enVision+ Texas Mathematics, Adv 06

Evaluation for 1.1a
Materials include a scope and sequence outlining the TEKS, ELPS, and concepts taught in the course.

The Scope and Sequence tab, located under the “Browse” heading in the Navigation Guide, includes four files for school years of various lengths (135, 165, 180, and 210 days). Each file includes “Topics” (course concepts), broken into individual lessons that indicate the Texas Essential Knowledge and Skills (TEKS) and the English Language Proficiency Standards (ELPS) are addressed. For example, the 165-day scope and sequence document describes topic 1, “Use Positive Rational Numbers,” as a 10.5-day lesson that teaches TEKS 6AM.3A, 6AM.3B, 6AM.3C, 6AM.3F, using ELPS 1C, 2B, 2D, 3C, 3F, 3G, and 4C. This same topic spans nine days, 10.5 days, and 15.5 days in the 135-day, 180-day, and 210-day scope and sequence files, respectively.

Additionally, the vertical Scope and Sequence in the end matter of the enVision+ Texas Mathematics 2028 Advanced Grade 6 teacher’s edition includes Advanced Grades 6–8, Geometry, and Algebra II, noting that “The statements in the Scope and Sequence are from the TEKS. The chart is organized to show progressions across similar strands.” For example, the scope and sequence chart includes 17 strands, such as “Numbers and Operations” and “Circles,” highlighting the strands taught in each course and showing that six of the 17 strands are taught in enVision+ Texas Mathematics 2028 Advanced Grade 6.

Evaluation for 1.1b
Materials include suggested pacing (pacing guide/calendar) to support effective implementation for various instructional calendars (e.g., varying numbers of instructional days - 165, 180, 210).

The Pacing Guides under the “Browse” heading include four files with pacing guides for school years of varying lengths (135, 165, 180, and 210 days) that list the topics, the number of days each topic should take to teach, and the TEKS covered. The days allotted for each topic vary by instructional calendar length. Specifically, topic 10, “Financial Math,” requires 12 days under the 180-day instructional calendar and requires nine days under the 165-day instructional calendar.

Likewise, four separate scope and sequence documents under the “Browse” heading cover school years of varying lengths (135, 165, 180, and 210 days) and suggest the number of days to allot for each topic. For example, topic 1, “Use Positive Rational Numbers,” recommends nine instructional days and two assessment days for the 135-day calendar, and 15.5 instructional days and 3.5 assessment days for the 210-day calendar.

Evaluation for 1.1c
Materials include an explanation for the rationale of unit order as well as how concepts to be learned connect throughout the course.

The article “Advanced Grade 6 Content Organization Rationale” states that “Topics are sequenced to support mathematics progressions that connect new content to prior knowledge,” and explains that the TEKS recur in multiple lessons within and across topics to reinforce more connections.

Furthermore, the Program Overview of the teacherʼs edition contains a graphic organizer that explains how the content grouping builds mathematical connections. It states that topics 5, “Integers and the Coordinate Plane,” and 6, “Integer Operations,” are grouped because “Students extend the number system to integers, identifying a number, its opposite, and its absolute value. Students represent absolute values on number lines and use these relationships of distance and direction, as well as other models, to add and subtract integers. They generalize these understandings to fluently add, subtract, multiply, and divide integers.”

Evaluation for 1.1d
Materials include protocols with corresponding guidance for unit and lesson internalization.

The Teacher’s Topic Internalization Protocol document in the Internalization Protocols and Observation Tools contains “Scope & Sequence and Pacing,” “Preparing to Teach the Topic,” “Assessing the Topic,” and “Gather and Prepare Materials,” providing detailed steps to internalize each unit and lesson. For example, in the “Scope & Sequence and Pacing” section, step 1 guides educators to “Consult your scope and sequence to place this topic in the context of the year-long curriculum,” and step 3 of the “Preparing to Teach the Topic” section guides the educator to “Identify relationships between lessons and how they build toward successful TEKS coverage.”

Additionally, the teacher’s edition provides a Math Background for each topic; topic 1, “Use Positive Rational Numbers,” guides the educator through key concepts, balance and coherence, process standards, and linguistic accommodations for the unit, for example. Furthermore, the teacher’s edition offers side-by-side guidance for lesson internalization for each page of the lesson, which corresponds to the student edition.

Evaluation for 1.1e
Materials include resources and guidance for instructional leaders to support teachers with implementing the materials as designed.

Several links in the Internalization Protocols and Observational Tools contain resources for instructional leaders to support educators in implementing the materials as designed, including step-by-step guidance on topic and lesson internalization. For example, the Instructional Leaders’ Lesson Internalization Protocol document contains a three-part Guidance section, one of which is “Implementation,” which suggests “facilitat[ing] discussions about fundamental concepts and skills addressed in the lesson and how these are assessed through the lesson’s practice and the Exit Tickets.”

Furthermore, the Internalization Protocols and Observational Tools provide the Instructional Leaders’ Student Work Analysis Protocol, a document with step-by-step guidance on analyzing student work. Specifically, guidance is provided for implementation to coach educators to “identify not just which TEKS are addressed, but which specific breakout of the TEKS” so educators can “Sort and Categorize Student Work” to “create meaningful categories based on solution strategies, common misconceptions, levels of reasoning, [and] completeness of solution.”

Evaluation for 1.2a
Materials include comprehensive unit overviews that provide the background content knowledge and academic vocabulary necessary to effectively teach the concepts in the unit.

The teacherʼs edition includes a Math Background section at the beginning of topic 2, “Analyze Geometric Relationships,” with guidance on key concepts such as “Find Missing Measurements” and “Find Area,” and academic vocabulary, including Linguistic Accommodations for terms such as scale factor, decompose, and kite. For example, the guidance states that the “Vocabulary in Context” includes volume, with a math meaning of “the amount of space occupied by a three-dimensional object as measured in cubic units” and other definitions, such as “the degree of loudness or the intensity of a sound.”

Moreover, the Math Background: Coherence section explains how concepts connect to what students in grade A6 learned previously, such as “Properties of Shapes” and “Volume” in grade 5, and “Graph Points on the Coordinate Plane” and “Evaluate Algebraic Expressions” earlier in grade A6. It also specifies how concepts such as “Scale Drawings” and “Area of a Triangle” in topic 2 lead to concepts such as “Solve Measurement Problems” and “Area and Surface Area” in grade A7.

Evaluation for 1.2b
Materials contain supports for families in both Spanish and English for each unit with suggestions on supporting the progress of their student.

The online platform includes a “Family Engagement” section for every topic, giving families access to each lessonʼs concepts in the unit. For example, in lesson 1-5, “Let’s Build: Multiplying Fractions,” families can watch two tutorial videos and review key concepts such as multiplying unit fractions. Families can change the language from English to Spanish by toggling the language selection at the top right of the window.

Additionally, each topic includes a “Home-School Connection” letter in English and Spanish with detailed support and suggestions that families can use to help their student understand and practice the topic’s concepts. For example, “Home-School Connection Topic 1” provides an activity that challenges families to help their student identify the operation (multiplication or division) used with two fractions to find a solution. The letter suggests, “Help your child recognize that the problem 5 divided by 1/2 means ‘How many halves are in 5?’ and understand the quotient, 10, is greater than the dividend.”

Evaluation for 1.3a
Materials include comprehensive, structured, detailed lesson plans that include daily objectives, questions, tasks, materials, and instructional assessments required to meet the content and language standards of the lesson (aligned with the TEKS and the ELPS).

The enVision+ Texas Mathematics advanced grade 6 lesson plan, lesson 1-1, “Let’s Investigate: Place Value Patterns in Whole Numbers and Decimals,” correlates with TEKS 6AM.3F and ELPS 1C for the mathematics objectives to “Use whole numbers to estimate decimal solutions” and “Add, subtract, and multiply decimals,” and the language objective to “Explain and justify, orally and in writing, how working with whole numbers and decimals involves similar reasoning with place value.” The lesson includes questions such as “How do you decide where the decimal is in the product?” and uses a “Quick Check” from the Assessment Sourcebook that aligns with the objectives.

Additionally, lesson 6-2, “Adding Integers,” includes adding positive and negative integers and modeling integer addition in real-life applications as daily objectives, which correlates with TEKS 6AM.3D and 6AM.3E, and ELPS 4C. In the lesson, students use number lines to identify additive inverses and opposite integers and answer questions, including “What do you notice about the number lines?” An Exit Ticket to assess achievement in adding integers prompts students to identify the total change in temperature after two drops in temperature during the day, aligning with the objective.

Evaluation for 1.3b
Materials include a lesson overview listing the teacher and student materials necessary to effectively deliver the lesson, and the suggested timing for each lesson component.

The “Texas Advanced Grade 6 Lesson Plans” document contains every lesson plan for the course. The plans specify the number of days allotted for each lesson, then break them into detailed steps and the minutes required for each step. The document also includes the educator and student materials needed for each lesson. For example, lesson 1-2, “Let’s Build: Add, Subtract, and Multiply Decimals,” indicates the lesson should take one day, lists four steps and the number of minutes needed for each step, and lists lesson presentation slides, decimal grids and decimal cube manipulatives for students as instructional materials.

Similarly, each topic begins with a Topic Overview that presents a chart of the lessons, citing the objectives, materials needed for students and for educators to deliver each lesson effectively, and the number of days allotted for each lesson within the topic. For example, “Topic 2 Overview, Analyze Geometric Relationships” includes “Let’s Investigate: Strategies for Finding Areas of Polygons,” which should take two days and requires “pattern blocks, grid paper, colored pencils, and scissors.”

Evaluation for 1.3c
Materials include guidance on the effective use of lesson materials for extended practice (e.g., homework, extension, enrichment).

Most lessons in the teacher’s edition include a “Lesson Quick Check” formative assessment with guidance for educators on extended practice based on student performance, including suggested resources. For example, if a student scores 0–3 points, the educator is guided to use “Reteach to Build Understanding Activity” or “Additional Practice” as an extension. If a student scores 4 or 5 points, the teacher should use the “Enrichment Activity” as an extension.

As mentioned previously, the “Texas Advanced Grade 6 Lesson Plans” document includes lesson plans for the entire course. A blue plus sign indicates opportunities for extended practice in each step of the lesson; in lesson 6-3, “Let’s Build: Subtract Integers,” “Step 2: Visual Learning,” a blue plus sign labeled “Extend Student Thinking” provides a challenge for students to add three integers, where two of the integers have the same sign, for example.

Evaluation for 2.1a
Materials include a variety of instructional assessments at the unit and lesson level (including diagnostic, formative, and summative) that vary in types of tasks and questions.

Materials in grade A6 include various instructional assessments at the unit and lesson levels, including diagnostic, formative, and summative, that vary in task and question types. For example, in the Assessment Sourcebook, each lesson includes an Exit Ticket that serves as a formative assessment. There is also a “Mid-Topic Performance Task” summative assessment for each topic to use midway through the unit, which exhibits multiple-choice, open-ended, fill-in-the-blank, and graphing question types, and a diagnostic “Readiness Assessment” at the beginning of each topic, with varied question types including problem-solving, computational, and real-world problems.

Likewise, the Assessment Sourcebook includes a Quick Check for each lesson with varied questions and tasks, such as multiple-choice, open-ended, and graphing. The materials include summative assessments at the end of each topic, such as “Topic 1 Performance Task Form A,” which provides students with a chart displaying weights, in ounces, of various ingredients in decimals and asks them to solve real-world problems such as “Describe a meal that you could put together that provides four servings of both a main course and a side dish. Explain how you know that it will not exceed the weight limit of the box,” for a box that can hold 2.75 pounds, or 44 ounces.

