Math Journeys Texas, Grade 4

Evaluation for 1.1a
Materials include an alignment guide outlining the TEKS, ELPS, and concepts covered, with a rationale for learning paths across grade levels (vertical alignment) and within the same grade level (horizontal alignment) as designed in the materials.

The materials in Math Journeys include a TEKS Correlation Guide outlining the TEKS corresponding to each of the lessons, including both the content and process standards for educators. Additionally, the scope and sequence includes both the TEKS and ELPS organized into different sections called “Journeys” in the program.

The Journey Instructional Guide explains how each journey connects to prior and future learning through both vertical and horizontal alignment. For example, the guide describes how the grades K–2 journey focuses on identifying numbers and counting from 1–20 then expands into counting, comparing, ordering, and estimating numbers through 100 in grade 1. The guide also explains how instruction progresses from concrete representations to increasingly abstract models of number sense.

Evaluation for 1.1b
Materials include an implementation guide with usage recommendations and strategies for effective educator use in various contexts, such as just-in-time supports, advanced learning, or as a course.

The materials in Math Journeys include a Program Guide which provides comprehensive guidance for implementation, including a three-step framework of “build, consolidate, and practice.” Each lesson starts with an educator-led small group lesson followed by logsheets for students to complete independently with educator feedback. There is the option for additional practice in Atlas Math, an adaptive digital practice platform. Additionally, the “Lesson Internalization and Planning” section provides guidance on planning for learner variability, including explanations for multiple entry points, just-in-time supports, extensions, and multiple representations and modes of expression.

The Journey Instructional Guide explains how each journey connects to prior and future learning through both vertical and horizontal alignment. For example, the guide describes how the grade K–2 journey, focused on identifying numbers and counting from 1–20, expands into counting, comparing, ordering, and estimating numbers through 100 in grade 1. The guide also explains how instruction progresses from concrete representations to increasingly abstract models of number sense.

Evaluation for 1.1c
Materials include a TEKS correlation guide with recommended skill entry points based on diagnostic assessment results.

The materials in Math Journeys include a TEKS Correlation Guide with the recommended skill entry points based on the diagnostic assessment results. The journeys and lessons align to the TEKS content and process standards. The TEKS Correlation Guide provides educators with guidance for implementing instruction by connecting lessons and activities to specific standards within each journey.

The placement test report sample for grade 4 provides the TEKS assessed on the diagnostic assessment and proposed grouping structures for students. The “Proposed Group Distribution” section of the report suggests grouping based on the assessment data and a recommended entry point for each group. The report also includes a page for each student which provides a snapshot of the student’s current understanding and a suggested entry point for each individual student. The materials do not include a sample grade 4 report.

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

The materials in the Math Journeys Program Guide, “Lesson Internalization and Planning” section, include protocols with corresponding guidance for both unit and lesson internalization. The “Preparing for a Journey” section provides unit internalization guidance by helping educators to understand how the journey builds on prior learning. The Program Guide also allows educators to review lesson objectives and mastery criteria, identify scaffolds and enrichment opportunities, gather materials, and analyze learning progressions before instruction.

Additionally, the Program Guide, “Lesson Internalization and Planning” section, includes three internalization guides to use with each component of the lessons. The “Preparing for a Building Session,” “Preparing for a Consolidating Session,” and “Preparing for an Additional Practise Session” sections guide educators in internalizing the lesson to clarify the objectives, anticipate student thinking and misconceptions, prepare materials, identify formative assessment opportunities, analyze student responses, and use progress-monitoring data to inform future instruction.

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

The materials in Math Journeys include resources to help instructional leaders with the implementation of the program. In the Program Guide, the “Hands on Partnership” section provides examples of how a math success manager can partner with and provide guidance for instructional leaders. The examples include “facilitation of [professional learning community] sessions, student work and progress monitoring analysis, aligning instructional practices, and developing a long-term tier two math model.”

Additionally, the on-demand video resources in the training materials provide guidance on specific strategies like “Let’s Add by Breaking Down.” The videos can help educators and leaders who may need more guidance on a specific strategy. The Math Journeys Coaching Guide includes look-fors, predictable struggles, and ready-to-use rubrics for instructional leaders during observations. These materials are broken into four phases for implementing the intervention including “Getting Math Journeys Going,” “Shifting the Instruction,” “Adapting to the Classroom,” and “Assessing Learning.” For example, the look-fors clearly outline that in the “Shifting the Instruction” phase “conclusions and formalizations come after the exploration not before.” This level of specificity provides guidance for instructional leaders to support educators with implementation.

Evaluation for 1.2a
If designed to be static, materials include detailed lesson plans with learning objectives, teacher and student materials, lesson components with suggested timeframes, and assessment resources aligned with the TEKS and ELPS.

Detailed lesson plans in Math Journeys include the TEKS-aligned learning objectives, student materials, educator resources, and lesson components. For example, Journey C2, station 4 identifies the learning objectives, “establish connections between quotative division and partitive division” and “develop understanding of the relationship between division and multiplication when one factor is unknown.” In the Program Guide, the “Inside a Journey” section provides suggested timeframes for lesson components. Additionally, the C2 Lesson-Level ELPS Companion includes ELPS alignment. A table provides an ELPS focus, language objective, language demand, expected oral output, and expected written output for each lesson. For example, Journey C2, station 4 provides sentence stems stating, “I made ___ groups of ___ with ___ left over. / We shared ___ into ___ groups, so each group got ___.”

The Program Guide details assessment resources that include diagnostic and progress-monitoring tools that support instructional decision-making. The digital practice progress reports identify student performance for each of the TEKS, while formative assessments such as logsheets, embedded lesson assessments, and end-of-journey assessments provide additional measures of student understanding. The answer key provides the TEKS and ELPS alignment for each question.

Evaluation for 1.2b
If designed to be adaptive, materials include responsive learning objectives, lesson components with suggested timeframes, and assessment resources aligned with the TEKS and ELPS.

Math Journeys includes adaptive lesson plans that provide TEKS-aligned learning objectives, student materials, educator resources, and lesson components. For example, Journey C2, station 4 includes the objectives, “establish connections between quotative division and partitive division” and “develop understanding of the relationship between division and multiplication when one factor is unknown.” The lesson plan identifies adaptive activities through the “Activities Added to Individualized Practice After Session Completed” section. Additionally, in the Program Guide, the “Inside a Journey” section provides suggested timeframes. While the lesson plans include key vocabulary and cognates, the Lesson-Level ELPS Companion provides additional ELPS alignment and support. For example, a table includes the ELPS alignment for each adaptive lesson stating, “3.C, the student follows the spoken prompt, and 3.C, the student comprehends the written prompt.”

The materials include multiple assessment resources described within the Program Guide, including diagnostic and progress-monitoring tools. The digital practice progress reports provide TEKS-aligned assessment data by identifying student performance on each of the TEKS standards. This data supports instructional decision-making and monitors student growth. Additionally, the formative assessments called logsheets, embedded in the lesson plans and the end-of-journey assessments, align to the TEKS and can be found in the answer key.

Evaluation for 1.2c
Materials contain support for families in Spanish and English for each unit, with suggestions on supporting the progress of their student(s).

Each journey includes a family letter that provides information about student learning, key vocabulary from the unit, questions families can discuss at home, and suggested practice activities to support student progress. For example, the Journey C1 family letter recommends a simple array-building activity using buttons, coins, beans, or crackers and prompts families to discuss rows, columns, and totals. The platform translates the family letter, which is available in both Spanish and English to support communication with families.

Family support resources include a family letter for each cluster (A–F) in both English and Spanish. The “What Are We Achieving Together?” section explains the learning within the cluster and highlights instructional models and strategies. In addition, the “How Can We Help from Home?” section provides games and activities for families to complete at home. For example, Family Letter C, “Multiplication Thinking,” includes suggestions for practicing with the area model and multiplication diagram and provides both a visual representation and directions for the activity.

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

In the Program Guide, the “Assessment and Progress Monitoring” section includes the definition and intended purpose of the assessments. For example, the materials explicitly define the diagnostic assessment as a tool used at the start of instruction to identify prior knowledge and to determine in which journey to place students. Formative assessments, such as warm-ups, logsheets, and embedded checks for understanding, can inform real-time instructional adjustments. Finally, the materials define summative assessments, such as the end-of-journey tests and cumulative progress reports, as opportunities to evaluate students’ mastery of targeted skills and determine readiness for advancement within the instructional pathways.

The Placement Test Implementation Guide provides further details of the definition and purpose of the diagnostic or placement test. For example, the guide states, “The placement test is designed to provide educators with a clear and actionable understanding of students’ mathematical knowledge and skills.” Additionally, the materials explain that educators can use the placement test for “identifying unfinished learning, setting instructional priorities, and determining which students may benefit from targeted instruction.”

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

The materials in Math Journeys include guidance to ensure consistent and accurate administration of the instructional assessments. The Placement Test Implementation Guide provides clear, detailed guidance for educators on administering the diagnostic assessment consistently and accurately. This guidance includes the estimated test length, the instructor’s role during administration, and the suggested materials. For example, the “Administration Guidance” section clarifies that “In grades 3–5, students complete the assessment independently in a self-paced format” and that “paper and pencil are recommended to support student thinking and working memory during the assessment.” This level of detail ensures that all educators administer the diagnostic assessment in a consistent and accurate manner across classrooms.

