HMH Into Math Texas, Grade 4

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

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


The scope and sequence charts include the TEKS and ELPS alignments for each module and lesson and identify the concepts taught throughout the instructional year. Additional materials further support this alignment through TEKS and ELPS correlations and “Correlations by Lesson” documents that break down concepts taught by module. In the “Discover” tab, 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 to remain 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 using 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 materials include clear explanations of the unit rationale and lesson order, and the materials 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 deepen their understanding of place value before extending this understanding to decimal representations and equivalence. The materials also explain how place-value concepts progress from representing whole numbers to regrouping, comparing, ordering, and rounding numbers and how the program prepares students to apply problem-solving models throughout the year.

Additionally, fraction instruction progresses from comparing fractions using visual models to generating equivalent fractions and applying common multiples and comparison strategies to compare and order fractions. Division concepts also build from exploring division patterns and estimating quotients to representing division with arrays and equations, interpreting remainders, and applying area models and the distributive property to support flexible problem-solving and real-world application.

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

The materials include guidance that supports 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 to help educators understand the learning sequence in 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 flow charts, 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, and 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 a “Module Internalization Tool” and “Lesson Internalization Tool” that 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 in preparing for and internalizing 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 materials include resources and guidance documents for instructional leaders to support educators with 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.

Additionally, the materials include look-for checklists designed for the first 30 days of instruction and reflection tools and guiding questions to support classroom visits, targeted feedback, and ongoing instructional implementation conversations with educators.

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 materials include comprehensive unit overviews that provide the background content knowledge and academic vocabulary necessary 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, examples, summaries, and explanations of mathematical concepts to support conceptual understanding and procedural fluency. For example, there is guidance for teaching multiplication with one-digit numbers through multiplication strategies, algorithms, rounding, estimation, place-value patterns, and problem-solving. The course also supports instruction in addition and subtraction of whole numbers and decimals through explanations related to place value, standard algorithms, regrouping, and decimal alignment.

Additionally, the module overviews identify prior learning, current module concepts, and future learning connections related to place value, comparing numbers, rounding, decimals, and operations to help educators connect mathematical ideas across the course. The materials further support educator understanding through a “Teacher’s Glossary” section that includes academic vocabulary introduced in each module and mathematical definitions 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 materials include supports for families through “Family Letters” in both English and Spanish for each module. The letters 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 place value, fractions, multiplication, estimation, and mental math through hands-on practice and real-world objects such as measuring cups. Additionally, the letters include guidance for families on how to support student progress using online resources such as “Practice Plus” and activities that reinforce in-class learning outside school.

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 materials include comprehensive, structured, and detailed lesson plans that provide daily objectives, instructional tasks, required materials, guiding questions, and instructional assessments aligned with the TEKS and ELPS. Each lesson includes learning and language objectives, instructional routines, discussion opportunities, practice tasks, and assessment opportunities aligned with 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 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 fraction circles, fraction tiles, number lines, digital fraction bars, foam base 10 blocks, and digital manipulatives to support conceptual understanding. The materials include lessons such as “Represent Addition of Fractions,” “Use Benchmarks to Evaluate Reasonableness,” and “Explore Multiplication Patterns with Tens, Hundreds, and Thousands” that have clearly defined learning objectives connected to mathematical reasoning and problem-solving.

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 and integrate the 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 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 materials identify manipulatives and resources needed for instruction and conceptual understanding, including fraction circles, plastic rainbow fraction tiles, foam two-color counters, digital fraction bars, foam base 10 flats, rods, units, plastic base-ten cubes, and digital base 10 blocks.

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. In 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, the 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 materials include guidance for effective extended practice through differentiated activities, independent practice opportunities, homework, and enrichment resources connected to each lesson. All lessons provide 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 with 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 who require intervention, reinforcement, or enrichment.

Additionally, independent practice rotations, “Project-Based Learning” opportunities, and digital supports provide continued reinforcement and extended learning opportunities beyond the core lessons.

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 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 that provide 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 student readiness before each module, “Quick Checks” provide formative assessment opportunities during and after instruction, and Module Tests measure mastery of skills taught in 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 students to apply and extend mathematical understanding. The 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 the beginning, middle, and end of 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 materials clearly define and explain the purpose of assessment types throughout the program. The Implementation Guide and the Program Assessments at a Glance identify the 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 monitors student progress. For example, “Benchmark Assessments” determine student progress at strategic points during the school year, and “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 performance-based assessments that engage students in DOK Level 4 tasks that apply and extend mathematical understanding.

Additionally, Module Tests are summative assessments that measure student understanding of the TEKS and mathematical process standards taught in a module and 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 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 educators should administer assessments: before each module, using “Prerequisite Checks”; after instructional tasks, using “Quick Checks”; and at the end of each module, using Module Tests. The materials also provide suggested administration times and identify assessment formats, including digital assessments, editable files, and PDFs.

Additionally, the Module Test guidance includes information on the purpose, administration procedures, and recommended duration (approximately 45 minutes) as well as 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 materials include formative and summative assessments that align with the TEKS and instructional objectives of the lessons and modules. Assessment resources include item guides, answer keys, and assessment documentation identifying TEKS alignment, DOK levels, lesson connections, and scoring guidance to ensure that assessments measure the intended learning outcomes.

The materials demonstrate alignment among 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 fraction concepts and operations. Similarly, module 6 assessments align with the TEKS that address multiplication concepts and problem-solving. Formative assessments such as “Quick Checks” and “Practice on Your Own” tasks directly connect to lesson learning goals and TEKS-based instructional objectives.

Assessment item guides identify the TEKS assessed, DOK levels, and aligned instructional tasks, and Module Test answer keys identify the correct response for each item, provide the 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 with the corresponding TEKS and specific lessons and provides evidence of alignment among the assessments, course standards, and instructional content.

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

The 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, explaining, solving, justifying, and applying mathematical concepts.

The materials include formative and summative assessments that progress from foundational skills to increasingly complex reasoning and problem-solving. For example, assessment tasks require students to decompose fractions, solve addition and subtraction problems with mixed numbers, evaluate reasonableness using benchmark fractions, multiply using powers of 10, apply expanded notation and area models, and solve multi-digit multiplication problems using a variety of strategies aligned with the grade-level TEKS. Lesson assessments include DOK Levels 1–3 questions that support increasingly complex thinking and allow educators to monitor student understanding across varying levels of rigor.