Evaluation for 2.1b
Materials include the definition and intended purpose for the types of instructional assessments included.

The Assessment Guide includes the definitions and intended purposes for diagnostic, formative, and summative assessments in a chart that indicates the “why” and “when” for each type of assessment. For example, the guide states for diagnostic assessment: “Why: Diagnose students’ readiness for learning by assessing prerequisite content. When: Before instruction,” and goes on to suggest in “Instructional Outcomes Informed by Assessment Results” that educators “develop individual study plans, make grouping decisions, and prescribe specific activities to fill gaps in understanding of prerequisite content.”

Additionally, the “How to Assess” section of the Assessment Sourcebook explains how observation assessments, portfolio assessments, performance-based tasks, and program assessments evaluate student readiness, understanding, complex thinking, and content mastery.

Evaluation for 2.1c
Materials include teacher guidance to ensure consistent and accurate administration of instructional assessments.

The Grade A6 Assessment Guide includes guidance for educators to ensure consistent and accurate administration of instructional assessments. For example, the section “How to Administer Assessments” guides educators to “Prepare for Assessments” by understanding and scheduling time for the assessment, ensuring a proper testing environment, and actively “Monitor Assessments,” including making accommodations for emergent bilingual students.

Similarly, lesson 3-11ʼs “Explore and Share” provides explicit educator directions for moving through the observational assessment, including questions such as “How do students choose a conversion factor to use?” and “How do students critique Linus’s reasoning?” during the “Monitor, Select, and Sequence” section when assessing a small group of students, providing guidance for monitoring and responding to student understanding during the activity.

Evaluation for 2.1d
Diagnostic, formative, and summative assessments are aligned to the TEKS and objectives of the course, unit, or lesson.

In the Assessment Sourcebook, all assessment types throughout the course include answer keys that indicate the TEKS aligned to each question and to each lessonʼs and unitʼs objectives. For example, each topic begins with a “Readiness Assessment” as a diagnostic tool for the specific TEKS and objectives students will learn during the unit. Throughout the topic, formative assessments such as the Quick Check and Exit Ticket gauge students’ achievement of the TEKS and objectives aligned with the individual lessons in the unit. Finally, the “Topic Assessment” and “Topic Performance Task” serve as summative assessments that align with all the unit’s TEKS and objectives.

Additionally, the course begins with the “Readiness Assessment” to gauge students’ performance on the specific objectives they learned in grade 5, aligned to grade 5 TEKS, such as 5.3E and 5.6B. Throughout each topic in grade A6, formative assessments such as the Exit Ticket measure students’ achievement of the TEKS and objectives aligned with each unit’s lessons. Finally, the “Topic Performance Task” serves as a summative assessment that aligns with all the unit’s TEKS and objectives. For example, lesson 3-3 has an objective for students to “Use multiplication and division to find equivalent ratios” using TEKS 6AM.6A, 6AM.5C, and 6AM.5D, with a “Quick Check” that assesses the objective and the TEKS with a question requiring students to “Select all the correct statements that tell if the operation will give an equivalent ratio for 8 to 20.” The “Topic Performance Task” for Topic 3 aligns with the objectives and the TEKS within all lessons in the topic, including 6AM.6A.

Evaluation for 2.1e
Instructional assessments include TEKS-aligned items at varying levels of complexity.

Each assessment in the Assessment Sourcebook includes an answer key showing the DOK level and the TEKS for each item-type question. For example, “Topic 1 Assessment” includes eleven questions (one DOK-1, eight DOK-2, and two DOK-3), some of which are multiple choice, while others are open-ended or multi-select.

Additionally, each lesson provides TEKS-aligned items at varying levels of complexity. For example, the lesson 7-1 “Understand and Represent Exponents” assessment requires students to interpret exponents, connect exponents to repeated multiplication, evaluate exponents, and apply concepts in context, demonstrating varying cognitive complexity, while remaining aligned to TEKS 6AM.3G.

Evaluation for 2.2a
Instructional assessments and scoring information provide guidance for interpreting student performance.

For each lesson, students answer five questions in an online Quick Check that is automatically scored and, based on student performance, assigned a remediation, practice, or enrichment activity. The teacherʼs edition provides a scoring interpretation for the Quick Check that guides educators to “use the Assessment Sourcebook version to prescribe targeted remediation or enrichment.” For example, the scoring interpretation advises educators to use “Reteach to Build Understanding” and “Additional Vocabulary Support Activities” for students who score 0–3 points.

Additionally, instructional assessments and scoring information guide educators in interpreting student performance on the Exit Ticket in lesson 5-2, “Understand Integers.” The guidance includes identifying common errors and interpreting misconceptions, helping educators determine why students may be struggling and how to respond instructionally.

Evaluation for 2.2b
Materials provide guidance for the use of included tasks and activities to respond to student trends in performance on assessments.

Each assessment in the Assessment Sourcebook includes an answer key with an “Intervention Lesson” tied to each question or task to support students with the concept. For example, if a student does not answer the questions in the Quick Check for lesson 3-8 correctly, the educator is guided to use “Intervention Lesson M28” for that student.

Moreover, there is usage guidance for tasks and activities to respond to student assessment performance trends in lesson 5-5, “Understand Absolute Values of Rational Numbers.” The lesson includes a “Prevent Misconceptions” section connected to the Exit Ticket that identifies a likely student misconception and provides specific reteaching guidance educators can use when assessment results reveal that misunderstanding.

Evaluation for 2.2c
Materials include tools for teachers to track student progress and growth, and tools for students to track their own progress and growth.

Grade A6 materials include charts for educators to track student progress and growth, and charts for students to track their own progress and growth. For example, the “Student Progress and Growth Teacher Tool” allows educators to mark student progress and growth according to the TEKS for each lesson, using symbols such as “✖ Working on it,” “- Almost there,” and “+ Got it.” Likewise, students can track their own progress and growth “Before Topic,” “During Topic,” and “After Topic” using the “Student Progress and Growth Tracker” for each lesson objective with selections such as “I Can,” “With Help,” and “Not Yet.”

Furthermore, in the “Self-Assessment” section of the Exit Ticket, students assess their understanding of the mathematics and language objectives using a structured rating scale, which provides a progress-monitoring tool and helps educators identify areas in which additional support may be needed. For example, in the Exit Ticket for lesson 1-6, “Understand Division with Fractions,” students select one of three self-assessment indicators, such as “Got it,” “Need help,” and “Working on it,” to respond to the math goal “I can investigate division with fractions.”

Evaluation for 3.1a
Materials include teacher guidance for differentiated instruction, activities, and paired (scaffolded) lessons for students who have not yet reached proficiency on grade-level content and skills.

The section “Support Student Understanding” in lesson 3-4, “Comparing Ratios,” of the teacher’s edition recommends presenting another ratio problem to students who need more practice, suggesting educators set up two ratios to represent “Farm Stand A” and “Farm Stand B” and their prices for an ear of corn. The materials then guide the educator to help students use the ratios to decide which farm stand has the better price for 12 ears of corn. The educator uses a set of three questions to guide students to understand that the farm stand offering a lower price when both ratios represent 12 ears of corn has the better deal.

Additionally, the Exit Ticket in “Let’s Build” lessons includes educator guidance with specific suggestions for students who have not yet reached proficiency. For example, in lesson 4-3, “Let’s Build: Represent Percents Greater Than 100 or Less Than 1,” the teacher’s edition states: “If students choose A as the answer, they may have confused the square root with dividing in half,” in interpreting the Exit Ticket results. Also, the “Support Student Understanding” strategies in “Step 2: Visual Learning,” aligned with the second lesson example, are designed for students who have not yet reached proficiency in representing percents using 10✕10 grids. Finally, the Quick Check assessment at the end of the lesson helps educators interpret results to assign appropriate remediation, such as “Reteach to Build Understanding” or “Additional Vocabulary Support” activities for students who have not yet reached proficiency with the lesson.

Evaluation for 3.1b
Materials include pre-teaching or embedded supports for unfamiliar vocabulary and references in text (e.g., figurative language, idioms, academic language).

Grade A6 materials include pre-teaching and embedded supports for unfamiliar vocabulary and references within the text in the Linguistic Accommodations at the beginning of each topic in the teacher’s edition. For example, in the “Context Setting Vocabulary” section of topic 3, pre-teaching supports state, “Students will encounter the following terms in this topic; be sure they understand their meaning to make sense of the problem situation.” The section continues with terms for each lesson, such as blood type and dealership in lesson 3-2; items 9–14 of the “Practice and Problem-Solving” section include a chart listing the various blood types and how many donors of each blood type are represented, then has students complete a variety of ratio comparisons between numbers of donors of the different blood types and the total number of donors. Likewise, the “Vocabulary in Context” section of Linguistic Accommodations provides meanings of the word rate, including the “Math Meaning” as “a special type of ratio that compares quantities with unlike units of measure,” and “Other Meanings” as “an amount of money paid for something.”

Additionally, the “Language Support” section of lesson 3-4, “Let’s Build: Compare Ratios,” in the teacher’s edition provides embedded support for unfamiliar academic and mathematical vocabulary by explicitly connecting the terms bar diagram, double number line, and ratio table to visual representations used in the lesson. Students discuss how the terms relate to the visual representations and complete the sentence frame “I think using a _____ to compare ratios to solve problems is most efficient because ______,” using the vocabulary in context. The page also includes structured Math Talk discussion prompts that require students to use ratio language to compare two visual models and the ratios they represent.

Evaluation for 3.1c
Materials include teacher guidance for differentiated instruction, enrichment, and extension activities for students who have demonstrated proficiency in grade-level content and skill.

Lesson 5-1, “Let’s Build: Classify Numbers,” in the teacher’s edition prompts the educator to ask, “How does the data set change if the robot is supposed to measure each person’s height, in feet, in each room of an office building?” in the section “Early Finishers,” providing an extension for students who finish their work early.

Additionally, “Let’s Build” lessons provide guided examples to help students learn each lesson’s concepts and include an “Extend Student Thinking” section with educator guidance to extend one example for students who have demonstrated proficiency in the concept. For example, in “Topic 3: Understand and Use Ratio and Rate,” the “Extend Student Thinking” section of “Example 3 Try It!” guides educators to “Challenge students to determine the ratio for the multiplication relationship if the number of hours doubles for each number of miles. Encourage students to think about new patterns that emerge.”

Evaluation for 3.2a
Materials include explicit (direct) prompts and guidance to support the teacher in modeling and explaining the concept(s) to be learned.

The teacher’s edition uses scripted questions to provide direct instructional guidance, including sample responses and discussion prompts for “Example 3: Find the Base or Height of a Parallelogram” in lesson 2-5, “Let’s Build: Find Areas of Parallelograms Including Rhombuses.” Specifically, the example is split into two parts that ask students to calculate the height and the base of a parallelogram when given the area, using the formula A = b ✕ h. The educator is prompted to ask, “What are you trying to find in Part A? Part B?” and “Could you use the same process to find the base or height of a rhombus? Explain.” The materials offer guidance for struggling students (“Support Student Understanding”) and extensions for those who understand (“Extend Student Thinking”). For example, the “Support Student Understanding” section notes that “Students may need additional practice finding the base or height of a parallelogram when the area is known,” and provides guiding questions the educator can use to support students with another example.

