8th Grade - Gateway 1

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for assessing grade-level content. The assessments are aligned to grade-level standards. Examples include:

• In Unit 5, Mid-Unit Assessment, Problem 5 provides a scenario in which bacteria is doubling over time. Students determine if the given population of bacteria is a function of the number of days. Students also determine if the function is linear and explain their reasoning (8.F.1). The problem reads, “Lin counts five bacteria under a microscope. She counts them again each day for four days, and finds that the number of bacteria doubled each day—from 5 to 10, then from 10 to 20, and so on. Is the population of bacteria a function of the number of days? If so, is it linear? Explain your reasoning.”
• In Unit 7, End-of-Unit Assessment, Problem 2, students select all expressions that equal  $$6^{-10}$$. The five choices are all variations of using the properties of exponents (8.EE.1).

Assessments are located on each Unit Page under the Assessments tab for each of the first eight units. Unit 9 is an optional unit and has no assessments. Assessments are limited to seven problems, but these are often broken into multiple prompts and assess numerous standards. Units 1 and 5 contain Mid-Unit Assessments for a total of 10 summative assessments.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for spending a majority of instructional time on major work of the grade.

• The number of units devoted to major work of the grade, including assessments and supporting work, is 7 out of 8, which is approximately 88 percent.
• The number of non-optional lessons devoted to major work of the grade, including assessments and supporting work, is 103 out of 121 total non-optional lessons, which is approximately 85 percent.
• The number of days devoted to major work, including assessments and supporting work, is 118 out of 149 days, which is approximately 79 percent.

A lesson-level analysis is most representative of the instructional materials because this calculation includes all lessons with connections to major work with no additional days factored in. As a result, approximately 85 percent of the instructional materials focus on major work of the grade. An analysis of days devoted to major work includes 18 days for review and assessment, but the materials do not indicate which items to use for the review.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 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. Multiple lessons in the Grade 8 curriculum incorporate supporting standards in ways that support and/or maintain the focus on major work standards. Examples of the connections between supporting work and major work include the following:

• In Unit 5, Lesson 17, students use volume formulas for three-dimensional figures to explore aspects of functions. Students compare functions presented as equations, graphs, and/or tables when exploring linear and nonlinear relationships. In the lesson Warm-Up, students determine if two quantities are functions by examining the distance a truck traveled in relation to the amount of gas used (8.F.1). Activities 3 and 4 address 8.F.1, 8.F.B, and 8.G.9 as: “Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height.” The lesson Cool-Down connects 8.G.C and 8.F.1 as students explore the height and volume of cylinders. In Lesson 18, students explore the effect on the volume of cylinders in functions where two variables are scaled. Students use volume formulas for cylinders and graphs of the functions in this exploration.
• In Unit 6, the first three lessons introduce students to associations in two variables and how to display and analyze those associations using scatter plots (8.SP.2). In the Lesson 4 Warm-Up, students estimate the slope of a given line that lies close to, but not directly on, two points. During lesson activities, students state the meaning of the slope in the given contexts and also whether outliers have an effect on the slope of the line of best fit. Lessons 5 through 8 continue to build students' understanding of lines and slope (8.EE.5) to build an understanding of scatter plots and bivariate data.
• In Unit 8, Lesson 3, Activities, students assess whether different rational numbers are solutions for square roots. In Lesson 4, students find solutions to functions that include square roots, and they estimate solutions to square roots in order to develop a deeper understanding of square roots as numbers and not just as solutions to the side length of a square when the area is known (8.NS.2, 8.EE.2).

The instructional materials for Kendall Hunt’s llustrative Mathematics 6-8 Math, Grade 8 meet expectations that the amount of content designated for one grade level is viable for one year.

The suggested amount of time and expectations of the materials for teachers and students are viable for one school year as written and would not require significant modifications. As designed, the instructional materials can be completed in a school year.

• The provided scope and sequence found in the Grade 8 Course Guide includes materials for 149 instructional days. There are 121 non-optional lessons, 18 assessment days (16 summative), and 10 optional lessons.
• 118 of these non-optional lessons are designed to address grade-level standards. 3 non-optional lessons provide problem contexts and activities that prepare students for the unit or connect work from prior grades.
• 4 of the optional lessons are present throughout the first eight units, and Unit 9 is an optional unit which includes 6 lessons.
• Units 1-8 are comprised of 11 to 22 lessons, and each lesson is designed for 45-50 minutes. Within each unit, lessons contain a Warm-Up, two or three Activities, a Lesson Synthesis, and a Cool-Down. Guidance regarding the number of minutes needed to complete each component of the lesson is provided in the Course Guide.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for 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. The instructional materials also relate grade-level concepts explicitly to prior knowledge from earlier grades.

The materials are intentionally designed to address the standards the way they are laid out in the progressions, and the Grade 8 Narrative in the Course Guide describes how the standards and progressions are connected for educators. Units begin with lessons connected to the standards from prior grades that are relevant to the current topic. Standards from the grade level and prior grades, and standards that will be addressed later in the year are identified in the sections as “addressing,” “building on,” and “building towards,” respectively. For example:

• In Unit 3, Lesson 1, the Lesson Narrative states, “This lesson is the first of four where students work with proportional relationships from a Grade 8 perspective.” In the lesson, 7.RP.2 is identified as the standard that the lesson is “building on,” the standard the lesson is “addressing” is 8.EE.B, and the standard the lesson is “building toward” is 8.EE.5. According to the CCSSM progressions, the study of proportional relationships is a foundation for the study of functions. After Lesson 4, students move to linear relationships. The Lesson 5 Overview states, “After revisiting examples of proportional relationships in the previous lessons, this lesson is the first of four lessons that moves from proportional relationships to linear relationships with positive rates of change.”

