HMH Into Math Texas, Grade 5

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

The program materials include multiple scope and sequence resources that clearly outline the Texas Essential Knowledge and Skills (TEKS), the English Language Proficiency Standards (ELPS), and concepts taught throughout the course. The materials include yearlong scope and sequence documents located within the Program Overview and the Implementation Guide.


The scope and sequence charts include a TEKS and ELPS alignment for each module and lesson and identify the concepts taught throughout the instructional year. Additional program materials further support this alignment through TEKS and ELPS correlations as well as “Correlations by Lesson” documents that break down concepts taught by module. Within the “Discover” tab under the program dropdown, the scope and sequence also outlines the TEKS, ELPS, and lesson concepts to support instructional planning and pacing.

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 materials provide a “Scope and Sequence with Yearly Pacing” for 165-day and 210-day instructional calendars. The guidance provided in the calendar “moves step by step through each module and shows how each module covers the instructional sequence and standards, and then lists each Module Review and Module Test.” These options support effective implementation while remaining aligned with the TEKS, ELPS, and course concepts.

The materials provide guidance on extending the 165-day pacing calendar to 180 days by adding “one extra day to each lesson where a Readiness Standard is first taught.” Educators can use the extra day to provide targeted reinforcement, practice, or challenges based on formative assessments, “Quick Checks,” and observations.

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 program materials include clear explanations of the unit rationale and lesson order, and they explain how concepts connect throughout the course. The Teacher’s Guide includes a “Module Order Rationale” that outlines the sequence of modules and describes how the organization reflects natural mathematical progressions and students’ cognitive development.

The materials explain how concepts build across modules and lessons to develop deeper mathematical understanding. For example, students begin with decimal place value to the thousandths, which provides the foundation for representing, comparing, ordering, and rounding decimals before applying these skills to computation and real-world problem-solving. The materials also explain how decimal concepts progress from representing decimal addition and subtraction with concrete models to assessing reasonableness and applying efficient computation strategies.

Additionally, fraction instruction progresses from representing fraction sums and differences to adding and subtracting fractions with unlike denominators using equivalent fractions and benchmark numbers before applying properties and equations to solve problems. The materials further explain how multi-digit multiplication and division strategies support later work with decimal operations and provide a foundation for fraction multiplication and division, demonstrating the conceptual connections that support multistep problem-solving throughout the course.

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

The program materials include guidance that support educators with unit and lesson internalization through comprehensive planning supports, structured instructional routines, and embedded educator guidance. The Teacher’s Guide provides planning support that reviews the standards, outlines what students will learn in each module, explains the module flow, identifies the TEKS and ELPS, and includes background information to support instructional preparation and lesson internalization.

The materials also include resources that help educators understand the sequence of learning within each module and prepare for daily instruction. The “Before You Teach” sections outline the mathematical understanding needed for each lesson and explain what students will learn. Planning tools such as the “Editable Lesson Plan,” pacing guidance, module flowcharts, and lesson sequencing explanations support educator preparation.

Additionally, the “Step Through a Module” section of the Implementation Guide provides guidance for internalizing module and lesson components, while the “Teacher Background” sections include mathematical rationale, examples, summaries, and explanations of the importance of concepts to deepen educator understanding before instruction. The materials also include the “Module Internalization Tool” and “Lesson Internalization Tool,” which provide protocols to support module and lesson internalization through guiding questions such as “What concepts, strategies, or procedures will students learn?” and “What do the lesson and language objectives expect from students?” Educators can use the tools individually or collaboratively as note-taking or discussion guides to prepare for and internalize upcoming modules and lessons.

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

The program materials include resources and guidance documents to help instructional leaders support educators in implementing the program as designed. The materials provide implementation supports through resources such as the Professional Learning and Coaching Guide, which includes instructional support, implementation guidance, pacing recommendations, and professional learning resources to assist leaders in supporting effective program use.

The materials also include tools that support classroom observations, reflection, and coaching conversations. For example, the Look-Fors and Reflection Guide provides classroom observation indicators, implementation expectations, reflection questions, and suggested discussion topics to help instructional leaders monitor instruction and support educator planning. The “Look-Fors” section includes indicators for the first 30 days of instruction, such as implementing “Benchmark Assessments” and the Classroom Panel to deliver Essential Sessions during whole-group instruction.

Additionally, the “Reflection Guide” section includes guiding questions for classroom visits and coaching conversations, such as identifying indicators of classroom success, examining supporting evidence, and determining areas where additional support or resources may be needed. These resources support targeted feedback, instructional coaching, and ongoing monitoring of program implementation across classrooms.

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 program materials include comprehensive unit overviews that provide the background content knowledge and academic vocabulary educators need to effectively teach mathematical concepts. The materials include educator guidance that supports concept development by outlining prior learning connections, strategy examples, instructional explanations, and academic vocabulary at the beginning of each unit.

The materials provide educator background information, TEKS alignment, examples, summaries, explanations, and instructional guidance to support conceptual understanding and procedural fluency. For example, the materials include guidance for teaching fraction addition and subtraction through concepts such as common denominators, estimation, rational numbers, mixed numbers, pictorial models, and properties of operations. The materials also support instruction in decimal division through explanations related to estimation, standard algorithms, pictorial models, place value, and partial quotients.

Additionally, the module overviews identify prior learning, current module concepts, and future learning connections related to decimals, place-value relationships, rounding, comparing decimals, fractions, and operations, helping educators connect mathematical ideas across the course. The materials further support educator understanding through a “Teacher’s Glossary” section that includes academic vocabulary introduced within each module, along with definitions and illustrations to support instructional delivery and mathematical understanding.

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 program materials include family supports through “Family Letters” for each module in both English and Spanish. The materials support communication and family engagement by explaining unit learning goals, introducing key academic vocabulary, and providing guidance for supporting student learning at home.

The “Family Letters” include lesson-specific online resources, home activities, and suggestions for reinforcing mathematical concepts through real-world applications and mathematical discussions. For example, the letters provide activities that support concepts such as decimal place value, decimal multiplication, fraction operations, and problem-solving through hands-on practice and everyday situations, including shopping and recipes.

Additionally, the letters include guidance for families on how to support student progress using online resources and activities that reinforce in-class learning outside of school. The “Family Letters” also provide background information on mathematical concepts and suggestions for monitoring and supporting students’ academic progress throughout each module.

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 program materials include comprehensive, structured, and detailed lesson plans that provide daily objectives, instructional tasks, required materials, guiding questions, and instructional assessments aligned to the TEKS and ELPS. Each lesson includes learning and language objectives, instructional routines, discussion opportunities, practice tasks, and assessment opportunities aligned to the lesson goal. Additionally, the “Before You Teach” section at the beginning of each lesson supports educator planning through TEKS and ELPS correlations, lesson objectives, questions, tasks, and assessments designed to support student mastery.

The lesson materials also include hands-on instructional supports and embedded opportunities for student discourse and formative assessment. For example, lessons identify manipulatives and materials such as base 10 blocks, number lines, money models, fraction circles, plastic rainbow fraction tiles, and fraction bars to support conceptual understanding. Lessons such as “Understand Decimal Multiplication Patterns” and “Represent Fraction Sums and Differences” include clearly defined learning and language objectives focused on mathematical reasoning, problem-solving, and the use of academic vocabulary.

The materials also incorporate routines such as “Spark Your Learning,” “Learn,” “Assess,” “Three Reads,” “Collect and Display,” “Compare and Connect,” “Turn and Talk,” and “Quick Checks” to support student engagement, mathematical discussions, and ongoing assessment of understanding while integrating TEKS process standards and ELPS expectations for communication, reasoning, speaking, and writing.

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 program materials include lesson overviews that identify the educator and student materials necessary for effective lesson implementation and provide suggested timing for instruction. The materials include lesson overviews and “Before You Teach” sections that outline the manipulatives, resources, and instructional materials needed for each lesson and instructional group.

The lesson materials include clear instructional routines, hands-on supports, and embedded opportunities for student discourse and formative assessment.

The materials also provide suggested time allocations for lesson components, including approximately 45 minutes for Essential Sessions and additional time for differentiation and practice opportunities. Within the “Step Through the Module” and “Before You Teach” sections, the materials outline timing expectations for lesson routines such as learning goals, “Spark Your Learning,” guided instruction, and assessment activities. Additionally, lessons include instructional structures and routines such as “Learn” and “Assess” to support pacing, lesson delivery, and student engagement throughout instruction.

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

The program materials include guidance for effective extended practice through differentiated activities, independent practice opportunities, homework, and enrichment resources connected to each lesson. Each lesson provides a clear overview of the instructional components, including lesson objectives, academic vocabulary, required materials, and resources needed to support effective lesson implementation.

The materials also include differentiation and practice sections that guide educators in selecting appropriate next steps based on student needs. For example, the “Program Activity Report” provides guidance for selecting resources aligned to student performance, including opportunities for review, additional practice, or extensions. The differentiation and practice sections included after lessons guide educators through instructional decisions and questioning strategies to support students requiring intervention, reinforcement, or enrichment.

Additionally, independent practice rotations, project-based learning opportunities, and digital supports are included to provide continued reinforcement and extended learning opportunities beyond the core lesson.