Additionally, each “Let’s Build” lesson includes explicit prompts and guidance to help educators model and explain the lesson’s mathematical concepts. For example, in the teacher’s edition, “Step 2: Visual Learning” of lesson 8-2 “Let’s Build: Apply Properties of Equality” begins with an example, exemplar student responses, and explicit prompts for the educator to ask, “What do the Addition and Subtraction Properties of Equality state?” “What does the Multiplication Property of Equality state?” and “How does the Division Property of Equality differ from the other three properties?” When guiding students to the individual “Try It!” problem that accompanies the example, educators are prompted to ask students, “What math was done to both sides of the first equation to result in the second equation?” Before students move on to the “Practice and Problem Solving” section of the lesson, “Talk About Math Ideas” offers explicit guidance for the educator to “Provide groups with algebra tiles and pan balances as available. Listen and watch for students using the different representations, as well as equations, to model the properties of equality and to write equivalent expressions. Encourage group members to take turns modeling each property of equality using each available method.”

Evaluation for 3.2b
Materials include teacher guidance and recommendations for effective lesson delivery and facilitation using a variety of instructional approaches.

Grade A6 materials include educator guidance and recommendations for effective lesson delivery and facilitation using various instructional approaches across three lesson types. Each lesson type breaks instruction into four distinct steps that guide educators in delivering comprehensive instruction involving investigation, connection, problem-solving, and differentiation. For example, the teacher’s edition describes the three lesson types that are used within each topic on the “3 Types of Lessons” page. There are “Let’s Build” lessons in each topic that guide educators in delivering lessons through student exploration, followed by a set of guided examples with recommendations to assist students as they progress into the independent practice and problem-solving section, and end with guidance and recommendations for next steps in the assessment portion of the lesson. “Let’s Investigate” lessons within each topic guide educators in leading student investigations through problem-based learning, making connections through student choice, individual application through practice and problem-solving, and ending with guidance for assessment and differentiation. Finally, the teacher’s edition describes the “Let’s Model in 3 Acts” lessons as “an investigation into a real-world situation presented with an engaging video,” in which educators guide students to brainstorm questions and predict answers, model the problem to find an answer and discuss solution strategies, and then reveal the answer to the main question to have students “discuss any difference between their conjectures and the answer and revise their models as needed.”

Additionally, “Let’s Build” lessons in the teacher’s edition incorporate a variety of instructional approaches and guide educators in delivering effective instruction. For example, lesson 10-3, “Let’s Build: Understand Credit Reports,” begins by having students explore how credit affects trip planning by analyzing trip-related expenses and considering which purchases they might make using credit, connecting mathematical concepts to a real-world financial context. The “Monitor, Select, and Sequence” section guides the educator to observe students at work and asks, “What scenarios do students consider when thinking about how to pay for the trip?” and the Connect Solution Strategies recommends a whole-class structure to “Select solutions for students to present that propose paying for the trip on credit. Then have any students share who propose that Blythe save cash for the trip by placing other expenses on the credit card” and “Have students discuss how each strategy for paying for the trip affects Blythe’s budget.” In “Step 2: Visual Learning,” educators and students examine the components of a credit report and discuss how borrowers and lenders use this information. Through guided explanations, tables, visual representations, and targeted questioning, students build conceptual understanding before applying it during practice activities. This progression from real-world exploration to guided instruction and independent application demonstrates how the lesson supports effective teaching and learning through multiple instructional approaches.

Evaluation for 3.2c
Materials support multiple types of practice (e.g., guided, independent, collaborative) and include guidance for teachers and recommended structures (e.g., whole group, small group, individual) to support effective implementation.

The “Practice in enVision+ Texas Mathematics” page in the Program Overview uses a chart to identify multiple types of practice, including whole-group, small-group, and individual practice, that connect to various program components such as “Math Talks,” “Fluency Practice,” “Topic Review,” and “Hands-On Games” using guided, independent, and collaborative practice structures. For example, the page guides the educator in using different structures, such as a whole-group structure to “promote mathematical discourse and classroom conversations as students practice and apply new and previously learned concepts.” At the same time, the chart recommends whole-group and small-group structures for Math Talks, using guided or collaborative practice, and recommends whole-group, small-group, or individual structures for Topic Review, using guided, collaborative, or independent practice.

Additionally, in lesson 8-2, “Let’s Build: Apply Properties of Equality,” the teacher’s edition recommends that “Step 1: Explore and Share” be completed in small-group and whole-group instructional structures, guiding educators in the “Monitor, Select, and Sequence” section to observe small groups of students collaborating to decide if four separate computations to either side of a simple addition equation will change the result of the equations, before using the “Connect Solution Strategies” as a whole class to share results and discuss. Then, “Step 2: Visual Learning” guides educators in engaging the whole class in a discussion on using the properties of equality to write equivalent equations and in modeling three examples through guided practice before leading into “Step 3: Practice and Problem-Solving,” during which students work independently on individual practice problems in the student edition or using the adaptive practice on the Savvas Realize digital platform.

Evaluation for 3.3a
Materials include teacher guidance on providing linguistic accommodations for various levels of language proficiency [as defined by the English Language Proficiency Standards (ELPS)], which are designed to engage students in using increasingly more academic language.

Grade A6 materials include a “Linguistic Accommodations” section in the teacher’s edition that provides guidance and support for topic vocabulary. The English Language Proficiency Standards (ELPS) chart at the bottom of the section’s second page identifies the specific ELPS targeted in each lesson, meeting the needs of all five language proficiency levels and aligning the ELPS modalities to the lesson content. For example, the ELPS chart in “Topic 3: Understand and Use Ratio and Rate” indicates the targeted ELPS in each lesson, such as Speaking 2B in lesson 1, Listening 1C in lesson 3, Reading 3F in lesson 5, and Writing 4C in lesson 7, which are designed to engage students in using increasingly academic language. Specifically, lesson 5-7, “Let’s Investigate: Use Rates to Compare,” guides the educator in the “Targeted ELPS Support” aligned to Writing 4C to “Write compare and comparison on the board. Ask [pre-production level] students to name which is the root word” and answer the question “There are five black pens and four blue pens. Is that a compare or a comparison?” The section also guides educators to have intermediate students use the two words to fill in the blanks in the sentences “I [compare] unit rates to find which is greater. A sentence gives the [comparison],” and to ask advanced students to “write definitions for compare, comparison, and unit rate” and “write two math statements involving unit rates.”

Additionally, “Topic 4: Understand and Use Percent” includes a “Linguistic Accommodations” page that provides educators with explicit implementation guidance aligned with the ELPS, such as an ELPS alignment table that maps specific listening, speaking, reading, and writing standards to each lesson across the topic, providing educators with a clear implementation road map for delivering instruction. Specifically, in lesson 4-1, “Let’s Build: Understand Percent,” the materials provide educators with targeted ELPS support for using ELPS Speaking 2E with pre-production, beginning, intermediate, high intermediate, and advanced levels of emergent bilingual students. The support guides educators to “Have [Pre-Production and Beginning] students work with a partner to describe the strip diagrams in Example 2,” and ask pre-production students to use words and numbers to describe the strip diagrams, and beginning students to use words to describe the sizes of each part of the strip diagrams. The guidance tells educators to “Have [intermediate, high intermediate, and advanced] students complete Example 2,” and ask intermediate students to elaborate on the relationship between one half and 50 percent, high intermediate students to speak about the relationships between the different strip diagrams and their percentages, and advanced students to elaborate on the steps they took to find the fraction of the whole that matches the percent of the whole.

Evaluation for 3.3b
Materials include implementation guidance to support teachers in effectively using the materials in state-approved bilingual/ESL programs.

The “Linguistic Accommodations: Understand and Use Ratio and Rate” page of the teacher’s edition guides educators on language development opportunities within the topic using sections such as “Topic Vocabulary Support,” “Metalinguistic Transfer to Spanish,” “Cognates,” and “Academic Vocabulary” to support multilingual learners. The page includes specific guidance for language integration to use the materials in state-approved bilingual programs effectively, such as “Activities in the topic offer diverse and engaging experiences with math vocabulary to build confidence and support for all, but particularly for Emergent Bilingual (EB) students in developing both their academic and language skills.”

Additionally, each “Let’s Build” lesson offers callout boxes and notes in the teacher’s edition to guide educators in effectively using the materials in state-approved bilingual programs, such as “Language Support,” “Language Routine,” and “Emergent Bilingual Student Support.” For example, in lesson 9-3, “Let’s Build: Solve Inequalities Using Addition and Subtraction,” “Language Support” guides educators in having students review the language of inequalities using a number line on the floor as students stand on a number while pointing in a specific direction and stating which inequality they are representing, and the “Language Routine: Compare and Connect” section guides educators in having pairs of students compare the graphs of inequalities from two examples, then helping them make connections between the inequalities and their graphs. The “Emergent Bilingual Student Support” note in “Step 3: Practice and Problem-Solving” includes implementation guidance for item 10 to “Review the difference between [the greater-than-or-equal-to symbol] and [the greater-than symbol] as well as vocabulary from the problem” and to ask, “What does maximum mean?”

Evaluation for 3.3c
Materials include embedded guidance for teachers to support emergent bilingual students in developing academic vocabulary, increasing comprehension, building background knowledge, and making cross-linguistic connections through oral and written discourse.

Grade A6 teacher’s edition materials include “Linguistic Accommodations” for each topic, with embedded educator guidance to support emergent bilingual students in oral and written discourse and to develop academic vocabulary by reinforcing previously learned terms, defining current concept vocabulary, and previewing upcoming vocabulary. These resources provide background knowledge, cross-linguistic connections, and guidance on Spanish transfer (including transferable terms, nontransferable terms, and cognates) to increase comprehension throughout the topic. For example, the “Linguistic Accommodations” for “Topic 5: Analyze Geometric Relationships” provides a “Topic Vocabulary Support” section that develops academic vocabulary to increase comprehension while building background knowledge, and guides the educator to “be sure [students] understand [context-setting vocabulary] meaning to make sense of the problem situation,” highlighting the use of scale factor for the topic, and terms for individual lessons, such as side length in lesson 2. It also includes guidance on cognates, such as volume/volumen, for cross-linguistic connections students will use in oral and written discourse, as well as false cognates, such as large and largo. Finally, each topic includes an ELPS chart, broken down by lesson, to guide educators in using the English Language Proficiency Standards, particularly for oral and written discourse, such as in lesson 1, “Targeted ELPS Support: Listening 3E,” which guides the educator to “Read Example 1 aloud. Direct [pre-production] students to the drawings of triangles. Point to the first picture and ask students to point to the side that is 8 units long. Say: If that is side BC, give me a thumbs- up.” It also guides educators to “Have [advanced] students read Example 3. Draw triangle ABC. Ask: Is side BC included between angles A and B?” Later, in lesson 3, the “Targeted ELPS Support: Writing 4B” section guides the educator to review example 1 and “Have [pre-production] students complete these sentence stems: Non-overlapping angles that share a ray are _____ . Angles opposite each other are _____,” and continues with varied ways to engage the students in written discourse for other levels, such as “Define skewed. Have [high intermediate] pairs write a description of a skewed intersection,” and ask, “How small would angle ABC need to be for the intersection to be skewed?”

