The instructional materials reviewed for Discovery Education Math Techbook Grade 8 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 8.EE.B explores the connections between proportional relationships, lines, and linear equations to build conceptual understanding.

• In Concept 7.1 students work with proportional relationships that are represented by tables, graphs, equations, and verbal descriptions (8.EE.5). Through these multiple representations, students develop their understanding of the connections between proportional relationships, lines, and linear equations. Concept 7.1 also includes investigations that offer students the opportunity to make connections between similar triangles, constant of proportionality, rate of change, and slope. Students develop their understanding of these connections through discussions with their classmates and by completing multiple hands-on activities (8.EE.6).
• In Concept 7.3 students continue to develop their understanding of the connections between proportional relationships, lines, and linear equations as they work with non-proportional relationships that include a constant rate of change and are represented by equations, graphs, tables, and verbal descriptions (8.EE.5). Students complete investigations that help them make the connections between the different representations, and they also work with situations that result in graphs of horizontal and vertical lines. As they work with the various representations, students are asked to create new ones both manually and with technology.

Standard 8.F.1 develops understanding that a function is a rule that assigns to each input exactly one output. Opportunities to develop key mathematical concepts of functions are found in the following examples:

• In Concept 7.2 students begin to understand functions by examining situations represented with tables of data to see if they can write a rule that uses one column of data as input and produces the corresponding values in the second column of data as output. At this time, students are also presented with tables of data for which a consistent rule cannot be written. As the Concept progresses, understanding of a function as a rule that assigns exactly one output to each input is further developed as students investigate functions represented by graphs and equations, and this Concept concludes by having students match three different representations of a function (verbal, graphical, and tabular) together (8.F.1).
• In Concept 7.4 the understanding of a function is expanded as students are presented with non-linear functions represented by tables, graphs, and verbal descriptions (8.F.3).

Cluster 8.G.A builds over several Concepts an understanding of congruence and similarity through different representations and tools.

• In Concept 3.1 students have opportunities to verify the properties of rotations, reflections, and translations as they transform real-world figures, such as stop signs, and geometric figures, such as triangles and line segments, with and without coordinates. They perform these transformations with different tools, such as transparencies and a dynamic Geometry tool.
• In Concept 3.2 students continue to use various tools both on and off a coordinate plane as they develop understanding of congruence by experimenting with different combinations of rigid transformations. Students either create a new figure given an initial figure and a sequence of rigid transformations or create a sequence of rigid transformations that maps a preimage onto its image.
• In Concept 3.3 the materials include dilations as a possible transformation, and students get the opportunity to develop their understanding of similarity using similar tools and activities used when developing congruence in Concept 3.2.
• In Concept 6.1 students use transformations, along with manual and virtual tools, to develop an understanding of the relationships between interior and exterior angles within a triangle.
• In Concept 6.2 students use transformations and different tools to understand relationships between pairs of angles that are formed when parallel lines are intersected by a transversal, and students also examine relationships between pairs of angles that are formed when lines that are not parallel are intersected by a transversal.
• In Concept 6.2 students develop an understanding of the angle-angle criterion for similarity of triangles using their understanding of angle pairs formed by parallel lines and a transversal along with the relationship of interior and exterior angles within a triangle.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 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 in which 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 8.EE.7 addresses solving linear equations in one variable.

• In Concept 8.2 Practice, the Coach and Play sections present students with opportunities to solve linear equations in one variable. The Coach section offers scaffolded feedback to students if they incorrectly answer a question for two incorrect answers, and on the third incorrect answer, the feedback shows students how to solve the equations. The Play section allows students to earn rewards for correctly answering questions that involve solving linear equations in one variable.

Standard 8.G.9 addresses using formulas for the volume of cylinders, cones, and spheres.

• In Concept 11.1 students explore the formulas for the volumes of cylinders, cones, and spheres. Students develop procedural skills with them by completing numerous problems where they are asked to find the volume of objects with given dimensions or find a missing dimension given the volume and other dimensions.
• In Concept 11.1 Extension students are provided with more opportunities to develop procedural skill with the formulas for volumes of special shapes that are either composed of cylinders, cones and spheres or created by modifying a cone.
• In Concept 11.1 Apply students continue to use the three formulas as they use them to solve real-world problems involving the volumes of cylinders, cones, and spheres.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 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 materials have Extension and Apply questions with each Concept, and many of these allow students the opportunity to apply procedural skills and understandings in non-routine ways or within unique contexts. The Introduction section for most Concepts establishes real-world contexts in which students apply the skills and understandings of the Concept.