The End-of-Journey Test: Instructions and Guide provides guidance for the consistent and accurate administration of instructional assessments. This resource provides standardized proctoring protocols, including recommended timeframes and materials, alongside specific interpretation tables to ensure mastery.

Evaluation for 2.1c
Digital assessments include printable versions and accommodations, including text-to-speech, content and language supports, and calculators, that educators can enable or disable to support individual students.

Educators can access the end-of-journey and formative assessments in both digital and print versions. For example, in grade 4, the C4 end-of-journey assessment includes an End-of-Journey Test: Instructions and Guide that provides a link to download printables including the C4 end-of-journey assessment, answer key, and instructions guide.

The materials do not allow educators to enable and disable text-to-speech, content and language supports, or calculators for individual students. Students can access text-to-speech and the educator can switch the language of the assessment to English or Spanish. In the Program Guide, the “Accommodations in Assessment” section states that the assessment platform provides embedded text-to-speech functionality, content and language supports, and specific guidance as to when students can use calculators. For example, the guide states, “Calculators are not provided within the digital interface; instead, they are provided and managed by educators during test administration.” Additionally, the content and language supports provide the ability to switch the language between English and Spanish, but do not provide other content or language supports. The guide clearly explains the embedded accommodations, but does not allow the educator to enable or disable the accommodation.

Evaluation for 2.1d
Materials include diagnostic assessments with TEKS-aligned tasks or questions, including interactive item types with varying complexity levels.

The materials include diagnostic assessments with TEKS-aligned tasks or questions. Additionally, a sample diagnostic assessment and placement report for grade 4 provides results for 24 distinct assessment tasks mapped directly to TEKS 4.2–4.5 and mathematical process standards 4.1.A–4.1.G.

The Placement Test Implementation Guide provides evidence of item types and different levels of complexity for grades 3–5. For example, the assessments include multiple-choice, numerical-response, and open-response questions. The guide includes an example of each question type along with a visual and an explanation of each item type. The guide also describes assessment questions at different levels of complexity, including Depth of Knowledge (DOK) 1, Factual Knowledge; DOK 2, Procedural Knowledge; and DOK 3, Reasoning and Problem Solving.

Evaluation for 2.1e
Materials include a variety of formative assessments with TEKS-aligned tasks or questions, including interactive item types with varying complexity levels.

The materials include a variety of formative assessments with TEKS-aligned tasks including interactive item types with varying complexity levels in the logsheets for each lesson. For example, the logsheets in Journey C4 include text entry, numeric entry, writing equations and completing visual representations. The logsheets also require students to compare equations, answer word problems, and think critically around division. For example, the logsheet for Journey C4, station 14 provides questions with numbers shown on dice and asks students to “Write the division equation with the greatest remainder and the division equation with the smallest remainder in each case.” The logsheet for Journey C4, station 15 includes a word problem that asks students to write a sentence for the problem, “In the auditorium, there are 98 seats arranged in rows of 14 seats each. How many rows of seats are there?”

A digital assessment item bank includes questions from the digital formative checkpoints that educators can view. The questions span more than two levels of complexity and progress from basic arithmetic recall to multistep fractional word problems and comparative justification tasks. For example, Journey Test C4 asks the open-response question, “What is the result of this operation? 136 divided by 4.” In contrast, Journey Test D1 asks a multiple-choice question with four number lines, “Which number line has a point plotted to represent 6/2?” Furthermore, these checkpoints feature more than two unique interactive item formats, incorporating standard single-select buttons, multi-select criteria boxes, and interactive text-entry fields containing custom stacked fraction lines.

Evaluation for 2.2a
Instructional assessments include scoring information and guidance for interpreting student performance, including rationale for each correct and incorrect response.

The materials in Math Journeys include a placement test report sample for grade 4 providing detailed scoring information and guidance for interpreting student performance. For example, the “Results: General Overview” section shows a colorful graphic comparison of low, medium-low, medium-high, and high percentages of students in the program, the region, and the class. The report includes student-by-student scoring information with a chart that shows student performance by content strand and process standard. Additionally, the recommended starting journey includes guidance for interpreting student performance along with suggested instructional focus points for each student.

Each end-of-journey test includes a downloadable answer key that includes rationales for correct and incorrect responses. For open-ended item types the answer key also provides rationales for common incorrect answers. In grade 4, the answer key for the end-of-journey test for C2 provides the TEKS 4.4.F for question 1 and provides the correct response as 34 with the rationale, “Because 4 × 34 = 136, 136 ÷ 4 = 34.” The answer key then outlines incorrect answers of 30 and 33 and a rationale which states, “These quotients are too small because 4 × 30 = 120 and 4 × 33 = 132, both less than 136.”

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 placement test report sample provides information that supports the educator in using the results from assessments to support the purposeful planning of tasks and activities for the appropriate TEKS. For example, the report includes a chart that lists the TEKS, the process standard, the percentage correct for the class, and the percentage correct for students using the program. Educators can use the class trends to prioritize the necessary TEKS. Additionally, the “Placement in Journeys” section groups students in the class into proposed group distributions and provides an appropriate journey entry point.

Each end-of-journey assessment includes an End-of-Journey Test Answer Key and Formative Guide with explicit guidance for educators and specific instructional activities to address the student trends and misconceptions identified on assessments. In the assessment guidance matrix for the C3 test, the text connects trends in multi-digit multiplication performance to targeted instructional next steps. For example, to address trends in which students omit place value or partial products in multi-digit equations (such as multiplying two-digit by two-digit numbers), the materials instruct educators to use area models, base 10 blocks, or place value charts to clearly isolate and reinforce each partial-product step.

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 a general progress report for the digital practice portion that allows educators to track student progress and growth over time and overall performance by topic and student expectation. For example, the report provides information including total hours spent, the number of activities worked on, the overall grade, and the grade by the TEKS strand for each student.

The materials do not include tools for students to track their own progress and growth over time. Students complete their journey through the digital Atlas Math Applet Library, which provides specific activities for students. A vertical progress bar appears immediately after students complete one of the journeys. However, the progress bar disappears as soon as students complete the activity, which limits students’ ability to track their own meaningful progress and growth over time.

Evaluation for 2.2d
If designed to be static, materials provide prompts and guidance to support educators in conducting frequent checks for understanding at key points throughout each lesson or activity.

The materials in Math Journeys provide prompts and guidance to support educators in conducting frequent checks for understanding throughout each lesson. For grade 4, in Journey C4, station 2, the materials include embedded guidance for educators to check for student understanding at key points during instruction. In activity 1, students explore dividing 35 counters that represent cows into 5 stables, each with 7 cows. The expandable prompts guide educators to ask, “What does it mean to distribute the tiles equally?” and “Could you explain your strategy for making sure each sheet gets the same number of tiles?” This check for understanding allows educators to correct any misconception before continuing with the remainder of the lesson.

Additionally, the materials include scripted questions and PowerPoint presentations for educators. For example, in Journey B5, station 17, “Productive Practice: Addition and Subtraction II,” the goal of the lesson is to strengthen students’ additive operations. The lesson begins with prompts for educators to ask students to draw 2 abacuses with 5 columns. Additional prompts guide the educator to ask students to represent 1,017 on the first abacus and then multiply by 10 and represent that number on the second abacus to compare the representations. Additional lesson guidance explains that the goal is for students to notice that, because you multiplied by 10, the value of each digit in the second result represents 10 times what the same digit represents in the first result.

Evaluation for 2.2e
If designed to be adaptive, materials provide frequent checks for understanding at key points throughout each lesson or activity.

The materials provide frequent checks for understanding during digital practice in the Atlas Math Applet Library, but do not adjust the level of difficulty based on students’ responses. For example, in applet A105, “Real World Division Situations,” students complete a division word problem supported by sentence stems such as, “The flowers will be planted in ___ pots,” “After planting, she will have ___ flowers,” “In total she planted ___ flowers,” and “In each pot she has to plant ___ flowers.” Students receive immediate feedback through a progress indicator that displays a green check mark for correct responses and a question mark for incorrect responses. However, the next question does not change based on student performance, and the activity continues with the same sequence of questions.

The digital materials provide linear interactive checks for understanding with scaffolding hints but are not adaptive, as they fail to dynamically alter lesson progression, task complexity, or mathematical rigor in response to student errors. For example, when logged in as a student completing the grade 4 digital division task applet A118.1, “Long Division,” intentionally submitting incorrect inputs triggers sequential text scaffolding hints related to multiplication facts, but the underlying problem structure remains identical. If a user inputs wrong choices across multiple steps, the system permits three distinct correction attempts per procedural stage before automatically generating and displaying the correct numerical answers for the user. Because the platform resolves the mathematical division steps for the student rather than recalibrating numerical rigor, modifying pacing, or deploying automated visual models inside the active learning window, the materials fail to deliver true adaptive personalization across their grade 4 digital activities.

Evaluation for 3.1a
Materials include explicit educator guidance for lessons or activities scaffolded for students who have not yet reached proficiency in prerequisite or grade-level concepts and skills.