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.

The program materials include a variety of instructional assessments at both the lesson and unit levels, including diagnostic, formative, summative, and performance-based assessments. The assessment system includes “Benchmark Assessments,” “Prerequisite Checks,” “Quick Checks,” Module Reviews, Module Tests, and Module Projects, providing multiple opportunities to monitor student progress and measure mastery of grade-level content.

The materials include varied task types and questioning structures to assess student understanding in multiple ways. For example, “Prerequisite Checks” assess readiness before each module, “Quick Checks” provide formative assessment opportunities during and after instruction, and Module Tests measure mastery of skills taught within each module through a variety of item types, including multiple-choice, open-ended, drag-and-drop, and fill-in-the-blank questions.

Additionally, Module Projects engage students in Depth of Knowledge (DOK) Level 4 tasks that require the application and extension of mathematical understanding. Lessons also include opportunities for students to demonstrate learning through estimation tasks, mental math strategies, area models, equations, written explanations, concrete and pictorial models, and real-world problem-solving activities, allowing students to communicate their understanding using multiple representations. The materials also include “Benchmark Assessments” administered at strategic points during the school year to identify misconceptions and monitor student growth over time.

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

The program materials include clear definitions and explanations of the intended purposes for the various assessment types used throughout the program. The Implementation Guide and the Program Assessments at a Glance resource identify formative and summative assessments and provide information regarding their purpose, benefits, administration, access, and recommended timing throughout the school year.

The materials explain how each assessment supports instruction and helps educators monitor student progress. For example, “Benchmark Assessments” determine student progress at strategic points during the school year, while “Prerequisite Checks” assess previously learned concepts and help determine student readiness for a module. “Quick Checks” provide formative data after instruction to identify students who need support, additional practice, or enrichment. Module Projects are described as performance-based assessments that engage students in DOK Level 4 tasks that apply and extend mathematical understanding.

Additionally, Module Tests are identified as summative assessments that measure student understanding of the TEKS and process standards taught within a module and are used to monitor mastery of content and mathematical practices. The materials also include “Data-Driven Support” tables that help educators analyze prerequisite understanding, determine readiness for instruction, and make decisions regarding reteaching and reassessment based on student performance data.

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

The program materials include educator guidance that supports the consistent and accurate administration of instructional assessments. Assessment resources provide clear information regarding the purpose, timing, format, duration, and administration of assessments to promote consistent implementation across classrooms.

The materials include guidance for administering both formative and summative assessments. For example, the Program Assessments at a Glance resource identifies when assessments should be administered, including before each module for “Prerequisite Checks,” after instructional tasks for “Quick Checks,” and at the end of each module for Module Tests. The materials also provide suggested administration times and identify assessment formats such as digital assessments, editable files, and PDFs.

Additionally, the Module Test guidance includes information on the purpose, administration procedures, and recommended duration of approximately 45 minutes, along with support sections such as “What Should I Do?” “What More Can I Do?” and “What Can I Aspire to Do?” to assist educators with implementation. The “Quick Checks” and “Learning Goal” resources provide educator guidance for collecting formative assessment data, monitoring mastery of lesson objectives, and determining appropriate instructional next steps based on student performance.

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

The program materials include formative and summative assessments that align with the TEKS and instructional objectives of the lessons and modules. Assessment resources include assessment documentation, item guides, and answer keys that identify TEKS alignment, DOK levels, lesson connections, and scoring guidance to ensure assessments measure the intended learning outcomes.

The materials demonstrate alignment between assessment tasks, lesson objectives, and the TEKS throughout the program. For example, module 4 assessments, including the “Prerequisite Check,” Module Review, Module Test, and Module Project, align with the TEKS related to decimal multiplication concepts and operations. Similarly, module 6 assessments align with the TEKS that address fraction addition and subtraction concepts. Formative assessments, such as “Quick Checks” and “Practice on Your Own” tasks, are directly connected to lesson learning goals and TEKS-based instructional objectives.

Assessment item guides identify the TEKS assessed, DOK levels, and aligned instructional tasks, while Module Test answer keys identify the correct response for each item, provide TEKS correlations, include point values and scoring rationales, and support accurate interpretation of student performance. Additionally, the diagnostic “Benchmark Assessments” include answer keys with a TEKS correlation table that aligns each assessment item to the corresponding TEKS and specific lessons, providing evidence of alignment between the assessments, course standards, and instructional content.

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

The program materials include TEKS-aligned assessment items at varying levels of complexity across lessons and modules. Assessment resources incorporate multiple DOK levels and provide opportunities for students to demonstrate understanding through modeling, estimating, explaining, solving, justifying, and applying mathematical concepts.

The materials include formative and summative assessments that progress from foundational skills to more complex reasoning and problem-solving. For example, assessment tasks require students to represent decimal multiplication using pictorial models, solve multistep decimal multiplication problems, assess reasonableness through estimation, represent fraction sums and differences, solve problems with unlike denominators, and apply rational number concepts in real-world contexts. Lesson assessments include DOK Level 1–3 questions that support increasingly complex thinking and allow educators to monitor student understanding across varying levels of rigor.

Additionally, assessment answer keys identify TEKS alignment and DOK levels for assessment items, while “Practice on Your Own” tasks provide TEKS-aligned opportunities for independent application of skills. Module Projects provide opportunities for students to engage in DOK Level 4 tasks that extend mathematical application, reasoning, and problem-solving. These assessment components ensure students demonstrate learning across multiple levels of cognitive complexity.

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

The program materials include instructional assessments and scoring information that provide guidance for interpreting student performance and using assessment data to inform instruction. The assessment system includes diagnostic, formative, and summative measures, including “Prerequisite Checks,” “Quick Checks,” Module Reviews, Module Tests, and Module Projects, which provide multiple opportunities to monitor student understanding and mastery of the grade-level TEKS and lesson objectives.

The materials support educators in analyzing assessment results and determining appropriate instructional next steps. For example, Module Reviews identify key vocabulary, concepts, and skills for students to review prior to the Module Test, while the accompanying “Data-Driven Support” tables connect assessment results to the specific TEKS, content areas, and recommended reteaching resources or lessons. Assessment answer keys provide correct responses, scoring guidance, explanations of student responses, and rubrics for open-ended items to support accurate interpretation of student performance.

Additionally, Module Projects include scoring rubrics that evaluate mathematical understanding, reasoning, and completeness across multiple proficiency levels. When assessments are administered digitally, educators can access class and student performance data to identify trends, determine mastery levels, and select targeted intervention, on-level support, or enrichment activities to address student needs.

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

The program materials provide guidance for using instructional tasks and activities to respond to student trends in assessment performance through differentiated support and data-driven instructional planning. The materials include “Data-Driven Support” resources, “Differentiation and Practice” guidance, assessment reports, and targeted instructional recommendations that help educators address student learning needs identified through assessment data.

The materials guide educators in analyzing assessment results and selecting appropriate instructional responses based on student performance. For example, educators are directed to review “Program Activity Reports” and assessment data to determine student mastery and identify students who would benefit from small-group instruction, reteaching, additional practice, or enrichment. The “Differentiation and Practice” resources provide a structured process for grouping students and assigning targeted supports, including “Almost There” reteach activities, “On Track” practice activities, “Ready for More” or “Extension” tasks, “Practice Plus” homework, and personalized adaptive practice through Waggle.

Additionally, “Data-Driven Support” tables connect assessment results to specific reteaching resources, lessons, and activities available in digital and printable formats. Assessment guidance also includes performance thresholds that help educators determine appropriate instructional pathways and respond to patterns in student understanding, misconceptions, and learning gaps.

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.

The program materials include educator-facing and student-facing tools that support monitoring student progress and growth throughout the school year. Educators can use resources such as the “Whole-Class Proficiency and Growth Tracker,” “Program Activity Reports,” “Assessment Reports,” “Standards Reports,” “Growth Reports,” and TEKS proficiency reports available through the Ed tool to monitor student performance, identify trends, and inform instructional decisions.

The materials also include tools that support student goal setting, self-assessment, and reflection. Students use “Learning Goal” routines during lessons to assess their understanding of lesson objectives and reflect on their progress toward meeting learning goals. At the beginning of each lesson, students preview an “I Can” statement and revisit it at the conclusion of the lesson to evaluate their level of understanding.

Additionally, the “Student Learning Tracker” allows students to establish goals, reflect on growth at the beginning, middle, and end of the year, track “Module Test” scores, and compare assessment results to personal goals. These resources provide ongoing opportunities for both educators and students to monitor growth, celebrate progress, and make informed decisions about future learning.

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 materials include educator guidance for differentiated instruction, activities, and scaffolded supports for students who have not yet reached proficiency on grade-level content and skills. The materials provide educators with targeted intervention resources, small-group activities, prerequisite support, and scaffolded instructional strategies designed to address learning gaps while maintaining access to grade-level mathematics.

