8th Grade - Gateway 1

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.