The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding and include the use of visual representations, interactive examples, and different strategies.

Cluster 6.RP.A addresses understanding ratio concepts and using ratio reasoning to solve problems.

• In Concept 1.2 students begin to develop their understanding of ratio concepts and rate reasonings as they explore different representations of ratios and rates in the context of rabbits and weasels. Students represent ratios with tape diagrams, double number lines, and tables of values. Students also compare ratios using these different representations. (6.RP.1, 6.RP.3a)
• In Concept 4.1 students continue to develop their understanding of rate reasoning as they identify the unit rate for a given context. Finding the unit rate is coupled with the different representations from Concept 1.2, and students work to solve problems using unit rate and the different representations. Students also discuss how scenarios can have more than one unit rate, and they compare unit rates. (6.RP.2, 6.RP.3b)
• In Concept 4.2 students enrich their understanding of ratios and rates as they examine converting between units of measurement in contexts such as length, volume, and money. Students also add to their understanding by making complex conversions in order to determine if a person could outrun different animals. (6.RP.3d)
• In Concept 4.3 students add to their understanding of ratios and rates as they use them to examine percentages and to solve problems finding the percent of a quantity or finding the whole, given a part and the percent. (6.RP.3c)

Cluster 6.NS.C addresses applying and extending previous understandings of numbers to the system of rational numbers.

• In Concept 5.1 students begin to apply and extend their previous understandings of numbers to rational integers by learning how to plot negative integers on horizontal and vertical number lines. Through the number lines, students also begin to compare rational integers, and the students also begin to understand opposites as they explore an interactive applet that has them place objects on a tray in a position that is opposite of objects already on the tray. Positions on the tray are identified by rational integers along a horizontal number line. (6.NS.C.5)
• In Concept 5.2 students expand their understandings from Concept 5.1 as they begin to work with all rational numbers and not just integers. Students continue to use horizontal and vertical number lines for the purposes of plotting, comparing, and ordering rational numbers. (6.NS.6c)
• In Concept 5.3 students further develop their understanding of rational numbers as they build upon the idea of opposites to explore and compare absolute values, and they play an interactive game where they choose numbers for absolute value in order to throw boomerangs and defeat attacking skeletons. Students choose the absolute values based on the skeletons’ positions on a horizontal number line as the boomerang travels through those positions that have the chosen absolute value. (6.NS.6a, 6.NS.7)
• In Unit 6 students are introduced to the coordinate plane with four quadrants so that they can explore and understand how rational numbers are used to locate points on a coordinate grid and how real-world objects, such as maps, can be identified when superimposed on a coordinate grid. Students also develop an understanding of how rational numbers on a coordinate grid can be used to calculate lengths, perimeters, and areas of geometric shapes whose vertices are identified by coordinate pairs of rational numbers. (6.NS.6b, 6.NS.8)

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, the Practice section is designed to give students opportunities to develop procedural skills and fluency in each Concept, and the Practice section has Coach and Play sections that allow the students to choose how they want to work. The Coach section provides ten guided practice questions, and the Play section is independent practice with at least 15 questions.

Standard 6.NS.2 addresses students being able to fluently divide multi-digit numbers using the standard algorithm.

• In Concept 2.1 students investigate the standard division algorithm with multi-digit numbers during the Discover section of the Concept. In the Coach section of Practice, there are problems that highlight students using division to solve a problem, and there are other problems that highlight students being asked to determine when division is needed to solve a problem.
• In Concept 2.1 Extension students build procedural fluency in dividing multi-digit numbers using the standard algorithm as they examine division problems for patterns.

