8th Grade - Gateway 1

The instructional materials for Discovery Education Math Techbook Grade 8 Mathematics meet the expectation for assessing grade‐level content and, if applicable, content from earlier grades. The Unit Assessments that are included in the Teacher View were reviewed for Discovery Education Math Techbook Grade 8 Mathematics. Overall, there were no assessment questions addressing standards from above Grade 8.

Examples of grade-level assessment items include the following:

• Unit 1 question #11- Students show their knowledge of 8.EE.1 by dragging and dropping tiles into an equation to create equivalent expressions.
• Unit 4 question #7- Students demonstrate their knowledge of 8.G.7 by trying to informally prove the converse of the Pythagorean Theorem by producing supporting evidence.
• Unit 5 question #14- Students demonstrate their knowledge of 8.EE.A by applying their understanding of scientific notation to solve a real-world application problem to determine the approximate number of liters of water that go over the Horseshoe Falls in the Niagara River in 1 year and explain how they found their solution.

The instructional materials for Discovery Education Math Techbook Grade 8 Mathematics meet the expectations for spending the majority of class time on the major clusters of each grade. Overall, approximately 80 percent of the instructional time is spent on major work of 8th grade.

To determine this, three perspectives were evaluated: 1) the number of units devoted to major work, 2) the number of instructional periods/sessions devoted to major work, and 3) the time devoted to major work. Time devoted to major work is the most reflective for this indicator because it specifically addresses the amount of class time spent on concepts.

Evidence was collected from the Content Pages, Scope and Sequence, Table of Contents, and the Course Overview.

• Units – Approximately 9 complete units out of 11 units, which is approximately 82 percent, is spent on major work.
• Sessions (Excluding Extensions)– Approximately 99 sessions out of 117, which is approximately 85 percent, is spent on major work.
• Time (Excluding Extensions)– Approximately 5,350 minutes out of 6,650 minutes, which is approximately 80 percent, is spent on major work.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 Mathematics meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Overall, many Concepts that address supporting work integrate it with major work so that students are given more opportunities to engage with the major work of the grade.

Some examples of how the Concepts that address supporting standards integrate them with major work are:

• In Concept 2.3 students have the opportunity to identify irrational numbers and compare their sizes and locations on a number line, supporting standards 8.NS.1,2 by evaluating square roots of small perfect squares and cube roots of small perfect cubes, major standard 8.EE.2.
• In Concept 10.2 students have the opportunity to model a bivariate relationship with a linear model and use the linear model to answer questions in the context of the bivariate data set, supporting standards 8.SP.2,3. Modeling a bivariate relationship with a linear model and using the model to answer questions in terms of a context enhances students' opportunities to construct a function to model a linear relationship between two variables and interpret the rate of change and initial value of the function in relationship to the situation being modeled, major standard 8.F.4.
• In Concept 11.1 students have the opportunity to work with and solve problems using the formulas for the volumes of cones, cylinders, and spheres, supporting standard 8.G.9. Since the formulas for the volumes of cones, cylinders, and spheres involve squared and cubed terms, these problems enhance students' opportunities to use and evaluate square and cube roots, major standard 8.EE.2.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 Mathematics meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades.

The Scope and Sequence document states that one period is equivalent to approximately 50 instructional minutes. According to the times given in each of the Units, there are approximately 133 50-minute class periods without Extension Lessons. Since there are eleven Unit Assessments, the total number of days would increase to approximately 144 days if one day is allotted for each Unit Assessment. The Scope and Sequence document also states that “the instructional time does not include in-class time for the extensive, independent student work available in the Practice and Apply sections that many teachers may wish to incorporate as part of their daily instruction.”

There are 26 Practice and 26 Apply sections in the materials as each of the 26 Concepts in the materials has a Practice and Apply section. Adding one day to complete the Practice and Apply sections for each Concept brings the total number of instructional days to approximately 170 50-minute class periods. Overall, with 170 50-minute class periods, the instructional materials meet the expectations for this indicator.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 Mathematics meet the expectations for the material being consistent with the progressions in the Standards. The materials develop according to the grade-by-grade progressions in the Standards, and the content from prior or future grades is clearly identified and related to grade-level work.

At the beginning of each Concept, there is a section called Progressions and Standards, and in this section, there are three parts titled During Previous Instruction, Through The Investigations In This Concept, and During Subsequent Instruction.