Additionally, assessment resources include a variety of TEKS-aligned item types, such as multiple-choice, multi-select, text-entry, fill-in-the-blank, and open-ended responses. Assessment answer keys identify TEKS alignment and DOK levels for assessment items, and 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 that 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 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 that 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, and 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 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. “Data-Driven Support” tables, “Differentiation and Practice” guidance, assessment reports, and targeted instructional recommendations 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, guidance directs educators 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” activities, “Practice Plus” homework, and personalized adaptive practice through Waggle.

Additionally, “Data-Driven Support” tables connect assessment results to specific reteaching resources, lessons, and activities that are 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 materials include educator-facing and student-facing tools that support monitoring student progress and growth throughout the school year. 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 “Ed” help educators monitor student performance, identify trends, and make instructional decisions.

The materials also include tools to 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 goals. At the beginning of each lesson, students preview an “I Can” statement and revisit it at the end 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 lessons for students who have not yet reached proficiency on grade-level content and skills. Lessons include “15+ minutes of suggested Differentiation and Practice opportunities” and a “Prerequisite Check” with a “Data-Driven Support” table to help educators identify unfinished learning and strengthen students’ prerequisite skills before instruction.

A corresponding educator-led small-group activity in each lesson supports students who have not yet demonstrated proficiency with lesson content. The “Teacher-Led Small Group Activities” include differentiated levels of support, such as “Almost There,” “On Track,” and “Extension,” allowing educators to select instructional responses based on student performance data. The “Almost There: Multiply Using Expanded Notation” small-group activity provides explicit educator guidance, guiding questions, structured instructional steps, area models, and repeated practice opportunities to support multiplication concepts. The “Almost There: Compare Fractions Using Visual Models” activity provides reteaching support through educator-guided instruction and printable fraction strips. “Intervention Activities” provide scaffolded support through multiple representations, such as fraction circles, fraction tiles, number lines, base 10 blocks, area models, place-value charts, and expanded notation. Step-by-step guidance supports conceptual understanding before students apply grade-level skills independently. The “Comparing Multi-Digit Numbers” intervention activity guides students through number comparisons using base 10 blocks, number lines, and place-value charts and provides explicit think-aloud supports and examples before independent practice.

Additional reteaching opportunities include “Teacher-Led Small Group Activities,” “Collaborative Groups,” “Games,” and “Independent Practice.” Structured guidance and targeted supports address learning gaps and maintain access to grade-level content.

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 pre-teaching and embedded supports for unfamiliar vocabulary, academic language, and unfamiliar references in text. The “Mathematical Language & Emergent Bilinguals” sections provide vocabulary routines, response scaffolds, discussion supports, ELPS strategies, and “Mathematical Language Routines” that help students understand mathematical concepts and communicate mathematical thinking using precise academic language.

The materials provide dedicated vocabulary instruction for new terms such as mixed number through student-friendly definitions and the “Words to Learn” routine. Guidance supports the introduction and reinforcement of academic vocabulary before instruction and includes support for understanding, pronouncing, and using mathematical terms during lessons. The materials recommend the “Words to Learn” routine for terms such as mixed number, associative property of addition, and commutative property of addition when students need support to describe their thinking. Embedded supports help students understand unfamiliar concepts and references through questioning prompts, visual models, discussion routines, “Turn and Talk” opportunities, and representations such as fraction strips and concrete models. Mathematical vocabulary connects visual representations and prior learning experiences to strengthen comprehension.

Additional support includes ELPS teaching strategies that help construct meaning from unfamiliar language and references. In module 10, lesson 4, educator guidance explains how to use context to construct the meaning of figurative language such as idioms, hyperbole, and personification that may arise during student discussions.

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 “15+ minutes of suggested Differentiation and Practice opportunities,” and the “Data-Driven Support” tables provide guidance for using assessment results to identify students who are ready for additional challenge and deeper application of content.

A corresponding small-group activity in each lesson extends learning for students who have demonstrated proficiency in the lesson objectives. The “Ready for More: Compare Fractions Using Visual Models” activity provides an educator-led small-group lesson in which students create fraction cards with visual models, compare and order fractions, justify reasoning, and identify equivalent fractions. Explicit educator guidance, materials lists, and implementation procedures support facilitation of the activity.

Additional extension opportunities through Module Projects provide DOK Level 4 tasks. Projects such as “The Geometry Park” and “Fraction Festival” require students to apply grade-level mathematics in authentic, real-world contexts, analyze information, justify reasoning, develop solutions, and complete complex problem-solving. In “The Geometry Park” Module Project, students design a park using geometric attributes, and in the “Fraction Festival” Module Project, students create pricing displays and special deals using fraction and decimal concepts and mathematical thinking through written and oral responses.

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 instructional support, worked examples, questioning strategies, visual representations, and structured routines to support the clear explanation and modeling of mathematical learning.

Educator prompts guide instruction and mathematical discussions. Module 4, lesson 3, “Represent Addition of Fractions” includes questions such as “What is the problem about?” “What numbers are important? Why are they important?” and “What question is being asked?” The materials provide guidance through the “Analyze, Plan, Solve, Justify, and Evaluate” process to support the modeling of mathematical thinking and problem-solving.

Additional instructional guidance supports discussions, addresses misconceptions, and connects concrete, pictorial, and abstract representations. The materials include possible student responses, visual models, “Turn and Talk” opportunities, “Compare and Connect” routines, DOK-leveled questions, and supports for emergent bilingual students and diverse students. Sentence frames, gestures, visuals, fraction strips, and color-coded fraction tiles support understanding of concepts such as equivalent fractions and fraction addition. The materials also include explicit prompts for checking understanding and explaining reasoning. “Quick Checks,” “Learning Goals,” and discussion routines provide opportunities for students to explain mathematical thinking, justify solutions, and describe mathematical relationships using academic vocabulary.

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. Instructional routines, planning supports, discussion protocols, and strategies that promote conceptual understanding, procedural fluency, mathematical discourse, and problem-solving support educators throughout the materials.

Additional instructional guidance supports discussions, addresses misconceptions, and connects concrete, pictorial, and abstract representations. Possible student responses, visual models, “Turn and Talk” opportunities, “Compare and Connect” routines, and DOK-leveled questions support emergent bilingual students and diverse students. Sentence frames, gestures, visuals, fraction strips, and color-coded fraction tiles support student understanding of concepts such as equivalent fractions and fraction addition. The materials also include explicit prompts for checking understanding and explaining reasoning. “Quick Checks,” “Learning Goals,” and discussion routines provide opportunities for students to explain mathematical thinking, justify solutions, and describe mathematical relationships using academic vocabulary.