Additionally, in “Topic 9: Understand and Solve Inequalities,” the materials include embedded guidance within the Linguistic Accommodations page, such as “Topic Vocabulary Support” for educators to develop academic vocabulary, increase comprehension, and build background knowledge through oral and written discourse using topic terms, including inequality, boundary, and reverse, with the “Math Meaning” and “Other Meanings” of the vocabulary as found in the lesson. Similarly, the materials provide a list of “Context-Setting Vocabulary” for each lesson, such as average weight, greatest weight, and round in lesson 5, and guide educators to “be sure [students] understand their meaning to make sense of the problem situation.” Embedded guidance for making cross-linguistic connections through oral and written discourse is included in the “Metalinguistic Transfer to Spanish” section of the page, with a list of cognates, such as graduation/graduación, adventure/aventura, and semester/semestre, and false cognates, including red, which refers to the color of ripe strawberries, tomatoes, and chilies in English, but means “web or net” in Spanish. “Targeted ELPS Support” sections embedded in lesson instruction guide educators in supporting emergent bilingual students through oral and written discourse. In lesson 9-2 “Let’s Build: Represent Solutions of Inequalities,” the “Targeted ELPS Support” aligned to Writing 4F guides educators to develop academic vocabulary, build background knowledge, and increase comprehension by encouraging students to describe or explain key information orally and in writing, including the variable, inequality symbol, and solutions of the inequality.

Evaluation for 3.3d
If designed for dual language immersion (DLI) programs, materials include resources that outline opportunities to address metalinguistic transfer from English to the partner language.

This guidance is not applicable because the program is not designed for dual language immersion (DLI) programs.

Evaluation for 4.1a
Practice opportunities over the course of a lesson and/or unit (including instructional assessments) require students to demonstrate depth of understanding aligned to the TEKS.

Each topic includes lessons with a “Practice and Problem-Solving” section that challenges students’ depth of understanding in solving problems aligned to the lesson’s TEKS, and ends with a Topic Assessment that uses scaffolded, DOK-leveled questions aligned to the topic’s TEKS to assess student learning. For example, lesson 7-1, “Let’s Build: Understand and Represent Exponents” uses a variety of questions aligned to TEKS 6AM.3G in the “Practice and Problem-Solving” section, such as questions requiring students to calculate expressions with exponents, real-world questions for which students apply and analyze exponent calculations, and questions that require students to demonstrate depth of understanding to compare and contrast expressions with exponents. The Assessment Sourcebook provides a Topic Assessment for topic 7, with questions requiring students to demonstrate depth of understanding by analyzing exponent expressions and solving a real-world evaluation aligned to 6AM.3G, and problems that require students to simplify one-variable expressions and evaluate and identify multiple one-variable expressions with the same solution, demonstrating depth of understanding of TEKS 6AM.7C.

Additionally, at the beginning of lesson 2-1 “Let’s Build: Draw Triangles with Given Conditions,” students use rulers, compasses, straws, and scissors to explore which combinations of three side lengths can form a triangle, activating conceptual thinking and mathematical reasoning before formal instruction begins with TEKS 6AM.1C and 6AM.1E. During “Step 2: Visual Learning,” students engage with five sequenced examples that increase in complexity, from drawing and comparing triangles with given side lengths using Desmos, to determining possible side lengths of triangles, to extending their understanding by drawing triangles using a combination of side lengths and angle measures, to analyzing triangles given only three angle measures. The embedded “Try It” prompts for each guided example require students to justify their thinking. Next, the “Do You Understand/Do You Know How” section and the “Key Concept” sections further consolidate understanding by requiring students to articulate the three sets of conditions that always determine a unique triangle. During “Practice and Problem-Solving,” students engage with practice exercises spanning DOK levels 1–3, including higher-order thinking problems such as items 17 and 22, which require students to draw multiple triangles under given conditions and explain their reasoning, demonstrating a depth of understanding of TEKS 6AM.12D.

Evaluation for 4.1b
Questions and tasks progressively increase in rigor and complexity, leading to grade-level proficiency in the mathematics TEKS.

“Let’s Build: Classify Numbers” in lesson 5-1 of the teacher’s edition uses scaffolded steps, including “Explore and Share” and “Visual Learning,” which guide educators in using various strategies and visual models such as Venn diagrams and concentric circles to represent and classify numbers. Next, the students use a number line to locate whole numbers, integers, and rational numbers. The “Practice and Problem-Solving” questions begin with classifying whole numbers, integers, and rational numbers in a Venn diagram, and increase in rigor, requiring students to communicate mathematical reasoning for number classifications and solve real-world problems such as “A bank account has a balance of $0, a withdrawal of -$50, a bank transfer of -$2.50, and a deposit of $20.50. Classify each amount as a whole number, an integer, and/or a rational number,” leading to proficiency in TEKS 6AM.2A.

Additionally, lesson 2-1, “Let’s Build: Draw Triangles with Given Conditions,” includes questions and tasks that increase in rigor and complexity from foundational exploration through higher-order reasoning. Students begin at a concrete, exploratory level in “Step 1: Explore and Share,” testing combinations of wood pieces to determine which can form a triangle and building an intuitive understanding of the triangle inequality before any formal rule is introduced. In “Step 2: Visual Learning,” the five examples provided systematically increase in complexity. Specifically, the first example guides students to use Desmos to observe and compare triangles with given side lengths and identify that they are identical regardless of orientation. Each consecutive example increases in rigor, including asking students to determine possible triangle side lengths based on the rule that the sum of the two shorter sides must be greater than the longest side, then to draw triangles given the side lengths and an angle measure to evaluate if multiple triangles can be formed, before moving into more complex guided examples. In example 4, students use non-routine reasoning to analyze why two side lengths and a non-included angle do not always determine a unique triangle, and use geometry software to explore multiple possibilities. Finally, in example 5, students reason that three angle measures without side lengths cannot determine a unique triangle formation and justify their answers using proportional reasoning and equations. The embedded “Try It!” prompts reinforce this progression by requiring students to write and solve equations to find missing angle measures, write three side lengths that do and do not form a triangle, and explain their reasoning, leading to grade-level proficiency in TEKS 6AM.12D.

Evaluation for 4.2a
Materials demonstrate coherence across units by explicitly connecting patterns, big ideas, and relationships between mathematical concepts.

Each topic in the teacher’s edition begins with guiding pages to prepare educators to teach its concepts, such as the “Math Background: Coherence” page, which connects the topic’s concepts to patterns, big ideas, and relationships that students previously learned in grade A6, as well as how they connect to concepts that students will learn later in grade A6. For example, the “Math Background: Coherence” page in “Topic 3: Understand and Use Ratio and Rate” explains in the “Earlier in Advanced Grade 6: Computations with Rational Numbers” section, “In Topic 1, students developed fluency in adding, subtracting, multiplying, and dividing rational numbers,” which connects to “Topic 3: Ratio Reasoning” because “students use ratios to describe multiplicative relationships and learn to reason about ratios by using equivalent ratios.” Students will also learn about rates, create equivalent rates, compare rates, and solve unit rate problems, which demonstrates coherence in the section “Later in Advanced Grade 6: Understand and Use Percent,” which explains, “In Topic 4, students will understand percent as a rate in which the comparison is to 100,” and “In Topic 11, students will use percents as they summarize data distributions.”

Additionally, each lesson contains a “Coherence” section that explicitly maps what students learned previously within the topic and across prior units, what they are learning in the current lesson, and where their learning is headed later in the topic and course. For example, the “Coherence” section in lesson 5-1, “Classify Numbers,” explains a cross-unit connection to the big idea from the previous topic, in which students added, subtracted, multiplied, and divided rational numbers to solve problems. Building directly on that prior unit’s work with rational numbers, this lesson connects the pattern to classifying numbers as whole numbers, integers, and/or rational numbers using a Venn diagram, and locating numbers on a number line based on their classification. Looking ahead, the “Coherence” section notes that students will locate, order, and compare relationships between integers and rational numbers using a number line, and use rational numbers to locate points on the coordinate plane. In lesson 5-2, “Understand Integers,” the “Coherence” section notes that students will connect patterns, big ideas, and relationships between mathematical concepts by reading, writing, comparing, and ordering rational numbers later in the topic.

Evaluation for 4.2b
Materials demonstrate coherence across units by connecting the content and language learned in previous courses/grade levels and what will be learned in future courses/grade levels to the content to be learned in the current course/grade level.

Grade A6 materials demonstrate coherence across units by connecting the content and language learned in previous courses/grade levels and what will be learned in future courses and grade levels to the content to be learned in the current course and grade level at the beginning of each topic in the teacher’s edition. For example, “Topic 11: Display, Describe, and Summarize Data” begins with several pages that prepare educators to teach the topic’s lessons, including “Math Background: Coherence” and “Linguistic Accommodations.” The Math Background: Coherence page is separated into three columns, “Look Back,” “Topic 11,” and “Look Ahead.” The “Look Back” section connects the content learned in grade 5 and earlier in grade A6 to the content students are learning in topic 11, demonstrating the mathematical language that connects the concepts, such as “Grade 5: Students displayed numerical data using line plots, which are also called dot plots. Students’ experiences creating and analyzing these data displays began their examination of data distributions,” and “Earlier in Advanced Grade A6: In Topic 1, students developed fluency with adding, subtracting, multiplying, and dividing decimals.… These computational skills will enable students to compute statistical measures.” The “Topic 11” section headers include the mathematical terms data displays and data distribution in reference to the content students will learn throughout that topicʼs lessons, and the “Statistical Measures” section describes how students will “determine the mean, median, and mode, which are measures of center” and “learn how to find the first and third quartiles of a data set.” Finally, “Look Ahead” explains how the content and language students will learn in topic 11 connect, including “In Advanced Grade 7, students will learn how to use samples to draw inferences about a population and to compare populations. They will use measures of center and measures of variability from random samples to make informal inferences comparing two populations.”

Additionally, each lesson begins with a “Coherence” section that maps the progression of mathematical content and a “Vocabulary” section that indicates whether language is being introduced or reinforced from prior learning. For example, the “Coherence” section on the first page of lesson 2-5, “Find Areas of Parallelograms Including Rhombuses,” in the teacher’s edition notes that students used the formula for the area of a rectangle to solve problems in grades 4 and 5, connecting to deriving the formula for the area of a parallelogram to solve real-world and mathematical problems. “Reinforce Vocabulary” connects the mathematical language previously learned, including “area of a parallelogram, parallelograms, and rhombuses.” Lesson 2-9, “Solve Problems Involving Volume of Prisms,” uses the “Coherence” section to connect the content, such as “find the volume of right rectangular prisms and triangular prisms,” and vocabulary, including cube, fractional edge length, right rectangular prism, and volume, to the content students will learn in advanced grade 7, such as solving real-world and mathematical problems involving the volume of more complex three-dimensional figures.

Evaluation for 4.2c
Materials demonstrate coherence at the lesson level by connecting students' prior knowledge of concepts and procedures from the current and prior grade level(s) to new mathematical knowledge and skills..

Grade A6 materials demonstrate coherence at the lesson level by connecting students’ prior knowledge of concepts and procedures from the current and prior grade levels to new mathematical knowledge and skills, as outlined in the “Coherence” section on the first page of each lesson in the teacher’s edition. For example, the first page of lesson 7-1, “Let’s Build: Understand and Represent Exponents,” explains, “In Grade 5, students learned how to use exponents to denote powers of 10,” which connects to the lesson because “students extend their understanding of exponents to include other bases” in grade A6. Likewise, the section highlights that students will “learn how to write and evaluate expressions with whole-number exponents,” which connects later in the topic, where students will “write and evaluate both numerical expressions and algebraic expressions that contain exponents.”