Cluster 8.F.B addresses students using functions to model relationships between quantities.

• In Concept 7.3 Apply 1 students choose a country and research the country’s average monthly temperatures over a six-month period. Students create two separate tables of data to help them determine if Celsius is a function of Fahrenheit, Fahrenheit is a function of Celsius, both tables represent functions, or neither table represents a function. Students create graphs for the tables of data and write equations to model the data in each of the graphs.
• In Concept 7.3 Apply 2 students use functions to determine which package would be the best choice when throwing a party for 8 guests and themselves.
• In Concept 8.1 Apply 1 students use functions to model the relationship between the length of the humerus bone and either a male’s or female’s height, and they also test the functions created against their own measurements and the measurements of three classmates.

Standard 8.EE.8c addresses students solving real-world and mathematical problems leading to two linear equations in two variables.

• In Concept 9.1 Extension students solve a real-world problem that leads to three equations in three unknowns. Students are presented with a scenario where a person invests three different amounts of money in three different types of accounts, and the students determine how much money was invested by the person in each account.
• In Concept 9.1 Apply 1 students use a system of two equations with two unknowns to answer various questions about flight plans that include finding the speed of the wind, the length of the flight in miles, and the length of the flight in minutes. In Apply 2 students use a system of two equations with two unknowns to determine which of two pets would be cheaper to own.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 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 example.
• 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 5.2 includes the following:

• Conceptual understanding includes “discover and understand the rules and methods for adding and subtracting numbers in scientific notation”, “discover and understand the rules and methods for multiplying and dividing numbers in scientific notation”, and “discover and understand how the properties of exponents can make operations with scientific notation more quick and efficient”;
• Fluency includes “add and subtract numbers in scientific notation with both like and unlike degrees with precision and efficiency”, “multiply and divide numbers in scientific notation with precision and efficiency”, and “rewrite the sum, difference, product, or quotient in scientific notation”; and
• Application includes “apply the rules for performing operations with scientific notation to solve a variety of real-world problems.”

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 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 3 MPs 3, 5, and 7 are identified as the focus MPs on the Progressions and Standards tab in Concepts 3.1, 3.2, and 3.3. In Concept 3.3 there are four sessions, and the following MPs are addressed in each of the four sessions respectively: MPs 3, 5, and 8 (Sessions 1 and 2); MPs 3 and 5; and MPs 3, 5, and 7.

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

• MPs 1 and 2: In Concept 10.1 Investigation 1 students make sense of a problem to determine if there is any relationship between their height and the length of their hand span. Students determine that they need to create a scatter plot of the data for the students in their class, and they persevere in solving the problem by collecting the data, graphing it, and analyzing it to see if they can identify the existence of a relationship. Students engage in reasoning abstractly and quantitatively as they consider what labels, ranges, and scales to use for each axis in the scatter plot.
• MP4: In Concept 7.4 Apply 4 students research gravity on the moon, cite evidence that explains any data, graph the data, discuss their findings, compare and contrast linear and nonlinear functions, identify rates of change and y-intercepts, and explain if they would rather play baseball on Earth or the moon and justify their answer with the information that has been gathered.
• MP5: In Concept 3.2 Investigation 5 students are presented with pairs of congruent figures to determine a rigid transformation or sequence of rigid transformations that maps one image onto the other. To complete this Investigation, the materials include a dynamic geometry tool, but students still get to engage with using appropriate tools strategically as they have to determine which rigid transformations to use within the dynamic tool in order to map one image onto the other.
• MP7: In Concept 5.1 Investigation 1 students look for and make use of structure as they identify compact ways to write large numbers using scientific notation.
• MP8: In Concept 1.1 Investigations 1 through 4, students look for and express regularity in repeated reasoning as they develop and use rules for simplifying expressions with integer exponents based on completing various hands-on and virtual activities.