“Differentiation and Practice” resources include additional recommendations for transitioning among whole-group instruction, educator-led small groups, collaborative learning, and independent practice. Student performance data helps inform instructional groupings and aids educators in selecting appropriate learning experiences based on student needs. The Implementation Guide further supports effective lesson delivery by recommending instructional routines and structures that promote discourse, exploration, collaboration, and the independent application of mathematical concepts.

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 provide educator guidance and recommended instructional structures for effective implementation. The program incorporates guided instruction, collaborative learning, hands-on exploration, and independent practice through whole-group, small-group, partner, and individual settings.

The lessons include educator-guided instruction, collaborative mathematical discussions, independent problem-solving tasks, and “15+ minutes of suggested Differentiation and Practice opportunities.” The materials incorporate manipulatives and visual models such as fraction circles, fraction tiles, base 10 blocks, and area models and include routines that support mathematical discourse and conceptual understanding. Opportunities for student practice include independent practice activities, educator-led small-group instruction, collaborative group tasks, games, and mathematical routines. The “On Track: Represent Multiplication” small-group activity engages students in collaborative learning by creating multiplication problems, representing products using base 10 blocks, and discussing solution strategies with peers. Materials lists, step-by-step procedures, and recommendations for organizing and facilitating small-group instruction support implementation. Instructional routines such as “Compare and Connect” in the Teacher’s Guide support transitions from whole-group instruction to student-led discussions. Opportunities for peer discourse and justification support students in comparing fraction bars and number lines when identifying decimal values.

“Practice Plus” activities for independent practice and homework provide additional learning opportunities. The Implementation Guide outlines instructional structures that include whole-group instruction, collaborative groups, educator-led small groups, and independent learning. Further implementation support, such as the “Lesson Pacing Guidance,” includes recommended instructional time allocations for whole-class instruction, collaborative practice, independent work, and small-group intervention or enrichment.

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 to support students at varying levels of English language proficiency and promoting the use of increasingly academic language. Embedded emergent bilingual supports, ELPS-aligned strategies, language-development routines, and differentiated scaffolds support students’ use of content-specific vocabulary when engaging in mathematical discourse and sharing understanding.

The “Mathematical Language Routines” and “Emergent Bilinguals” sections provide vocabulary-development routines, response scaffolds, discussion supports, and other routines that support the development of mathematical understanding and academic language. The materials provide structured opportunities for explaining reasoning and communicating mathematical thinking through speaking, listening, reading, and writing activities aligned with the ELPS. An “Emergent Bilinguals: Supporting All Language Learners” section provides differentiated supports for students at multiple proficiency levels, including pre-production/beginning, intermediate, and high intermediate/advanced. For pre-production/beginning students, supports include visuals, icons, models, sentence stems, and opportunities to act out story problems using pictures and representations. Support for intermediate students includes sentence starters and opportunities to explain mathematical thinking using both a home language and English. For high intermediate/advanced students, word banks and structured discussions support independent academic discourse, opportunities for peer feedback, and increasingly sophisticated mathematical vocabulary.

Module 6, lesson 3 includes sentence frames such as “The _____ Property helped because _____,” and a word bank containing terms such as commutative, associative, regroup, double, and product. Additional supports include choral reading, structured partner discussions, and academic language routines that encourage the use of precise mathematical vocabulary in oral and written responses. The “Language Growth for All Learners” guidance explains that varying levels of scaffolding allow educators to select supports based on students’ language proficiency and promote increasingly independent and academic mathematical communication.

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 help educators effectively use the program with emergent bilingual students in bilingual and English as a Second Language (ESL) settings. Lesson-level language objectives aligned with the ELPS, language development routines, vocabulary supports, and instructional scaffolds integrate language acquisition into mathematics instruction.

The “Language Growth for All Learners” resources provide observable and measurable language objectives for each lesson and explain how instruction embeds language development. The materials include supports such as the “Words to Learn” routine, vocabulary cards, “Mathematical Language Routines,” formative language assessment tools, and scaffolded opportunities for speaking, listening, reading, and writing about mathematics. The Language Development Resource Guide provides implementation guidance for supporting emergent bilingual students across the program. The materials provide embedded supports in “Key Routines,” Essential Sessions, “Spark Your Learning” tasks, ELPS supports, instructional tasks, and formative assessments. Lesson-level ELPS supports throughout instruction provide point-of-use language development opportunities and maintain access to grade-level mathematics content.

The “Mathematical Language & Emergent Bilinguals” section includes additional implementation guidance and practical strategies such as partner structures, discussion routines, vocabulary-development supports, sentence stems, word banks, and collaborative learning protocols. Strategic partner assignments promote participation, language development, and academic discourse among emergent bilingual students. Vocabulary-development routines, response scaffolds, discussion supports, and ELPS-aligned language activities support access to mathematical content while students develop academic language. These resources support the facilitation of mathematical communication, comprehension, and language growth in 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 for educators to support emergent bilingual students as they develop academic vocabulary, increase comprehension, build background knowledge, make cross-linguistic connections, and engage in oral and written discourse. The lessons incorporate ELPS-aligned supports, vocabulary routines, discussion protocols, sentence stems, word banks, and “Mathematical Language Routines” to promote both language development and mathematical understanding.

Visual word banks, vocabulary routines, sentence starters, sentence frames, and opportunities to use content-specific language in speaking and writing support academic vocabulary development. The materials strengthen comprehension and background knowledge through routines such as “Three Reads,” “Collect and Display,” and “Stronger and Clearer Each Time” and ELPS teaching strategies that support understanding of mathematical contexts and connections to prior knowledge. The lessons include opportunities for partner discussions, collaborative learning activities, written explanations, and mathematical reasoning using vocabulary such as numerator, denominator, equivalent fraction, and measurement terms. Differentiated language supports such as the “Supporting All Language Learners” framework allow educators to choose scaffolds based on individual language needs and encourage communication through gestures, visuals, oral explanations, and written responses.