Additionally, the “Coherence” section of the teacher’s edition in lesson 9-2, “Represent Solutions of Inequalities,” demonstrates coherence by connecting students’ prior knowledge of concepts and procedures from grade 5, using the greater-than, less-than, and equal-to symbols to compare whole numbers, fractions, and decimals to the new mathematical knowledge and skills of the current lesson to write inequalities to represent mathematical and real-world situations, and solving inequalities later in the topic. Lesson 9-4, “Solve Inequalities Using Multiplication or Division,” connects students’ prior knowledge of graphing solutions of inequalities and solving them using the addition and subtraction properties of inequality covered previously to the new mathematical knowledge and skills of using the multiplication and division properties of inequality and graphing solutions on a number line.

Evaluation for 4.3a
Materials provide spaced retrieval opportunities with previously learned skills and concepts across lessons and units.

Grade A6 materials provide spaced retrieval opportunities with previously learned skills and concepts across lessons and units, as demonstrated in a Venn diagram on the “Space and Interleaved Practice in enVision+ Texas Mathematics” page in the Program Overview, which compares the sections within the materials that offer spaced and interleaved practice, noting that Math Talks and Fluency Practice provide spaced retrieval opportunities. For example, the Math Talk section in lesson 1-7, “Let’s Build: Divide Fractions,” of the teacher’s edition asks students to identify mistakes in the steps used to divide the mixed number two and a half, by the fraction four-fifths, providing spaced retrieval of changing a mixed number to an improper fraction, dividing a fraction by a fraction, and changing an improper fraction to a mixed number, which are skills and concepts previously learned in grades 4 and 5. The “Fluency Practice” at the end of “Topic 1: Use Positive Rational Numbers” provides spaced retrieval practice for the skills and concepts of adding, subtracting, multiplying, and dividing rational numbers that students learned throughout the topic.

Additionally, the student edition includes a “Review What You Know” section for each topic that uses spaced retrieval to activate prior knowledge and lets students practice the prerequisite skills for that topic. For example, “Review What You Know” in “Topic 3: Understand and Use Ratio and Rate” asks students to recall vocabulary and skills related to equivalent fractions, equations, units of measure, and measurement conversions. The “Item Analysis” table maps these exercises to TEKS standards 3.3F, 4.3C, 6AM.9B, 4.8A, and 5.7A. “Review What You Know” in “Topic 4: Understand and Use Percent” requires students to retrieve skills related to rates, decimal computation, and equivalent ratios, and the exercises are mapped to TEKS standards 6AM.2A, 6AM.5B, 6AM.3F, and 6AM.2E, providing spaced retrieval opportunities across lessons and units.

Evaluation for 4.3b
Materials provide interleaved practice opportunities with previously learned skills and concepts across lessons and units.

Grade A6 materials provide interleaved practice opportunities with previously learned skills and concepts across lessons in “Practice and Problem-Solving” and units in “Review What You Know.” For example, the “Practice and Problem-Solving” section of lesson 10-8, “Let’s Build: Understand Income Tax,” provides interleaved practice across lessons to calculate income tax, such as directly multiplying percent tax rates and whole dollar amounts, using a table to determine tax rates and subtracting, multiplying, and adding values to calculate tax amounts, and using proportions to solve income tax amounts. The “Review What You Know” section of “Topic 10: Financial Math” provides interleaved practice across topics, including the “Vocabulary” section with cloze sentences to use words including equivalent, percent, percent equation, and unit price, the “Operations with Decimals” section, in which students multiply and divide money decimal amounts, the “Equivalent Rational Numbers” section, when students convert rational numbers between decimals and percents, and the “Solve Percent Problems” section, in which students solve word problems involving money and percentages.

Additionally, each “Let’s Build” lesson includes practice and problem-solving as part of the four-step instructional process, providing students with opportunities to review and apply previously taught concepts while engaging with new mathematical content. For example, the “Practice and Problem-Solving” section in lesson 10-8, “Understand Income Tax,” allows students to recall skills in multiplying decimals and percents while applying them to the new topic of income tax. Students strengthen their understanding of previously learned concepts while developing new mathematical knowledge as they connect prior knowledge to unfamiliar contexts.

Evaluation for 5.1a
Questions and tasks require students to interpret, analyze, and evaluate models and representations for mathematical concepts and situations.

The “Let’s Model in 3 Acts” lessons in each topic of the teacher’s edition require students to create models to solve problems involving mathematical concepts and situations, and to interpret, analyze, and evaluate their models to find efficient solutions. For example, lesson 7-5’s “Let’s Model in 3 Acts: The Field Trip” presents students with a 30-second video about an educator preparing for a field trip. After students watch the video, they are tasked with writing a question they can solve using a model, guided by questions including “After watching the video, what is the first question that comes to mind? Write the Main Question you will answer. Make a prediction to answer this Main Question. Explain your prediction,” followed by a number-line model with boxes on the ends labeled “Too small” and “Too large.” In the second part of the lesson, students receive the detailed information needed to calculate the total cost of a field trip to the museum, including bus fare, museum admission prices based on participant age, and number of students and chaperones attending. Students then interpret how their model can be used with the given information to find the total cost of the field trip and analyze the problem’s variables to create and solve an algebraic equation. After the answer is revealed in the third part of the lesson, students evaluate their model using questions 11 and 12: “Does your answer match the answer in the video? If not, what are some reasons that would explain the difference? Would you change your model now that you know the answer? Explain.”

Additionally, in lesson 2-5, “Find Areas of Parallelograms Including Rhombuses,” students engage with visual and geometric models, including decomposed parallelograms, rectangles, and rhombuses, and reason about what those representations show rather than simply applying a formula. In “Step 1: Explore and Share,” students interpret a grid-based model of a parallelogram and analyze the relationship between its area and the area of a rectangle. The guidance in the teacher’s edition prompts the educator to ask, “How do students find the area of each shape?” and directs students to notice that shifting points on the parallelogram produces a rectangle with the same area, requiring students to analyze the geometric transformation and evaluate whether the two figures are equivalent in area. In “Step 2: Visual Learning,” students evaluate a visual decomposition model showing how a parallelogram can be rearranged into a rectangle. The educator asks, “How can you tell that the parallelogram and the rectangle have the same area?” and “Why is the slanted side of the parallelogram not considered its height?,” requiring students to interpret the model and evaluate common misconceptions about what height means in the context of a parallelogram. In “Step 3: Practice and Problem-Solving,” students draw and analyze a rectangle and a parallelogram with the same base and height, then evaluate whether their areas are equal, connect the visual model back to the formula, and justify their reasoning.

Evaluation for 5.1b
Questions and tasks require students to create models to represent mathematical situations.

Materials in Grade A6 include questions and tasks that require students to create models to represent mathematical situations in the “Let’s Model in 3 Acts” lessons in each topic of the teacher’s edition. For example, lesson 11-7, “Let’s Model in 3 Acts: Vocal Range,” presents students with a 37-second video about two contestants on a singing show who each scored 24 points from three judges. After students watch the video, they are tasked with writing a question they can solve using a model, guided by prompts including “After watching the video, what is the first question that comes to mind? Write the Main Question you will answer. Make a prediction to answer this Main Question. The person who should win this competition is ________.” Students then justify their predictions. In the second part of the lesson, students receive detailed information about the weekly scores that three judges gave each contestant for six weeks. Students then create a statistical model to represent the given information to determine which contestant should win the competition.

Additionally, the student edition directs students to create and use mathematical models in lesson 1-5, “Let’s Build: Multiply Fractions,” to represent fraction multiplication situations and explain the meaning of the resulting products. In “Step 2: Visual Learning,” students create area models, number-line models, and fraction representations to show the product of fractions and mixed numbers. The lesson asks students to model multiplication situations, determine how the model represents the factors and product, and verify that different models produce the same result. In example 2, students use an area model to represent fraction multiplication, and in example 3, they use a model to visualize the product of mixed numbers before connecting the model to the multiplication algorithm.

Evaluation for 5.1c
Questions and tasks provide opportunities for students to apply conceptual understanding to new problem situations and contexts.

Grade A6 materials include questions and tasks that give students opportunities to apply conceptual understanding to new problem citations and contexts in the “Let’s Investigate” lessons throughout the topics. For example, lesson 7-6, “Let’s Investigate: Equivalent Expressions” in the student edition begins with a guided activity, “Healthy Harvesting,” using algebra tiles to represent two gardens with 10 plants and two more added weekly, and a third garden with two plants and four more added weekly. Students write equations to represent the situation and determine how many weeks it will take for the three gardens to have the same number of plants. Then, students use algebra tiles to create new garden arrangements, solidifying their understanding that the gardens remain equivalent even when arranged differently. Next, students select a new problem from the “Choose a Problem” section that contains four similar contexts, including selecting different expressions that are equivalent using variables L and W to represent the perimeter of a rectangle, selecting equivalent expressions that represent the extension of a square patio by x ft, determining if two sets of algebra tiles arranged differently represent the same expression, and using algebra tiles to determine if three expressions containing the variable x are equivalent to apply the conceptual understanding to new problem situations and contexts. A guiding question asks students, “How was the strategy you used in Choose a Problem the same as your strategy in Healthy Harvesting? How were your strategies different?”

Additionally, in lesson 9-1, “Connecting Equations to Inequalities,” students apply their conceptual understanding of equations to new problem situations involving inequalities across a variety of real-world contexts. In “Step 1: Investigate,” students apply what they know about writing and solving equations to new contexts, such as a teen logging driving hours to earn a permit and a teen saving money to buy a bicycle. Questions in the teacher’s edition include “How did you translate the word problems into equations?” and “How did you know where to insert the words ‘at least’?” In “Step 2: Connect,” students apply their understanding to new contexts in four real-world situations, including solar panel production, a school field trip budget, a bakery muffin cost, and a cell phone plan comparison. Questions guide student thinking, such as “How does the wording of the problem help you write an inequality?” and “Is $5,000 a solution for this problem?,” requiring students to apply their understanding of boundary values and solution sets to evaluate new situations.

Evaluation for 5.2a
Materials provide tasks that are designed to build student automaticity and fluency necessary to complete grade-level tasks.

Grade A6 materials provide tasks designed to build the automaticity and fluency students need to complete grade-level tasks, such as the “Fluency Practice” at the end of each topic in the student edition. For example, “Topic 5: Rational Numbers and the Coordinate Plane: Fluency Practice” builds automaticity in adding and subtracting rational numbers to simplify coordinates and plot them on the coordinate plane. Students use this fluency later in lessons that require them to plot coordinates in the grade-level tasks of “Topic 8: Represent and Solve Equations.”

Additionally, materials in the topic 3 activities in the Differentiation Library Overview include Hands-On Games such as “Assembly Required,” in which students “solve problems to find the value of the card,” and “Bounce: Equivalent Ratios” to maintain fluency skills used in the “Topic 3: Understand and Use Ratio and Rate.” The materials also include digital games described as “Fluency Games: Practice to achieve and maintain fluency” and a fluency practice activity focused on “Add[ing] and subtract[ing] decimals,” giving students additional opportunities to build the automaticity and fluency needed to complete tasks throughout the topic.

Evaluation for 5.2b
Materials provide opportunities for students to practice the application of efficient, flexible, and accurate mathematical procedures within the lesson and/or throughout a unit.

