The instructional materials reviewed for Discovery Education Math Techbook Grade 7 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 7.EE.A addresses understanding how the properties of operations can be used to generate equivalent expressions.

• In Concept 5.3 students develop an understanding of how the properties of operations can be used to generate equivalent expressions through expanding and factoring linear expressions (7.EE.1). In addition to working with abstract, algebraic expressions, students have the opportunity to visually represent the expressions they are manipulating through the use of virtual algebra tiles. With these tiles, students can also represent positive and negative coefficients and constants.
• In Concept 6.1 students have the opportunity to further their understanding of using the properties of operations to generate equivalent expressions in the context of writing and solving linear equations (7.EE.2). In addition to working with abstract, algebraic equations, students have the opportunity to visually represent the expressions from each side of the equation with virtual algebra tiles, and the tiles can also represent positive and negative coefficients and constants.

Cluster 7.NS.A addresses applying and extending previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. This cluster is introduced and developed in Unit 1.

• In Concept 1.1 students understanding of operations with rational numbers is developed by presenting them with different contexts that promote using positive and negative numbers, and within these contexts, the students have to add and subtract the rational numbers in order to answer questions. Students are given the opportunities to use horizontal and vertical number lines as they answer the questions. In some contexts, the number lines are labeled for them, and in other contexts, the students have to label the number lines on their own (7.NS.1).
• In Concept 1.2 students continue to develop their understanding of operations with rational numbers as they complete an activity about walking either forwards or backwards. In this activity, students place a number line on the floor of their classroom with tape, label the number line, and walk along the number line to certain points based on directions given to them (7.NS.1).
• In Concept 1.3 understanding of operations with rational numbers is further developed as students use an interactive applet to help a superhero rescue a cat by multiplying rational numbers. In addition to using a number line in the applet, the materials also develop understanding by allowing students to control the number of jumps the superhero makes and the size and direction of each jump. Also, in Concept 1.3 understanding of division with rational numbers is developed by building upon what students already know about the relationship between multiplication and division with other types of numbers and by using horizontal and vertical number lines (7.NS.2).

The instructional materials reviewed for Discovery Education Math Techbook Grade 7 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. The Practice section has Coach and Play sections that allow the students to choose how they want to work. The Coach provides ten guided practice questions, and the Play section is independent practice with at least 15 questions.

Cluster 7.NS.A addresses students developing procedural skills with adding, subtracting, multiplying, and dividing rational numbers.

• Concept 1.2, students build procedural skill with adding and subtracting rational numbers within the context of having a bank account. Students also build procedural skill with adding and subtracting rational numbers playing different card games where red cards represent negative numbers and black cards represent positive numbers. In Practice, the Coach and Play sections present students with more opportunities to develop procedural skill with adding and subtracting rational numbers. The Coach section offers feedback to students if they incorrectly answer a question, and the Play section allows students to earn rewards for correctly answering questions that promote procedural skill with adding and subtracting rational numbers.
• Concept 1.3, students build procedural skill with multiplying and dividing rational numbers as they examine different scenarios that involve writing numerical equations with multiplication or division of rational numbers. The students also build procedural skill in using the distributive property to aid in multiplying and dividing rational numbers.

Standard 7.EE.1 addresses students being able to add, subtract, factor, and expand linear expressions with rational coefficients, and 7.EE.4 expects students to develop procedural skills with solving linear equations and inequalities.

• In Concept 5.3, students practice expanding expressions with the aid of virtual, algebra tiles, and at the same time, they add and subtract linear expressions while determining which expressions are equivalent to each other. Students also build procedural skill with factoring linear expressions in order to create equivalent expressions.
• In Concept 6.1, students continue to build procedural skill with adding, subtracting, factoring, and expanding linear expressions within the context of solving linear equations. Students are given the opportunity to use virtual, algebra tiles while solving linear equations, and they also get to build procedural skill with solving linear equations that involve coefficients and constants written as fractions and decimals.
• In Concept 6.2, students get the opportunity to use virtual algebra tiles while building procedural skill with solving linear inequalities. Students practice solving linear inequalities within the context of moving boxes up an elevator and taking a class field trip to the zoo. Students also practice solving linear inequalities that involve multiplying or dividing one side of the inequality by a negative number and examining the effects of that.