The materials include explicit support for cross-linguistic connections through translanguaging practices that encourage students to plan and draft responses in a home language before they move to English through sentence frames and word banks. These embedded supports promote the development of academic vocabulary, comprehension, background knowledge, cross-linguistic connections, and meaningful oral and written discourse for emergent bilingual students.

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 with the TEKS. The lessons incorporate conceptual investigations, mathematical discussions, problem-solving activities, formative assessments, and performance tasks that require students to represent, explain, justify, 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 4, lesson 8, the “Quick Check” requires students to add and subtract mixed numbers with like denominators using strip diagrams and number lines to solve contextual problems, represent their thinking visually, compare quantities, and explain relationships between amounts. The results inform instructional decisions and provide targeted support based on individual student needs. Additional assessments such as the Module Test for module 2 measure student understanding of decimal-place value, and challenge activities such as “Represent Tenths as Fractions and Decimals” require students to apply grade-level concepts in engaging problem-solving situations.

The materials also provide cumulative assessment opportunities through Module Tests and “Project-Based Learning” experiences. For example, the Module Test for module 10 requires students to analyze multistep problems, create mathematical models, and solve problems using multiple operations. These assessment opportunities require students to demonstrate conceptual understanding, mathematical reasoning, and application of 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. Students build conceptual understanding as learning progresses 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 the lessons and modules. In module 4, lesson 3, students begin with a contextual problem in “Spark Your Learning” before progressing through the “Analyze, Plan, Solve, Justify, and Evaluate” process. Students analyze information, develop solution strategies, represent fraction addition using models and number lines, write equations, justify their reasoning with evidence, evaluate the effectiveness of their strategies, and critique mathematical misconceptions. The lesson concludes with a “Practice on Your Own” activity that provides evidence of student mastery. Similarly, module 4, lesson 4 extends learning by requiring students to represent fraction subtraction using multiple representations, explain connections between fraction circles and fraction strips, solve problems using models and equations, and communicate their reasoning through mathematical discussions.

The materials continue this progression across modules by building from foundational concepts to increasingly complex applications. In module 2, students progress from identifying decimals represented by pictorial grids to explaining why equivalent decimals represent the same value. In module 3, instruction progresses from comparing fractions with visual models to using comparison symbols, generating equivalent fractions, and applying common multiples to determine equivalence. The progression culminates in “Project-Based Learning” module projects in which students apply grade-level concepts 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 among 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. In module 7, lesson 4, students discuss a “Turn and Talk” prompt that asks, “What place value did you round each factor to when determining your estimate?” to connect prior learning about place value and estimation to multiplication. Additional prompts ask students, “How can you use base-ten blocks to represent the problem and show how multiplication and addition are related?” and “Discuss how using the Distributive Property to find the partial products helps you solve the problem faster.” These discussions reinforce relationships among addition, place value, multiplication, estimation, and the distributive property and extend previously learned strategies to new mathematical situations.

The Teacher’s Guide supports coherence across modules by explaining how concepts build toward future learning. Fraction instruction connects part-whole relationships, equivalent fractions, visual models, computation strategies, and problem-solving to prepare students for fraction operations and rational numbers. Multiplication instruction builds from place-value understanding and estimation to multi-digit operations, algebraic reasoning, and proportional relationships. Division instruction further connects multiplication, estimation, quotient strategies, and interpreting remainders in multistep problems. 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 Teacher’s Guide and “Before You Teach” sections connect prior learning to current instruction and explain how current concepts prepare students for future mathematics. The materials connect previously learned concepts, such as composing and decomposing fractions, describing fraction equivalence, comparing and ordering numbers using place-value understanding, and representing equivalent fractions, to current learning involving decimal place value, fraction decomposition, and multiplication strategies. The materials reinforce previously introduced mathematical language, including place value, unit fraction, equivalent fractions, compare, order, tenths, and hundredths, while introducing new grade-level vocabulary and representations.

The materials also prepare students for future learning by explicitly identifying concepts and mathematical language that will be developed in later grades. For example, module 4, lesson 1 extends prior understanding of composing and decomposing fractions to prepare students for adding and subtracting fractions with unlike denominators in grade 5. Module 2 develops decimal place-value concepts to prepare students to represent, compare, and order decimals beyond hundredths, perform operations with decimals, and connect fractions, decimals, and percents in future grades. The “Teacher Background” and “What Will Students Learn in This Module?” sections explain how current learning supports future concepts such as benchmark fractions, estimation, multi-digit operations, algebraic reasoning, proportional relationships, and real-world problem-solving.

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” tables before introducing new mathematical concepts and procedures.

The materials connect prior knowledge to new learning through instructional routines, scaffolded questions, and multiple representations. “Prerequisite Checks” and “Data-Driven Support” tables help educators identify unfinished learning and provide targeted intervention before new instruction begins. The lessons about fractions build on prior understanding of fraction models, equivalence, and part-whole relationships before extending learning to fraction addition, subtraction, and problem-solving. Similarly, in module 7, lesson 1, students build on prior knowledge of multiplication properties, place value, and one-digit multiplication to develop an understanding of multiplying by multiples of 10. “Spark Your Learning” prompts students to select an appropriate multiplication strategy, connect multiplication facts to place-value reasoning, and determine how multiplying by tens affects the product.

The materials continue this progression by extending familiar concepts and procedures to new mathematical situations. Educator guidance addresses common errors by reinforcing place-value concepts as students align partial products and apply multiplication strategies. Scaffolding questions prompt students to connect familiar strategies to new concepts, and intervention supports strengthen prerequisite understanding before students work with larger numbers, decimal operations, and increasingly complex multiplication procedures. These instructional supports help educators connect prior knowledge to new learning and develop 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 and introduce new content. “Practice Plus” and the “Train Your Memory” section provide ongoing opportunities to revisit place value, multiplication, fractions, geometry, number relationships, money, and fluency skills from earlier lessons, previous modules, and prior grade levels. Students review concepts through activities such as comparing multi-digit numbers, solving multiplication word problems, identifying lines of symmetry, applying place-value concepts, and completing fluency practice. Cumulative assessments also revisit foundational concepts. For example, the Module Test for module 8 requires students to apply previously learned concepts involving division patterns, quotients, place value, and problem-solving and demonstrate current grade-level understanding.

Module Projects incorporate additional spaced retrieval opportunities by requiring students to apply previously learned skills and concepts as well as current learning in DOK Level 4 tasks. These cumulative opportunities reinforce mathematical understanding across lessons and units and give 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. New learning integrates previously taught content, and students apply multiple concepts and strategies in the same lesson, practice activity, and assessment.