Throughout the lesson plans, the materials embed scaffolded supports through instructional support boxes designed for students who have not yet reached proficiency. In Journey C2, station 3, the “Wrap-Up” section includes clickable support boxes that provide targeted intervention strategies for educators. For example, the support boxes prompt educators to “Ask students to write the number of rows and columns that make up the set of dots” to reinforce both procedural and conceptual understanding. The materials also suggest tactical modifications, such as directing educators to “Provide students with colored tiles to represent each rectangular arrangement associated with each multiplicative frame.” These embedded supports give educators immediate access to differentiated instructional strategies.

The materials also provide explicit educator guidance through structured lesson activities. The grade 4, Journey C3, station 7 supports multidigit multiplication for students working toward grade-level proficiency by directing educators to use an area model grid to decompose 14 × 16 into four smaller sections (100, 60, 40, and 24). The Mathematical Language Routine (MLR) 7 instructs educators to ask students to connect each section’s dimensions to its area and explain how the partial products combine to equal 224. Additionally, support indicator 1E guides educators through specific questioning sequences, including “Could you explain how breaking the grid into sections helps you solve the operation?” and “Do you think there might be another way to organize these sections?” The materials then direct educators to repeat these same steps with additional examples, including 17 × 17 and 19 × 11. This supports a scaffolded transition from visual representations to abstract numerical equations.

Evaluation for 3.1b
Materials include explicit educator guidance for language supports, including pre-teaching and embedded supports for developing academic vocabulary and unfamiliar references in text.

Each lesson begins with a “Key Vocabulary” section and includes a Journey Language Bank for every journey to support the development of academic language. The “Key Vocabulary” section presents key content vocabulary, English–Spanish cognates, false cognates, and multiple-meaning words. Embedded within the materials, the Journey Language Bank promotes the development of academic vocabulary through routines, discourse, and writing activities. In grade 4, in the C4 Journey Language Bank, the pre-teach/embedded language routine support bank provides three routines that identify when to use the routine, what actions educators should take, and the lessons where each routine is most effective. For example, in lessons C4.1–C4.5, educators should task students with routine 1, “Name the Division Situation Before Solving,” before students begin “a sharing or grouping task.” Educators should ask, “What is known, what is found, and whether the problem is about the number of groups or the amount in each group.” The guidance directs educators to “Have students restate the situation with precise language before they compute.” This routine supports educators in developing students’ academic vocabulary as it relates to solving division problems.

In grade 4, lesson D2.4, “Exploring Multiples,” embeds language supports that address unfamiliar references as they arise during instruction. The lesson incorporates the MLR6, “Three Reads,” protocol at the point of use. Educators should display a real-world calendar during the first read while asking questions such as “What dates are we looking at?” and “What do you notice about the day of the week?” These prompts guide educators in facilitating conversations about the calendar’s narrative and physical structure across tier-differentiated language proficiency tracks before students begin mathematical computation. Additionally, in the Journey Language Bank, the “Unfamiliar References” section provides a “Quick Bridge Before the Lesson” section that guides educators on supporting unfamiliar references. In Journey C1, station 5, the word order has multiple meanings and is an unfamiliar reference. The materials suggest to “Write 6, 12, 18 on the board and point from left to right. ‘Here order means smallest first, then bigger. It is not an order you give to a person.’ Students read the three numbers in order and say the smallest first.”

Evaluation for 3.1c
Materials include explicit educator guidance for enrichment and extension activities for students who have demonstrated proficiency in grade-level and above grade-level content and skills.

The lessons in Math Journeys include extension tabs with enrichment activities. For example, in the grade 4, Journey C4, station 2, “Wrap-Up” section, the extension tab provides a task that states, “Ask students to distribute 19 colored tiles into 2 groups, 3 groups, 4 groups, and so on.”

The Enrichment and Extension Guide provides guidance for which students receive extension work and how to facilitate each extension logsheet as well as the extension tips embedded in the lessons. For example, the Journey C3 Enrichment and Extension Guide provides tables for the nine lessons within the journey that include extension logsheets. The table includes information on the intent of the task, what to listen for, and defines what counts as mastery. Additionally, in Journey C3, station 5, the table explains that the task is for students to place number cards in each blank resulting in an even product. Then students place results on the number line and consider “which part of the number line has the most results?” The information in the table guides educators to listen for “Parity talk before any multiplying: which card placements guarantee an even product and why? Then magnitude talk at the line. Students explain why the results crowd into one region in terms of the size of the factors the cards produced.” This guide provides explicit guidance for enrichment and extension activities.

Evaluation for 3.1d
Digital materials include accommodations, including text-to-speech, content and language supports, and calculators that educators can enable or disable to support individual students.

The materials include digital assessments and applets. Students can access text-to-speech accommodations and digital help supports in Atlas Math, but educators cannot enable or disable the supports to assist individual students. The Program Guide explains the content and language supports for changing the language between English and Spanish, but provides educators with no other content and language supports. Additionally, the Program Guide states, “Calculators are not provided within the digital interface; instead, they are provided and managed by educators during test administration.”

Additionally, the lessons in the applet include a speaker button for text-to-speech and a help button to provide assistance when needed. These features are available to all students. However, educators cannot enable or disable supports for individual students.

Evaluation for 3.1e
Materials include educator guidance on offering options and supports for students to demonstrate understanding of mathematical concepts in various ways, such as perform, express, and represent.

In grade 4, Journey C4, station 6 creates multiple opportunities for students to demonstrate their understanding of the lesson topic through discussion, manipulatives, and writing. The “Warm-Up” section includes guided discourse and a blue educator-support box that provides explicit facilitation guidance. Students model long division with manipulatives, while educators receive step-by-step support to help students understand distribution as division. For example, the guidance states that educators should “Accept any ideas that involve distributing more than 1 snap cube in each distribution. While some distributions allow for grouping more snap cubes, encourage students to select the distribution they find most comfortable. The main goal of this activity is not to find the most efficient strategy, but to understand how the distributions are represented.” The lesson also incorporates written responses, with the final logsheet prompt asking students, “What do each pair of division equations have in common?”

Across multiple modalities, grade 4, Journey D2, station 5 embeds educator guidance that allows students to demonstrate their understanding through verbal explanations and visual representations. Process standard support boxes, including standard 1F, direct educators to ask students to analyze relationship chains and choose whether to explain or map how numbers are related. To facilitate these modalities, the lesson plan scripts exact questions for educators, including, “Could you explain how the numbers in your chain are related to each other?” and “In the chain 12 – 4 – 20, what factor-and-multiple relationship does the 4 have with each of its neighbors?” These embedded prompts provide educators with structured support for eliciting and assessing student thinking.

Evaluation for 3.2a
Materials include explicit (direct) prompts and guidance for educators to build knowledge by activating prior knowledge, anchoring big ideas, and highlighting and connecting key patterns, features, and relationships through multiple means of representation.

In grade 4, Journey D2, station 6 uses a warm-up with explicit prompts and guidance to help educators anchor big ideas and activate prior knowledge. Students explore factors by examining hundreds charts that display multiples of three and identifying patterns that emerge when the charts are rearranged. Before defining factors formally, students connect these observations to their understanding of multiples, prompting them to draw on prior knowledge as the lesson concept develops.

The cluster overview C, “Multiplicative Thinking,” establishes a conceptual framework for multiplicative thinking which outlines the cognitive shift from additive reasoning to multiplicative scaling. The overview connects big ideas through multiple representations by directing educators to use rectangular arrays and fixed-interval jumps on a number line to visualize multiplication as iterated addition. In addition, dedicated text boxes containing mathematical background information identify specific learning milestones and guide educators through a progression that includes introducing multiplication tables, connecting inverse operations through multiple strategies, and transitioning students from visual models to the abstract standard algorithm.

Evaluation for 3.2b
If designed to be static, materials include educator guidance for effective lesson delivery and facilitation using various instructional approaches.

Math Journeys incorporates varied instructional approaches throughout, including hands-on exploration, whole-group and small-group discourse, and independent consolidation. Lessons move from guided discovery, in which educators facilitate exploration, to formalization of mathematical concepts. For example, in grade 4, Journey D3, station 9 engages students in representing decimals and fractions on a number line by exploring intervals and responding to prompts such as “Do you notice a pattern in how the fractions are spaced on the number line?” and “Could you explain how understanding the structure of the number line helps you locate where the red flag should be placed?” Together, these prompts and activities support students in formalizing representations of decimals and fractions on a number line.

Each lesson plan provides educators with explicit directions and guidance to make each lesson effective and engaging. For example, Journey C4, station 6 begins with a warm-up using a whole-group discussion. The development activity involves whole-group instruction using manipulatives to model division. The final section is the wrap-up, where students practice individually on logsheets. Each section of the lesson provides explicit guidance for the educator with step-by-step instructions and prompts.

Evaluation for 3.2c
Materials include multi-tiered intervention methods for various types of practice and structures and educator guidance to support effective implementation.