For example, lessons include at least minutes of suggested Differentiation and Practice opportunities. Lessons begin with a “Prerequisite Check” that helps educators identify unfinished learning and use data-driven instructional supports to address gaps before new content is introduced. The materials also provide scaffolded learning experiences through manipulatives, visual models, place-value representations, fraction circles, fraction tiles, number lines, benchmark fractions, vocabulary routines, and “Mathematical Language Routines” that support conceptual understanding and access to grade-level content. Each lesson includes a corresponding educator-led small-group activity designed to support students who have not yet demonstrated proficiency. For example, the “Almost There: Compare and Order Decimals” activity provides explicit educator guidance, questioning strategies, place-value charts, and scaffolded instruction to help students compare and order decimals. Similarly, intervention activities provide step-by-step support through visual models, guided practice, and structured problem-solving processes that build understanding before students independently apply grade-level skills.

Additionally, the materials embed scaffolded supports throughout the materials. For example, algebra comparison problems are broken into clearly sequenced steps that guide students from representing unknown quantities with models to writing and solving equations. Other reteach activities use base 10 manipulatives and visual representations to support conceptual understanding of decimal operations and multiplication concepts. The materials also provide educators with guidance for reteaching through small-group instruction, collaborative learning opportunities, games, and independent practice activities informed by assessment data. These resources demonstrate substantial evidence of differentiated instruction, targeted intervention, prerequisite support, and scaffolded learning experiences for students who have not yet reached proficiency.

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).

The materials include embedded supports for unfamiliar vocabulary, academic language, and mathematical concepts to help students access and communicate grade-level content. Dedicated vocabulary routines, ELPS supports, response scaffolds, and “Mathematical Language Routines” help students develop an understanding of mathematical terminology and use academic language during instruction.

For example, the “Mathematical Language & Emergent Bilinguals” section includes a “Words to Learn” routine that explicitly introduces and reinforces content-specific vocabulary such as common denominator, common factor, common multiple, equivalent fractions, mixed number, overestimate, and underestimate. Additional modules introduce terms such as prime number, composite number, divisibility, and divisible through dedicated vocabulary sections, student-friendly definitions, and vocabulary cards designed to support understanding and retention. The Teacher’s Guide provides embedded recommendations for using the “Words to Learn” routine before instruction begins and includes supports for students who need additional language development. The materials also incorporate “Mathematical Language Routines” such as “Three Reads,” “Discussion Supports,” and “Compare and Connect” to help students comprehend mathematical contexts, discuss strategies, explain solutions, and use academic vocabulary in both oral and written communication.

Additional embedded supports help students make meaning of unfamiliar concepts through visual representations, arrays, factor pairs, questioning prompts, discussion routines, common error guidance, and sample responses. ELPS Teaching Strategies further support vocabulary development by prompting students to use academic language in explanations and written responses. For example, the materials encourage students to use terms such as estimate, decimal point, and difference when explaining their mathematical reasoning. The materials provide substantial support for developing academic and content-specific vocabulary and helping students understand unfamiliar mathematical concepts and references.

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.

The materials include educator guidance for differentiated instruction, enrichment, and extension activities for students who have demonstrated proficiency in grade-level content and skills. Each lesson includes opportunities for advanced learners to extend their understanding through educator-led small-group activities, higher-level problem-solving, and real-world applications of mathematical concepts.

For example, the “Ready for More” small-group activities provide extension opportunities designed for students who are ready for additional challenges. In the “Ready for More: Solve Problems Using Area Models” activity, students solve a multistep real-world problem involving decimal operations, represent their thinking using area models, explain their reasoning, and justify their solutions. The activity further extends learning by having students create similar problems, exchange problems with peers, represent the situations using area models, and solve them, promoting higher-order thinking, mathematical communication, and application of concepts beyond the core lesson expectations. The Teacher’s Guide also includes at least 15 minutes of suggested Differentiation and Practice opportunities within each lesson, while the “Data-Driven Support” section helps educators identify students who are ready for more advanced tasks and deeper application of learning. In addition, every lesson includes a corresponding small-group activity designed to extend learning for students who have demonstrated proficiency with the lesson objectives.

The materials further support enrichment through “Project-Based Learning: Module Projects,” which are designated as DOK Level 4 tasks. These projects engage students in authentic, real-world investigations that require them to synthesize concepts, apply mathematical reasoning, and solve complex problems. For example, projects such as “Taking Out the Trash” require students to analyze data, estimate costs, evaluate environmental impacts, and develop evidence-based proposals, providing meaningful opportunities for advanced application of grade-level mathematics.

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 materials include explicit prompts and guidance to support educators in modeling and explaining mathematical concepts. The Teacher’s Guide provides direct questioning strategies, worked examples, instructional routines, representations, and educator prompts that support clear explanations of grade-level content and mathematical reasoning.

For example, the “Module 4 Teacher Background” section includes mathematical explanations, concept summaries, worked examples, and “Why This Is Important” discussions that support educator understanding of decimal multiplication concepts. Similarly, the “Module 6 Teacher Background” section provides detailed explanations, examples, and instructional guidance related to estimating with fractions, finding common denominators, and adding and subtracting fractions and rational numbers. Lesson overviews identify learning objectives, instructional representations, and recommended manipulatives that support direct modeling and explanation of concepts. The materials provide explicit educator prompts throughout lessons. For example, in module 1, lesson 1 “Understand Thousandths,” the guidance directs educators to ask questions such as “Which place does the 6 occupy in 0.36?” “How much is one thousandth worth as a numeral?” “Do trailing zeros change the value? Why or why not?” and “Explain how place determines each term’s value.” The materials also include possible student responses and explanations that support educators in modeling mathematical reasoning and clarifying misconceptions.

Additional guidance supports educators in using instructional routines such as “Three Reads,” “Turn and Talk,” “Defend,” “Use Representations,” and “Stronger and Clearer Each Time.” DOK questions further support educator modeling by prompting students to explain, justify, and reason about mathematical concepts at increasing levels of complexity. The materials also provide place-value charts, expanded notation models, examples of student thinking, Universal Design for Learning (UDL) supports, ELPS supports, and guidance for addressing common misconceptions. These embedded prompts, models, and instructional supports provide comprehensive guidance for educators as they model and explain mathematical concepts.

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

The materials include educator guidance and recommendations for effective lesson delivery and facilitation using a variety of instructional approaches. Guidance includes instructional routines, questioning strategies, discourse opportunities, language supports, and UDL recommendations that promote student engagement and mathematical understanding through multiple pathways.

For example, the module 4 Teacher’s Guide and the “Mathematical Language & Emergent Bilinguals” section provide planning supports, language-development routines, instructional recommendations, and visual models to support conceptual understanding of decimal multiplication. Similarly, module 6 includes vocabulary routines, response scaffolds, and “Mathematical Language Routines” that support students in developing an understanding of fraction concepts through discussion, modeling, and problem-solving. The Teacher’s Guide directs educators to implement instructional approaches such as the “Three Reads” routine, “Compare and Connect,” “Stronger and Clearer Each Time,” “Turn and Talk,” and “Process Standard” routines to facilitate mathematical discourse and deepen understanding. Lessons incorporate direct instruction, guided questioning, collaborative learning, written explanations, real-world problem-solving, and opportunities for students to justify their reasoning and compare solution strategies. For example, module 2, lesson 7 includes ELPS supports, emergent bilingual strategies, and UDL recommendations that encourage students to evaluate solution processes, create remainder interpretation cards, reflect on their learning, and connect mathematical concepts to authentic contexts.

Additional supports, such as UDL strategies, emergent bilingual scaffolds, prior-knowledge connections, and visual representations, provide educators with recommendations for engaging students through varied instructional methods. The “Implementation Guide” further emphasizes the use of instructional routines and discourse structures throughout lessons to support effective lesson delivery and student engagement. These resources demonstrate that the materials provide comprehensive educator guidance and recommendations for facilitating instruction through a variety of 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 materials support multiple types of practice and include guidance for educators on implementing instruction through a variety of instructional structures. Lessons provide opportunities for guided instruction, collaborative learning, independent practice, and educator-led small-group instruction while offering clear recommendations for whole-group, partner, small-group, and individual learning settings.

For example, the module 4 Teacher’s Guide includes educator-guided modeling, collaborative discussions, independent problem-solving, and “15+ minutes of suggested Differentiation and Practice opportunities” designed to reinforce learning. Similarly, module 6 incorporates guided instruction using fraction circles, fraction tiles, and visual models; collaborative learning through “Mathematical Language Routines” and “Turn and Talk” discussions; and independent practice through problem-solving activities and differentiation resources. Students engage with mathematical concepts through whole-group instruction, partner interactions, small-group support, and individual application, using models, equations, estimation tasks, and real-world problem-solving. The materials also provide structured differentiation through “Almost There,” “On Track,” and “Ready for More” activities, along with educator-led small-group lessons that include materials lists, implementation guidance, and opportunities for guided, collaborative, and independent practice. “Practice Plus” resources extend learning through independent practice and homework opportunities, while Module Projects engage students in collaborative, real-world applications of mathematics that may be completed in whole-group or small-group settings.

Additionally, the Implementation Guide and “Lesson Pacing Guidance” provide recommended instructional frameworks and time allocations for whole-class instruction, collaborative groups, independent practice, and educator-led small groups. These resources ensure students engage in multiple types of practice while providing educators with clear guidance for implementing a variety of instructional structures.

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.