Standard 6.NS.3 addresses students being able to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

• In Concept 3.2 Discover students develop fluency in adding, subtracting, multiplying, and dividing multi-digit decimals in the context of money, using the standard algorithms for each operation. Multiplying decimals is represented by an area model to help students connect and develop procedural skill with the standard algorithm for multiplying decimals and show how dividing by decimals can be related to dividing by whole numbers to help students develop procedural skill in using the standard algorithm for dividing decimals.
• In Concept 3.2 Practice Coach students further develop their procedural skills with the standard algorithms for multiplying and dividing decimals. In the Coach and Play sections, students use the standard algorithm for adding, subtracting, multiplying, and dividing decimals in order to strengthen their fluency in these operations with multi-digit decimals.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectation for teachers and students spending sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. The Introduction section for most concepts establishes real-world contexts in which students apply the skills and understandings of the Concept while Extension and Apply questions provide students with the opportunity to apply procedural skills and understandings in non-routine ways in unique contexts.

Standard 6.RP.3 addresses students using ratio and rate reasoning to solve real-world and mathematical problems.

• In Concept 1.1 students begin to apply ratio and rate reasoning to solve problems as they are presented with videos and pictures of animals and asked to state different comparisons about the pictures. Students then compile survey data gathered from their classmates and write ratios that describe the responses from the survey. Then, students use ratios to design their own strand of lights for a light company.
• In Concept 1.2 students use different representations of ratios- such as tape diagrams, double number lines, and tables- to solve non-routine or novel real-world problems, such as predators and their prey, animals living in a garden, and mixing paints to produce certain shades of colors.
• In Unit 4 students apply their knowledge of rates to solve various real-world problems along with converting units of measurement and working with percents.

Standard 6.NS.1 addresses students solving word problems involving division of fractions by fractions.

• In Concept 3.1 students use division of fractions by fractions to solve real-world problems in various contexts that include adjusting recipes for making food, using ribbon and string to create different objects, and distributing supplies for camping and hiking.

Standards 6.EE.7 and 6.EE.9 addresses students writing and solving linear equations in order to solve real-world and mathematical problems.

• In Concept 8.1 students write and solve linear equations to solve problems that involve saving money from recycling, determining the number of wolves living in separate packs, determining the amount of money made by collecting rocks, studying volcanoes, and determining the profit made by selling fresh fish.
• In Concept 9.1 students examine a situation that involves a county fair and roller coasters. Using the roller coasters, students explore the relationship between independent and dependent variables, and the exploration of the relationship leads to writing equations. Then, the story leads into using a problem involving popcorn and another involving the games at a fair to help students create tables and write equations.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately.

Each Concept includes Discover, Practice, and Apply sections.

• Discover includes Introduction, Investigation, Summary, and Extension sections that give students the opportunity to build conceptual understanding of the mathematics and practice procedural skills, typically in the context of a real-world problems.
• Practice focuses on procedural skills with a Coach section that provides student support to develop fluency, for example, leading students through solving an algorithmic problem and giving immediate feedback; as well as a Play section where students demonstrate procedural fluency without support.
• Apply includes extended tasks based on real-world applications.

In the Model Lesson section of the teacher materials, Progressions and Standards includes a diagram that identifies for teachers the balance of conceptual understandings, procedural fluencies, and applications that should emerge from each Concept in a Unit. For example, Concept 6.2 includes the following:

• Conceptual understanding includes “explain how absolute value can be used to calculate distances on a coordinate plane” and “explain how the coordinate plane relates to attributes of two-dimensional geometric figures.”
• Fluency includes “use absolute value to find distance between points that lie on the same vertical or same horizontal plane” and “plot vertices of a geometric figure based on a description of attributes.”
• Application includes “use absolute value of coordinates to determine distance between points on a horizontal line or on a vertical line to determine area and perimeter of rectangles for a real world context” and “use coordinates to describe and identify two-dimensional geometric figures for a real world context.”

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectation for identifying and using the Standards for Mathematical Practice (MPs) to enrich the mathematics content within and throughout the grade. Overall, the MPs are identified in different places throughout the materials, and the MPs enrich the content as students make sense of problems, reason about the mathematics, and use different models and tools to complete the problems.