Math Journeys incorporates multiple tiers of intervention through varied practice structures and embedded educator guidance. In grade 4, Journey D2, station 6 provides opportunities for students to practice finding factors independently through logsheets and collaboratively through whole-group development activities in which students make rectangles and identify their factors. Embedded blue support boxes help educators to facilitate these activities. For example, one blue box notes, “Students may still be more familiar with the concept of multiple than with factor. If so, they might rephrase the previous observation by indicating that the area is a multiple of the rectangle’s dimensions.” These embedded supports help educators implement the lesson effectively while addressing student misconceptions.

The materials provide small-group, collaborative, and individual intervention methods across various types of practice and instructional structures to support struggling learners. For example, Journey D2, station 8 explicitly splits the core content into segmented learning environments to deliver targeted, small-group remediation alongside universal independent settings. Within the small-group structure, the curriculum alters the type of student practice by providing diverse tactile and linguistic scaffolds. These include dry-erase magic boards to map out common factors using a problem-solving model and multilevel language sentence stems. In the independent practice phase, the instructional approach transitions to a specialized, scaffolded-support version of the standard student logbook. This enables struggling learners to receive further multitiered practice variations. Students use devices to reference a digital number chart widget and color code common multiples before transferring calculations to the page. Furthermore, the materials provide the required educator-facing implementation guidance by embedding specific scripts, explicitly directing the educator to facilitate process standards 1F and 1B using exact guiding prompts such as “Is there any pattern in how often the common multiples appear?” and “What strategy did you use to decide which numbers are factors of both 18 and 30?” These prompts actively scaffold student understanding through the intervention tasks.

Evaluation for 3.2d
Materials include enrichment and extension methods that support various forms of engagement, and guidance to support educators in effective implementation.

The materials provide clickable extension tabs throughout the lessons. In grade 4, Journey C4, station 6, the instructions guide the educator to extend the practice of dividing as they “Ask students to check if they get the same remainder when dividing by 9 as when dividing by 3. The answer is yes.” These tabs, embedded in the wrap-up, provide guidance on enrichment and extension. Additionally, most lessons provide an extension version of differentiated logsheets.

For some lessons Math Journeys includes extension logsheets for enrichment while other students work on the traditional logsheet. These extension activities incorporate more puzzle-like tasks that encourage productive struggle and student discovery. For example, in Journey D2, station 6, the extension logsheet asks students to read about perfect, deficient, and abundant numbers. Students then classify numbers into these categories and identify patterns among them. These enrichment opportunities promote deeper mathematical thinking through exploration and pattern discovery.

Evaluation for 3.2e
Materials include prompts and guidance to support educators in providing timely feedback during lesson delivery.

Embedded throughout Math Journeys, educator prompts and guidance support educators in providing timely feedback during lesson delivery. Within each lesson, specific prompts and facilitation guidance enable educators to pause at key points to check for understanding, address misconceptions, and provide immediate, meaningful feedback. For example, in Journey C4, station 4, the warm-up presents four division equations and directs educators to “ask students to justify why each of the four items could be the one that doesn’t belong.” The lesson also provides a sample response for each equation and specifies how students should respond by noting, “When justifying which one doesn’t belong, expect students to provide multiple answers.” These embedded supports equip educators with the guidance needed to deliver responsive instruction and keep students progressing toward the lesson objectives.

In section 4, “Anticipated responses/misconceptions/redirection bank” of the journey language bank, the materials include lesson-specific misconceptions and provide guidance on how to respond with feedback and correction. For each misconception, the educator receives its possible meaning and suggested redirection. For example, in Journey C4, station 5, a common misconception is that a “student gives only the quotient and ignores whether the remainder matters in the context.” The materials suggest that this is because “the student may see the equation but not yet interpret the story,” and instruct the educator to ask “What does the remainder mean here?” and determine together how to interpret the remainder.

Evaluation for 3.3a
If designed to be static, materials include educator guidance on providing and incorporating linguistic accommodations for all 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.

Some lessons provide embedded linguistic accommodations for all levels of the ELPS. For example, in Journey D4, station 3, development activity 2, the lesson embeds emergent bilingual (EB) student supports. The lesson asks students if the two students in the problem will eat less than a whole quesadilla and requires them to write the amount they eat together as an equation. The EB student supports state to keep the quesadilla pieces visible and provides expected actions or statements for each proficiency level aligned to pre-production, beginning, intermediate, high intermediate, and advanced. For example, the supports state that students at the pre-production level may “point to each sixth of the quesadilla while a partner counts.” A student at the advanced level may say, “___ sixths is greater than one whole because the numerator exceeds the denominator.”

Additionally, each journey includes a Lesson-Level ELPS Companion that organizes key language components in a table. The table identifies the ELPS focus, language objective, language demand, expected oral output, expected written output, and suggested use within each lesson. This resource offers language-development guidance throughout each journey.

Evaluation for 3.3b
If designed to be adaptive, materials include embedded linguistic accommodations for all 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 digital resources incorporate accessibility features, including text-to-speech, visuals of manipulatives, and a support button for content. However, they do not provide linguistic accommodations. For example, in applet 123, “Interpreting Fractions Graphically,” when students misidentify an equivalent fraction or the colored part, clicking the help button displays a green line that partitions the shape into the required number of parts. Students can use this visual support to identify an equivalent fraction. However, the materials do not include linguistic accommodations when students do not understand due to language barriers.

In the Program Guide, the “Overview” section explains the structure as requiring the educator to provide accommodations during the “Build” and “Consolidate” portions of the lesson so that the student learns the academic vocabulary before they reach the adaptive portion of the lesson. Therefore, there are no linguistic accommodations embedded in the adaptive material.

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

To support implementation with EB students, Math Journeys includes a Lesson-Level ELPS Companion for each journey. According to the materials, educators can use this guide to “add a compact ELPS layer without crowding the main guide.” The document identifies the language objective, language demand, expected oral and written outputs, and suggested usage. For example, in Journey C4, station 5, “Division Word Problems”, the Lesson-Level ELPS Companion recommends that educators “Use this section to keep productive practice language rich. Prompt students to explain not only what they did, but why that first step was efficient.” The guide also identifies the expected oral output as, “I distributed 5 cubes to each group first because 5 x 8 is easy.” This targeted guidance supports educators in effectively implementing the materials with EB students.

The materials include a Texas Bilingual/ESL Implementation Guide with examples and structured instructions for integrating each state-approved model in the classroom. Across the elementary grade band, the curriculum provides a dedicated, standalone Texas ESL/Bilingual Implementation Guide to help educators navigate the distinct structural and pedagogical demands of the state-approved language programs. The guide outlines clear, actionable educator-facing strategies, explicitly detailing distinct instructional pathways and facilitation methods for ESL content-based, ESL pull-out/inclusionary coordination, transitional bilingual early exit, and transitional bilingual late exit program models. By providing customized classroom implementation blueprints tailored to these state-approved frameworks, rather than generic, one-size-fits-all directives, the materials fully equip educators to promote language development alongside mathematical content mastery.

Evaluation for 3.3d
Materials include embedded guidance 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 in each lesson to support EB students in developing academic vocabulary, increasing comprehension, building background knowledge, and making cross-linguistic connections through both oral and written discourse. In grade 4, Journey C4, station 4, development activity 1 includes linguistic support with prompts to develop oral language. The guidance provided explains that students distribute tiles for division and “partner pairs say aloud what the new numbers mean. For example, partner pairs would say ‘We give 5 tiles to each person,’ ‘We distribute 20 tiles in total,’ and ‘55 tiles are remaining.’” The guide then states, “Prompt students to replace everyday words with precise terms—distribute instead of given, remainder instead of left over, and quotient for the total each person receives.” This embedded linguistic guidance supports EB students to acquire language and content knowledge for success.

Additionally, the materials include embedded guidance to support EB students to develop academic vocabulary and increase comprehension through oral and written discourse. For example, Journey D1, station 12 systematically embeds MLR7, “Compare and Connect,” to build mathematical communication skills in proximity to content delivery. To increase oral comprehension, the facilitation text directs educators to first prompt students to describe what a visual fraction model shows before introducing or naming an abstract mathematical rule. This approach requires students to verbally negotiate meaning before calculating numbers. Furthermore, to develop academic vocabulary across both spoken and text-based mediums, the routine requires partner pairs to verbally speak a full, rigorous comparison statement, such as “three fourths is greater than two fourths because three parts is more than two parts.” This enables students to process the linguistic demand before physically recording the corresponding written mathematical symbol. This instructional design ensures that EB students successfully internalize the meaning and structure of fraction relationships prior to abstract procedural tasks.

Evaluation for 3.3e
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 throughout learning pathways (including instructional assessments) require students to demonstrate depth of understanding aligned to the TEKS.

Math Journeys provides practice opportunities throughout each journey and includes instructional assessments that require students to demonstrate a deep understanding aligned to the TEKS. For example, in Journey D3, station 9, “Practicing Fractions and Decimals,” students engage in progressively more complex tasks. In activity 1, students determine flag values on an equal-interval number line by reasoning about fractional distances. Next, students complete formative assessments called logsheets, showing their ability to represent and convert decimal fractions. The lesson plans explicitly include process standard 1F, “analyze relationships to connect mathematical ideas,” showing the depth required for students to analyze the relationship of intervals and fractional distances on the number line in both the practice and the assessment.