The materials include educator guidance for providing linguistic accommodations that support students at varying levels of English language proficiency and promote the use of increasingly academic language. The “Mathematical Language & Emergent Bilinguals” and “Supporting All Language Learners” sections provide differentiated scaffolds aligned to the ELPS and designed to support students as they progress toward more sophisticated mathematical communication.

The materials provide explicit accommodations for multiple proficiency levels, including pre-production/beginning, intermediate, and high intermediate/advanced. For example, lessons encourage beginning students to use their home language, visuals, simple sentences, and mathematical representations to communicate understanding. Lessons guide intermediate students in discussing mathematical reasoning in English and in composing written responses using complete sentences and content-specific vocabulary. Lessons expect high intermediate/advanced students to explain ideas independently using academic language and mathematical terminology supported by targeted word banks. The materials also include sentence stems, response frames, vocabulary routines, and “Mathematical Language Routines” such as “Three Reads,” “Compare and Connect,” “Discussion Supports,” and “Stronger and Clearer Each Time” to support academic language development.

Additionally, emergent bilingual teaching strategies provide guidance for educators on using linguistic accommodations such as home-language supports, structured partner discussions, sentence frames, and vocabulary scaffolds. For example, students may discuss solutions in their home language before transitioning to English responses, and the materials encourage advanced students to justify solutions using academic vocabulary such as divisor, quotient, estimate, and pounds. These supports provide a clear progression from scaffolded communication to increasingly independent oral and written mathematical explanations, supporting students at varying levels of English language proficiency.

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

The materials include implementation guidance to support educators in effectively using the program with emergent bilingual students in bilingual and ESL settings. The “Language Growth for All Learners” and “Mathematical Language & Emergent Bilinguals” sections provide observable, measurable language objectives aligned with the ELPS, along with vocabulary routines, language-development supports, response scaffolds, and opportunities for mathematical discourse embedded throughout instruction.

The materials provide guidance through resources such as “Language Growth for All Learners” and the Language Development Resource Guide, which explain how educators can access language supports throughout lessons and instructional routines. Guidance provides educators with observable and measurable language objectives, ELPS connections, visual vocabulary supports, formative language assessment tools, and scaffolds that support students at varying levels of language proficiency. The materials include supports such as the “Words to Learn” routine, vocabulary cards, sentence stems, word banks, “Mathematical Language Routines,” and discussion protocols designed to promote mathematical communication and academic discourse.

Additionally, the Language Development Resource Guide highlights language supports embedded throughout “Essential Sessions,” “Spark Your Learning” activities, instructional tasks, formative assessments, and ELPS-specific teaching strategies. Resources such as the “4Ls (Look, Lean, Lower, and Listen)” routine provide explicit guidance for supporting student participation and collaborative conversations. The materials also identify ELPS supports throughout lessons, enabling educators to plan and implement targeted language development opportunities while maintaining access to grade-level mathematics content. These resources provide comprehensive guidance for integrating language supports within bilingual and ESL instructional settings.

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.

The materials include embedded guidance to support emergent bilingual students in developing academic vocabulary, increasing comprehension, building background knowledge, making cross-linguistic connections, and engaging in oral and written discourse. To support language development and comprehension of mathematical concepts, the “Mathematical Language & Emergent Bilinguals” sections provide vocabulary-development routines; response scaffolds; sentence stems; word banks; and “Mathematical Language Routines” such as “Three Reads,” “Compare and Connect,” “Discussion Supports,” and “Stronger and Clearer Each Time.”

The materials support academic vocabulary development through routines such as “Words to Learn” and structured opportunities for students to use content-specific language when explaining mathematical reasoning. Discussion routines, visual representations, gestures, and connections to prior mathematical learning support comprehension and background knowledge. Students engage in oral discourse through partner discussions, structured academic conversations, and opportunities to justify solutions using precise mathematical language. Written discourse is supported through sentence stems, written explanations, equations, and reasoning tasks that require students to communicate mathematical understanding using academic vocabulary.

The materials also provide support for cross-linguistic connections. For example, emergent bilingual guidance encourages students to speak and write in their home language while using mathematical models to support explanations before communicating their thinking in English. These supports help students leverage linguistic resources from their primary language while developing academic language proficiency in mathematics.

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.

The materials provide practice opportunities and instructional assessments that require students to demonstrate depth of understanding aligned to the TEKS. Lessons incorporate conceptual investigations, mathematical discussions, estimation activities, formative assessments, and performance tasks that require students to represent, explain, justify, analyze, and apply mathematical concepts using multiple strategies and representations.

The materials provide opportunities for students to demonstrate conceptual understanding through “Quick Checks,” Module Reviews, Module Tests, challenge activities, and “Project-Based Learning” experiences. In module 15, lesson 3, the “Quick Check” requires students to determine the volume of rectangular prisms using unit cubes and the formula V = Bh. Students calculate base area, determine volume, explain why cubic units are appropriate, and connect formulas to concrete representations of layered unit cubes. The results inform instructional decisions and targeted support based on individual student needs. Additional assessments, such as the module 15 test, measure student understanding of volume concepts, while challenge activities such as “Use Unit Cubes to Build Three-Dimensional Figures” require students to analyze figures, compare volumes, solve open-ended problems, and justify their reasoning.

The materials also provide cumulative assessment opportunities through Module Tests and “Project-Based Learning” experiences. For example, the module 4 test requires students to multiply decimals using area models, decimal grids, and open-ended word problems that emphasize conceptual understanding and mathematical reasoning. These assessment opportunities require students to demonstrate conceptual understanding, mathematical reasoning, and application of the grade-level TEKS rather than procedural fluency alone.

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

The materials include questions and tasks that progressively increase in rigor and complexity to support grade-level proficiency in the mathematics TEKS. The materials build conceptual understanding through a progression of learning experiences that move from concrete models and foundational skills to strategic thinking, justification, problem-solving, and real-world application.

The materials sequence questions and tasks to develop increasingly sophisticated mathematical thinking throughout lessons and modules. In module 15, lesson 3, students begin by using unit cubes and layers to develop an understanding of volume before progressing to determining the volume of rectangular prisms using the formula V = Bh. The “Quick Check” requires students to determine base area, calculate volume, explain why cubic units are appropriate, and justify their reasoning through open-ended responses. Students also self-assess their understanding before completing the “Practice on Your Own” activity, which provides evidence of mastery of the lesson TEKS. Similarly, in module 4, students progress from using pictorial models and place-value reasoning to multiply decimals to explaining solution strategies, justifying the placement of the decimal point using estimation and place-value evidence, and responding to increasingly rigorous DOK questions.

The materials continue this progression across modules by building from foundational concepts to increasingly complex applications. For example, in module 5, lesson 2, students progress from modeling decimal division with base 10 blocks to representing division with pictorial models and area models while explaining quotients using place-value reasoning. Module reviews reinforce vocabulary, concepts, and skills before module assessments, while “Project-Based Learning” Module Projects culminate learning through DOK Level 4 tasks that require analysis, reasoning, justification, and authentic problem-solving.

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

The materials demonstrate coherence across units by explicitly connecting patterns, big ideas, and relationships between mathematical concepts. The materials connect previously learned concepts to new learning so that mathematical understanding develops across lessons and modules.

The materials connect mathematical concepts through instructional routines, discussion prompts, and opportunities for students to apply prior learning in new contexts. For example, in module 4, lesson 2, students receive a “Turn and Talk” prompt, “Discuss with another student how area model decimal multiplication is related to partial products multiplication strategies,” to connect prior multiplication strategies to decimal multiplication. Additional prompts ask students to explain how adding decimals using base 10 blocks is like adding whole numbers using base 10 blocks and to describe the difference between an estimate and an exact answer. These discussions reinforce relationships among place value, multiplication, estimation, decimal operations, the distributive property, and visual models while extending previously learned strategies to new mathematical situations.

The educator guide supports coherence across modules by explaining how concepts build toward future learning. Decimal instruction connects whole-number multiplication, place value, estimation strategies, mathematical models, and real-world applications to prepare students for rational numbers and proportional reasoning. Fraction instruction builds from equivalent fractions and number relationships to fraction operations, estimation, and algebraic thinking. Planning guidance also connects representing fractions as products of whole numbers and unit fractions to later work with dividing whole numbers by unit fractions and unit fractions by whole numbers. These connections reinforce mathematical relationships across units and help students apply previously learned concepts to increasingly complex mathematical ideas.

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.

The materials demonstrate coherence across grade levels by connecting previously learned content and mathematical language to current learning while preparing students for future mathematical concepts. Educator resources identify prior learning, current grade-level expectations, and future learning so that concepts, representations, and vocabulary build progressively across grade levels.

The educator guide and “Before You Teach” sections connect prior learning to current instruction and explain how current concepts prepare students for future mathematics. Previously learned concepts such as whole-number multiplication, whole-number division, place-value relationships, equivalent fractions, representing decimals, and identifying tenths and hundredths are connected to current learning involving decimal operations, fraction operations, and decimal place value through the thousandths. Previously introduced mathematical language, including place value, equivalent fractions, compare, order, tenths, hundredths, quotient, and division, is reinforced while new grade-level vocabulary, representations, and reasoning strategies are developed.

