8th Grade - Gateway 1

The materials reviewed for enVision Mathematics Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The materials contain diagnostic, formative, and summative assessments. Each Topic includes a Topic Readiness Assessment, Lesson Quizzes, Mid-Topic Checkpoint, Mid-Topic Performance Task, Mid-Topic Assessment, Topic Performance Task, and Topic Assessment. Even-numbered Topics include a Cumulative/Benchmark Assessment. In addition, teacher resources include a Grade Level Readiness Assessment and Progress Monitoring Assessments. Assessments can be administered online or printed in paper/pencil format. No above-grade-level assessment items are present. 

Examples of grade-level assessment items aligned to standards include: 

• Topic 1, Assessment Form A, Problem 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, Problem 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.” A graph showing the height and horizontal distance of the shot put is provided. (8.F.5)

• Topic 6, Assessment Form A, Problem 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.” A coordinate plane showing four figures in various orientations is provided. (8.G.2)

• Topics 1 - 8, Cumulative/Benchmark Assessment, Problem 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.8b)

The materials reviewed for enVision Mathematics Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

The “Solve & Discuss It!” section presents students with high-interest problems that embed new mathematical ideas, connect prior knowledge, and provide multiple entry points. Example problems provide guided instruction and formalize the mathematics of the lesson frequently using multiple representations. The “Try It!” sections provide problems that can be used as formative assessments following example problems and the “Convince Me!” sections provide 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 of extensive work with grade-level problems to meet the full intent of grade-level standards include:

• In Topic 1, Lesson 1-1, 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?” In Lesson 1-2, 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?” In Lesson 1-2, Practice & Problem Solving, Problem 11, students identify rational and irrational numbers from a list of decimals, whole numbers, fractions and radicals. “Lisa writes the following list of numbers. 5.737737773…, 26, \sqrt{45}, -$$\frac{3}{2}$$, 0, 9. a. Which numbers are rational? b. Which numbers are irrational?” Students engage in extensive work with grade-level problems to meet the full intent of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number).

• In Topic 1, Lesson 1-5, 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.” In the Practice & Problem Solving, Problems 1 - 17 students solve equations involving variables squared and cubed. For example, Problem 16, “Find the value of c in the equation c$$^3$$ = 1,728.” In Problem 22, students apply both squares and cubes to a real-world scenario, “The Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot. How much will the hardwood floors cost?” Students engage in extensive work with grade-level problems to meet the full intent 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. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational).

• In Topic 3, Lesson 3-6, Try it!, students create a graph when given a verbal description, “Haru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes, and then continues in the same direction at a slower pace for 30 more minutes. Sketch a graph of the relationship of Haru’s distance from home over time.” In Practice & Problem Solving, Problem 10, students chose the best scenario to describe a graph, “Which description best represents the graph shown? (A) People are waiting for a train. A train comes and some people get on. The other people wait for the next train. As time goes by, people gradually leave the station. (B) One train arrives and some people get off the train and wait in the station. (C) People are waiting for a train. Everyone gets on the first train that comes. (D) People are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.” Students are provided a graph showing a scenario between total people and time. In Practice & Problem Solving, Problems 8, 9, 11, and 12 students sketch a graph based on the verbal description of two quantities. For example, in Problem 8, “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.” Students engage in extensive work with grade-level problems to meet the full intent of 8.F.5 (Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally).

• In Topic 6, Lesson 6-4, Try It!, students create an additional sequence of transformations for the original triangle that was mapped onto an image from Example 3. In Practice & Problem Solving, Problem 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.” Lesson 6-4 Quiz, Question 5, students perform a transformation on coordinate vertices, “Figure ABCD has vertices A(1,1), B(1,4), C(4,4) and D(4,1). What are the coordinates of the vertices of Figure A’B’C’D’ after a reflection across the line x = -2 and a translation of 3 units up?” Students engage in extensive work with grade-level problems to meet the full intent of 8.G.3 (Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates).

The materials reviewed for enVision Mathematics Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each 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 the 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 materials reviewed for enVision Mathematics Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so that supporting standards/clusters are connected to the major standards/clusters of the grade. Examples of connections include:

• In Topic 4, Mid Topic Checkpoint, Problems 3-5, students create a function for the line of best fit, extrapolate a value, and describe the relationship between the data for a scatter plot representing a set of test scores after various hours of studying. “3. Use the y-intercept and the point (4, 90) from the line on the scatter plot. What is the equation of the linear model? 4. Predict Adam’s test score when he studies for 6 hours. 5. Describe the relationship between the data in the scatterplot.” Students are provided a scatter plot of Adam’s test scores vs. time studying. This connects the supporting work of 8.SP.A (Investigate patterns of association in bivariate data) to the major work of 8.F.B (Use functions to model relationships between quantities).

• In Topic 7, Lesson 7-3, Practice & Problem Solving, Problem 14, students use an approximation of an irrational number to determine if a ramp matches recommendations. “It is recommended that a ramp have at least 6 feet of horizontal distance for every 1 foot of vertical rise along an incline. The ramp shown has a vertical rise of 2 feet (and a distance of 21 feet along the incline as shown in an image). Does the ramp shown match the recommended specifications? Explain.” These connect the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers.) to the major work of 8.G.B (Understand and apply the Pythagorean Theorem).

• In Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 10, students find the radius of a cone when given the volume. “The volume of the cone is 462 cubic yards. What is the radius of the cone? Use \frac{22}{7} for x.” This 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.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. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational ).