Embedded instructional activities allow students to demonstrate a depth of understanding and mastery aligned to the TEKS in the formative assessments. For example, in Journey D4, station 4, “Subtracting Fractions I,” the materials require grade 4 students to demonstrate a depth of understanding aligned to TEKS 4.3.E. The materials provide practice opportunities that begin with using models to subtract fractions with like denominators. The examples begin with students using fraction circle manipulatives to model removing parts and writing the subtraction equation. As the lesson progresses, the depth of practice opportunities increase as students subtract fractions greater than one. The materials require that the students explain subtracting from a whole in two different ways. The materials dive to an even deeper understanding when they provide practice opportunities in a real-world context. Each station or lesson includes a formative assessment known as the logsheet. The logsheet for station 4 requires students to subtract fractions at three different levels of understanding: with a given model, by creating their own model, and in writing. The logsheet, used as an instructional assessment, demonstrates the depth of understanding because it requires students to apply their understanding of subtracting fractions to a new situation rather than replicate practiced examples.

Evaluation for 4.1b
Questions and tasks, including enrichment and extension materials, increase in rigor and complexity, leading to grade-level and above grade-level proficiency in the mathematics TEKS.

Educators guide students through questions and tasks that are intentionally designed to increase in rigor and complexity, guiding students toward grade-level proficiency in the mathematics TEKS. Scaffolded lessons build conceptual understanding and support student progression toward mastery. In grade 4, Journey D4, station 6, “Reviewing Fractions on the Number Line,” students transition from using fraction bars, to constructing a zero-to-one number line, to jumping forward and backward on the number line, building proficiency for TEKS 4.3.G. In activity 1.1, students use 1/5 fraction bar pieces to construct equal divisions on a number line and explain why just five iterations produce one unit. Then in activity 1.2, students mark tenths on the same number line to explore the idea of equivalent fractions, moving forward and backward on the number line, and going past the whole. Next, in activity 2, students decompose a jump of 6/10 into two parts to pass one. Finally, the wrap-up increases complexity further with tasks in which students must analyze all possible landing points from two jumps of unknown direction and work backward from an endpoint to determine which direction to move.

Students tackle questions and tasks, including enrichment and extension materials, that grow in rigor and complexity. These tasks enable them to reach and exceed grade-level proficiency in the mathematics TEKS. Journey D3, station 8, “Ordering Decimals,” progressively builds rigor throughout the lesson. Students place decimal points to make comparisons true, then order all possible decimal numbers formed from three given digits, using systematic place value reasoning at grade level. Students advance to expressing and comparing 8.31 and 13.8 as decimal fractions (831/100 and 1,380/100), justifying why a number with more decimal places can be smaller. In task 4 of the wrap-up, students identify all valid digit arrangements, analyze why trailing zeros disqualify certain options, and order the results. Finally, the educator provides a differentiated extension logsheet to further challenge students. The logsheet requires students to compare numbers and circle the greater length, while the extension logsheet requires students to use the provided digit cards to create a number to the left and to the right, that is closest to the given whole number. This extension pushes the DOK providing the enrichment and extension to reach and exceed grade-level proficiency.

Evaluation for 4.2a
Materials demonstrate coherence across concepts horizontally within the grade level by connecting patterns, big ideas, and relationships.

The materials provide a horizontal layout that builds on previous journeys and lessons within a cluster. In grade 4, cluster D, “Rational Numbers,” links factoring numbers, divisibility, and fraction equivalence in a coherent horizontal concept to compare fractions. In Journey D3, station 4, “Comparing and Converting Fractions,” students use grid paper to determine which denominators to use to find equivalent fractions for fifteenths by seeing how they can equally divide the 15 squares into equal groups. Then, students connect factoring to creating equivalent fractions to compare fractions. This lesson directly builds on the multiples and divisibility initially introduced in Journey C3 and the conceptual understanding of equivalent fractions introduced in Journey D1 and previous lessons in “Journey D3.”

In grade 4, the lesson plans within a journey demonstrate a coherence across concepts horizontally by connecting patterns, big ideas, and relationships. For example, Journey C3 builds understanding of multiplication by beginning with models and transitioning to algorithms over 16 lessons. Lesson 13, “Multiplication through the Standard Algorithm I,” provides multiple examples of horizontal coherence of concepts. Activity 1 begins by asking students to solve a three-digit by one-digit multiplication problem using the area model they learned in lesson 6. Students then use the area model example to identify patterns and transition to partial products. Students use partial products in lesson 15 where they identify patterns and transition from partial product to standard algorithm. Students also use the same patterns identified in lesson 13 for three-digits time one-digit multiplication, and in lesson 14 where they apply those patterns to two-digits times two-digits. In this way students connect the big ideas and relationships taught in previous lessons.

Evaluation for 4.2b
Materials demonstrate coherence vertically across concepts and grade bands, including connections from grade K–6, by connecting patterns, big ideas, and relationships.

The cluster overview outlines connections from grades K–6 by providing the background knowledge students developed in previous clusters and describing how the next cluster will build upon the current cluster. For instance, cluster overview C, “Multiplicative Thinking,” includes a learning progression that explains that in the previous cluster “students have moved away from counting by ones to thinking in groups,” and “established the part-whole logic where students understood a total quantity can be broken down and rearranged.” The document then outlines that in this cluster “Students begin to see that adding repeatedly is slow, so they transition to working with equal groups.” The progression finally explains that the next cluster takes the rectangular array and skip-counting strategies from this cluster and “applies them to partitioning in sharing.” This progression shows the connecting patterns, big ideas, and relationships.

Additionally, the D3 Learning Brief demonstrates vertical coherence across multiple grade levels to support a wide continuum of mathematical learning. This document outlines a unified conceptual progression by mapping a comprehensive sequence of 21 content standards across a specific cluster, including six grade 3 TEKS, six grade 4 TEKS, three grade 5 TEKS, five grade 6 TEKS, and one grade 7 TEKS. To manage varying student entry points and address potential learning gaps, the document features a diagnostic section that explicitly balances students who have ascertained prerequisite mathematical concepts from four previously listed foundational pathways against those who require active reinforcement on core structural skills. The overview features pictures and brief lists to show the specific skills students learn across the different levels. This pictorial progression traces student expectations as they develop intermediate strategies to compare fractions with differing numerators and denominators, work with decimal fractions to connect them directly to decimal numbers, and ultimately discover percentages to perform simple calculations. By embedding this extensive vertical scope at the beginning of the pathway within the broader cluster, the materials ensure that foundational fractional reasoning scales seamlessly across multiple grade levels rather than being taught in isolation.

Evaluation for 4.2c
Materials demonstrate coherence across lessons or activities by connecting students’ prior knowledge of concepts and procedures to the mathematical concepts to be learned in the current grade level and future grade levels.

Math Journeys demonstrates coherence across lessons by connecting students’ prior knowledge of concepts and procedures to new mathematical concepts. Each lesson begins with a warm-up that activates students’ prior knowledge and connects to concepts in the current lesson. For example, in Journey D3, station 2, “Generating Equivalent Fractions: Length Model,” students build a fraction wall with fraction pieces and identify which fractions are equivalent to 2/3. This directly connects to their previous knowledge of equivalent fractions.The lesson plans explicitly guide the educator stating, “Students should suggest four-sixths and eight-twelfths because, looking at the wall, these fractions align with two-thirds at the same point.” Then, activity 1 guides students to see that 2/3 is equivalent to 4/6 because both the numerator and the denominator are multiplied by two, linking the visual model to multiplicative procedures. Finally, in activity 2 and activity 3, the lesson plans guide educators to help students recognize that equivalence is found by multiplying or dividing the numerator and denominator by the same number, linking it to simplifying fractions. This coherence in concept of finding equivalent fractions through multiplying helps students solidify their knowledge of equivalence and links to future work they will do in adding and subtracting fractions with unlike denominators.

In grade 4 the materials demonstrate coherence vertically across concepts and grades within each journey. For example, Journey C4 takes the multiplicative relationship and the multiplicative frame representation learned in grade 3 and applies it to multi-digit multiplication in grade 4. Specifically, Journey C4, station 10, activity 2 grounds students in the concept of division by using a multiplicative frame before introducing the area model for three-digit division. The activity then returns to the multiplicative frame once the quotient has been determined. The multiplicative frame is a fundamental representation throughout the entire Math Journeys program. The grade 5 materials also utilize multiplicative frames as part of Journey D5, station 9, to assist students in multiplying fractions and during Journey D5, station 11, to assist students in dividing fractions.

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

The Math Journeys materials offer spaced retrieval opportunities by revisiting previously
learned concepts across different learning pathways. In grade 4, Journey C4, station 9 asks students to retrieve multiplication facts to complete the warm-up activity by adding from a multiplication table. Furthermore, in activities 1.1 and 1.2 students apply previously learned concepts to distribute snap cubes into equal groups by applying known quotients and deriving nearby facts. Finally, activity 2 includes a word problem for students to apply division reasoning using concepts first established in Journey C2.