The Warm-Ups in lessons frequently work with prior-grade standards in ways that support learning of grade-level problems and make connections to progressions from previous grades. For example:

• In Unit 3, Lessons 1 and 2 begin with looking at graphs and comparing proportional relationships (7.RP.2). The Lesson 1 Warm-Up has students compare graphs of proportional relationships with the same scale. The Lesson 2 Warm-Up includes different scales. Lessons 3 and 4 continue work with graphing proportional relationships with attention to graph features such as scale. Students pay attention to graph features in preparation for grade-level work with linear equations and graphs of linear equations.
• The Unit 8, Lesson 3 (6.EE.4, 5.NF.4) Warm-Up includes practice with multiplying fractions (5.NF.4) which is essential in the lesson for estimating square roots. This connection is explicit in the teacher materials that state, “The purpose of this warm-up is for students to review multiplication of fractions in preparation for the main problem of this lesson: estimating solutions to the equation $$x^2$$= 2. For example, 3⁄2 · 3⁄2 = 9⁄4, which is a value close to 2 so 3⁄2 is a value close to the $$\sqrt2$$.”

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems.

In the Course Guide under Lessons and Standards, there is a table which reflects the mathematics in the materials. All grade-level standards are represented across the 9 units. Tasks are aligned to grade-level work and are connected to prior-grade knowledge. For example:

• In the Course Guide, the Narrative for Unit 5 describes multiple ways that function understanding is developed by building upon prior-grade work, including input/output patterns in Grades 4 and 5, independent and dependent variables in Grade 6, work with ratios and proportions in Grades 6 through 8, and work with geometric measurement as early as Kindergarten. Students explore linear relationships built from work early in Units 1 and 2 with similarity of triangles. The unit develops from exploring input/output patterns in order to develop a formal understanding of the definition of a function in Lessons 1 and 2, to exploring different representations of functions in Lessons 3 through 7. This work includes verbal descriptions, equations, tables, and graphs. Multiple representations are used in each lesson. The lessons near the end of the unit work with volume formulas. Lesson 22, the final lesson in the unit, explicitly connects work with functions to volume and includes exposure to non-linear functions.
• In Unit 4, Lesson 13 consists of problems where students use graphs of systems of linear equations to reason about the solutions to equations. The first activity has students match the graph with the system of equations, then students work on a digital graph to find solutions to a system of equations. In the second activity, the students look at six different graphic representations of systems of equations and discuss the meaning of having no solution, one solution, or infinite solutions. In the Cool-Down, students look at the graphic representation of a system of equations and give possible solutions based on what the graph looks like. There are also five practice problems that are connected to the lesson.

A typical lesson has a Warm-Up, one or more Activities, Synthesis, and a Cool-Down. Additionally, every lesson provides practice problems that can be used as independent or group work. Some lessons also provide an “Are you ready for more?” problem. These problems are an opportunity for students to explore grade-level mathematics in more depth and often make connections between the topic in the lesson and other concepts at grade level. They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 8 meet expectations for fostering 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, including:

8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.

• The Unit 4 overview describes how work in solving linear equations builds on work from Grades 6 and 7 with equations and expressions in one variable, then further builds to algebraic methods for solving equations. The Lesson 10 learning goals state: “Make connections between ordered pairs in tables and graphs,” and “Given descriptions of two linear relationships, interpret ordered pairs in contexts, focusing on when or where the same ordered pair makes each relationship true.” The Lesson 13 learning goals are also connected to the cluster heading, including: “Understand that a system of equations can have no solutions, one solution, or infinitely many solutions.” “Connect features of the graphs to the number of solutions the system has.” and “Start solving systems algebraically.”

8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

• The Unit 5 overview details the progression of geometric measurement since Kindergarten and lays out how the unit will extend concepts of volume to deepen students' understanding of the geometric features of three-dimensional shapes such as radii, bases, and heights. In addition, the unit integrates student work with functions as they develop a conceptual understanding of formulas for volume. All Unit 5 lessons include real-world or mathematical problems. Sample learning goals aligned to the cluster heading include: “Find volumes of cones in mathematical and real-world problems” in Lesson 15 and “Solve a variety of mathematical and real-world problems about the volume of a cylinder, cone, or sphere” in Lesson 21.

The 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. Multiple examples of tasks connecting standards within and across clusters and domains are present.

• In Unit 5, Lesson 8, Activities 2 and 3, students model real-world problems with functions by interpreting the start value and the slope (8.F.B). This includes understanding linear equations as describing linear functions and involves comparing different representations of a function, with an emphasis on equations and graphs, but also includes verbal/written descriptions of the real-world scenario (8.F.A).
• In Unit 8, Lesson 2, 8.EE.A and 8.F.B are connected as students take measurements of area and side lengths, collect data, and determine y = $$x^2$$ as a function that models the relationship between the area of a square and a side length. Students make estimates of side lengths given the area of a square and begin to use square root symbols and terms.
• In Unit 3, Lesson 8, 8.G.A and 8.EE.B are connected. The Warm-Up task reviews 8.G.A.1 where students must identify lines that have been translated from a given line. The rest of the lesson addresses 8.EE.B with an emphasis on students using their understanding of translations to derive y = mx + b from a graph of y = mx. The series of tasks use contexts to help students make connections to the equation of a line and how translations can be helpful in interpretation in context.