The materials also prepare students for future learning by explicitly identifying concepts and mathematical language that will be developed in later grades. Decimal operations and place-value understanding are extended to prepare students for adding and subtracting decimals through the thousandths, while decimal multiplication and division prepare students for future work with rational numbers, proportional reasoning, algebraic concepts, ratios, rates, percents, scale, and unit conversions. The “Teacher Background” and “What Will Students Learn in This Module?” sections explain how current learning supports future mathematical reasoning and problem-solving through increasingly sophisticated applications of decimals, fractions, and division.

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..

The materials demonstrate coherence at the lesson level by connecting students’ prior knowledge of concepts and procedures to new mathematical learning. Lessons activate previously learned skills through “Prerequisite Checks,” “Spark Your Learning” activities, educator questioning, and “Data-Driven Support” resources before introducing new mathematical concepts and procedures.

The materials connect prior knowledge to new learning through instructional routines, scaffolding questions, and multiple representations. “Prerequisite Checks” and “Data-Driven Support” resources help educators identify unfinished learning and provide targeted intervention before new instruction begins. Lessons build on prior understanding of whole-number multiplication, place value, decimal concepts, equivalent fractions, number relationships, and fraction representations before extending learning to decimal operations, fraction operations, estimation, and problem-solving. In module 13, lesson 2, students build on prior knowledge of multiplication, equal groups, arrays, area, and layered models to develop volume formulas. Students first determine the number of cartons in one crate using multiplication before extending that understanding to calculate the volume of rectangular prisms using layers of equal groups.

The materials continue this progression by extending familiar concepts and procedures to new mathematical situations. “Spark Your Learning” activities include scaffolding questions that prompt students to apply familiar strategies when determining point locations on a coordinate plane. “Turn and Talk” discussions encourage students to explain how models support their understanding of volume, while educator guidance recommends using manipulatives, visual models, choral responses, and home-language supports to strengthen connections between prior knowledge and new concepts. These instructional supports help educators connect prior knowledge to new learning while developing conceptual understanding throughout each lesson.

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

The materials provide spaced retrieval opportunities for previously learned skills and concepts across lessons and units. Lesson routines, cumulative practice, formative assessments, module assessments, and performance tasks revisit previously taught content, reinforcing learning over time.

The materials incorporate “Prerequisite Checks,” “Practice Plus,” “Train Your Memory,” Module Reviews, Module Tests, and “Project-Based Learning” experiences to reinforce previously learned concepts while introducing new content. “Practice Plus” and the “Train Your Memory” section provide ongoing opportunities to revisit decimal multiplication, division, multistep problem-solving, whole-number operations, fluency, estimation, and problem-solving strategies from earlier lessons, previous modules, and, at times, prior grade levels. Students review concepts through activities such as multiplying decimals with visual models, solving multistep word problems, applying multiplication fluency, and connecting whole-number operations to decimal reasoning. Foundational concepts are also revisited in cumulative assessments. For example, the module 8 test requires students to apply previously learned vocabulary, multiplication strategies, area models, estimation, and problem-solving while demonstrating current grade-level understanding.

Module Projects incorporate additional spaced retrieval opportunities by requiring previously learned skills and concepts to be applied alongside current learning in authentic, DOK Level 4 tasks. These cumulative opportunities reinforce mathematical understanding across lessons and units while giving educators ongoing evidence of student learning to inform instructional decisions.

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

The materials provide interleaved practice opportunities with previously learned skills and concepts across lessons and units. Previously taught content is integrated with new learning so that students apply multiple concepts and strategies within the same lesson, practice activity, and assessment.

The materials incorporate interleaved practice through lesson activities, “Practice Plus,” and the “Train Your Memory” section. Throughout lessons, previously learned concepts such as place value, whole-number multiplication, decimal operations, equivalent fractions, benchmark fractions, mixed numbers, and fraction representations are integrated with new grade-level content. In module 4, lesson 2, students apply place-value reasoning, whole-number multiplication strategies, and pictorial models while representing multiplication with decimals and whole numbers. These concepts are revisited throughout the module as students use estimation, area models, and place-value strategies to solve increasingly complex decimal multiplication problems. Similarly, in module 6, lesson 5, students apply prior understanding of equivalent fractions to rewrite fractions with common denominators before extending that understanding to fraction addition and subtraction with unlike denominators. Benchmark fractions, equivalent fractions, and fraction representations continue to be reinforced throughout subsequent lessons.

“Practice Plus” and the “Train Your Memory” section incorporate additional interleaved practice by combining concepts from multiple lessons and units within the same set of practice problems. Students engage with data analysis, coordinate graphing, area of composite figures, multiplication fluency, coordinate-plane graphing, and fraction operations within a single activity, requiring different concepts, representations, and procedures to be selected and applied across tasks. For example, students identify points on a coordinate plane while also subtracting fractions with unlike denominators and writing answers in simplest form. These opportunities reinforce previously learned skills and concepts across lessons and units while strengthening flexible mathematical thinking.

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

The materials provide questions and tasks that require students to interpret, analyze, and evaluate models and representations for mathematical concepts and situations. Students engage with visual models, graphs, tables, number lines, and equations to explain mathematical relationships, justify solutions, and connect multiple representations throughout instruction.

The materials engage students in interpreting, analyzing, and evaluating mathematical models across a variety of lessons. In module 6, lesson 1, students interpret pictorial fraction models, analyze how equivalent fractions create common denominators, and evaluate whether their models accurately represent fraction equations and solutions while explaining the relationship between visual models and symbolic notation. Similarly, in module 4, lesson 1, students interpret decimal representations of money, analyze place-value relationships, and evaluate whether their written representations accurately model the value while justifying their reasoning.

Additional lessons require students to interpret, analyze, and evaluate a variety of mathematical representations. In module 10, lesson 4, students analyze a multistep problem, create a graph and table to identify a pattern, and evaluate how using a graph or rule helps verify their solution. In module 15, task 2, students represent the base of a rectangular prism with a pictorial model, analyze the relationship between base area and height to develop the volume formula, and evaluate how the model supports their understanding of volume. Students also interpret number lines in module 1 by plotting decimals to the thousandths and explaining the value of each digit, strengthening their ability to connect visual models with mathematical concepts.

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

The materials provide questions and tasks that require students to create models to represent mathematical situations. Students regularly use concrete manipulatives, pictorial representations, and mathematical equations to model real-world situations and demonstrate conceptual understanding before applying symbolic procedures.

The materials intentionally require students to create mathematical models across multiple modules. In module 4, lesson 7, students create area models to represent decimal multiplication, connect the models to place-value relationships, and record equations that match their representations. Similarly, in module 6, lesson 5, students use fraction bars and fraction tiles to create pictorial models of fraction addition with unlike denominators before connecting the models to symbolic equations and explaining why equivalent fractions are needed.

Additional tasks provide students with opportunities to create models across a variety of mathematical contexts. Students use base 10 blocks and pictorial models to represent decimal subtraction before recording matching equations, and they construct area models to solve division situations, such as sharing 3,420 bronze coins equally among 20 museums or dividing 114 bottles of paint among 19 students. Other lessons prompt students to use base 10 blocks to model decimal quantities, create pictorial representations, and explain how their models accurately represent the mathematical situation, reinforcing the use of models throughout instruction.

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

The materials provide questions and tasks that give students opportunities to apply conceptual understanding to new problem situations and contexts. Students build conceptual understanding through models and guided instruction before applying their learning to unfamiliar problems, real-world situations, and increasingly complex mathematical tasks.

The materials intentionally develop conceptual understanding before asking students to apply it in new contexts. In module 4, students progress from using objects, area models, place-value reasoning, and estimation to solve decimal multiplication problems involving money, measurement, and multistep situations while explaining how place-value relationships support their strategies. Similarly, in module 6, students build conceptual understanding of fraction addition and subtraction with equivalent fractions, fraction bars, fraction tiles, and pictorial models before applying that understanding to estimate, solve, and justify solutions to contextual problems involving unlike denominators.

Additional lessons require students to transfer conceptual understanding to new mathematical situations. In module 13, lesson 2, students develop the volume formula for a right rectangular prism by creating a pictorial model of the base, determining its area, and explaining how the base dimensions relate to the prism’s volume. Other tasks ask students to compare triangles to identify shared attributes using geometric reasoning, represent area by drawing a grid before applying the formula (A = L x W), and extend prior learning by applying multiplication concepts to mixed fractions in the “Ready for More” activity.

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

The materials provide tasks that are designed to build student automaticity and fluency necessary to complete grade-level tasks. Students engage in repeated practice with conceptual models, efficient strategies, and increasingly complex problems that strengthen computational accuracy while supporting flexible mathematical thinking.

The materials intentionally develop automaticity and fluency across multiple modules. In module 4, lessons 5–8, students repeatedly solve decimal multiplication problems using area models, place-value reasoning, estimation, and equations before transitioning to more efficient computational methods. Similarly, in module 6, lessons 5 and 6, students repeatedly generate equivalent fractions, model fraction addition and subtraction with unlike denominators, estimate for reasonableness, and justify solutions using fraction bars, fraction tiles, pictorial models, and equations. These repeated experiences strengthen automaticity with foundational concepts while building fluency with grade-level fraction operations.