The lesson components provide space retrieval opportunities for both concepts and skills throughout the program. For example, in Journey D3, station 9, activity 1 requires students to use their conceptual understanding of needing like units when adding decimals. Students learned this concept previously in Journey D3, station 6. This concept is reinforced with short activities over several days. Journey D3, station 9 provides another example of spaced retrieval opportunities for the skill of comparing and ordering decimals. Students first learn this skill in Journey D3, station 8, use it again during the Journey D3, station 9, wrap-up, and again during Journey D3, station 10, activity 3.

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

The lessons in Math Journeys offer interleaved practice by integrating previously learned concepts and skills across the journeys. The materials include a cluster overview which outlines the progression of concepts and skills that connect across the journeys within the cluster. For example, the cluster plan in cluster overview C, “Multiplicative Thinking,” outlines how students first learn multiplication strategies like equal groups, skip counting, and arrays in Journey C1. Students then continue to build on these strategies and concepts by connecting them to division concepts in Journey C2. Then, the materials bridge the strategies and algorithms in Journey C3 to master multiplication strategies with interleaved practice integrating content previously learned in Journey C1. Finally, Journey C4 provides students the opportunity to master division strategies by bridging the strategies and concepts previously learned in Journey C2 to the algorithm. This cluster plan in the cluster overview outlines how the concepts connect throughout the journeys or learning pathways.

In grade 4 the materials provide routine interleaved practice opportunities by mixing notation, computation, and inequality concepts across pathways and lessons. In the Journey D3, station 6, “Introducing Decimals and Decimal Fractions,” wrap-up activity, students complete three distinct sections to practice flexible strategy selection. The first section requires students to write equivalent fractions and decimals to the hundredths place. The subsequent section immediately shifts gears, forcing students to perform active decimal addition involving tenths and hundredths. In the final section, students evaluate a colored hundredth grid model. This multi-layered task requires students to generate four separate equivalent representations in fractional tenths, fractional hundredths, decimal tenths, and decimal hundredths. Students then utilize these values to actively order the decimals using greater than and less than symbols. This rapid alternation between representation, calculation, and inequality notation weave together multiple distinct operations or skills.

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

Embedded activities throughout the lessons require students to interpret, analyze, and evaluate models and representations. In Journey D4, station 4, the warm-up requires students to interpret, analyze, and evaluate two area model fraction representations by justifying the amount of pizza both students ate. The warm-up includes two images of the pizza remaining. One is circular with three-fourths remaining and the other with three rectangular pieces of pizzas remaining, and the total amount unknown. The lesson plans provide guidance through a sample student response. Students should recognize that one-fourth of the circular pizza was eaten, but an unknown amount of the rectangular pizza was eaten because his pizza is rectangular, and its original length is unknown. The materials also provide two guiding questions, “Could you explain why we can’t compare the pieces just by looking at the fractions?” and “Does the size of the whole pizza matter when comparing fractions?”

Throughout the lessons, students actively interpret, analyze, and evaluate models and representations of mathematical concepts and situations. In grade 4 students use pattern blocks to compose and decompose fractions. Journey D4, station 2, requires students to interpret a model when asked to identify which fraction is represented by a specific pattern block in a given shape. After students practice interpreting the pattern block models the educator asks “which color in the projected shape represents a unit fraction.” Students then analyze a given set of blocks. As they become more comfortable in analyzing a given model they next create their own. In activity 1.2 students then evaluate their own model as they “explain why that shape is valid for representing the fraction assigned to them.”

Evaluation for 5.1b
Questions and tasks provide opportunities for students to create concrete models and pictorial representations to represent mathematical situations.

Math Journeys includes questions and tasks that provide opportunities for students to create concrete models and pictorial representations to represent mathematical situations throughout the lessons. For example, Journey D4, station 4, “Subtracting Fractions 1,” activity 1.1, includes opportunities for students to manipulate fraction pieces to subtract fractions with common denominators. The activity asks students to represent 9/10 and physically remove 7/10. The activity then connects the concrete model to represent subtracting with common denominators. Then, the logsheets require students to draw a pictorial representation of their fraction pieces and write the matching equation 9/10 - 7/10 = 2/10. The lesson plans include questions to support connecting the representations to the mathematical situation prompting, “Could you explain what happens to the denominator every time we subtract fractions?” and “Do you notice any pattern that helps us solve fraction subtraction?”

The progression of lessons provides ample opportunities for students to create their own models and representations. In grade 4, during Journey D4, station 6, students iterate a 1/5 fraction bar along a strip of paper 10 times to create two wholes. This activity enables students to gain familiarity with fifths, which are new to grade 4, and practice with fractions greater than one. Students are then asked to label the iteration marks thus creating a representation of a fifths number line which they then subdivide to create a tenths number line as well.

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

The materials include questions and tasks with opportunities for students to apply conceptual understanding to new problem situations and contexts. One example of a task in Journey D4, station 7, requires students to continue with the previously learned conceptual understanding of jumping forwards and backwards on a number line. Students now apply this concept to a plant growth word problem which requires students to apply fraction addition in a new measurement context. Students receive information about a plant that was 3/8 of an inch tall last week and grew 4/8 of an inch this week. Students solve “how tall is it now” with a number line that is sectioned into eighths. This lesson bridges the conceptual understanding of jumping forward by fractional amounts on a number line to real-world problems. Additionally, the lesson includes a drop down support to guide process standard 1A, “Apply mathematics to problems,” with questions such as, “Could you explain how the jumps on the Number Line diagram show the plant’s growth?” and “How does finding 3/8 + 4/8 on the Number Line diagram help you answer the question about how tall the plant is now?” The embedded questions help students apply conceptual understanding to the new context.

Students engage in lessons with opportunities to apply conceptual understanding to new problem situations and contexts. For example, in the Journey D4, station 3, “Adding Fractions 1,” wrap-up, students have the opportunity to apply the conceptual understanding that the value of a fraction depends on the unit to a new, real-world situation. The problem states that Madison claims that three glasses, each 1/3 full of juice, combine to fill exactly one whole glass. Students must transfer the concept that fractions depend on the size of the unit to determine why the claim is false when the glasses are different sizes.

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

The Atlas Math digital applet practices are embedded in the lesson warm-ups or used as extra practice for students and provide tasks designed to build students’ fluency and automaticity in performing math operations. For example, applet A289 asks students to add and subtract fractions with the same denominator across seven denominator families of halves, thirds, fourths, sixths, eighths, tenths, and twelfths. These applets build fact fluency and automatic retrieval by providing the volume of targeted practice across the previously learned skills of adding and subtracting fractions with like denominators at grade level.

The lessons incorporate regular routines and practices to build student automaticity and fluency with grade-level multi-digit calculation skills. In the Journey C3, station 8, warm-up, the lesson incorporates a dedicated multiplicative frame number string routine designed to establish automaticity. Prior to utilizing models, students evaluate five sequential frames with missing whole values. Students must instantly retrieve the basic single-digit fact of 2 x 3 from memory to scale up larger products. Students apply this automatic fact recall to determine hidden totals for 20 x 3, 2 x 30, 300 x 2, and 200 x 3. The lesson then shifts to concrete models to build strategic computational fluency. For the final multiplicative frame, the educator demonstrates how to use physical base 10 blocks to calculate the largest expanded product. Students utilize these base 10 blocks to visualize place-value scaling and accurately solve the multi-digit calculation. This transition from instant memory-based retrieval to structured base 10 decomposition ensures students develop both fact automaticity and flexible calculation fluency within a single routine.

Evaluation for 5.2b
Materials provide opportunities for students to practice the application of efficient, flexible, and accurate mathematical procedures throughout learning pathways.

The materials provide efficient, flexible, and accurate mathematical procedures throughout the learning pathways. In grade 4, Journey D3, station 3, “Comparing Fractions,” requires students to use process standard 1C when they “select the technique that makes comparing 3/4 and 5/8 most efficient.” This requires students to evaluate and choose among common denominator, common numerator, and benchmark strategies. Additionally, the lesson plans provide explicit questions for educators to ask to assist students in practicing the application of efficient, flexible, accurate procedures. The questions within this lesson state, “Could you find a strategy to create equivalent fractions that makes comparing 3/4 and 5/8 easier?” and “Why might converting 3/4 to 6/8 be a more efficient technique than finding a common numerator for this problem?” These specific questions guide students toward efficient and accurate mathematical procedures, while the original task allows for flexibility.

The lessons build in a variety of efficient, flexible, and accurate mathematical procedures and provide students with many opportunities to practice them. Grade 4, Journey D4, station 5, introduces additive part-part-whole frames used with fractions. This procedure is flexible, allowing students to build on their experience using it with whole numbers in previous grades and apply it to a new concept. It is also efficient as it allows the student to choose which operation (addition or subtraction) is most efficient for any given equation. During the warm-up students have the opportunity to prove the accuracy of the additive part-part-whole frames when they use the inverse operation to verify their solution.

Evaluation for 5.2c
Materials provide opportunities for students to evaluate mathematical representations, models, strategies, and solutions for efficiency, flexibility, and accuracy throughout learning pathways.