The materials incorporate interleaved practice through lesson activities, “Practice Plus,” and the “Train Your Memory” section. The lessons integrate previously learned concepts such as fraction equivalence, fraction composition, place value, multiplication facts, repeated addition, multiples of 10, and estimation with new grade-level content. In module 4, lesson 1, students apply prior understanding of unit fractions and fraction decomposition and represent fractions as sums using concrete models, pictorial models, equations, and written explanations. The module revisits these concepts as students add, subtract, rename, and estimate with fractions and mixed numbers. Similarly, in module 6, lesson 3, students apply place-value understanding and multiplication facts and use area models and the distributive property to develop increasingly advanced multiplication strategies. Expanded notation, estimation, mental math, and the standard algorithm continue to reinforce these concepts.

“Practice Plus” and the “Train Your Memory” section incorporate additional interleaved practice by combining concepts from multiple lessons and units in the same set of practice problems. Students engage with division, place value, multiplication strategies, decimal comparison, fraction ordering, and computation fluency in a single activity, selecting and applying different concepts, representations, and procedures across tasks. For example, students describe mental math strategies for multiplication while ordering fractions and solving problems involving multiple mathematical concepts. These opportunities reinforce previously learned skills and concepts across lessons and units and strengthen 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 interpret mathematical situations using visual models, analyze the relationships among multiple representations, and evaluate how those representations support mathematical reasoning and problem-solving.

The materials engage students in interpreting, analyzing, and evaluating mathematical models throughout the program. Module 4 develops these skills as students interpret fraction models, analyze how visual representations support multiple fraction decompositions, evaluate whether equations accurately match the models, and explain their reasoning. Similarly, module 6 requires students to interpret multiplication models; analyze relationships among place-value strategies, number patterns, and visual representations; and evaluate which representations most effectively support solving multiplication problems.

Additional lesson tasks reinforce these expectations through a variety of mathematical representations. In module 2, lesson 2, students interpret number lines by plotting and labeling fractions and decimals to the hundredths and analyze how the representations show equivalent values. In module 10, lesson 2, students interpret a strip diagram representing a multistep problem, analyze how the diagram connects to equations used to solve the situation, and evaluate how the strip diagram and equations support their mathematical reasoning. These opportunities consistently require students to interpret, analyze, and evaluate mathematical models and representations across lessons.

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 construct concrete and pictorial models to represent mathematical concepts, connect those models to equations, and use multiple representations to strengthen conceptual understanding.

The materials engage students in creating mathematical models throughout the program. In module 4, students build concrete models and pictorial representations of fractions, decompose and recombine fractions in multiple ways, write equations that match their models, and compare representations to explain fractional relationships. Similarly, module 6 requires students to create base 10 block models and other representations to model multi-digit multiplication situations, compare the representations, and explain how each model supports the product.

Additional lesson tasks provide opportunities for students to create models using a variety of mathematical representations. In module 2, lesson 1, students select tools such as base 10 blocks to build decimals, draw matching visual models, and connect mixed numbers and decimals by labeling tenths and explaining their reasoning. “Practice on Your Own” activities extend this work by asking students to create visual models for given decimals, and module 12, lesson 3 requires students to use manipulatives or digital tools to draw and label lines that represent mathematical situations. These opportunities consistently require students to create models that represent mathematical concepts and support problem-solving.

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. Rather than simply repeating procedures, students extend their understanding by applying mathematical concepts through multiple representations, real-world scenarios, and mathematical reasoning.

The materials engage students in applying conceptual understanding throughout the program. Module 4 guides students to build and decompose fractions with concrete models before representing the same fraction in multiple ways, verifying equivalence, and explaining their reasoning in new situations. Similarly, module 6 develops conceptual understanding of multiplication through base 10 blocks, area models, the distributive property, and expanded notation before students apply these representations to increasingly complex multiplication problems using multiple strategies and algorithms.

Additional lesson tasks require students to apply their understanding in new mathematical contexts. In module 4, lesson 3, students solve a real-world measurement problem by representing fraction addition on a number line, writing an equation, and explaining their reasoning using evidence from the model. “Ready For More” small-group activities extend learning by asking students to create real-world problems from a given set of equations, represent the situations with strip diagrams, and discuss common mathematical relationships with peers. Other lessons ask students to solve fraction word problems using pictorial models and strip diagrams before writing matching equations and justifying their solutions, which reinforces the application of conceptual understanding in a variety of contexts.

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 designed to build the automaticity and fluency students need to complete grade-level tasks. Repeated opportunities to apply mathematical concepts, practice efficient procedures, and solve increasingly complex problems help students develop both computational automaticity and mathematical fluency.

The materials build automaticity and fluency throughout the program. In module 4, students repeatedly build, decompose, and represent fractions using concrete models, drawings, and equations before independently solving additional fraction tasks, which strengthens fluency with fraction relationships and representations. Similarly, module 6 provides repeated opportunities for students to solve multiplication problems using base 10 blocks, area models, partial products, and the standard algorithm, which helps them develop automaticity with place-value reasoning and multiplication strategies and build fluency with multi-digit multiplication.

Additional practice opportunities reinforce these skills across the program. “Practice on Your Own” activities require students to repeatedly use the standard algorithm to divide four-digit numbers by one-digit divisors, evaluate the reasonableness of quotients, solve multistep word problems, and analyze completed division problems for accuracy. “Fluency Practice” provides repeated practice with multi-digit division using both standard long division and division equations, and “Practice Plus” activities strengthen automaticity with equivalent fractions and decimals through pictorial models, expanded form, and standard decimal notation. These repeated practice opportunities help students develop the automaticity and fluency necessary to successfully complete grade-level mathematical 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 in lessons and throughout each 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 engage students in practicing efficient, flexible, and accurate mathematical procedures throughout the program. In module 4, students solve fraction problems using concrete manipulatives, pictorial representations, and equations before decomposing fractions in multiple ways, verifying equivalent solutions, and explaining why different approaches are valid. Similarly, module 6 guides students through multiplication using base 10 blocks, area models, expanded notation, partial products, and the standard algorithm and encourages them to compare strategies and determine which procedures are most efficient for a given problem.