Additional practice opportunities reinforce automaticity and fluency throughout the program. “Practice on Your Own” activities develop multi-digit division by having students estimate quotient digits, adjust their estimates, justify their reasoning, and solve real-world problems. “Fluency Practice” provides repeated long division using standard algorithms and equations, while “Practice Plus” includes repeated decimal addition and subtraction with regrouping, estimation, and reasonableness checks. Other lessons ask students to evaluate estimated quotient digits before completing long division, helping students build computational efficiency while applying flexible strategies to grade-level tasks.

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.

The materials provide opportunities for students to practice applying efficient, flexible, and accurate mathematical procedures within lessons and throughout the module. Students regularly solve problems using multiple strategies, compare solution methods, justify their reasoning, and select the approach that best fits the mathematical context.

The materials intentionally develop efficient, flexible, and accurate procedures across multiple modules. In module 4, lessons 2–8, students solve decimal multiplication problems using objects, area models, place-value reasoning, estimation, and the standard algorithm while comparing strategies and evaluating the efficiency of different approaches. Similarly, in module 6, lessons 5–9, students solve fraction addition and subtraction problems using visual models, equivalent fractions, estimation, and symbolic procedures while explaining the strategies they selected and comparing them with alternative methods.

Additional opportunities reinforce these expectations across a variety of mathematical concepts. Students solve decimal division problems using the standard algorithm, partial quotients, place value, estimation, area models, and number sense while selecting appropriate strategies and comparing solution methods. Other lessons ask students to multiply whole numbers by unit fractions using models and real-world contexts, simplify answers, justify why multiplying by a unit fraction results in a smaller product, and reinforce prior learning through “Train Your Memory.” Small-group activities also prompt students to model situations with strip diagrams, giving students repeated opportunities to apply efficient, flexible, and accurate mathematical procedures in meaningful contexts.

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.

The materials provide opportunities for students to evaluate procedures, processes, and solutions for efficiency, flexibility, and accuracy within the lesson and throughout the module. Students regularly compare strategies, analyze the reasonableness of solutions, justify their thinking, and reflect on the effectiveness of different mathematical approaches.

The materials intentionally embed opportunities for students to evaluate mathematical procedures across multiple modules. In module 4, lessons 6–8, students compare area models, place-value reasoning, estimation, and the standard algorithm to determine which strategies produce accurate decimal products and which are most efficient for a given situation. Similarly, in module 6, lessons 6–9, students compare strategies for finding common denominators, critique and correct solution methods, identify errors, and explain why different approaches produce equivalent results while reflecting on the effectiveness of their chosen procedures.

Additional lessons extend these evaluation opportunities across other mathematical concepts. Students compare multiple strategies for dividing decimals by evaluating estimation, base 10 blocks, partial quotients, and the standard algorithm before explaining which approach is most efficient and accurate. Other tasks ask students to interpret remainders as fractions in real-world division situations, compare their processes with alternative methods, determine decimal placement in division using the standard algorithm, and justify why their solutions are reasonable. Throughout the materials, students explain their reasoning, evaluate their mathematical processes, and refine their strategies using evidence from their work.

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

The materials contain embedded supports for educators to guide students toward increasingly efficient approaches. Educator guidance includes instructional routines, discussion protocols, questioning strategies, and scaffolds that help students compare solution methods, refine their thinking, and transition from conceptual models to more efficient mathematical procedures while maintaining conceptual understanding.

The materials intentionally guide students toward increasingly efficient strategies across multiple modules. In module 4, lessons 2 through 8, educators transition students from decimal area models and place-value reasoning to more efficient multiplication procedures through routines such as “Compare and Connect,” “Stronger and Clearer Each Time,” and embedded discussion supports. Similarly, in module 6, lessons 5–9, instruction progresses from fraction bars, fraction tiles, and pictorial models to symbolic procedures for finding common denominators and solving fraction operations. Throughout both modules, educator prompts encourage students to compare strategies, justify their choices, and determine which approaches are most efficient.

Additional embedded supports appear throughout the Teacher’s Guide. Lessons include routines such as “Three Reads,” “Compare and Connect,” and “Collect and Display,” along with UDL supports, common error interventions, and scaffolds for multilingual learners that help students refine their mathematical thinking. In module 8, lesson 1, educators guide students from pictorial models to division equations while addressing misconceptions and comparing solution strategies. Other lessons prompt students to compare model-first and equation-first approaches, discuss how mental math improves efficiency, and explain why one strategy is more effective than another, helping students develop increasingly efficient problem-solving approaches across the curriculum.

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

The materials explicitly state how the conceptual and procedural emphasis of the TEKS are addressed. The Teacher’s Guide explains how students first develop conceptual understanding through models, visual representations, and mathematical reasoning before applying increasingly efficient procedures and algorithms. Module overviews and educator background sections consistently describe how conceptual understanding serves as the foundation for procedural fluency.

The materials make these connections explicit across multiple modules. In module 4, the planning guide, module overview, and educator background explain that students develop conceptual understanding of decimal multiplication through place-value reasoning, objects, and area models before progressing to efficient computational procedures. Likewise, module 6 explains that students build conceptual understanding of fraction operations using fraction bars, fraction tiles, equivalent fractions, and visual models before transitioning to symbolic procedures for adding and subtracting fractions with unlike denominators. Throughout these lessons, students connect conceptual models to efficient procedures while explaining how the representations support their computations.

The Teacher’s Guide further reinforces this progression through module overviews and introductory guidance. In module 1, the “Teacher Background” explains how students develop an understanding of place-value relationships before applying that knowledge to writing, rounding, comparing, and ordering decimals. Similarly, module 13 explains that students first understand the relationship between the area of a rectangular base and the volume of a rectangular prism before applying formulas to determine perimeter, area, and volume. The “Grade Five Focus Areas” and the section titled “How Does the Module Support the Conceptual and Procedural Emphasis of the TEKS?” also explain that readiness standards are intentionally sequenced so conceptual understanding develops before procedural fluency, ensuring students can justify strategies and apply them accurately and flexibly.

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 materials include questions and tasks that require students to use concrete models and manipulatives, pictorial representations, and abstract representations, as required by the TEKS. Throughout the lessons, students progress from using hands-on materials to visual models and symbolic equations, helping them connect multiple representations while developing conceptual understanding and procedural fluency.

The materials consistently incorporate this progression across multiple modules. In module 4, lessons 2 through 8, students represent decimal multiplication using manipulatives, create area models, and connect those models to multiplication equations and algorithms while explaining how each representation supports the solution. Similarly, in module 6, lessons 5 through 9, students use fraction bars and fraction tiles, create fraction drawings and number lines, and write fraction addition and subtraction equations with unlike denominators while explaining how the concrete and visual models support the abstract procedures.

The materials also provide lesson-specific opportunities for students to move among concrete, pictorial, and abstract representations. In module 8, lesson 1, task 2, students “make a pictorial model to represent the situation” and explain how the model represents the dividend, divisor, and quotient of the corresponding division equation. Another lesson has students use objects to represent the addition of fractions with unlike denominators before writing multiple equations to model the solution. Likewise, the learning goal in lesson 4.2 explicitly states that students will represent and solve decimal multiplication problems using objects, pictorial models, place value, and whole-number multiplication, reinforcing the progression from concrete experiences to abstract reasoning.

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 materials include supports for students in connecting, creating, defining, and explaining concrete and representational models to abstract concepts, as required by the TEKS. Throughout the lessons, students use manipulatives and visual representations to build conceptual understanding before connecting those models to symbolic equations and algorithms. Educator prompts consistently encourage students to explain how each representation models the mathematical concept and supports the abstract procedure.

The materials provide these supports across multiple modules. In module 4, lessons 2 through 8, students represent decimal multiplication with place-value manipulatives, create area models, and connect those representations to multiplication equations while explaining how place-value relationships determine the placement of the decimal point. Similarly, in module 6, lessons 5 through 9, students use fraction bars, fraction tiles, drawings, and number lines to model fraction addition and subtraction, connect those models to symbolic equations, and explain how equivalent fractions support each step of the computation.

The materials also include lesson-level opportunities for students to connect models to abstract reasoning. In module 7, lesson 1, task 1, students use counters to model the number of cats with striped tails, draw a matching representation, and “describe how you can solve for the number of cats that have striped tails. Use evidence from your model and write for your teacher.” Likewise, in module 6, lesson 1, task 1, students “show how you can use objects to represent the situation” before connecting their model to an equation. Another lesson has students use base 10 blocks to model the weight of six apples, draw a quick picture of the model, and explain how regrouping in the model connects to the symbolic decimal multiplication process.

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

The materials provide opportunities for students to develop academic mathematical language through the intentional use of visuals, manipulatives, and embedded language supports aligned to mathematical concepts. Throughout the materials, students engage with concrete models, visual representations, and structured discourse routines that encourage them to communicate mathematical ideas using precise vocabulary and academic language.