The instructional materials reviewed for Discovery Education Math Techbook Grade 8 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 4.3 Investigation 4 students make sense of a problem as they try to determine if two planes are within one nautical mile of each other on radar, realizing that they can use the Pythagorean theorem to determine the distance between the planes. Students first analyze the paths of the planes two-dimensionally, and they persevere in solving the problem as they have to complete their analysis in three-dimensional space after correctly completing it in two-dimensional space.
• MP2: In Concept 9.1 Investigation 2 students reason abstractly and quantitatively as they answer questions about different race scenarios. The students reason abstractly as they create different mathematical representations of the race scenarios in order to perform various computations, and they reason quantitatively as they re-contextualize what they see in the abstract representations in order to determine the outcomes of the race scenarios and how those outcomes unfolded.
• MPs 4 and 5: In Concept 7.1 Investigation 4 students model with mathematics as they try to put together a triathlon team that can break the world record. Students use data for different athletes swimming, biking, and running times- represented in various ways- and examine many different combinations of athletes to determine which combination is the fastest. As students examine the times of the athletes, they can choose to represent data from the athletes with an equation, table, or graph, and they can choose to create these representations manually or with technology.
• MPs 7 and 8: In Concept 2.2 Investigation 1 students look for and make use of structure while completing Using a Spreadsheet as they examine the prime factorizations of the denominators of fractions to determine whether or not the fraction has a decimal expansion that terminates or repeats. In the same Investigation while completing Converting Fractions to Decimals, students look for and express regularity in repeated reasoning as they create the decimal expansions for two sets of fractions and compare the sets of fractions based on the decimal expansions created using the standard algorithm for division.

The materials reviewed for Discovery Education Math Techbook Grade 8 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:

• In Concept 1.1 Investigation 6 students analyze the work of a fictitious student for possible errors because two different expressions simplify to the same value. Students determine which expressions are correct and explain their reasoning.
• In Concept 4.3 Investigation 1 students construct an argument as they estimate the distance between two locations on a grid. They also assess their own estimate by analyzing the reasoning and estimate of a fictitious student.
• In Concept 6.1 Introduction students discuss their conjectures about relationships among angles with another student. Through the discussions, students are expected to refine their original conjectures based on the critiques given to them by others.
• Unit 6 Assessment Problem 12 asks students to use their knowledge of rigid transformations to justify their answers in one part of the problem. Then, they explain a relationship among the measures of the three angles of a triangle.

The materials reviewed for Discovery Education Math Techbook Grade 8 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 are some examples of the materials assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others:

• The Concept 1.1 Investigation 3 Instructional Notes state, "Have students work in small groups to make a conjecture for this activity. Students should explain their conjecture using examples to their group members. Group members are asked to challenge any conjectures and/or examples that they believe are incorrect or only partially correct."
• In Concept 3.1 Introduction students determine if the conclusions of two fictitious students are correct or incorrect. The Instructional Notes and the Teacher Note for the Introduction both alert teachers that this activity is a place where teachers can assist students in constructing viable arguments and analyzing the arguments of others.
• In Concept 6.2 Investigation 1 the Teacher Note guides teachers to "facilitate a discussion about the relationships between the different pairs of angles in the diagram and help students come to the conclusion that every pair is either congruent or supplementary." The Teacher Note includes the following questions to assist teachers in facilitating the discussion: “How many angles are formed by a transversal intersecting a pair of parallel lines, and how could you describe the angles formed when the transversal is perpendicular to the parallel lines?”

The materials reviewed for Discovery Education Math Techbook Grade 8 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 1.1 Session 1 Introduction the Instructional Notes state that teachers should "model precise mathematical language and then ask students to explain their thinking in a similar way." In the Concept 1.1 Investigation 4 Instructional Notes, teachers are reminded "as students engage in these example problems, continue to review the vocabulary associated with the expressions (expansion or expanded form, exponential form, standard form)."
• The Concept 2.1 Session 3 Instructional Notes state that "after students complete Another Number Riddle, have them discuss their results with a partner. Listen to student responses as they justify their answers and provide ideas to help refine their mathematical communication."
• The Concept 3.1 Session 4 Instructional Notes state, "you may want to have students make a table in their notes, such as the one below, to help them remember the vocabulary introduced in this concept." The column headings for the table given in the Instructional Notes are "Vocabulary Term," "Formal Definition," "Description in Your Own Words," and "Example Image."

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 could be asked to refer 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 Common Misconceptions, the materials will state that "Students may have difficulty with the vocabulary" when appropriate.