Additional lesson activities reinforce these expectations across a variety of mathematical contexts. In “Practice on Your Own” activities, students apply mental math, multiplication properties, and multiple solution strategies to solve real-world problems, determine unknown values, and justify their reasoning. Other lessons require students to rewrite expressions using the commutative, associative, and distributive properties; compare the efficiency of different multiplication strategies; and communicate their reasoning using partial products and place-value models. “Reteach” activities further strengthen procedural flexibility by guiding students through partial quotients with rectangular models and step-by-step procedures, which helps them apply efficient, flexible, and accurate mathematical procedures across lessons.

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 in lessons and throughout each module. Students compare multiple strategies and representations, justify their reasoning, and reflect on the effectiveness of different mathematical approaches to solve problems.

The materials engage students in evaluating mathematical procedures throughout the program. In module 4, students create multiple fraction decompositions, determine whether each procedure accurately represents the same fraction, justify why different equations model the same quantity, and explain which representation most effectively communicates their reasoning. Similarly, module 6 guides students in comparing area models, base 10 block representations, partial products, and the standard algorithm; evaluating which procedures are most efficient for a given multiplication problem; and explaining why different strategies produce the same product.

Additional lesson activities reinforce these evaluation skills in a variety of mathematical contexts. Students estimate and solve problems using the standard algorithm, compare exact and estimated solutions, and justify the reasonableness of their answers. Structured problem-solving tasks ask students to analyze information, develop and justify solution plans, reflect on the effectiveness of their strategies, and consider whether a different approach would be more efficient. “Practice on Your Own” activities include prompts such as “Evaluate Process” that encourage students to explain how they evaluated their solution process, and educator guidance supports discussions comparing different representations and strategies to determine their efficiency, flexibility, and accuracy.

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, questioning strategies, and scaffolds that help students refine their thinking, compare solution methods, and transition from concrete representations to increasingly efficient mathematical procedures while maintaining conceptual understanding.

The materials embed instructional supports throughout multiple modules to help educators guide students toward increasingly efficient strategies. In module 4, students begin by modeling fractions with manipulatives before transitioning to drawings and equations, and educator prompts encourage discussion about why one representation may be more effective than another. Similarly, module 6 guides instruction from base 10 block and area models to partial products and increasingly efficient multiplication procedures. Educator notes, “Compare and Connect,” and “Discussion Supports” prompt students to compare strategies, determine when one method is more efficient, and explain why different procedures produce equivalent results.

Additional embedded educator guidance reinforces increasingly efficient approaches across a variety of mathematical concepts. In module 9, educators prompt students to connect area models to partial quotients, discuss multiple ways to break apart dividends, and explain how multiplication facts support more efficient division. Lessons also include routines such as “Three Reads,” “Turn and Talk,” and “Quick Check,” Universal Design for Learning (UDL) supports, discussion prompts, and questions that require students to compare strategies, address misconceptions, justify efficient methods, and refine their problem-solving approaches throughout instruction.

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

The materials explicitly state how they address the conceptual and procedural emphasis of the TEKS. Educator guidance explains how students’ conceptual understanding develops through concrete models, visual representations, and mathematical reasoning before they apply increasingly efficient procedures aligned with the TEKS. The module overviews, planning guidance, and educator background information consistently outline this progression.

The materials explicitly connect conceptual understanding to procedural fluency across multiple modules. In module 2, the “Teacher Background” section explains that students develop conceptual understanding by recognizing decimals as alternate representations of fractions with denominators of 10 and 100 before procedurally representing, comparing, and ordering decimals and applying the problem-solving model. Similarly, module 4 explains how students build conceptual understanding of fractions through manipulatives, fraction strips, visual models, and decomposition before transitioning to symbolic fraction equations and procedures. Module 6 follows the same progression by developing multiplication concepts with base 10 blocks and area models before introducing partial products and the standard algorithm.

Additional evidence appears throughout the educator guidance and module overviews. Module 9 explains that students first develop a conceptual understanding of division through equal groups, multiple representations, and interpreting remainders before progressing to procedures using arrays, area models, equations, partial quotients, place-value strategies, and the standard algorithm. The “Grade Four Focus Areas” section also explains how the curriculum intentionally sequences conceptual understanding before procedural knowledge by prioritizing readiness standards, building mathematical connections, and preparing students for success in grade 5 and later mathematical learning.

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. Students regularly move among hands-on models, visual representations, and symbolic equations to develop conceptual understanding before applying abstract mathematical procedures.

The materials intentionally connect concrete, pictorial, and abstract representations throughout fraction and multiplication instruction. In module 4, lesson 1, students use fraction manipulatives to model fractions; create drawings, such as strip diagrams and fraction models; write equations that represent multiple fraction decompositions; and explain how the representations are connected. Similarly, in module 6, students use base 10 blocks to model multiplication, create area models and partial-product drawings, and solve the same problems using symbolic multiplication procedures, which reinforces connections among all three representations.

Additional tasks require students to apply multiple representations in meaningful mathematical contexts. Students use fraction models to represent 5/6 of a pound of chicken divided into 1/6-pound patties, explain the meaning of each unit fraction, and write addition equations to represent the situation symbolically. Other lessons ask students to draw and label strip diagrams for fractions, such as 7/10; use concrete or digital tools to create additional models; and write matching equations that demonstrate equivalent fractional relationships, which reinforces the progression from concrete and pictorial models 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 to connect, create, define, and explain concrete and representational models to abstract (symbolic, numeric, and algorithmic) concepts, as required by the TEKS. Educator guidance consistently moves students from hands-on models and visual representations to symbolic notation and prompts students to explain the mathematical connections among each representation.

The materials intentionally support this progression across multiple modules. In module 4, lesson 1, students build fraction models with manipulatives, create drawings of the same fraction, connect both representations to symbolic fraction equations, and explain how the numerator, denominator, and equivalent decompositions are represented across each model. Similarly, in module 6, students use base 10 blocks, area models, and partial-product drawings to model multiplication before connecting those representations to the standard algorithm and explaining how each step reflects place-value relationships.

Additional lessons continue to bridge concrete, representational, and abstract thinking. Students use base 10 blocks to model division, draw quick pictures, connect both representations to symbolic division, and explain why regrouping is necessary and how it relates to multiplication. Other tasks require students to model multiplication with base 10 blocks, create drawings, write matching equations, and explain how place value is represented across the models. Students also use number lines to model elapsed time before connecting their visual representations to subtraction strategies, which strengthens students’ understanding of how concrete and representational models support abstract mathematical procedures.