Grade A6 materials provide opportunities for students to practice the application of efficient, flexible, and accurate mathematical procedures within lesson 5-4, “Let’s Build: Represent Rational Numbers on the Number Line.” For example, the “Vocabulary Support” in the teacher’s edition guides the educator in explaining to students that it is more efficient to plot rational numbers on a number line by first renaming them as mixed numbers. Then, in the examples, students practice flexible mathematical procedures, including using number lines in horizontal and vertical orientations to plot rational numbers, using a number line to compare positive and negative rational numbers accurately, and using a number line to order negative temperatures from warmest to coldest efficiently.

Additionally, in lesson 10-4, “Construct Net Worth Statements,” the student edition provides multiple opportunities to practice efficient, flexible, and accurate procedures in organizing assets and liabilities and calculating net worth. In “Step 1: Explore and Share,” students apply a problem-solving procedure flexibly to a new financial context in which they determine how Julia can earn $165 to buy a BMX bike in three weeks using different approaches, such as determining how much Julia already has and planning the same jobs each week, or calculating earnings week by week using different job combinations. In “Step 2: Visual Learning,” students practice creating efficient tables to accurately organize financial assets and liabilities across three examples. In example 1, students create two tables to record Roger’s assets and liabilities and compare totals. In example 2, students use a more efficient way to organize Mr. and Mrs. Lopez’s assets and liabilities into a single table with separate “Assets” and “Liabilities” columns, then calculate totals. Example 3 provides students with a total amount of assets and liabilities and asks them to determine net worth.

Evaluation for 5.2c
Materials provide opportunities for students to evaluate procedures, processes, and solutions for efficiency, flexibility, and accuracy within the lesson and throughout a unit.

Materials in Grade A6 provide opportunities for students to evaluate procedures, processes, and solutions for efficacy, flexibility, and accuracy within the “Let’s Model in 3 Acts” lessons in each topic of the teacher’s edition. For example, lesson 2-8’s “Let’s Model in 3 Acts: Trapezoid Table Tiles” presents students with an 18-second video about students using small square tiles to fill a large trapezoid shape on a table. After students watch the video, they are tasked with writing a question they can solve using a model, guided by questions including “After watching the video, what is the first question that comes to mind? Write the Main Question you will answer. Make a prediction to answer this Main Question,” followed by a question asking students to justify their predictions. In the second part of the lesson, students receive detailed information about the dimensions of the square tiles and the trapezoid’s height and bases. Students then use various procedures and processes to determine how many small tiles it will take to fill the large trapezoid area. After some time, the educator reveals the solution, and students compare and evaluate their procedures, processes, and solutions for efficacy, flexibility, and accuracy, including possible errors, in small groups.

Additionally, the materials give students opportunities to analyze and compare different representations of solutions, evaluate misconceptions, and discuss the appropriateness of equations, inequalities, and number-line models in lesson 9-1, “Let’s Investigate: Connecting Equations to Inequalities.” Students evaluate correct and incorrect solutions in “Sample Student Work” and justify their mathematical choices to ensure accuracy, and evaluate efficiency through the discussion of different representations and solution methods. In “Step 2: Connect,” students evaluate procedures and processes for accuracy by comparing equations, inequalities, and number-line representations, and discuss whether a situation should be modeled with an equation or an inequality. The teacher’s edition asks, “How do you know which inequality symbol to use?” and directs students to use keywords and mathematical reasoning to choose appropriate solution methods. The lesson supports evaluation of procedures and processes for flexibility through multiple representations of solutions, such as in “Step 1: Investigate,” in which students are asked, “What different ways can you think of to represent the solutions on a number line?” The lesson also presents sample student work showing equations, inequalities, and number-line models, and asks students to compare similarities and differences between equations and inequalities. Students are encouraged to represent the same situation using different mathematical approaches. The lesson supports evaluating solutions for accuracy by analyzing correct and incorrect student solutions. The “Select and Sequence Students’ Solutions” section directs educators to begin with correct work, then show number lines that do not represent all possible solutions and address misconceptions, and finally show number lines that correctly graph all possible solutions. “Student Work 3” incorrectly represents all possible solutions, and “Student Work 4” correctly represents all possible solutions. Students analyze these examples and discuss why certain representations are accurate, and others are not.

Evaluation for 5.2d
Materials contain embedded supports for teachers to guide students toward increasingly efficient approaches.

Grade A6 materials include embedded supports to guide educators as they lead students through increasingly efficient approaches in the “Step 1: Explore and Share” and “Step 2: Visual Learning” sections of “Let’s Build” lessons. For example, the teacher’s edition embeds support for the educator in lesson 6-2, “Let’s Build: Add Integers,” to guide students as they look for patterns in adding positive and negative integers. In “Step 1: Explore and Share,” students discuss strategies for making predictions or reasoning so they will realize that “both strategies indicate that if you add a negative number that is less than the opposite of the positive number, the result will be negative,” and revisit adding two negative integers after using a number line in “Step 2: Visual Learning” to verify their prediction. Next, students use two-color counters to model adding integers with different signs, and finally, a number line to model the abstract algorithm for adding integers with different signs. Before students move on to individual practice, the embedded questions “How is adding negative integers the same as adding positive integers?” and “How can you model the sum of integers with different signs?” guide students toward the most efficient approaches.

Additionally, in lesson 3-3, “Let’s Investigate Patterns in Equivalent Ratios,” the teacher’s edition provides embedded guidance in “Step 1: Investigate” to help educators monitor students’ thinking, sequence solution strategies, and connect equivalent-ratio representations during an activity in which students evaluate sample student answers to a problem involving ratios. The guidance directs educators to “Start with a correct solution that uses equivalent ratios to determine the number of additional buses needed per district (Student 1 Work). Then show solutions that show incorrect strategies (Student 2 and Student 3 Work). Finally, show other correct solution strategies such as using a double number line (Student 4 Work) and connect those representations to using equivalent ratios to determine the number of additional buses needed per district.” The guidance continues in “Step 2: Connect” through “Present and Analyze” questions such as “How can you use multiplication...” and “How can you use division...” Finally, in “Talk About Math Ideas,” embedded support advises educators to listen for discussions about the relationship between patterns and equivalent ratios, guide students toward increasingly efficient approaches, and ask, “How can you use patterns in numbers from data found in tables to find equivalent ratios?”

Evaluation for 5.3a
Materials explicitly state how the conceptual and procedural emphasis of the TEKS are addressed.

Grade A6 materials state how the conceptual and procedural emphases of the TEKS are addressed in the “Math Background” pages at the beginning of each topic in the teacher’s edition. For example, the “Math Background: Balance” page in “Topic 8: Represent and Solve Equations” states the conceptual emphasis in the “Conceptual Understanding” section, “Understand Equations and Properties of Equality”: “Students learn the conceptual prerequisites for solving equations. Students understand the definition of an equation, use substitution to determine the solution to an equation, and identify how properties of equality generate equivalent equations. A concept is that terms on either side of the equal sign (=) must balance” (6AM.8A). In the “Procedural Skill and Fluency” section, “Solve Addition and Subtraction Equations” states, “Students use the concepts of equations and equality to solve addition and subtraction equations. Students recognize that one operation has an inverse relationship with the other” (6AM.8D).

Additionally, the teacher’s edition addresses the conceptual emphasis of TEKS 6AM.3D and 6AM.3E through the “Essential Understanding” and “Conceptual Understanding” sections on the first page of lesson 6-2, “ Let’s Build: Add Integers,” stating that “Adding integers requires either adding the absolute values when the integers have the same sign or subtracting the absolute values when the integers have different signs and interpreting the sign of the sum.” The materials also emphasize understanding additive inverses and opposite integers using number lines, helping students develop conceptual understanding of integer operations rather than relying solely on procedures. The materials address the procedural emphasis (1 pt) of TEKS 6AM.3D and 6AM.3E by stating that students will “add integers with the same signs and different signs, identify additive inverses and opposite integers using models such as number lines, and use integer addition to model real-world situations.”

Evaluation for 5.3b
Questions and tasks include the use of concrete models and manipulatives, pictorial representations (figures/drawings), and abstract representations, as required by the TEKS.

The Program Overview describes the progression of concrete to pictorial (representational) to abstract models in the article “Concrete, Representational, Abstract (CRA) Instructional Model,” noting, “Explore and Share in Let’s Build lessons and Let’s Investigate lessons asks students to solve a problem that embeds new ideas and to solve it any way they choose. Students might use concrete materials (counters, geoboards, AngLegs, algebra tiles) [or] pictorial representations (number lines, arrays, grids, strip diagrams). Students have access to digital manipulatives through the Brainingcamp program embedded in the materials for use as concrete models, and the Brainingcamp Live feature allows educators to view each student’s work with the digital manipulatives in real time.”

Additionally, lesson 1-7, “Let’s Build: Divide Fractions,” in the student edition includes concrete models and manipulatives, such as fraction strips, fraction tiles, strips of paper, and area models, and students use visual fraction models to represent and solve fraction division problems. The lesson also directs students to construct and analyze models during “Step 1: Explore and Share” and “Step 2: Visual Learning” activities. The lesson includes pictorial representations, such as bar graphs, area models, fraction diagrams, number lines, and visual representations of real-world contexts. In example 1, students use an area model to divide fractions. Additional tasks require students to interpret and create visual fraction models, represent division situations graphically, and connect models to solution strategies. Finally, the lesson includes abstract representations, such as symbolic fraction division expressions, multiplication by reciprocals, mixed-number computations, estimation, and equivalent numerical expressions. Students rewrite mixed numbers as fractions, multiply by reciprocals, solve symbolic division problems, and justify solutions using mathematical notation.

Evaluation for 5.3c
Materials include supports for students in connecting, creating, defining, and explaining concrete and representational models to abstract (symbolic/numeric/algorithmic) concepts, as required by the TEKS.

The Program Overview describes the progression of concrete to pictorial (representational) to abstract in the article “Concrete, Representational, Abstract (CRA) Instructional Model”: “Explore and Share in Let’s Build lessons and Let’s Investigate lessons asks students to solve a problem that embeds new ideas and to solve it any way they choose. Students might use concrete materials (counters, geoboards, AngLegs, algebra tiles) [or] pictorial representations (number lines, arrays, grids, strip diagrams)....When students share and discuss solutions, rich conversations deepen conceptual understanding of connections between concrete, pictorial, and abstract.” The article explains in the “Visual Learning” section that “The Visual Learning in a Let’s Build lesson provides explicit instruction that connects students’ work in the Explore and Share to new ideas taught and shows pictures of concrete materials, drawings of concrete materials, diagrams, or some combinations of those to represent math concepts. The Visual Learning connects these representations to abstract numbers, symbols, and procedures.” Students have access to digital manipulatives through the Brainingcamp program embedded in the materials, which serve as concrete models they can connect to abstract concepts. The Brainingcamp Live feature also allows educators to view each student’s work with the digital manipulatives in real time. Moreover, the Desmos digital program offers representational models that students can create to connect to abstract concepts. In each Let’s Build lesson, the progression from “Explore and Share” to “Visual Examples” supports students in connecting and creating concrete and representational models to connect with abstract concepts. Then, in the final guided examples, students discuss how to define and explain the abstract concepts they connect to the concrete and representational models.