Students evaluate the efficiency, flexibility, and accuracy of mathematical representations and concrete models throughout the learning pathways. For example, grade 4, Journey D4, station 2, “Representing and Decomposing Fractions,” provides students with visual representations of pattern blocks and asks them to determine what fraction each pattern block color represents. The warm-up concludes with students evaluating whether the blue portion of the pattern blocks is equivalent to 1/2 by examining the relationship between the numerator and denominator. The lesson plans provide guidance directing students to “verify this equivalence using manipulatives.” This task requires students to evaluate mathematical representations and solutions with concrete models for accuracy. Throughout the lesson students evaluate representations by determining the fraction that each pattern block represents and decomposing images into fractions. Students also use concrete models to compose using the unit fraction provided. These activities require students to evaluate the efficiency, flexibility, and accuracy of mathematical representations and concrete models.

Throughout Journey D4, station 6, students actively evaluate mathematical representations, models, strategies, and solutions for efficiency, flexibility, and accuracy. In activity 1.1 students work in pairs to create a number line divided into fifths by iterating a 1/5 fraction bar. Then students verify their diagram against applet A271 to check for accuracy. As students work, they discover that every fifth aligns with an even-numerator tenth, prompting them to evaluate the equivalence and accuracy of two representational systems, fifths and tenths. During the class discussion, students explain why they believe certain fractions are equivalent, assessing their reasoning by referencing the visible alignment on both the paper strip and the applet.

Evaluation for 5.2d
Materials contain guidance to support students in selecting increasingly efficient approaches to solve mathematics problems.

Educators guide students to select increasingly efficient approaches to solve math problems. In Journey C4, station 9, students explore division by dividing 100 snap cubes into six equal groups. The lesson plan guides educators to help students select efficient strategies such as “when dividing 100 snap cubes into six equal towers, they may choose to start with groups of 10—building 6 towers of 10 first, then distributing the remaining 40 cubes—rather than distributing one cube at a time.” In activity 2 students explore dividing items into parts. For example, they separate 78 pears into 3 baskets by first dividing 60 into 3 and then 18 into 3. The materials provide step-by-step guidance as well as questions to help students analyze the steps including, “Could you explain how the two division equations are connected?” The step-by-step guidance and questions help students to consider efficient approaches when solving mathematics problems.

Throughout the pathways, the materials deliver explicit educator guidance to support students in selecting increasingly efficient mathematical approaches. In the Journey C3, station 4, developmental activity, the educator displays a visual depiction of tricycles arranged in equal clusters alongside two isolated units. The educator prompts students to calculate the total number of wheels across all vehicles. To establish an efficient computational path, the guidance instructs the educator to actively discourage individuals from counting wheel values by ones. The curriculum guides students to evaluate the grouping structure to deduce a total of 17 vehicles through process standard 1A. To facilitate this strategic progression, the educator utilizes two embedded prompts, “Can you explain your strategy for counting the tricycles efficiently?” and “How does knowing there are 17 tricycles with 3 wheels each help you find the total number of wheels?” The lesson scales this efficiency by introducing an area model representation to calculate 17 x 3. To support EB students, the materials incorporate MLR8 discussion supports. This language routine displays a dedicated reference chart alongside three distinct sentence frames to help individuals articulate their procedural choices. The sentence frames are, “I can decompose ___ into ___ and ___” for EB students; “When I multiply ___ × ___, I get ___ because ___” for expanding students; and “To find the product of ___ × ___, I first calculated ___ × ___, which equals ___. Then I added ___ × ___, which equals ___, for a total of ___” for bridging students. This systematic support guides students away from visual item counting and scaffolds their transition toward more efficient, abstract multi-digit multiplication strategies.

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

The Math Journeys lesson plans explain how the TEKS are addressed both conceptually and procedurally. For example, Journey D4, station 3, “Adding Fractions I,” includes a “Before Starting” note explicitly stating the procedural scope that “it is sufficient to express fractions greater than 1 as mixed numbers,” and clarifying exactly what procedural understanding should be present at this stage. The warm-up establishes the conceptual emphasis of needing the same sized units when combining fractions through the use of fraction manipulatives. The lesson plans address both the conceptual and procedural emphases of TEKS 4.3.E in the summary, learning objectives, and activity sequence.

The educator-led lessons include guidance to help educators address the conceptual and procedural emphasis of the TEKS. In Journey D1, station 6, the learning objectives explicitly state the emphasis on both concepts and procedures for TEKS 4.2.E and 4.2.G. Learning objective B states that students will “associate the value of each coin to the decimal and the fraction of a dollar it represents.” This emphasizes decimals as a part of a whole using the concept of money. Learning objective C provides a procedural emphasis when students “formalize equivalences between decimal fractions and express them in decimal form.”

Evaluation for 5.3b
Questions and tasks provide opportunities for students to use concrete models, pictorial representations, and abstract models as required by the TEKS.

Within the lesson sequence students have opportunities to use concrete objects, pictorial models, and abstract strategies. Grade 4, Journey D4, station 7, “Adding and Subtracting Fractions on the Number Line,” includes opportunities for students to physically move their finger along a paper-strip number line as a concrete model for fraction addition. Then, students sketch jumps on the provided number line diagrams in their logbooks as a pictorial representation. Finally, students record the operations symbolically, such as 3/8 + 4/8 = 7/8 as the abstract model. Students repeat the three steps again for subtracting fractions. This one lesson provides opportunities for students to use concrete models, pictorial representations, and abstract models as required by TEKS 4.3.E.

The lessons actively engage students in using concrete models, pictorial representations, and abstract models as required by the TEKS. The Journey D3, station 6, warm-up, requires students to use coins, pennies, and dimes as concrete models, prompting them to determine how many pennies make a dime and what fraction of a dollar each coin represents. Students shade portions of a 100-square grid as a pictorial model, actively discovering and visually confirming that 10/100 = 1/10 before recording their findings. The educator then introduces decimal notation, writing 1/10 as 0.1 and 1/100 as 0.01 as the abstract model, explicitly connecting concrete, pictorial, and abstract representations in a single session.

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 lesson plans provide comprehensive supports for students in connecting, creating, defining, and explaining concrete and representational models as they transition to abstract concepts. In Journey D4, station 2, “Representing and Decomposing Fractions,” students explore composing and decomposing unit fractions with pattern blocks as concrete models. In the warm-up students see a figure created with the blue parallelogram and green triangle pattern blocks. Students investigate “what fraction of the whole each green triangle represents.” The lesson plan explicitly provides guidance for connecting the concrete model to the abstract value of the fraction by stating, “Students should reason that since one parallelogram equals two triangles and each green triangle represents 1/8, then the parallelogram represents 2/8 (1/8 + 1/8 = 2/8).” At the representational level, in activity 2 students draw and color shapes using colors to show how they can decompose 3/4 into 2/4 and 1/4 or three 1/4 parts. Students additionally provide an equation for the colors used to compose 3/4, bridging the representational model to the abstract equation.

In Journey D3, station 2, educators use the lesson plan to guide students through structured opportunities to connect concrete and representational models to abstract concepts as required by the grade 4 TEKS. Students create, define, and explain these models to demonstrate their understanding of the concepts. In the warm-up the materials support the explicit definition and handling of concrete models by providing students with rectangular fraction pieces to manipulate. This task requires students to actively analyze mathematical equivalence by physically aligning different denominators to a fixed length. Students must recognize and suggest that 4/6 and 8/12 define equivalent values because they physically align perfectly at the same endpoint as 2/3. In the development activity, to ensure students explain and connect these physical manipulations to abstract mathematical principles, the materials incorporate process standard 1F. This prompts students to analyze the mathematical relationship between multiplication and equivalent fractions when examining the equivalence of 2/3 and 4/6. This provides a conceptual explanation and definition of multi-operational relationships, driven by two explicit student-facing prompts: “Could you find a rule that provides you a shortcut to create equivalent fractions?” and “What relationship do you see between the numerators and denominators of two-thirds, four-sixths, and eight-twelths?” Furthermore, the lesson explicitly targets mathematical communication for EB students using MLR8 discussion supports. These supports provide tiered sentence stems for students to explain equivalent fractions, such as “To change two-thirds to four-sixths, I multiplied the numerator and denominator by 2” and “Each third is divided into 4 equal parts to make twelfths, so you multiply both the numerator and denominator by 4 to show the same amount.” The subsequent parts of the lesson systematically transition from concrete tools to the creation of representational models. Here, students independently generate a pictorial model by partitioning and shading blank fraction models to represent specified fraction quantities. This activity forces students to visually define what each shaded section represents in relation to the whole. The lesson culminates in connecting these concrete and pictorial experiences directly to abstract notations. The tasks guide students to explicitly connect their visual sketches to written numerical equations, forcing a direct cognitive bridge between the visual area models, physical fraction lengths, and abstract fractional symbols.

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

The materials in Math Journeys provide consistent opportunities for students to develop academic mathematical language through the use of visuals, manipulatives, and language goals. The materials provide a cluster overview for each cluster and a “Cluster Plan” section that lists the language goals and academic vocabulary words. For example, cluster overview C, “Multiplicative Thinking,” includes language goals for reading, speaking, listening, or writing, along with academic vocabulary words for C1–C4. The reading language goal for C4 states, “students will analyze and sequence the area model or partial products model” and the writing goal notes, “students will record their problem-solving process, labeling each part of a multistep calculation.” A separate box then lists the academic vocabulary words partial product, area model, standard algorithm, estimate, round, long division, and place value disk. This section of the cluster overview helps educators internalize the academic vocabulary along with the language goals to provide opportunities for students to develop academic mathematical language.

