8th Grade - Gateway 1

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The assessments are aligned to grade-level standards. The instructional materials reviewed for this indicator were the Post-Tests, which are the same assessments as the Pre-Tests, both Form A and Form B End of Topic Tests, Standardized Practice Test, and the Topic Level Performance Task. Examples include: 

• Module 1, Topic 1, Standardized Test Practice, 8.G.2: Students use understanding of congruence in translations to determine the length of a side in an image. Question 2 states, “Blake drew square ABCD. Then, he drew the image of it, square A’B’C’D’, 2 centimeters to the right of the original figure. Line segment BC is 3 centimeters. How long is B’C’?” 

• Module 3, Topic 2, Performance Task, 8.SP.1-4: Student create scatter plots and lines of best fit to investigate relationships between age and height. For example, in Patterns in Bivariate Data: Growing Tall: “The average height of girls from age 8 to 15 is plotted below. The second plot shows the heights of professional players on a WNBA team. Two graphs are displayed with data. Describe the pattern of each data set. What do the patterns indicate about the connection between the age and height? Explain. Estimate lines of best fit and use them to make predictions if possible. What would you expect to be the average height of an 18 year old? A 40 year old? Your work should include: Description of patterns of data and an explanation of the connections; Equations for approximate lines of best fit; A description of the slope and y-intercept for each line; Use of the lines of best fit to predict a future value.” 

• Module 2, Topic 1, End of Topic Test Form A, 8.EE.5 & 6: Students compare proportional relationships represented in different ways (situation, equation, and graph) to find the best deal. Question 9 states, “During the week, Azim bought gas at 3 different gas stations. He bought 15 gallons at Joe’s for $41.85. The cost per gallon of gas at ZoomGas is given by the equation y=5+2.71x, where y is the total cost and x is the number of gallons of gas bought. The table shows the amount spent at Corner Gas. Which of the gas station provides the best deal for gas? Explain your reasoning.” 

• Module 4, Topic 1, End of Topic Test Form A, 8.NS.1: Students justify understanding of rational numbers. Question 5 states, “Tell whether each statement about number sets is true or false. If false, provide a counterexample: a. Every terminating decimal is a rational number. b. The set of natural numbers contains the set of integers. c. Zero is an integer. d. A square root is sometimes a rational number.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The design of the materials concentrates on the mathematics of the grade. Each lesson has three sections (Engage, Develop, and Demonstrate) which contain grade-level problems. Each topic also includes a performance task. 

• In the Engage section, the students complete one activity that will “activate student thinking by tapping into prior knowledge and real-world experiences and provide an introduction that generates curiosity and plants the seeds for deeper learning.” For example, in Module 5, Topic 1, Lesson 3 (609C), students calculate the number of times they have blinked in their lifetimes. The large numbers generated motivate the need to write large numbers with a more efficient notation. (8.EE.3, 4) 

• In the Develop section, the students do multiple activities that “build a deep understanding of mathematics through a variety of activities—real-world problems, sorting activities, worked examples, and peer analysis—in an environment where collaboration, conversations, and questioning are routine practices.” For example, Module 5, Topic 1, Lesson 4, Activity 4 (666) has students operate on and answer questions about numbers written in scientific notation and standard form. They choose appropriate units, compare relative sizes, and operate on numbers in different forms. (8.EE.3, 4) 

• In the Talk the Talk section, the students “reflect on and evaluate what was learned.” An example of this is Module 5, Topic 2, Lesson 4 (714), where students “solve for the volume of an irregular space.” (8.G.9) 

The end of each lesson in the student book includes Practice, Stretch, and Review problems. These problems engage students with grade level content. Practice problems address the lesson goals. Stretch problems expand and deepen student thinking. Review problems connect to specific, previously-learned standards. All problems, especially Practice and Review, are expected be assigned to all students. 

After the lessons are complete, the students work individually with the MATHia software and/or on Skills Practice that is included. 

• MATHia - Module 1, Topic 3 (127B and C): Approximately 160 minutes are spent in MATHia software, and students identify and classify angle pairs in a given figure containing lines cut by transversals. They use the Angle-Angle Similarity Theorem to verify that images are similar. 

• Skills Practice - Module 2, Topic 1 (167B and C): There are five problem sets for additional practice of these lesson skills: Identifying Proportional Relationships, Determining Slopes, Proportional and Nonproportional Relationships, and Transforming Linear Relationships.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the amount of time spent on major work, the number of topics, the number of lessons, and the number of days were examined. Review and assessment days were also included in the evidence. 

• The approximate number of topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 10 out of 12, which is approximately 83 percent. 

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 44 out of 54, which is approximately 81.5 percent. 

• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 116 out of 140, which is approximately 82.8 percent. 

The approximate number of days is most representative of the instructional materials because it most closely reflects the actual amount of time that students are interacting with major work of the grade. As a result, approximately 82.5 percent of the instructional materials focus on major work of the grade.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Supporting standards/clusters are connected to the major standards/clusters of the grade. Examples include: 

• In Module 2, Topic 2, Lessons 2, 3, & 4 Lines of Best Fit: Students determine the equation for the line of best fit scatterplot (8.SP.3) to analyze data, make predictions, and interpret the linear function in terms of the situation it models (8.F.4). 

• In Module 4, Topic 1, Lesson 3, Activity 3.2 & 3.3: What are Those!? The Real Numbers: Students estimate the value of the square root of a number that is not a perfect square by using the two closest perfect squares and a number line (8.NS.2) then use estimation to determine cube roots when the radicands are not perfect cubes and determine the volumes of cubes generating a list of numbers that are perfect cubes. (8.EE.2). 