The instructional materials reviewed for Discovery Education Math Techbook Grade 7 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 within each Concept, and many of these allow students the opportunity to apply procedural skills and understandings in non-routine ways and with unique contexts. Also, the Introduction section for most concepts establishes real-world contexts in which students apply the skills and understandings of the Concept.

Cluster 7.RP.A addresses students using proportional relationships to solve real-world and mathematical problems.

• In Concept 3.1 students apply ratios and unit rates to solve problems that involve a food truck business, shipping food versus buying locally, speed limits, and planning for walking trails in a city park.
• In Concept 3.3 students apply proportional relationships to solve problems about falling from a height, making predictions about where an animal that escaped from the zoo might be, and determining how many helium-filled balloons would be needed to lift different weights into the sky.
• In Concept 4.1 students are presented with real-world scenarios and are asked to apply proportional relationships in order to solve problems. The scenarios presented include constructing a garden, artwork, and the speeds of baseball pitches. Also, in Concept 4.2 students apply proportional relationships to examine scale models of houses, maps, and changing the size of photographs.

Standard 7.NS.3 addresses solving real-world and mathematical problems involving the four operations with rational numbers.

• In Concept 2.2 students apply operations with rational numbers as they develop strategies for working their way through a maze, buying submarine sandwiches for a party, delivering aid to villages and individuals, and planning their route for driving from one place to another.

The instructional materials reviewed for Discovery Education Math Techbook Grade 7 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 10.1 includes the following:

• Conceptual understanding includes “understand how and why complementary, supplementary, vertical, and adjacent angles are related.”
• Fluency includes “identify angles as complementary, supplementary, vertical, or adjacent,” “find missing angle measures using angle relationships,” and “write and solve equations to determine missing angle measures.”
• Application includes “solve multi-step equations involving angle measure in real-world and mathematical contexts.”

The instructional materials reviewed for Discovery Education Math Techbook Grade 7 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 7 MPs 3, 4, and 6 are identified as the focus MPs on the Progressions and Standards tab in both Concepts 7.1 and 7.2. In Concept 7.1 there are five sessions, and the following MPs are addressed in each of the five sessions respectively: MPs 3 and 6; MPs 2, 3, and 4; MPs 2 and 3; MPs 3 and 8; and MPs 1, 3, 7, and 8.

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

• MP1: In Concept 2.1 Investigation 1 students must make sense of criteria that are given to them as they select rational numbers to plot on number lines that meet the given criteria. The students must persevere in the Investigation as they are then asked to place numbers between given pairs of rational numbers, and the pairs of rational numbers include: 4 and 5; 2/5 and 3/5; -6.85 and -6.84; and 285.3274 and 285.3275.
• MP2: In Concept 12.1 Investigation 3 students consider building a water desalination machine that is in the form of a pyramid. Students reason abstractly as they calculate the surface area of the pyramid with given dimensions so that they can determine how much material is needed to build the machine. The students also reason quantitatively as they consider what changes would need to be made to the dimensions of the machine in order to provide enough drinking water for different numbers of people.
• MP4: In Concept 3.3 Apply 1 students choose a roller coaster, research it, explain how long they would be willing to wait in line for it, and express the amount of people that would be in line before them to have to wait that long. Then, students use an equation, table, graph, or other model to explain and justify their answer, and they also show the relationship between the number of people ahead of them and the wait time.
• MP5: In Concept 9.1 Investigation 1 students use appropriate tools strategically as they determine what materials they would use to design a simulation that would help them predict the probability of delays at an airport based on a given set of data.
• MPs 7 and 8: In Concept 4.2 Investigation 3 students make use of structure by using the relationship between length, width, and area of rectangles to complete a table of values that includes different scale models of a rectangular picture. Then, the students express regularity in repeated reasoning by developing rules that can be used to determine how changes in side lengths affect the area of the rectangular models determined by the different side lengths.