• During Previous Instruction identifies content from previous grades and explains how previous content is related to grade-level work. For example, Concept 9.1 includes the following: "During Previous Instruction, students have learned how to use variables to represent unknown quantities in equations (6.EE.6), and have represented and solved real-world situations using equations (6.EE.5,7). Students also have worked with independent and dependent variables (6.EE.9). In Grade 7, students extended these understandings to proportional relationships and solve equations algebraically and graphically (7.EE.4a, 7.RP.2a–d)."
• Through The Investigations In This Concept discusses grade-level content. For example, Concept 9.1 includes the following: "Through The Investigations In This Concept, students learn how to represent situations with two variables using a system of equations (8.EE.8). Students explore methods for solving a system of equations algebraically (through substitution and elimination), graphically, and by inspection (8.EE.8c). Students develop an understanding that a system of equations can have one solution, infinite solutions, or no solution, and they can understand the meaning of the solution in context (8.EE.8a)."
• During Subsequent Instruction describes how the grade-level content relates to content later in the current grade-level and future grade-levels. For example, Concept 1.2 includes the following: "During Subsequent Instruction, students will apply the properties of exponents when using scientific notation and solving rational and radical equations (8.EE.4, A-REI.2). Students will extend the properties of integer exponents to ... leading to logarithmic functions (N-RN.1,2). They will recognize and graph exponential and logarithmic functions (A-REI.11) and use the properties of exponents to transform expressions for exponential functions (A-SSE.3c)."

The instructional materials give all students extensive work with grade-level problems. Each Concept has three sections- Discover, Practice, and Apply- which contain grade-level problems. The following are included in the three sections:

• In the Discover section, there are at least two Investigations that enable all students to work with grade-level problems. There are Extension Lessons that take grade-level work and enhance it by utilizing new contexts or different tools. For example the Unit 4 Concept 4.2 Extension has students extending their grade-level standards into working with a Dynamic Geometry Tool to create triangles and observe what happens under given restraints. There is Additional Assistance in the Summary, and in the Additional Assistance, there are math explanations, extra videos, and blackline masters that provide more instruction for students who need it. For example, the Unit 8 Concept 8.2 Summary has definitions to assist in solving linear equations, videos to deepen understanding, and View Math Explanations that are on grade-level and offer extra practice.
• In the Practice section, there are the Coach and Play sections, and each of these include grade-level problems. The Coach section offers grade-level work that gives students guided practice and hints when needed. For example, the Unit 10 Concept 10.3 Practice Coach has ten grade-level questions with supports and feedback available as students work through creating and analyzing two-way tables.
• In the Apply section, there are two or three grade-level problems that offer real-world contexts in which students apply the content that has been learned in the current Concept. For example, Concept 6.2 has the following two problems in the Apply section: How Are Parallel Lines and Transversals Used in Stage Sets, which has students identify and explain the relationships between angles created when parallel lines are cut by a transversal in a real-world context, and How Do Intersecting Lines Make Such Beautiful Designs, which offers students opportunities to explore lines and angle relationships in art and architecture.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 Mathematics meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards. Overall, the materials include learning objectives that are visibly shaped by CCSSM cluster headings and problems and activities that serve to connect two or more clusters in a domain or two or more domains in the grade.

Examples of how the learning objectives are visibly shaped by the CCSSM cluster headings include:

• In Concept 3.2 one objective is "describe a sequence of transformation(s) needed to generate a similar image from a given pre-image," which is shaped by the cluster heading for 8.G.A, "Understand congruence and similarity using physical models, transparencies, or geometry software."
• In Concept 7.2 one objective is "represent and compare functions in multiple ways," which is shaped by the cluster heading for 8.F.A, "Define, evaluate, and compare functions."
• In Concept 9.1, one objective is "match a system of linear equations with one solution, no solution, and infinite solutions to graphs and to real-world contexts," which is shaped by the cluster heading for 8.EE.C, "Analyze and solve linear equations and pairs of simultaneous linear equations."

Examples of problems and activities that serve to connect two or more clusters in a domain or two or more domains in the grade include:

• Unit 4 Concept 4.1 connects 8.EE.A with 8.G.B as students solve equations of the type x^2 = p while investigating and solving problems with the Pythagorean Theorem.
• Unit 7, Concept 7.1 connects 8.EE.B with 8.F.A as students graph proportional relationships, explain the slope of a graph using similar triangles, and interpret the equation y = mx + b as defining a linear function.
• Unit 7, Concept 7.2 connects 8.F.A with 8.F.B as students explore characteristics of functions, determine when a table of values displays a linear relationship, and construct a function that models a linear relationship between two quantities.
• Unit 8, Concept 8.1 connects 8.EE.C with 8.F.B as students solve linear equations and analyze graphs to describe functional relationships.