The materials also develop academic mathematical language through visual representations and structured mathematical discussions. In module 14, lesson 1, students use coordinate planes, polygon models, comparison tables, and vocabulary boxes to connect terms such as polygon, heptagon, and nonagon to their visual representations. Students graph points, construct polygons, analyze attributes, compare figures, and participate in “Turn and Talk” discussions to explain their thinking and apply mathematical vocabulary in meaningful contexts.

The materials provide additional language-development support through embedded vocabulary routines and instructional resources. Vocabulary cards include definitions, visuals, and prompts such as “Define It,” “Discuss It,” and “Use It” to help students learn and apply mathematical terminology. In the Teacher’s Guide, the section titled “How Can I Improve All Students’ Academic Language Development?” incorporates routines such as “Words to Learn,” “Three Reads,” “Compare and Connect,” “Discussion Supports,” and “Stronger and Clearer Each Time” to strengthen students’ comprehension, fluency, and use of academic mathematical language throughout instruction.

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

The materials include embedded educator guidance that supports students’ development and use of academic mathematical vocabulary within the instructional context. Throughout the materials, educators are provided with routines, discussion prompts, and visual supports to help students develop and apply precise mathematical language as they engage in meaningful mathematical tasks and discussions.

The materials embed vocabulary development within mathematical content by providing educator guidance that connects academic language to models, representations, and mathematical reasoning. In module 4, “Multiply Decimals,” educators model decimal multiplication using objects, area models, and place-value strategies while students explain multiplication strategies using terms such as decimal, product, place value, area model, and estimate. Similarly, in module 6, “Represent Fractions Using the Same Whole,” educators use fraction circles, fraction tiles, fraction bars, and number lines to support students as they apply vocabulary such as equivalent fractions, benchmark fractions, numerator, denominator, and mixed numbers while representing fractions and justifying their reasoning. Additional lessons reinforce terms such as polygon, heptagon, nonagon, expanded notation, partial products, and distributive property through “Turn and Talk” routines, partner discussions, and explanations that require students to communicate their mathematical thinking using precise vocabulary.

The materials also provide embedded language supports and instructional routines that help students acquire and apply academic mathematical vocabulary. The Teacher’s Guide incorporates strategies such as “Turn and Talk,” ELPS supports, and emergent bilingual scaffolds that encourage students to elaborate on their ideas, create visuals to represent mathematical terms, and use sentence frames during discussions. The Language Development Resource Guide includes resources such as the “High-Utility Word” routine and “Quick-Teach” routine to strengthen vocabulary comprehension and application across content areas. Additionally, lessons incorporate the “Words to Learn” routine to introduce and reinforce domain-specific terms, such as income, tax, income tax, and payroll tax, ensuring that students develop the vocabulary needed to explain their mathematical thinking accurately in context.

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.

The materials include embedded educator guidance to support the application of appropriate mathematical language through vocabulary, syntax, and mathematical discourse. Throughout the materials, educators use sentence frames, discussion routines, exemplar responses, and questioning strategies to encourage students to explain and justify their reasoning using precise mathematical language during collaborative conversations and written responses.

The materials embed mathematical language development into content instruction by providing structured opportunities for students to explain, justify, and refine their thinking. In module 4, “Represent Multiplication with Decimals and Whole Numbers,” educators use area models, place-value representations, discussion prompts, and “Mathematical Language Routines” to support conversations about decimal multiplication strategies. Similarly, in module 6, “Represent Addition with Unequal Denominators,” educators use fraction models and discussion routines to help students explain equivalent fractions, compare strategies, and justify solutions using increasingly precise mathematical language. Across these lessons, students analyze visual models, respond to classmates’ ideas, participate in peer discussions, and strengthen their mathematical vocabulary, syntax, and discourse through structured conversations and written justifications.

The materials also provide embedded supports that promote productive mathematical conversations and model precise mathematical communication. Educators use routines such as “Three Reads,” “Turn and Talk,” and “Stronger and Clearer Each Time” to guide students in explaining their reasoning, interpreting remainders, comparing fractions, and refining mathematical explanations through peer discussions. Additional supports include “Words to Learn” routines, sentence frames, scripted questions, and exemplar responses that model the use of terms such as dividend, divisor, equivalent fractions, common multiple, quotient, and pictorial model. Tasks frequently require students to share their thinking with a partner, evaluate their solution processes, and use content-specific vocabulary during discussions and written responses, while educator guidance provides possible student responses and scaffolds to extend and strengthen mathematical discourse over time.

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

The materials appropriately integrate the TEKS process standards throughout instruction by embedding problem-solving, mathematical communication, multiple representations, and justification within lesson activities. Students routinely analyze problems, select appropriate tools, represent mathematical ideas in multiple ways, evaluate the reasonableness of solutions, and explain their thinking during partner and whole-group discussions. These opportunities are embedded throughout instruction, practice activities, and assessments.

The materials integrate the TEKS process standards within grade-level content by engaging students in mathematical reasoning, representation, and communication while they learn new concepts. In module 4, “Represent Multiplication with Decimals and Whole Numbers,” students analyze mathematical situations, select tools such as area models and place-value charts, create visual representations, estimate to evaluate reasonableness, and justify solution strategies. Similarly, in module 6, “Represent Addition with Unequal Denominators,” students use fraction circles, fraction bars, and number lines to solve real-world problems, analyze fraction relationships, communicate mathematical reasoning, and justify solutions through oral and written responses. These tasks consistently require students to apply problem-solving, representation, communication, and justification as they develop conceptual understanding.

The materials also embed the TEKS process standards through authentic tasks and assessments that require students to apply mathematical practices in meaningful contexts. Students create and analyze bar graphs, write and solve equations with unknown quantities, represent problems in multiple ways, and reflect on how changes to a problem affect the solution. In module 12, lesson 1, students write equations to represent situations, solve for unknown quantities, and show their steps, while module assessments continue to reinforce these practices. The Teacher’s Guide further explains how students learn individual steps of the problem-solving model and then apply all steps together in later lessons, demonstrating that the TEKS process standards are intentionally integrated throughout instruction, practice opportunities, and assessments.

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

The materials explain that the TEKS process standards are incorporated and connected throughout the course. They are embedded within the instructional design, lesson sequence, and assessment structure. The materials identify the applicable process standards and explain that students repeatedly engage in problem-solving, reasoning, communication, justification, and multiple representations as they progress through lessons, module assessments, and project-based learning experiences. Educators are guided to reinforce these standards through consistent instructional routines, mathematical discussions, and problem-solving experiences throughout the year.

The materials describe how the TEKS process standards are incorporated throughout instruction, assessments, and mathematical discourse within individual modules. For example, the module 4 and module 6 Planning Guides, TEKS Correlations, and Assessment Guides explain how students analyze problems, select mathematical tools, create representations, communicate reasoning, and justify solutions throughout lesson activities, formative assessments, Module Reviews, Module Tests, and “Project-Based Learning Module Projects.” These resources demonstrate that the process standards are intentionally embedded within instruction and repeatedly applied across multiple learning experiences.

The materials also provide a course-level description of how the TEKS process standards are connected throughout the curriculum. The “Mathematical Connections” section explains how concepts build across modules and integrate the process standards to strengthen reasoning, problem-solving, communication, and the use of representations. The “Correlations by Standard” chart and Teacher’s Guide identify the lessons in which each process standard (5.1A–5.1G) is addressed, illustrating how mathematical practices are embedded across multiple units and content strands. Additionally, “Process Standards” anchor charts in the Student Edition provide reflection questions and sentence stems that can be used throughout the year to help students articulate their thinking and make meaningful mathematical connections. Together, these resources provide educators with a coherent framework for consistently reinforcing the TEKS process standards throughout the course.

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

The materials include a unit-level description of how the TEKS process standards are incorporated and connected throughout each module. In the Teacher’s Guide, the section titled “How Are the Process Standards (PS) Connected to the Content in This Module?” explicitly identifies the lessons in which students apply specific process standards and explains how those standards support the mathematical content. For example, students apply mathematics to real-world problems (TEKS process standard 1.A), use a problem-solving model (TEKS process standard 1.B), select appropriate tools and techniques (TEKS process standard 1.C), and use representations (TEKS process standard 1.E) as they develop an understanding of addition, subtraction, multiplication, and fractions. This guidance helps educators understand how process standards are intentionally embedded and reinforced throughout the module.

The module overviews, Planning Guides, TEKS Correlations, and Assessment Guides further describe how the TEKS process standards are incorporated and connected throughout each unit. In module 4, students apply mathematical reasoning, problem-solving, communication, representations, and justification while developing decimal multiplication concepts. Similarly, in module 6, students engage in reasoning, communication, representation, and problem-solving as they estimate, represent, compare, add, and subtract fractions using multiple representations. Educators are guided to reinforce these practices through lesson discussions, visual models, collaborative tasks, formative assessments, and cumulative assessments, while students apply the same mathematical processes throughout the unit.

The materials also provide additional support that illustrates how the process standards are embedded and revisited across modules. Module overviews identify the lessons in which students apply specific process standards. Module overviews also explain how students first develop problem-solving skills and then apply them in later lessons involving numerical expressions, multiplication, division, area models, estimation, compatible numbers, and remainders. The “Process Standards” anchor charts further support implementation by providing reflection questions and prompts that encourage students to reflect on their choices of tools, representations, and strategies. This organized progression demonstrates that the TEKS process standards are intentionally integrated, connected to mathematical content, and reinforced throughout each module.