The MPs that are a focus for each Unit are identified under each Concept on a tab marked Progressions and Standards, and the MPs that are a focus for each Session appear on the session tab in a part labeled Standards for Mathematical Practice. For example, in Unit 12 MPs 1, 4, and 6 are identified as the focus MPs on the Progressions and Standards tab in both Concepts 12.1 and 12.2. For Sessions 1 and 2 in Concept 12.2, MPs 1 and 3 are listed as being addressed during the sessions.

Some examples of how the MPs are used to enrich the mathematics content include:

• MPs 1 and 2: In Concept 8.2 students engage in writing and solving inequalities. MP1 is used to enrich the content across the three Investigations as students make sense of various contexts in order to translate them into an inequality with appropriate constraints. MP2 is used to enrich the content as students reason to represent the symbols of inequalities (abstract) on a number line diagram with units from the context (quantities).
• MPs 4 and 5: In Concept 1.2 students engage in creating equivalent ratios. MP4 is used to enrich the mathematics content as students learn how to model ratios with tables, tape diagrams, double number lines, and graphs across the four Investigations of the Concept. MP5 enriches the content as students are asked which models they would use to solve different types of problems involving ratios in Investigation 4.
• MP7: In Concept 6.2 Investigation 1 teachers are reminded that the investigation is a chance for students to use the structure of finding the distance between two points on a number line when trying to find the distance between two points that have the same x-coordinate or the same y-coordinate.
• MP8: In Concept 3.1 Investigation 2 teachers are reminded to have students discuss how the fraction division they just completed compares to the fraction division students completed in Investigation 1. By looking for and using repeated reasoning across the investigations, the students use MP8 to strengthen their knowledge of dividing fractions.

The instructional materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectations for carefully attending to the full meaning of each practice standard. Overall, the materials, as a whole, address the full meaning of each of the Standards for Mathematical Practice (MPs).

Some examples of where the materials attend to the full meaning of the MPs include:

• MP1: In Concept 13.2 Apply 2 students are asked to redesign the packaging for a food product so that the new package holds the same amount of food as in the original package but uses less material. The students are also given three constraints for their new, redesigned package. In this problem, the students must first realize that holding the same amount of food refers to the volume of the new container and that using less material refers to the surface area of the package. Students then have to make sense of which dimensions are affected by the three constraints and how those constraints affect the dimensions. Once the students have made sense of the constraints, the students will have to persevere in finding a set of dimensions for their new package that meets all of the criteria given to them.
• MP2: In Concept 5.1 Investigation 1 students engage with plotting integers on a number line and comparing integer quantities in the contexts of temperatures and climbing heights relative to a fixed position. Students reason quantitatively as they are asked to put different sets of values in order from least to greatest while considering the context. The students reason abstractly by comparing pairs of integers free from context and by graphing sets of integers on horizontal or vertical number lines.
• MP 5: In Concept 11.2 Investigation 3 students are presented with different types of statistical plots, and they are asked to explain which measure of center and which measure of variability would best describe each plot. In addition to using specific measures of center and variability to strategically describe the plots, students also get to choose which tools they would use to help them calculate the measures of center and variability that they select.
• MPs 7 and 8: In Concept 4.3 Investigation 1 students explore the relationship between fractions, decimals, and percents through an interactive that uses an area model. As students select regions of the area model to shade in, the interactive model produces a fraction, decimal, and percent for the shaded-in area. Students use the structure of the rectangular area model, which includes 10-by-10 grids, to see the correspondence between the area model and the equivalent fraction, decimal, and percent. By changing the shaded area in the area model and recognizing the repetition in the relationship between the fraction, decimal and percent, students express the regularities of how a decimal is changed to a percent and how a percent is changed to a decimal.

The materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectation for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials provide multiple opportunities for students to explain their reasoning and to conduct error analysis of work.