Additionally, lesson 2-4, “Let’s Investigate Strategies to Find Areas of Polygons,” uses “Step 2: Connect” to support students in connecting and creating concrete and representational area models to abstract concepts by composing and decomposing polygons with pattern blocks and by drawing figures on a coordinate plane to determine area abstractly using addition and subtraction. Then, the materials guide the educator, using prompts in the teacher’s edition to help students explain how composing and decomposing pattern block models support the abstract understanding of a polygon’s area. In the “Choose a Problem: Present and Analyze” section, students explain how they use the concrete pattern blocks and representational area models to obtain the abstract area by composing and decomposing polygons. The “Key Concept” explains that students can decompose the representational model of a polygon into known shapes and add their areas, or draw a larger shape and subtract the areas of the regions that are not part of the polygon to find the abstract area.

Evaluation for 5.4a
Materials provide opportunities for students to develop academic mathematical language using visuals, manipulatives, and other language development strategies.

Grade A6 materials provide opportunities for students to develop their academic mathematical language using visuals and manipulatives within lessons, and other language development strategies within the “Advanced Grade 6: Additional Vocabulary Support” in the “Resources” tab of the Savvas Realize digital platform. For example, in lesson 5-1, “Let’s Build: Classify Numbers” of the teacher’s edition, students use visuals and digital manipulatives—including a list of numbers, a Venn diagram, and a number line—to write about and discuss how to classify different types of numbers. “Advanced Grade 6: Additional Vocabulary Support” provides worksheets to accompany the lesson, helping students develop their academic mathematical language through other language development strategies, such as cloze sentences, mapping diagrams, and Frayer Models.

Additionally, the materials provide opportunities to develop academic mathematical language through visuals, such as the interactive number line and visual representations in the Brainingcamp activity, which help students develop and discuss the concept of absolute value. Students can also develop mathematical language through the “Brainingcamp Manipulatives” activity, in which they move points on a number line to explore absolute value and compare positive and negative rational numbers. Finally, embedded educator discussion prompts in the teacher’s edition provide opportunities to develop academic mathematical language through other language development strategies that ask students to explain opposites, describe what the sign of a number indicates on a number line, and discuss why distance is always positive, requiring students to use mathematical vocabulary in context.

Evaluation for 5.4b
Materials include embedded teacher guidance to scaffold and support students' development and use of academic mathematical vocabulary in context.

Grade A6 materials include embedded educator guidance to scaffold and support students’ development and use of academic mathematical vocabulary in context within the Vocabulary Review section of the “Topic Reviews” at the end of each topic. For example, the Vocabulary Review in the teacher’s edition contains embedded educator guidance in topic 3, “Understand and Use Ratio and Rate: Topic Review,” to have students use the academic mathematical vocabulary terms, including ratio, unit price, and rate, in context orally and in writing to scaffold and support their development of the terms. The “Oral Language” embedded guidance directs educators to “Have a student volunteer write a ratio that includes units on the board” before students complete the page and “Then have the student volunteer ask the other students questions about the ratio, such as ‘Is this a speed?’ ‘Why is this a unit price?’ and ‘What kind of rate is this?’” and then repeat the activity with other student volunteers. After students complete the page, the “Writing in Math” embedded guidance suggests: “Have students work with a partner. Each partner writes a math question that uses a vocabulary term. Then students trade papers and give a written answer that uses the vocabulary term.”

Additionally, in the “Topic Review” of “Topic 2: Analyze Geometric Relationships,” the teacher’s edition provides Vocabulary Review guidance that directs educators to reinforce academic vocabulary through an “Oral Language” activity: “Before students complete the [review] page, you can reinforce oral language by using one or more of the following activities. Have students define the terms in their own words. Have students ask and answer math questions that use the words.” The guidance also includes a “Writing in Math” activity that prompts the educator to “Have students write an acrostic-style poem. First, they write each letter of a vocabulary word on a separate line going down the page. Next, they write a line of the poem beginning with the letter and related to the word.”

Evaluation for 5.4c
Materials include embedded teacher guidance to support the application of appropriate mathematical language to include vocabulary, syntax, and discourse to include guidance to support mathematical conversations that provide opportunities for students to hear, refine, and use math language with peers and develop their math language toolkit over time as well as guide teachers to support student responses using exemplar responses to questions and tasks.

Grade A6 “Let’s Build” lessons include embedded educator guidance to foster the appropriate academic mathematical vocabulary, syntax, and discourse that support mathematical conversations, providing opportunities for students to hear, refine, and use math language with peers, developing their math language toolkit over time, and “Sample Student Work” that guides educators to support students using exemplar responses to questions and tasks. For example, lesson 8-2, “Let’s Build: Apply Properties of Equality” in the teacher’s edition includes a “Language Support” section with educator guidance to “Have each student write an equation and an equivalent equation using one of the properties of equality. On the back, have each student write and describe, using math sentences, the property used and why. Then have students work with a partner to share their note cards. Have the first partner hold up their card and ask: What property did I use to make this equivalent equation? The other partner answers, describing the property used. If the partner answers incorrectly, the first partner shares the correct answer and explains why. Then the second partner takes a turn. After they both present their note cards, have students find new partners and repeat the activity,” supporting student discourse using appropriate academic vocabulary and syntax, providing opportunities for students to hear, refine, and use math language with peers, and developing their math language toolkit over time. In the “Explore and Share” section of the lesson, students begin with a simple equation, “4 + 8 = 12,” and make specific changes to the equation, including adding, subtracting, multiplying, and dividing either side of the equation by given values, and then determine whether the equation still remains true. The teacher’s edition provides “Sample Student Work” with exemplar responses, including “Nick shows his computations and tells which computations maintain equality. Amanda correctly completes each computation and then states whether the resulting equation is true.”

Additionally, lesson 8-7, “Let’s Build: Solve Two-Step Equations,” guides educators to use the Math Talk activity “Equation Detective,” which has students analyze a peer’s worked solution for errors and explain their reasoning aloud, supporting the application of appropriate mathematical language and building precise use of the term “isolate the variable” introduced in the “New Vocabulary” section. Students also build a graphic organizer of properties of equality, and the Language Support guidance directs educators to “Represent the first property on the board and explain your thinking process aloud,” then “Invite students to copy your example into their graphic organizer” and “Add rows for the other properties using proper mathematical language in the descriptions of each property,” reinforcing vocabulary and syntax as students write out how each property keeps an equation balanced. The “Language Routine: Information Gap” has partners exchange only partial information about a word problem, with guidance including “Partner A reads the card and decides what information is needed to solve the problem. Partner B asks what information Partner A needs. As Partner A answers, Partner B asks for justification. Both students then set up and solve the problem together.” This structure gives each student practice in hearing, using, and refining math language in discourse with peers, and supports them in developing their math language toolkit over time. “Talk About Math Ideas” and “Do You Understand” also include embedded exemplar responses, giving educators a clear model of the precision expected in student explanations. For example, in response to whether an operation could be performed on only one side of an equation, the sample answer explains, “No; an equation means the expressions on both sides of the equal sign are equal. So, you must perform the same operation on both sides of the equation to make sure both sides remain equal,” guiding educators to understand exactly how a complete and precise explanation should sound, giving them support for coaching students toward that same level of mathematical language.

Evaluation for 5.5a
TEKS process standards are integrated appropriately into the materials.

Grade A6 materials appropriately integrate the TEKS process standards, as discussed in the “Mathematical Process Standards in Instruction” section of the front matter of the teacher’s edition. For example, the article explains how the TEKS process standards are integrated into the materials through lessons, problems, and assessments in four sections with specific information, including “Mathematical Process Standards are infused and explicitly highlighted in lesson instruction using teal colored type.” The Savvas Realize digital program also integrates the TEKS process standards into the materials using a “Problem-Solving Model” that students can use to support problem-solving anytime.

Additionally, in the teacher’s edition, the “Mathematical Process Standards” page for “Topic 9: Understand and Solve Inequalities” identifies 6AM.1B and 6AM.1G as the process standards that are emphasized throughout the topic. The page provides educators with specific student behaviors to look for, such as representing situations in a variety of ways to check understanding and interpreting solutions represented on a number line back to the original inequality for 6AM.1B, and justifying when and how to use the properties of inequalities or identifying and locating mistakes when writing and solving two-step inequalities for 6AM.1G. The section also gives educators questioning strategies to use throughout the topic when students struggle with representing and solving inequalities, such as asking which quantity is the coefficient and which is the constant, or how a number line could be used to prove or disprove an inequality, giving educators concrete ways to build students’ reasoning and explanation skills as they work through the topic.

Evaluation for 5.5b
Materials include a description of how TEKS process standards are incorporated and connected throughout the course.

Grade A6 materials include a description of how the TEKS process standards are incorporated and connected throughout the course, as discussed in the “Mathematical Process Standards in Instruction” in the front matter of the teacher’s edition. For example, the article explains, “Special Emphasis on Two Process Standards per Topic: a TEKS Mathematical Process Standards page in front of every topic provides suggestions you may choose to implement as a way of giving special emphasis to two Process Standards in the topic.” It continues to describe how the process standards are connected by stating, “All math tasks evoke the selection, use, and management of multiple Process Standards,” and “Items that assess content also engage students in using Process Standards. The topic performance tasks are especially good opportunities to assess students’ proficiency with Process Standards.”

Additionally, the “165-Day Pacing Guide” of the teacher’s edition explicitly states that “TEKS Mathematical Process Standards are integrated throughout every Topic: 6AM.1A, 6AM.1B, 6AM.1C, 6AM.1D, 6AM.1E, 6AM.1F, 6AM.1G.” The pacing guide shows that the process standards are intentionally embedded across all 11 topics, while content TEKS and ELPS are identified for each topic, demonstrating that the mathematical process standards are consistently integrated throughout the entire grade 6 course rather than taught as isolated skills. The topic content pages in the front matter of the teacher’s edition corroborate this by listing the process standards for each lesson across the topics in bold.

Evaluation for 5.5c
Materials include a description for each unit of how TEKS process standards are incorporated and connected throughout the unit.

Every unit in the grade A6 teacher’s edition includes a description of how the TEKS process standards are incorporated and connected throughout each topic on the “Mathematical Process Standards” page at the beginning of the topic. For example, the “Mathematical Process Standards” page of “Topic 2: Analyze Geometric Relationships” describes how 6AM.1G is incorporated and connected throughout the topic, stating, “Mathematically proficient students construct arguments using accurate definitions and terminology related to the areas of parallelograms and triangles, explain how to find the area of a trapezoid and justify their solution using formulas and equations, and construct arguments to explain how to use the number of fractional cubes in a rectangular prism to find the volume.” The page details that 6AM.1F is incorporated and connected throughout the topic because “Mathematically proficient students determine how the areas of a parallelogram and a rectangle are related by decomposing the parallelogram and using the parts to compose a rectangle, analyze the structure of trapezoids and kites to determine how to decompose them into shapes for which the area formulas are known, and construct arguments to defend the application of formulas to new situations.”

Additionally, in the teacher’s edition, the “Mathematical Process Standards” page of the “Topic 5: Rational Numbers and the Coordinate Plane” topic overview emphasizes the process standards and explains that students develop proficiency through them as they engage in lessons throughout the topic. The materials describe how the TEKS process standards connect throughout the topic, showing how the highlighted process standards are reinforced across topic 5 through mathematical reasoning, communication, multiple representations, modeling, and problem-solving. The accompanying educator questioning and topic guidance connect these mathematical behaviors across the lesson sequence, showing that the process standards are intentionally developed throughout the unit rather than addressed as isolated skills.

Evaluation for 5.5d
Materials include an overview of the TEKS process standards incorporated into each lesson.