• In Module 5, Topic 2 Volume Problems with Cylinders, Cones, and Spheres: Students calculate volume (8.G.9) with measurements given in decimals and fractions which supports 8.EE.7b, solving linear equations with rational number coefficients.

The materials for Carnegie Learning Middle School Math Solution Course 3 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.

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 include: 

• In Module 2, Topic 1, Lesson 4, Transformations of Lines (8.EE.6, 8.G.1a, 8.G.1c)  “Students consider how vertically translating, dilating, or reflecting the line y=x affects its graph. They create a table with the results of the equations and use variables to generalize their thinking. Students graph lines using transformations of the line y=x. They also investigate how translating, reflecting, or rotating parallel lines maintains parallelism, while dilating does not.” Geometric transformations (8.G.A) is applied to the basic function y=x (8.EE.B). 

• Module 2, Topic 2, Lesson 1,  Using Tables, Graphs, and Equations (8.F.4, 8.EE.8a)  “Students interpret two problem situations involving total cost based on a unit rate and fixed charge, one with integers and the other with rational numbers. They create a table, graph, and equation to model each context. Students compare the two situations and interpret the point of intersection and values that lie on both sides of it to make decisions.” Students construct a function to model a linear relationship between two quantities (8.F.B) and compare two different proportional relationships represented in different ways (8.EE.B).

• In Module 3, Topic 1, Lesson 4, Describing Functions  (8.F.3, 8.F.4, 8.F.5). “Students explore linear and nonlinear functions through contexts. They recognize that all lines, except vertical ones, are linear functions. Students investigate a line graph to make sense of increasing, constant and decreasing functions. They use area and volume contexts as examples of a quadratic and cubic function, respectively. Students identify the domain and range of the functions.” Students interpret the equation y = mx + b as defining a linear function (8.F.A) to analyze the graphical behavior of linear and nonlinear functions (8.F.B). 

• In Module 4, Topic 2, Lesson 4, Side Lengths in Two and Three Dimensions (8.EE.2, 8.G.7), “Students calculate the lengths of the diagonals of rectangles and trapezoids on the coordinate plane. They draw a 3-dimensional diagonal inside a rectangular box and calculate its length by applying the Pythagorean on two planes. Students compute the area of complex figures where a hypotenuse also serves as a diameter. They compute the lengths of 3-dimensional diagonals in real-world contexts.” Square roots (8.EE.B) and the Pythagorean Theorem  are used to determine the length of a three-dimensional diagonal of a rectangular solid (8.G.B).

The materials for Carnegie Learning Middle School Math Solution Course 3 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 instructional materials clearly identify content from prior and future grade levels and use it to support the progressions of the grade level standards. The content is explicitly related to prior knowledge to help students scaffold new concepts. Content from other grade levels is clearly identified in multiple places throughout the materials. Examples include: 

• A chart in the Overview shows the sequence of concepts taught within the three grade levels of the series (FM-15). 

• The Family Guide (included in the student book) presents an overview of each Module with sections that look at “Where have we been?" and "Where are we going?” which address the progression of knowledge. 

• The Teacher Guide provides a detailed Module Overview which includes two sections titled, “How is ____ connected to prior learning?” and “When will students use knowledge from ___ in future learning?” 

  • Module 1 Overview- How is Transforming Geometric Objects connected to prior learning? (2D): “Transforming Geometric Objects builds on students’ long-developing geometric knowledge. In Kindergarten, students learned that an object’s name is not dependent on orientation or size, setting the foundation for similarity. Later, in grade 4, students identified lines of symmetry, lighting the way for the study of reflections and congruence. In this module, students also build on the grade 7 standards of operations with rational numbers, proportionality, scale drawings, uniqueness of triangles, and angles formed when two lines intersect. Students will use their knowledge of operations with rational numbers to determine the effects on coordinates of figures after transformations.” 

  • Module 3 Overview- When will students use knowledge from Modeling with Linear Equations in future learning? (374C): “As students learn in Modeling with Linear Equations, there is often more than one correct way to solve an equation. Throughout their study of algebra, students are expected to construct, solve, and graph equations to represent relationships between two quantities. This module provides students with opportunities to develop strategies focused solely on linear equations. As they continue on their mathematical journeys, they will encounter literal equations, polynomial equations, and trigonometric equations.“ 

• At the beginning of each Topic in a Module, there is a Topic Overview which includes sections entitled “What is the entry point for students?” and “Why is ____ important?” 

  • Module 4, Topic 2- Pythagorean Theorem (551D) - What is the entry point for students?: Students first studied right angles and right triangles in grade 4 and evaluated numeric expressions with whole-number exponents in grade 6. They learned to solve equations in the form x^2=p in the previous topic. Students have calculated distances on the coordinate plane. In previous grades, the two points were on the same horizontal or vertical line.“ 

  • Module 5, Topic 1- Exponents and Scientific Notation (609D) - Why are Exponents and Scientific Notation important?: “Students will continue to expand the complexity of powers that they can evaluate. In high school, students evaluate rational number exponents. Exponents and Scientific Notation prepares students for a more rigorous and abstract exposure in high school. Scientific notation will arise in students’ science courses in middle school and high school, particularly in the study of chemistry.”

• The Topic Overview also contains a table called “How does a student demonstrate understanding?”  The table lists what students should know and be able to do by the end of the topic.” 

• Each Lesson Resource has “Mixed Practice” at the end of each topic, a Mixed Practice worksheet provides practice with skills from previous topics and this topic. Spaced Review Fluency and problem solving from previous topics End of Topic Review Review problems from this topic.