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

The materials include an overview of the TEKS process standards incorporated into each lesson by identifying the applicable process standards and connecting them to lesson objectives and instructional activities. Each lesson highlights the mathematical processes students will use, such as problem-solving, selecting appropriate tools, creating representations, communicating mathematical reasoning, and justifying solutions. For example, lesson overviews identify opportunities for students to use a problem-solving model, communicate mathematical ideas using multiple representations, and explain solutions using precise mathematical language as they engage in grade-level mathematical tasks.

Lesson overviews such as those in module 4 and module 6 explicitly identify the TEKS process standards alongside the content standards, learning objectives, and language objectives. Students apply these process standards as they use area models, place-value charts, fraction circles, fraction bars, and equations to solve problems, justify strategies, communicate mathematical reasoning, and evaluate the reasonableness of solutions during collaborative and independent activities. The “What Will Students Learn in This Lesson?” section further highlights specific process standards such as applying mathematics to real-world situations (TEKS process standard 5.1A), communicating mathematical ideas using multiple representations (TEKS process standard 5.1D), creating and using representations (TEKS process standard 5.1E), and analyzing mathematical relationships (TEKS process standard 5.1F) within grade-level mathematical tasks.

The Implementation Guide and “Correlations by Lesson” charts provide a comprehensive lesson-by-lesson overview of the TEKS process standards incorporated throughout the curriculum. Each lesson is correlated with the applicable TEKS and corresponding process standards, enabling educators to quickly identify where students will engage in problem-solving, mathematical reasoning, communication, representation, tool selection, and justification. The materials also note when specific steps of the problem-solving model are used within lessons, providing educators with clear guidance for integrating and reinforcing the TEKS process standards consistently across daily instruction and throughout the course.

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

The materials provide opportunities for students to think mathematically, persevere through solving problems, and make sense of mathematics through open-ended problem-solving, multiple representations, and increasingly complex learning tasks. The course encourages students to analyze problems, select appropriate strategies and tools, represent mathematical ideas in multiple ways, justify their reasoning, and evaluate the reasonableness of their solutions. Instructional routines and problem-solving tasks promote productive struggle and provide opportunities for students to refine their thinking, explain their ideas, and make meaningful connections among mathematical concepts.

Throughout grade 5, students engage in structured problem-solving experiences that require them to analyze information, develop plans, solve problems using multiple representations, justify their reasoning, and evaluate the effectiveness of their strategies. Lessons prompt students to use area models, place-value charts, fraction circles, fraction bars, number lines, equations, and pictorial representations to solve increasingly complex tasks while refining their thinking through collaboration and reflection. Students analyze decimal multiplication situations, solve real-world fraction and division problems, compare multiple approaches, explain how representations support solutions, and determine whether their answers are reasonable. These opportunities encourage students to make sense of mathematical situations, revise strategies when necessary, and persevere through challenging tasks.

The materials consistently embed a problem-solving process within each lesson. The process encourages students to take time to understand problems before calculating, select and adjust strategies based on context, and determine whether solutions are reasonable. Grade-level mathematical word problems require students to read and interpret information, identify important details, represent situations in multiple ways, and justify their solutions using mathematical evidence. Through reflective questions that ask students to explain how representations model mathematical relationships, evaluate the effectiveness of their strategies, and describe why their solutions make sense, students regularly engage in mathematical thinking, perseverance, and sense-making while developing confidence in solving increasingly complex problems.

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.

The materials support students in understanding, explaining, and justifying that problems can be represented and solved in multiple ways. The course encourages students to use a variety of representations and strategies, compare different approaches, and explain why multiple methods can lead to the same solution. Instructional routines and mathematical discussions provide opportunities for students to communicate their reasoning, justify their choices, and develop flexibility in mathematical thinking.

Throughout grade 5, students demonstrate that mathematical problems can be represented and solved using multiple representations and solution methods. Students represent decimal multiplication with decimal models, repeated addition, multiplication expressions, and whole-number strategies; model fraction situations using fraction bars, circles, tiles, and number lines; and solve measurement-conversion problems by creating tables, equations, and visual models. Lessons regularly prompt students to compare representations, explain how models are alike and different, justify why alternative methods produce valid solutions, and evaluate the reasonableness of their answers. These experiences help students recognize that mathematical relationships can be represented and solved in different yet equivalent ways.

The materials consistently provide opportunities for students to explain and justify multiple approaches through routines such as “Show Another Way,” “Spark Discussions,” and “Turn and Talk.” Grade-level word problems encourage students to solve problems using several representations, reconnect solutions to decimal and fraction concepts, create labeled diagrams, and communicate their reasoning through written justifications and mathematical discussions. By engaging in meaningful conversations about the effectiveness of different strategies and representations, students develop conceptual understanding, mathematical discourse, and the ability to justify why multiple methods can be used to solve the same problem.

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.

The materials require students to make sense of mathematics through multiple opportunities to do mathematics, write about their reasoning, and discuss ideas with peers and educators. Throughout the materials, students actively solve problems, create mathematical representations, write to explain their thinking, and participate in collaborative discussions that promote mathematical reasoning and communication. Instructional routines encourage students to reflect on their strategies, ask questions, and refine their understanding through meaningful interactions with peers and educators.

Throughout grade 5, students engage in mathematical tasks involving decimal operations, fraction concepts, and place-value reasoning by using models, equations, base 10 blocks, drawings, and pictorial representations. Students estimate and calculate solutions, represent problems concretely and pictorially, explain how mathematical models support their thinking, and justify the reasonableness of their answers using precise mathematical vocabulary. Lessons also require students to write explanations about concepts such as decimal-point placement, common denominators, and the relationship between division and fractions, providing multiple opportunities to communicate mathematical understanding through writing.

The materials consistently embed opportunities for mathematical discourse through routines such as “Turn and Talk,” “Three Reads,” “Stronger and Clearer Each Time,” and “Mathematical Language Routines.” Students discuss how place value supports decimal operations, explain how equations and models represent situations, collaborate to identify and correct errors, and build on one another’s ideas through increasingly detailed explanations. By solving problems, writing about mathematical processes, and engaging in purposeful discussions with peers and educators, students deepen their conceptual understanding, strengthen their mathematical reasoning, and develop the ability to communicate mathematical ideas with precision.

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

The materials support educators in guiding students to share and reflect on their problem-solving approaches through explanations, arguments, and justifications. The materials include questioning strategies, discussion routines, and collaborative structures that encourage students to explain their thinking, compare solution methods, justify conclusions, and reflect on the effectiveness of their strategies. These opportunities promote meaningful mathematical discourse and support students in refining their reasoning through peer and educator interactions.

Throughout grade 5, students explain and justify their mathematical thinking using area models, pictorial models, equations, tables, and visual representations. Lessons require students to explain why solutions are reasonable, justify why expressions are equivalent, analyze numerical patterns, compare different representations, and evaluate the effectiveness of their problem-solving strategies. Project-based learning experiences also ask students to defend predictions, explain outcomes, and support conclusions using evidence and mathematical reasoning. These experiences encourage students to reflect on their thinking and communicate their reasoning in both written and oral forms.

The materials consistently embed routines such as “Turn and Talk,” “Three Reads,” “Compare and Connect,” “Critique, Correct, and Clarify,” and “Stronger and Clearer Each Time” to support student reflection and mathematical discourse. Guidance leads educators to ask follow-up questions, address misconceptions, prompt students to defend their reasoning, and encourage students to compare and evaluate different approaches. Through these structured opportunities for explanation, argumentation, and reflection, students strengthen their conceptual understanding and develop the ability to communicate and justify mathematical ideas using evidence-based reasoning.

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

The materials include guidance and prompts that support educators in providing explanatory feedback based on student responses and anticipated misconceptions. The materials provide scripted questions, sample responses, and instructional supports that help educators identify misconceptions, build on student thinking, and guide students toward conceptual understanding. The embedded guidance encourages educators not to simply correct student errors but to use errors as opportunities for discussion, reflection, and deeper mathematical reasoning.

Throughout grade 5, the materials anticipate common misconceptions related to decimal operations, fraction concepts, area models, and interpreting fractional parts of sets. Educator guidance includes “Spark Discussions,” “Addressing Common Errors,” and “Common Error” supports that identify likely mistakes, explain why students may make them, and provide targeted questions, visual models, and concrete examples to address misunderstandings. Students revise models, compare and order decimals, reinterpret area models, rewrite fractions using equivalent fractions, and explain what fractions represent within a set. The materials also provide possible student responses and follow-up prompts that help educators extend learning and encourage students to justify and refine their thinking.

The materials consistently support educators in facilitating meaningful mathematical discussions through reflective questioning, corrective instructional strategies, and evidence-based reasoning routines. Prompts guide educators to ask probing questions, address misconceptions in real time, emphasize key mathematical concepts, and help students connect visual representations to equations and mathematical relationships. By anticipating misconceptions and providing purposeful explanatory feedback, the materials equip educators to respond effectively to student reasoning while strengthening students’ mathematical understanding, communication, and problem-solving skills.