6th Grade - Gateway 1

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectation for assessing grade‐level content and, if applicable, content from earlier grades. The Unit Assessments included in the Teacher View were reviewed. Overall, there were no assessment questions addressing standards from above Grade 6.

Examples of grade-level assessment items include the following:

• Unit 1 question #10- Students show their knowledge of 6.RP.3 by finding equivalent ratios on a double number line that express cable modem download speeds.
• Unit 8 question #4- Students demonstrate their knowledge of 6.EE.7 by writing an equation that models a real-world problem and then explaining the meaning of each part of the equation.
• Unit 12 question #10- Students demonstrate their knowledge of 6.G.1 by applying their understanding of area and decomposing shapes to solve a real-world application problem based on the flag of Kuwait.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectations for spending the majority of class time on the major clusters of each grade. Overall, approximately 68 percent of the instructional time is spent on major work of 6th 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 7 complete units out of 13 units, which is approximately 54 percent, is spent on major work.
• Sessions (excluding Extensions)– Approximately 79 sessions out of 109, which is approximately 72 percent, is spent on major work.
• Time (excluding Extensions)– Approximately 4,240 minutes out of 6,250 minutes, which is approximately 68 percent, is spent on major work.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 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 for 6.G.A, a supporting cluster, are integrated with major work of the grade for 6.EE.2 and 6.RP.3 are as follows:

• In Concept 12.1 students have the opportunity to develop the formulas for the area of and to find the area of parallelograms and triangles, 6.G.1. This supporting work with area also integrates the major work of writing algebraic and numerical expressions, 6.EE.2a, and evaluating expressions for specific values of the variables, 6.EE.2c.
• In Concept 13.1 students are given the opportunity to create nets for three-dimensional figures and use those nets to calculate the surface area of the figures represented by the nets. By calculating the surface area of the figures, this Concept also provides an opportunity to evaluate numerical and algebraic expressions. This Concept uses supporting work from 6.G.4 to enhance the major work of 6.EE.2c.
• In Concept 13.2 students explore the effects of changing one or more dimensions of a three-dimensional figure on the volume of that figure. This supporting work of 6.G.2 enhances the major work of 6.RP.3d as students have to manipulate units appropriately as they multiply or divide quantities.

Some examples of how the Concepts for 6.NS.B, a supporting cluster, are integrated with major work of the grade for 6.RP.A and 6.EE.3 are as follows:

• In the Intro to Concept 2.1 students are asked to determine to which food bank they should make a donation. To make the decision, students have to be able to divide multi-digit numbers, 6.NS.2; use their result to show understanding of unit rates, 6.RP.2; and solve a unit rate problem, 6.RP.3b.
• In Investigation 4 of Concept 2.2 the materials have opportunities for students to write numerical expressions using the greatest common factor (GCF). Using the GCF connects to applying the properties of operations for generating equivalent expressions, and this Investigation integrates 6.NS.4 with 6.EE.3.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 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 125 50-minute class periods without Extension Lessons. Since there are thirteen Unit Assessments, the total number of days would increase to approximately 138 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 28 Practice and 28 Apply sections in the materials as each of the 28 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 166 50-minute class periods. Overall, with 166 50-minute class periods, the instructional materials meet the expectations for this indicator.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 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 7.2 includes the following: "During previous instruction, students learned how to solve multi-step real-world problems with whole numbers using the four operations, and write a number sentence/equation to represent the problem (4.OA.2, 4.OA.3). Grade 5 students worked more formally with expressions. They learned to express calculations written in words as numerical expressions and then evaluated and interpreted expressions (5.OA.2). They also learned to express and examine powers of 10 as whole number exponents and carry out order of operations, including expressions with parentheses, braces, and brackets (5.OA.1, 5.NBT.2). In the previous concept, students read and wrote algebraic expressions from math phrases and real-world contexts. They also learned some algebra vocabulary, such as term, coefficient, and variable in earlier concepts of this unit (6.EE.1, 6.EE.2a, 6.EE.2b, 6.EE.6)."
• Through The Investigations In This Concept discusses grade-level content. For example, Concept 11.2 includes the following: "Through the investigations in this concept, students will explore measures of variability as they describe data sets. They will learn how to calculate range, interquartile range (IQR), and mean average deviation (MAD), which will assist them in describing the overall shape of the distribution (6.SP.3, 6.SP.5d.). They will use these values to summarize and identify patterns within data sets as they relate to their real-world contexts (6.SP.5, 6.SP.5c). Students extend this understanding of IQR and MAD as they choose which measure of center best describes a data set based on the shape of the data distribution and the context (6.SP.5d)."
• 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 4.1 includes the following: "During subsequent instruction in this unit, students will discover percents as rates per 100 (6.RP.3c) and will also perform measurement conversions, viewing conversion factors as unit rates (6.RP.3d). ... In Grade 7, students will apply their knowledge of rates to determine unit rates derived from ratios of fractions (7.RP.1). ... In addition, they will apply unit rates to solve problems involving scale drawings (7.G.1)."

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 also Extension Lessons that take grade-level work and enhance it by utilizing new contexts or different tools. For example, the Unit 7 Concept 7.2 Extension 1 has students extend their grade-level work with evaluating expressions by using spreadsheets to evaluate an algebraic expression for any integer value. There is Additional Assistance in the Summary, with grade-level math explanations and extra videos that provide more instruction for students who need it, including blackline masters. For example, the Unit 9 Concept 9.1 Summary has definitions of dependent and independent variables, videos to deepen understanding, and blackline masters for 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 9 Concept 9.1 Practice Coach has ten grade-level questions with supports and feedback available as students work through the problems.
• 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 8.2 has the following three problems in the Apply section that use inequalities to describe different ranges: Can you Keep Solids Solid and Gases Gaseous, describes the temperature ranges for the three states of different substances; Could You Be a Nutritionist, describes a child's nutritional needs; and How Much Medicine Should You Give This Dog, describes the weight ranges for dogs that correspond to different doses of medicines.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 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 3.1 one objective states that students should "see the relationship between multiplication and division to explain why the standard algorithm for fraction division works." This objective is shaped by the cluster heading for 6.NS.A, "Apply and extend previous understandings of multiplication and division to divide fractions by fractions."
• In Concept 7.3 the objective is "identify and generate equivalent expressions using properties of operations," which is shaped by the cluster heading for 6.EE.A, "Apply and extend previous understandings of arithmetic to algebraic expressions."
• In Concept 11.2 one objective is "select appropriate statistics to describe a data set, based on the shape of the distribution and other factors," which is shaped by the cluster heading for 6.SP.B, "Summarize and describe distributions."

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:

• In Unit 2, the Investigations of Concept 2.1 connect 6.NS.B with 6.RP.A as students learn to divide multi-digit numbers using the standard algorithm and use that to solve unit rate problems.
• In Unit 6, Concept 6.2 connects 6.G.A with 6.NS.C as students graph polygons in all four quadrants of the coordinate plane and use the graphs to solve problems. In one portion of Investigation 2, students are given sets of coordinate pairs and are asked to match each set of coordinate pairs to the name of a shape.
• Unit 7 Concept 7.1 connects 6.EE.A with 6.EE.B as students learn to write numerical and algebraic expressions and use variables when solving a real-world or mathematical problem.
• Unit 10 Concept 10.2 connects 6.SP.A with 6.SP.B as students learn to describe a single distribution by its center, shape, and spread and then to summarize and describe multiple distributions with both graphs and statistical measures.