Some examples of students being prompted to construct viable arguments and/or analyze the arguments of others include:

• Unit 1 Assessment Problem 4 has students explain the difference between ratios and fractions and "describe how the process of making equivalent ratios is similar to the process of making equivalent fractions."
• Unit 4 Assessment Problem 10 has students analyze the arguments of two people as they calculated different unit rates for a problem, and the students construct a viable argument as they explain how the different unit rates could both be correct.
• In Concept 5.3 Session 3 students analyze the arguments of others that involve using absolute value to describe amounts of money owed to people, and the students use their analyses to construct a viable argument as to how the actual values of numbers compares to the absolute values of the numbers.
• In Concept 10.2 Investigation 1 students analyze data so that they can critique the hypothesis of two children playing a game, and after collecting data of their own, students construct their own hypothesis about the game.
• In Concept 12.2 Investigation 1 students construct an argument for how they found the area of a trapezoid, and they analyze the arguments of others because they are supposed to find a classmate that used a method different from them and compare the strategies that each student used.

The materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, teachers are given questions to ask during the Investigations that assist students in constructing viable arguments and analyzing the arguments of others.

The following examples illustrate how the materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others:

• In Concept 3.2 Session 1 students are asked to complete a problem during Investigation 1 entitled "Multiplying Money," and they are supposed to justify their answer to the multiple-choice problem. In the Instructional Notes for the Investigation, the teacher is given the following questions to ask students: “Why can’t choice A be correct?; What do you think the student who wrote choice B did wrong and why is this a common error?; What do you think the student who wrote choice D did wrong, and why is this a common error?” Each of these questions prompts students to analyze the argument on which each of the incorrect choices is based.
• In Concept 4.2 Session 2 Investigation 1 students complete "Find the Height in Yards," a problem that utilizes many different methods and models. The students explain how they solved the problem, and in the Instructional Notes for the Session, the teacher is given the following instruction, "After students complete each problem, have them discuss their solutions with a partner or in a class discussion, justifying their own work and critiquing that of fellow students.” This additional information for the teacher assists them in helping to engage the students in constructing a viable argument and analyzing the arguments of others.
• In Concept 11.1 Session 2 Investigation 1 students complete a task called "Analyze Methods" for finding the mean of a set of data. In the Instructional Notes for the Investigation, the teacher is given the following questions to ask students: “How are Christina’s and Davis’ strategies similar, and how are they different?; What steps did Davis use to find the mean?; How are the numerical expressions that model Davis’ strategy connected to the strategy that Group 3 used, and would this strategy work for any other groups?” Each of these questions prompts students to analyze the argument of another person.

The materials reviewed for Discovery Education Math Techbook Grade 6 Mathematics meet the expectation for attending to the specialized language of mathematics. Overall, the specialized language of mathematics is appropriately introduced and reinforced throughout the materials.

Some examples of attending to the specialized language of mathematics include:

• In Concept 5.2 Session 1 students are asked to "define what a rational number is with a partner, and then compare it to the definition in the glossary. Students will most likely be surprised that the definition includes the fact that all rational numbers can be written as fractions (when the denominator is not zero)."
• Concept 7.1 Investigation 1 states, "Have students complete Analyze. ... As a class, develop a definition of an algebraic expression and compare it to the glossary definition. Ask: How can you define an equation in terms of expressions? (An equation is made up of expressions that are equal to each other.)" In the next part of the Investigation, students work with a partner to determine the characteristics of a term in an expression, when terms are like each other, and when terms are not like each other.

The following are some examples of how the specialized language of mathematics is regularly addressed throughout the materials.

• The vocabulary terms for each unit are given in the Teacher Preparation for each Concept, and new vocabulary terms are often italicized or mentioned in a sentence.
• There is an Interactive Glossary that provides students with the definition of a word, an animation, and a video that uses the word in a real-world context. The glossary can be searched alphabetically or by Concept, and during lessons, students may be referred to the Interactive Glossary for assistance with the vocabulary.
• When there is a new vocabulary term, it is regularly used throughout the remainder of the unit to reinforce comprehension.
• In the Common Misconceptions, the materials will state that "Students may have difficulty with the vocabulary" when appropriate.