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 with 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.

Students use geometric models of points, lines, line segments, rays, endpoints, angles, and quadrilaterals; vocabulary boxes; and labeled diagrams to connect mathematical terms to their representations. Students identify and name figures, explain mathematical attributes, compare representations during “Turn and Talk” discussions, and apply precise mathematical vocabulary in context. Vocabulary routines such as “Say It,” “Define It,” “Discuss It,” “Use It,” and “Your Turn” require students to pronounce vocabulary, complete sentence frames, explain mathematical ideas, and strengthen their mathematical communication through speaking and writing.

The materials further support academic mathematical language development through intentional vocabulary instruction and discussion routines. The “Words to Learn” routine and vocabulary cards introduce and reinforce new mathematical terms, and instructional routines such as “Three Reads,” “Collect and Display,” “Compare and Connect,” “Discussion Supports,” and “Stronger and Clearer Each Time” provide repeated opportunities for students to hear, use, and refine academic language. Students also engage with manipulatives and visual models, such as number lines for solving elapsed time problems, and they discuss mathematical ideas with peers using precise vocabulary. These coordinated supports promote comprehension, fluency, and the accurate application 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 in the instructional context. Educators can use 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 Teacher’s Guide provides explicit educator support for introducing and reinforcing terms such as polygon, heptagon, nonagon, dividend, divisor, quotient, and remainder as students construct figures, analyze attributes, solve problems, and explain their reasoning during collaborative routines such as “Turn and Talk.” The materials prompt educators to provide definitions, model academic language, and encourage students to orally describe mathematical relationships and solution processes, which helps students move from recognizing vocabulary to accurately applying it in speaking and writing.

Additional supports further strengthen students’ use of academic mathematical vocabulary in context. During the “Three Reads” and “Discussion Supports” routines, educators encourage students to use content-specific terms when discussing plans, solutions, and processes and to challenge, critique, and elaborate on one another’s explanations. The Language Development Resource Guide provides routines such as the “High-Utility Word Routine” and “Quick-Teach Routine” to support vocabulary development across content areas. The materials also incorporate the “Words to Learn” routine to introduce and reinforce new terms such as profit when students describe their mathematical thinking. These embedded supports intentionally develop students’ academic mathematical language through repeated opportunities to hear, discuss, and apply precise vocabulary 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, sentence frames, discussion routines, exemplar responses, and questioning strategies help educators encourage students to explain and justify their reasoning using precise mathematical language during collaborative conversations and written responses.

The “Words to Learn” routine introduces academic vocabulary such as angle, right angle, straight angle, acute angle, and obtuse angle, and educators use guiding questions, exemplar responses, and mnemonic devices to model precise mathematical language. Students participate in “Turn and Talk” and “Stronger and Clearer Each Time” routines to explain their reasoning, hear and refine peers’ ideas, and revise their responses using increasingly accurate mathematical terminology. Additional supports, including anchor charts and possible student responses, help educators scaffold mathematical conversations and develop students’ mathematical language over time.

The materials also provide embedded guidance that supports mathematical conversations and students’ use of appropriate academic language in authentic contexts. The “Spark Discussion” and “Spark Your Learning” activities provide exemplar responses and sentence frames that encourage students to explain observations, justify strategies, and communicate conclusions using evidence. Students use content-specific vocabulary such as key, stem, leaf, pages, count, and at least when discussing data displays and mathematical representations with partners. These scaffolded opportunities allow students to hear, practice, refine, and apply mathematical language through peer discussions and written explanations, which strengthens both mathematical communication and conceptual understanding.

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 in 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. Instruction, practice activities, and assessments all embed these opportunities.

In “Use Representations,” students determine products and create pictorial models to communicate their thinking, which requires using multiple representations, connecting visual models to numerical expressions, and explaining mathematical ideas. Similarly, students select appropriate tools to measure side lengths, classify triangles by attributes, apply mathematical relationships to classify figures, and justify their reasoning using supporting evidence. These tasks engage students in problem-solving, representation, communication, and mathematical justification and deepen conceptual understanding.

The materials also demonstrate intentional integration of the TEKS process standards through assessments and module overviews. Module assessments incorporate content standards and opportunities for students to apply mathematical processes, and the Teacher’s Guide provides an overview of how each module and lesson embeds the process standards. For example, the materials explain how students repeatedly apply the steps of the problem-solving model and then use all of the steps to solve contextual problems. By consistently integrating the process standards into instruction, practice opportunities, and assessments, the materials support the development of mathematical reasoning, communication, and problem-solving skills throughout the course.

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

The materials explain how they incorporate and connect the TEKS process standards throughout the course by embedding the process standards in 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. The materials guide educators to reinforce these standards through consistent instructional routines, mathematical discussions, and problem-solving experiences throughout the year.

The “Correlations by Standard” documents map each process standard (4.1A–4.1G) to multiple lessons and content domains, which demonstrates that the number and operations, fractions, geometry, and other mathematical strands embed mathematical processes. This organization shows how students repeatedly engage in problem-solving, reasoning, representation, communication, and mathematical connections while developing grade-level concepts, which provides educators a cohesive framework for reinforcing these skills throughout the year.

The materials further describe how they connect the TEKS process standards throughout the course through the “Mathematical Connections” overview and supporting instructional resources. The Teacher’s Guide explains that mathematical connections, supported by the TEKS mathematical process standards, deepen understanding, promote transfer of learning, support coherence across topics, and strengthen problem-solving and reasoning. Additionally, the process standards anchor charts in the Student Edition include reflection questions and sentence stems that educators can incorporate into lessons throughout the year to help students articulate their thinking and connect mathematical meaning while developing problem-solving and critical-thinking skills.

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 each module incorporates and connects the TEKS process standards. 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 (PS 1.A), use a problem-solving model (PS 1.B), select appropriate tools and techniques (PS 1.C), and use representations (PS 1.E) as they develop an understanding of addition, subtraction, multiplication, and fractions. This guidance helps educators understand how the materials intentionally embed and reinforce the process standards.

The module overview identifies the specific process standards addressed in each lesson, including applying mathematics, using a problem-solving model, choosing tools, sharing ideas, using representations, analyzing relationships, and using precise mathematical language. The materials explain how students first develop individual process skills and then apply multiple standards together in later lessons when solving real-world problems. This progression demonstrates that the materials intentionally embed and reinforce the process standards to support mathematical reasoning, communication, and problem-solving throughout each module.