The “Item Analysis: Practice and Problem-Solving” answer key in the “Practice and Problem-Solving” section of the teacher’s edition includes an overview of the TEKS process standards incorporated into each lesson. For example, in lesson 2-2, “Let’s Build: Understand Angles and Side Lengths of Triangles,” the “Item Analysis: Practice and Problem-Solving” answer key provides an overview showing that 6AM.1B is integrated into example 1 and items 7, 12, 15, and 18; 6AM.1D is integrated into example 2 and items 8, 9, 14, and 15; and 6AM.1G is incorporated into example 3 and items 10, 11, 13, 16, and 17.

Additionally, the first page of each “Let’s Build” lesson in the teacher’s edition lists the TEKS content standards and the “Mathematical Process Standards” addressed in the lesson. For example, lesson 11-1, “Recognize Statistical Questions,” identifies “Mathematical Process Standards” 6AM.1B, 6AM.1C, 6AM.1D, 6AM.1E, and 6AM.1F as incorporated throughout the lesson. These process standards support students in acquiring and demonstrating mathematical understanding as they engage with the lesson content. This overview enables educators to easily identify the TEKS mathematical process standards throughout the lesson.

Evaluation for 6.1a
Materials provide opportunities for students to think mathematically, persevere through solving problems, and to make sense of mathematics.

Grade A6 materials provide opportunities for students to think mathematically, persevere through solving problems, and to make sense of mathematics in “Let’s Build” lessons, such as lesson 4-5, “Let’s Build: Solve Percent Problems.” In “Step 1: Explore and Share” of the teacherʼs edition, students think mathematically to use the meaning of percent and number sense to solve percent problems, including “What is 50% of 400? What is 5% of 400?” Students persevere through percent problems and make sense of mathematics in a whole-class setting using benchmark fractions and double number lines in example 1 of “Step 2: Visual Learning” to estimate, then solve for the part when given percentages and whole numbers, followed by individual “Try It!” problems.

Additionally, each topic includes a “Build G.R.I.T.” section that provides educators with concrete talking points and a modeled classroom dialogue to cultivate a growth mindset, resilience, self-efficacy, and time management skills as students work through multistep, real-world math tasks. The “Build G.R.I.T.” section of the teacher’s edition, “Topic 3: Understand and Use Ratio and Rate,” gives students opportunities to think mathematically about ratio and rate, encouraging educators to help students approach these problems with confidence and treat mistakes as stepping stones toward conceptual understanding rather than failures. It offers ways for educators to help students in building confidence, such as reminding students that a peer’s different approach or model is a chance to learn, building resilience to persevere through solving problems by having students check their solutions for accuracy and try different strategies such as drawing representations to make sense of mathematics, developing self-efficacy by having students discuss unit rates, unit price, and testing rates for accuracy with peers, and supporting time management through strategies such as setting up a table for unit conversions and using consistent labels. This is modeled through a sample dialogue in which the educator helps a student, Pavitra, reflect on where she went wrong in a ratio problem comparing soccer balls to footballs by prompting her to reread the problem and estimate whether her answer makes sense, rather than simply correcting the error.

Evaluation for 6.1b
Materials support students in understanding, explaining, and justifying that there can be multiple ways to represent and solve problems and complete tasks.

Grade A6 materials use the Math Talk and Explore and Share activities in the “Let’s Build” lessons to help students understand, justify, and explain that there are multiple ways to represent and solve problems and complete tasks. For example, lesson 8-2, “Let’s Build: Apply Properties of Equality,” in the teacher’s edition uses the Math Talk opening task, which presents a set of four equations with variables and different operations and asks students to determine, “Which one doesn’t belong?” Educator prompts such as “What do you know about two expressions that are joined by an equal sign? What does it mean for a number to be a solution to an equation? How could you check to see if 7 is a solution to each equation? How do you know when it’s not?” help students understand that there can be multiple ways to represent and solve problems and complete tasks. Next, the step 1 “Explore and Share” activity supports students in using a graphic organizer that shows four ways to manipulate the left side, right side, or both sides of the equation “4 + 8 = 12.” Using different whole-number computations, students work in small groups to explain and justify that multiple ways can represent and solve problems. The “Connect Solution Strategies” section of the teacher’s edition instructs educators to have the whole class discuss what they noticed about which computations kept the equations true and which made them false, then explain how the pattern could be used to predict the result for any computation instructions. “Sample Student Work” accompanying the activity shows how students can justify how and when computations kept the equations true or made them false.

Additionally, lesson 6-1, “Let’s Investigate: Relate Integers and Their Opposites” of the teacher’s edition supports students in understanding and explaining that there can be multiple ways to represent and solve problems and complete tasks by presenting additive inverses using multiple representations, including number lines, real-world contexts, symbolic notation, and authentic student work. In the “Select and Sequence,” “Connect Solution Strategies,” and “Present and Analyze” sections, educators compare multiple student approaches and facilitate discussions about different solution methods for problems involving integers and additive inverses. The lesson also supports students in explaining that there can be multiple ways to represent and solve problems and complete tasks through educator prompts, such as “How can you use opposites to help you solve?,” “Why does it make sense to think of the ground floor of the mall as 0?,” and “What integer represents Samuel’s withdrawal?” and by directing students to “Connect Solution Strategies,” participate in gallery walks, discuss sample student work, and compare different approaches for solving the investigation problems using mathematical vocabulary. The lesson further supports students in justifying that there can be multiple ways to represent and solve problems and complete tasks by requiring students to compare contextual, numerical, and number-line representations and defend which representations accurately model additive inverses, and instructs educators to “Encourage students to use the terms integer, opposite, additive inverse, and absolute value in their explanations,” reinforcing mathematical justification.

Evaluation for 6.1c
Materials are designed to require students to make sense of mathematics through multiple opportunities for students to do, write about, and discuss math with peers and teachers.

Grade A6 materials include a Math Talk in every “Let’s Build” lesson, giving students multiple opportunities to make sense of mathematics and discuss it with peers and educators. For example, in the teacher’s edition, lesson 9-4, “Let’s Build: Solve Inequalities Using Multiplication or Division” includes a “Math Talk: Algebra Strings” activity that shows the steps for simplifying a one-step, one-variable inequality problem by dividing both sides by three in each step until the coefficient of x becomes -1, with the instruction “Look for a pattern” on the first slide. The second slide reveals the last step, showing that the coefficient of x is no longer negative, but the inequality of x becomes negative, and it uses a question mark to require students to make sense of the mathematics and discuss as a whole class which symbol best represents the inequality. Examples 2 and 3 model the same concept and give students opportunities to do math with their educator. The “Connect” questions within multiple “Let’s Investigate” lessons, such as those in lesson 3-3, “Let’s Investigate: Patterns in Equivalent Ratios,” give students opportunities to make sense of mathematics and to do and write about mathematics with their peers and educators. For example, the teacher’s edition directs educators to group students who chose the same problem so they can discuss solution strategies as they do the required mathematics and write their strategies and solutions. Then the whole class answers questions about selecting strategies, representing ideas, analyzing relationships, and evaluating reasonableness, giving students opportunities to write about math with their educator.

Additionally, “Topic 9 Review: Vocabulary Review” in “Topic 9: Understand and Solve Inequalities” of the teacher’s edition provides an “Oral Language” section in which educators direct students to “work in groups to create crossword puzzles with visual clues and definitions” to discuss math with their peers and educator. In the “Writing in Math” section, students “write an inequality to represent the following situation,” “explain how they wrote their inequality,” “compare and contrast their solution” to that in the student edition; they write about math with their peers and educator and are encouraged to “draw models or graphs of their solutions and to use vocabulary terms to construct their explanations” to do math with their peers and educator.

Evaluation for 6.2a
Materials support teachers in guiding students to share and reflect on their problem-solving approaches, including explanations, arguments, and justifications.

Grade A6 materials include a Problem-Solving Model on the Savvas Realize digital program that supports educators in guiding students to share and reflect on their problem-solving approaches using guiding questions and a graphic organizer. For example, guiding questions for reflecting on their problem-solving approaches are broken into categories to support explanations, arguments, and justifications, such as “Analyze Given Information,” “Determine a Solution,” and “Justify the Solution and Evaluate its Reasonableness.” The Problem-Solving Model Recording Sheet is a graphic organizer that gives students space to share their problem-solving approaches, with sections that correspond to the guiding questions on the previous page. Students write the problem “Analyze Given Information” into “Need to Find” and “Given,” share their explanations and arguments in the “Formulate a Plan” section, which provides sample ways to represent problems and math tools to “Determine a Solution,” and share their justifications in the “Justify the Solution and Evaluate its Reasonableness” section of the recording sheet.

Additionally, in lesson 10-7, “Explore Making an Income,” the Math Talk activity asks students to compare two professions within the transportation industry: airline pilot and bus driver. Students discuss how both professions contribute to the same industry while recognizing differences in the training and education required for each career. The teacher’s edition directs the educator to pay particular attention to student observations such as “Airline pilots require more specialized training through formal education than bus drivers” and “Airline pilots require many more hours before they are certified to fly.” These observations help students justify their conclusion that airline pilots typically earn a higher income than bus drivers. This guidance supports students in sharing and reflecting on their problem-solving approaches, including explaining their reasoning, using evidence to support their arguments, and justifying how education and training can influence income.

Evaluation for 6.2b
Materials include prompts and guidance to support teachers in providing explanatory feedback based on student responses and anticipated misconceptions.

Grade A6 materials include a “Prevent Misconceptions” section for the Exit Ticket in every “Let’s Build” lesson, which supports educators with prompts and guidance for providing explanatory feedback based on student responses and anticipated misconceptions. For example, in lesson 1-7, “Let’s Build: Divide Fractions,” the Exit Ticket gives students four choices (3, 6, 24, and 32) for the correct answer to how many one-eighth-pound containers can be filled using three-quarters of a pound of almonds, and the “Prevent Misconceptions” section of the teacher’s edition guides the educator that if students selected 24, they may not have used the reciprocal of the divisor, and provides prompts, including “How do you modify the divisor when dividing fractions?” In the “Explore and Share” activity at the beginning of the lesson, students work in small groups to solve how many one-and-three-eighth-inch pieces can be cut from a five-and-one-half-inch strip. The “Monitor, Select, and Sequence” section of the teacher’s edition guides the educator to observe the students working and consider how they use a model and which operations they use to represent the problem, giving potential student responses to the question, including “Students may use a bar model to show the problem” and “Students who use models may solve the problem using repeated addition or subtraction.” Prompts with explanatory feedback, such as “How can you draw a picture that shows the problem?” and “Which mathematical operation would be helpful to solve the problem?” help educators anticipate misconceptions.

Additionally, lesson 7-1, “Let’s Build: Understand and Represent Exponents” in the teacher’s edition contains the section “Prevent Misconceptions,” which addresses item 2, guiding educators to “make sure students understand that the exponent should match the number of times that 4 is a factor, not the number of times it is multiplied.” To help students work through this misconception, the materials prompt the educator to ask, “What does the number 5 in the expression tell you?” and to have students “write five 4s and put multiplication symbols between them,” then ask, “Why do you only multiply 4 times when evaluating the expression?,” with the sample student response that “The first time you multiply, you use two 4s.” The materials also anticipate misconceptions around item 3, “Explain Math Ideas,” guiding educators to have students “Use the Identity Property of Multiplication to explain the value of 1 ✕ 1,” prompting the educator to remind students that “Any number times 1 is itself, so 1 ✕ 1 = 1.”