve this problem using the Pythagorean Theorem? Would the Pythagorean Theorem still apply if one or more of the three side lengths were not a whole number?”
• Student Edition, Glossary, “irrational numbers: An irrational number is a number that cannot be written in the form $$\frac{a}{b}$$, where a and b are integers and b ≠ 0. In decimal form, an irrational number cannot be written as a terminating or repeating decimal. Example, The numbers $$\pi$$ and $$\sqrt2$$ are irrational numbers.”

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 have a visual design (whether in print or online) that is not distracting or chaotic, and supports students in engaging thoughtfully with the subject.

The font size, graphics, amount of directions, and language used on student pages and in Digital Lessons is appropriate for students. Graphics promote understanding of the mathematics being learned. The digital format is easy to navigate and is engaging. There is ample “white space” for students to calculate and write answers in the student materials.

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 2021 Common Core Grade 8 provides a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

Each Topic Opener contains a Topic Planner that provides an overview of the Learning Objective, Essential Understanding, and Standards. The Content Overview Introduction also contains a breakdown of each Topic into lessons, objectives, and standards. Finally, the Teacher Edition Program Overview contains a Pacing Guide with Topic titles and number of instruction days required, “Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 to 2 days for the enVisionSTEM project and Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessment.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 contain some strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The online Teacher’s Resource Masters have Home School Connection Letters, in English and Spanish, for each Topic. The letters include information on the mathematical content and activities parents can do with their child to support the mathematical content. For example, Grade 8, Topic 8, Solve Problems Involving Surface Area and Volume, “Dear Family, Your child is learning to calculate the surface areas and volumes of cylinders, cones, and spheres. He or she will also solve related problems such as finding volumes of composite figures and determining a missing dimension of a three-dimensional figure. You can use the following activity to support your child’s understanding of surface area.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 contain explanations of the instructional approaches of the program and identification of the research-based strategies.

EnVision is based on research-based strategies. According to the Teacher Resource Program Overview, “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.” Additionally, the core instructional model is based in research, “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. 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. The enVision Mathematics instructional model is built on the interaction between these two instructional approaches.”

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 encourage students to monitor their own progress. 

Each Topic contains a Mid-Topic Checkpoint for students to monitor their understanding of concepts and skills taught in the first lessons of the Topic. Following the assessment students are asked, “How well did you do on the mid-topic checkpoint? Fill in the stars.” Three stars are provided.

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.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 provide opportunities for teachers to use a variety of grouping strategies.

Each lesson begins with a Problem-Based Learning activity which is introduced to the whole class. Then students break into small groups to work on the activity and come back together to discuss solutions and strategies as a whole class. Independent practice is found in the Problem & Practice Solving portion of the lesson. Icons in the Teacher’s Edition indicate whether the activity should be completed with Whole Class or Small Group.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 encourage teachers to draw upon home language and culture to facilitate learning. 

The Language Support Handbook provides research-based support strategies for English Language Learners, Academic Vocabulary Activities, a list of key vocabulary in 6 languages, and specific language support for each Topic Lesson. Digital and Student Edition Glossaries are in both English and Spanish. Assessments in Spanish can be accessed online. Each Topic’s Home-School Connection Letter explains the content of the Topic in English or Spanish.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 integrate technology including interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Examples include:

• The Math Practices and Problem Solving Handbook is an online reference available for students.
• Digital Desmos Activities provide embedded technology with engaging instruction of real-world content. 
• Visual Learning Animation Plus provides scaffold animations of learning with real aloud options to support English learners.
• Animated Glossary in digital resources provides math terms with support in English and Spanish.
• Math Practice Animations are online videos explaining the Practices and sample problems supporting the Practices.
• A variety of Interactive Math Tools are available online for students and teachers.
• Topic Readiness Tests and Lesson Quizzes taken online are automatically graded and remediation and enrichment activities are automatically assigned to students.

The digital instructional materials reviewed for enVision Mathematics Common Core Grade 8 are web-­based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are “platform neutral” (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. Examples include:

• Digital games that enhance fluency and provide opportunities for students to use procedural skills to solve problems are available online.  
• Virtual Nerd offers tutorials on a variety of math concepts with procedural skill emphasised.
• The online Readiness Assessment tab for each topic includes a Remediation link that has tutorials and opportunities for students to practice procedural skills using technology. 
• Fluency Practice Pages for each Topic are available online.

The digital materials reviewed for enVision Mathematics Common Core Grade 8 include opportunities for teachers to personalize learning for all students. Adaptive technology is not provided by digital materials. 

Digital materials include opportunities for teachers to personalize learning for all students. Examples include:   

• Teachers can select and assign individual practice items for student remediation based on the Topic Readiness assessment. If students take the test online it is automatically scored and students are automatically assigned enrichment or remediation activities.
• Teachers can create online classes and assignments for students.  
• Interactive Student Edition is accessible online and can be assigned to students.

The digital materials reviewed for enVision Mathematics Common Core Grade 8 can easily be customized for local use. Digital materials provide online materials for teachers to assign to students. Examples include:

• Interactive media lessons are accessible that cover all learning standards
• Lesson plans can be customized by day, week, or month or resequenced to match the district curriculum map.
• Outside content can be uploaded and Teacher Resource Masters can be customized.

The materials reviewed for enVision Mathematics Common Core Grade 8 include technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.). 

Teachers can create Online Discussion Boards and monitor student participation.