• In Topic 8, Lesson 8-4, Do You Know How?, Problem 4, students find the volume of a sphere to solve a word problem, “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 connects the supporting work of 8.G.C (Solve real-world and mathematical problems involving the volume of cylinders, cones, and spheres.) to the major work of 8.EE.A (Expressions and Equations Work with radicals and integer exponents).

The materials reviewed for enVision Mathematics Grade 8 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Examples from the materials include:

• In Topic 4, Lesson 4-5, Do You Know How, Problems 4-6, students determine percentages from a Two-Way Frequency Table, and estimate percentages for rational values for the data provided. “4. What percent of the people surveyed have artistic ability?  5. What percent of left-handed people surveyed have artistic ability? 6. What percent of the people who have artistic ability are left-handed?” Students are given a Two-Way Frequency Table illustrating the number of people who have artistic ability crossed with the dominant hand of those individuals. This connects the supporting work from 8.NS.1 (...Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.) with the supporting work from 8.SP.A (Investigate patterns of association in bivariate data).

• In Topic 5, Lesson 5-2, Practice & Problem Solving, Problem 9, students write and solve a system of equations to determine a break-even point for a real-world application. “The total cost, c, of renting a canoe for n hours can be represented by a system of equations. a. Write the system of equations that could be used to find the total cost, c, of renting a canoe for n hours. b. Graph the system of equations. c. When would the total cost for renting a canoe be the same on both rivers? Explain.” Students are provided an image of a sign that lists the canoe rental prices by river, River Y shows a price of $33 and River Z shows a cost of $5 per hour plus a deposit of $13. This connects the major work from 8.EE.C (Analyze and solve linear equations and pairs of simultaneous linear equations) to the major work from 8.F.B (Use functions to model relationships between quantities).

• In Topic 6, Lesson 6-10, Explore It!, students are asked to draw and label model representations of two flags and then to relate the sides and angles of the two flags. “Justin made two flags for his model sailboat. A. Draw and label triangles to represent each flag. B. How are the side lengths of the triangles related? C. How are the angle measurements of the triangles related?” Students are given an image of two triangular-shaped flags attached to a rope along one of their edges. The first has the angle not attached to the rope labeled as 46\degree, and the other triangle has the two angles along the rope labeled as  67$$\degree$$. In the bottom triangle, the side along the rope is 4 in. and one of the other sides is 5 in. This connects the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations) to the major work of 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software).

• In Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 16, students estimate the volume of a cone-shaped sculpture. “An artist makes a cone-shaped sculpture for an art exhibit. If the sculpture is 7 feet tall and has a base with a circumference of 24.492 feet, what is the volume of the sculpture? Use 3.14 for \pi, and round to the nearest hundredth” This connects the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers) to the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres).

The materials reviewed for enVision Mathematics Grade 8 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

The materials identify content from prior and future grade levels and use it to support the progressions of the grade-level standards. According to the Teacher’s Edition 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 of connections to future grades include:

• Topic 1, Topic Overview, Math Background Coherence, “Topic 1 How is content connected within Topic 1? Rational and Irrational Numbers In Lesson 1-1, students make the connection that repeating decimals are rational numbers because they can be written as fractions. In Lesson 1-2, students explore irrational numbers and recognize perfect squares. They learn that real numbers are either rational or irrational. Lesson 1-3 provides opportunities for students to compare and order rational and irrational numbers… Integer Exponents and Scientific Notation In Lesson 1-6, students multiply and divide exponential expressions with the same base and multiply exponential expressions with different bases. In Lesson 1-7, students use additional properties of exponents, such as the Zero Exponent Property and the Negative Exponent Property, to simplify exponential expressions…” Looking Ahead, “How does Topic 1 connect to what students will learn later?... Grade 9 Rational and Irrational Number Properties  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. Rational Exponents 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 4, Topic Overview,  Math Background Coherence, “Topic 4 How is content connected within Topic 4?... Frequency Tables In Lessons 4-4 and 4-5, students create and analyze two-way frequency and two-way relative frequency tables for paired categorical data. They learn to examine these tables and draw inferences about possible associations between the two data sets.” Looking Ahead, “How does Topic 4 connect to what students will learn later? High School … Frequency Tables In high school, students will continue their work with two-way frequency and two-way relative frequency tables for categorical data to understand joint, marginal, and conditional relative frequencies.”

• Topic 5, Topic Overview, Math Background Coherence, “Topic 5 How is content connected within Topic 5?... Solve Systems of Linear Equations Students are introduced to three different methods for finding the solution to a linear system. In Lesson 5-1, students analyze graphs of linear systems to determine the number of solutions to the system. Students progress into Lesson 5-2 where they graph a linear system, determine any intersection points, and check their solutions. In Lessons 5-3 and 5-4, students learn two different algebraic procedures for finding the solution of a linear system using substitution or elimination.” Looking Ahead, “How does Topic 5 connect to what students will learn later?… Algebra Represent and Solve Systems of Equations 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 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:

• In Topic 2, Lesson 2-1, Lesson Overview, Coherence, students, “combine like terms” and  “solve one- and two-step equations.” In Grade 7, students, “used variables to represent quantities” and “created simple equations to solve problems.”

• In Topic 6, Lesson 6-1, Lesson Overview, Coherence, students, “develop an understanding of translations, analyze the relationships between corresponding sides and angles of a preimage and its image, use a set of rules to translate figures on a coordinate plane” and “evaluate and describe translations.” In Grade 6, students, “drew polygons on the coordinate plane given coordinates of the vertices.”

• In Topic 8, Lesson 8-1, Lesson Overview, Coherence, students, “calculate the surface areas of cylinders, cones, and spheres.” In Grade 7, students, “found the surface areas of cubes and right prisms” and “calculated the area of a circle.”