The instructional materials reviewed for Discovery Education Math Techbook Grade 7 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:

• MPs 1 and 2: In Concept 1.2 Extension students have to make sense of problems in order to assign rational numbers to different real-world contexts, and they have to persevere in solving the problems as they are asked to adjust the numbers assigned as different characteristics of the contexts are changed. The students also reason abstractly as they manipulate numbers to meet the given characteristics for different contexts, and after the manipulations, the students quantify the new numbers in the original context for the problem.
• MPs 4 and 5: In Concept 9.2 Investigation 1 students model with mathematics as they create a probability distribution for the outcomes when playing the game rock-paper-scissors. Students can also adjust their distribution based on playing the game with different pre-determined strategies. As students create their distributions, they use appropriate tools strategically as they record both theoretical and experimental outcomes with organizers of their own choosing.
• MP7: In Concept 10.3 Investigation 3 students make use of structure by drawing auxiliary lines through complex polygons in order to decompose the complex polygon into triangles, trapezoids, or parallelograms. The areas for the triangles, trapezoids, or parallelograms are calculated, and by adding the areas together, the area of the original, complex polygon is found.
• MP8: In Concept 5.3 Investigation 1 students use virtual algebra tiles in Expanding Expressions to repeatedly practice multiplying a constant by a linear, binomial expression. Through the repeated reasoning from the interactive model, students end the investigation by expressing abstractly how to expand an expression that involves a constant multiplied by a linear, binomial term using the distributive property.

The materials reviewed for Discovery Education Math Techbook Grade 7 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.

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

• Unit 1 Assessment Problem 2 students explain why the subtraction of a positive number is the same as adding the inverse.
• In Concept 4.1 Investigation 4 students complete Analyze Another Student's Work. In this activity, students critique the work of a fictitious student and determine if the work is correct or not. Students critique the work to identify errors, analyze these errors, and construct arguments to arrive at a correct solution.
• In Concept 5.2 Investigation 1 students critique the reasoning of multiple pairs of fictitious students' work involving the order of operations with rational numbers. For each pair of fictitious students, students analyze the work to identify and explain what is correct and where there are errors. They construct an argument to support their analysis.
• In Concept 10.1 Investigation 2 students complete What's Your Strategy. In this activity, students construct an argument to explain the strategy they would use to play a game that involves using various angle relationships, such as vertical angles or supplementary angles. Students are also given the opportunity to critique other students' strategies as part of the activity.

The materials reviewed for Discovery Education Math Techbook Grade 7 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:

• In Concept 3.2 Session 3 teachers are reminded to have students “complete the Proportion and Justify Your Answer” in the Instructional Notes. Teachers "...monitor student progress, listening for appropriate mathematical reasoning. Reconvene the class and ask students to share their methods for finding the correct amount of water. Encourage students to offer constructive feedback about each method. Facilitate a discussion about how students can judge the reasonableness of their answers. Teachers may also wish to engage students in dimensional analysis by asking why gallons might be a more useful unit in this situation than cups.”
• In Concept 7.2 Session 1 students are directed to read and consider the on-screen questions that are part of the Intro. In the Instructional Notes for the Session, the following questions are suggested for teachers to promote further discussion: “How would you expect the data from Sulia’s class to be different if her argument were more compelling?; Would the contest be fairer if the seventh-grade student was picked from the bottom 50% of students in Sulia’s class, why or why not?; If the data distributions were changed so that the tallest student in the seventh-grade class was moved to the fifth-grade class, how would the arguments change?; and Would you still agree with Jordan?” These questions prompt students to analyze the arguments presented to them and help students develop a bank of questions to use when critiquing arguments.

The materials reviewed for Discovery Education Math Techbook Grade 7 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.

Examples of attending to the specialized language of mathematics include:

• Concept 8.1 Session 1 states, "Direct students to read the definition of probability, and have them complete Certain and Impossible Events."
• Concept 11.1 Session 1 states, "Students are also introduced to the terms diameter, radius, and circumference and consider how these parts may be related to each other."

The following are 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 directed 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.