The Teacher’s Guide further explains the integration of the process standards through the section titled “How are the Process Standards (PS) connected to the content in this module?” and the “Process Standards (PS) Anchor Charts.” These resources provide examples of how students communicate using diagrams, select appropriate tools, analyze relationships, and apply mathematics in real-world contexts as they learn new content. The materials also explicitly state how lessons integrate multiple process standards in specific modules, which allows educators to intentionally plan instruction and help students strengthen problem-solving skills through consistent engagement with the process standards across the module.

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

The materials include an overview of how each lesson incorporates the TEKS process standards 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.

The “Correlations by Lesson” and “Correlations by Standard” sections in the Implementation Guide provide a comprehensive overview of how the course incorporates the TEKS process standards. Each lesson identifies the corresponding process standards (4.1A–4.1G) alongside the content standards, which allows educators to see how daily instruction embeds problem-solving, selecting tools, communicating mathematical ideas, analyzing relationships, and justifying reasoning. The guide also identifies the specific lessons that address each process standard, giving educators a clear roadmap for intentionally integrating and reinforcing these mathematical practices across the curriculum.

The materials further demonstrate lesson-level integration of the process standards through the sections titled “What will students learn in this lesson?” and lesson-specific overviews such as “Multiply Using Partial Products.” These sections identify the content and process standards incorporated into instruction and explain how students will apply the standards during learning tasks. For example, students analyze mathematical relationships (4.1F) and display, explain, and justify mathematical ideas (4.1G) when using place value to decompose factors and solve partial products. This organization demonstrates that instruction intentionally embeds the TEKS process standards and consistently connects them to grade-level mathematical content.

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 materials encourage 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 4, 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 fraction circles, fraction bars, number lines, area models, equations, and pictorial representations to solve increasingly complex tasks and refine their thinking through collaboration and reflection. Students investigate fraction relationships, solve elapsed-time and multistep real-world problems, explain how remainders affect solutions, compare solution strategies, 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 in 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 why their answers make sense, narrate the impact of remainders, and evaluate the effectiveness of their strategies, students regularly engage in mathematical thinking, perseverance, and sense-making and develop 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 they can represent and solve problems in multiple ways. The materials encourage 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 4, students demonstrate that they can represent and solve mathematical tasks using multiple representations and solution methods. Students decompose fractions using fraction circles, fraction tiles, pictorial models, and equations; determine products using area models, arrays, equations, place-value charts, mental math, and basic facts; and solve real-world problems using multiple strategies and representations. Lessons regularly prompt students to compare methods; explain how representations are related; discuss the similarities and differences between strategies, such as rounding and using compatible numbers; and evaluate which approaches communicate their thinking most effectively. These opportunities help students recognize that they can represent and solve mathematical ideas 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,” “Turn and Talk,” and “Compare and Connect.” Grade-level word problems encourage students to solve problems using equations and alternative strategies, identify mathematical patterns, create pictorial models to justify solutions, and communicate their reasoning using precise mathematical language. By engaging in meaningful discussions and written explanations about the effectiveness of different strategies, students develop conceptual understanding, mathematical discourse, and the ability to justify why they can use multiple representations and solution methods 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 engage in solving problems, creating mathematical representations, explaining their thinking in writing, and participating in collaborative discussions. Instructional routines and mathematical discourse opportunities encourage students to clarify their reasoning, ask questions, and refine their understanding through meaningful interactions with peers and educators.

Throughout grade 4, students use hundred grids, base 10 models, arrays, strip diagrams, equations, and visual models to solve problems and communicate mathematical thinking. Lessons require students to represent decimals and fractions, solve multiplication and division problems, connect mathematical contexts to symbolic notation, and explain how equations and models represent their solutions. Students write explanations about what their models represent, narrate how they used specific strategies, justify the reasonableness of solutions, and describe how mathematical relationships support their thinking. These opportunities allow students to actively engage in mathematics and communicate their understanding through written responses and multiple representations.

The materials consistently embed opportunities for mathematical discussion through routines such as “Turn and Talk,” “Collect and Display,” “Share Ideas,” “Three Reads,” and “Mathematical Language Routines.” Students explain how equations represent problem situations, discuss efficient strategies, identify and correct errors with partners, critique and revise explanations, and refine their reasoning through collaborative conversations. By solving problems, writing about mathematical processes, and engaging in meaningful discussions with peers and educators, students develop conceptual understanding, mathematical reasoning, and the ability to communicate mathematical ideas effectively.

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 4, students explain and justify their mathematical thinking using fraction models, area models, equations, place-value representations, and visual models. Lessons require students to describe how they modeled situations, compare different solution strategies, explain why answers make sense, and support conclusions with evidence and reasoning. Students also participate in “Project-Based Learning” experiences that require them to justify decisions, write proposals, and defend solutions using mathematical representations and academic vocabulary. 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,” “Collect and Display,” “Compare and Connect,” “Critique, Correct, and Clarify,” and “Stronger and Clearer Each Time” to support student reflection and mathematical discourse. The materials guide educators to ask follow-up questions, address misconceptions, prompt students to defend their reasoning, and encourage them 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 prompts and guidance 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 rather to use errors as opportunities for discussion, reflection, and deeper mathematical reasoning.

Throughout grade 4, the materials anticipate common misconceptions related to fraction operations, multiplication and division relationships, place value, regrouping, and interpreting remainders. Educator guidance includes “Spark Discussions,” “Addressing Common Errors,” and “Common Error” supports that identify likely mistakes, explain why students may make those mistakes, and provide targeted questions, visual models, and concrete examples to address misunderstandings. Students revise models, analyze relationships among quantities, regroup numbers in different ways, and explain why remainders may represent fractional parts of a whole. The materials also provide possible student responses and follow-up prompts that help educators extend learning and encourage students to justify and compare their thinking.

The materials consistently support educators in facilitating meaningful mathematical discussions through reflective questioning, corrective instructional strategies, and evidence-based reasoning routines. Educators ask probing questions, address misconceptions in real time, emphasize key mathematical concepts, and help students connect new learning to prior knowledge. By anticipating misconceptions and providing purposeful explanatory feedback, the materials equip educators with ways to respond effectively to student reasoning and strengthen students’ mathematical understanding, communication, and problem-solving skills.