The lessons actively support students in developing academic mathematical language through visuals, manipulatives, and focused language strategies. In Journey D4, station 6, students use iterations of a 1/5 fraction bar to create a number line. Students progress to subdividing the 1/5 number line into 1/10 intervals. This lesson uses mostly student-focused discussion to discover the concept of equivalence. The lesson guide instructs educators to “lead a class discussion for them to share their strategies” or “lead a class discussion to conclude that since each fifth equals two-tenths, students always need twice as many tenths as fifths to represent the same number.” This lesson provides students the opportunity to practice using vocabulary such as denominator and equivalent using the number line they created.

Evaluation for 5.4b
Materials include embedded educator guidance to scaffold, support, and extend students’ use of academic mathematical vocabulary in context when communicating with peers and educators.

The Journey Language Bank for each journey provides the educator with embedded language support routines, along with discourse and writing supports, to scaffold, support, and extend students’ use of academic mathematical vocabulary when communicating with peers and educators. In grade 4, the C4 Journey Language Bank provides embedded supports to scaffold, support, and extend students’ use of academic vocabulary for division situations. In Journey C4, stations 1–5, the pre-teach/embedded language routine support bank requires educators to support students before they solve a grouping task, stating “ask what is known, what is found, and whether the problem is asking about the number of groups or the amount in each group.” Additionally, educators should restate the situation with “precise language before they compute.” The discourse-and-writing bank provides partner talk, quick write, and “revise your explanation” prompts, aligning with the suggested supports for Journey C2, stations 1–3. The prompts provide a partner talk stem which states, “We know ___, and we are finding ___. This is about ___ groups/___ in each group.” The materials prompt students to “write one sentence that explains the quotient and one sentence that explains the remainder” after solving the problems. These oral and writing communication prompts increase students’ use of academic mathematical vocabulary.

In grade 4, the embedded educator guidance in Journey C3, station 4 scaffolds the use of academic language through structured group counting and visual models. To introduce the concept, instructions direct the educator to display grouped and ungrouped tricycles. Instead of counting by ones, the educator demonstrates counting three groups of five plus two additional tricycles. To connect this scenario to process standard 1A, the educator applies mathematical reasoning to a real-life situation to determine the total number of wheels. Specifically, the educator poses two contextual questions to prompt student analysis: “Can you explain your strategy for counting the tricycles efficiently?” and “How does knowing there are 17 tricycles with 3 wheels each help you find the total number of wheels?” Once the class establishes grouping as a more efficient strategy than counting by ones, the educator guides students to calculate the final product using an area model. To support EB students during independent application, the instructions outline specific MLR8 discussion supports. Notes direct the educator to display a reference chart featuring an area model alongside targeted sentence frames. To reinforce key vocabulary such as product, factor, and total, these sentence frames assist students in expressing their mathematical thinking. For example, the educator prompts expanding students to demonstrate “When I multiply ___ × ___, I get ___ because ___.” Bridging students receive the prompt, “To find the product of ___ × ___, I first calculated ___ × ___, which equals ___. Then I added ___ × ___, which equals ___, for a total of ___.” Ultimately, this embedded guidance ensures that the educator systematically scaffolds, supports, and extends students’ formal mathematical terminology during peer discourse.

Evaluation for 5.4c
Materials include embedded guidance to support student application of appropriate mathematical language and academic vocabulary in discourse.

The lesson plans in Math Journeys include embedded guidance to support students in
applying appropriate mathematical language and academic vocabulary during discourse. Journey D4, station 7, “Adding and Subtracting Fractions on a Number Line,” includes guiding questions to prompt students to explain how a number line helps represent adding and subtracting fractions. For example, activity 1.1 asks students to solve the following situation: “A plant sprout was 3/8 in tall. During the last week, it grew 4/8 inches more. How tall is it now?” The lesson plans next prompt educators to ask, “Could you explain how the jumps on the number line diagram show the plant’s growth?” and “How does finding 3/8 + 4/8 on the number line diagram help you answer the question about how tall the plant is now?”

The classroom’s daily discourse and reasoning routines embed student application of academic language and precise mathematical vocabulary. In the Journey C4, station 6, warm-up, students actively engage in a comparative analysis task. The task requires students to formulate and articulate technical arguments defending both the similarities and differences between two visual arrangements showing four digital threes and two digital sixes. To explicitly encourage students to generate multiple mathematical justifications, the educator guidance embeds two intentional verbal prompts: “Could you explain why you think these images are the same or different?” and “What precise mathematical language can you use to justify your argument about these two arrangements?” These prompts require students to explain, defend, and justify their mathematical claims to the class using precise terminology. To support the educator in facilitating this high-level discourse, the materials provide multiple pre-drafted, suggested arguments that students may naturally produce during the discussion. This focus on student-led explanation continues into the core instruction, where students manipulate snap cubes to physically model division problems while executing the partial quotients strategy. As students work through these calculations, the lesson implements MLR2, “Collect and Display,” to capture and elevate student communication. Students must verbally explain each distribution step of the division problem, and the guidance explicitly directs the educator to record these student phrases next to the matching abstract numbers on the board. The framework ensures educators have an exact linguistic benchmark by embedding “I distributed 5 cubes to each group first because it’s easy to multiply by 8” and “I knew there would be 7 remaining because 95 − 88 = 7” directly within the lesson. Finally, the educator strategically organizes this collected student output so that critical mathematical terms, such as distribute, each group, remainder, and quotient appear alongside the corresponding subtraction steps of the algorithm.

Evaluation for 5.4d
Materials include embedded guidance to facilitate mathematical conversations allowing students to hear, refine, and use math language with peers.

The materials include prompts and guidance in the lessons for educators to facilitate math-related conversations. For example, in Journey D4, station 4, “Subtracting Fractions I,” the warm-up shows the amount of pizza two students have remaining and asks students if “they can determine what fraction of pizza each person has left.” The embedded supports explains the goal for students to recognize that the first student has 3/4 left because the pizza is circular and the part that was eaten can be seen, while the second student’s remaining amount is unknown because their pizza is rectangular and their original amount is unknown. Additionally, the embedded prompts assist educators in helping students refine and use math language. The prompts include, “Could you explain why we can’t compare the pieces just by looking at the fractions?” and “Does the size of the whole pizza matter when comparing fractions?” This warm-up provides an opportunity for students to listen to their peers’ arguments and evaluate their reasoning.

Journey D4, station 7, provides opportunities for students to practice using vocabulary related to number lines and fractions as they use number lines to subtract fractions with like denominators. The materials provide guidance for facilitating these student discussions in activity 1.1, prompting educators to ask “Could you explain how the jumps on the number line diagram show the plant’s growth?” and “Could you explain how you decided to break down the jump of 5/6?”

Evaluation for 5.4e
Materials include embedded guidance to anticipate a variety of student answers including exemplar responses to questions and tasks, including guidance to support and/or redirect inaccurate student responses.

Educators utilize embedded guidance to help anticipate a variety of student responses, including exemplar answers. Journey D3, station 6, “Introducing Decimals and Decimal Fractions,” anticipates that students may be familiar with money but not as familiar with decimal notation. Embedded prompts and explanations state, “although it’s not common in the context of money, in mathematics the fraction 1/10 is written as 0.1 in decimal form and is read as one-tenth. Similarly, instead of cents you will say hundredths.” The lesson plans also embed exemplar student responses such as, “expect students to use the grid to observe that, since there are 10 columns in total and only one equals one dime, you can also use the fraction 1/10.” ​These supports help students articulate their mathematical thinking using precise academic language and vocabulary.

The lesson plans provide guidance that helps educators anticipate a variety of student answers, provide exemplar responses to questions and tasks, and support or redirect inaccurate student responses. For example, in Journey D3, station 9, activity 1 presents multiple solution paths with two embedded exemplars stating, “A good strategy is to start at the green flag and jump forward 7/10” and “another option is to add 7/10 + 7/10 directly, then express the result as a mixed number (14/10).” The materials model the expected reasoning by providing an exemplar justification such as, “students should justify that the distance between 0 and 7/10 is 7/10.” The guide corrects inaccurate thinking by noting that “students should recognize that no fraction with a denominator of 10 can be placed between 2/10 and 3/10,” and connects to prior learning to address the error.

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

The Math Journeys materials appropriately integrate the TEKS process standards throughout instruction. For example, Journey D4, station 7, “Adding and Subtracting Fractions on the Number Line,” activity 2, explicitly tags process standard 1F, “Analyze relationships to connect mathematical ideas.” The lesson requires students to add and subtract fractions using a number line. Activity 2 asks students to solve how many gallons of water remain when 4/6 of a gallon of water is used from a cube containing 1 and 5/6 gallons. The materials explain, “Students engage in process standard 1F when they analyze the relationship between a fraction and the parts it can be decomposed into in order to subtract more efficiently on the Number Line diagram. When subtracting 5/6, they recognize that 5/6 = 1/6 + 4/6, which lets them jump back 1/6 to reach the whole number one and then jump back 4/6​ to complete the subtraction, connecting the decomposition to the structure of the number line.” Students explain their reasoning and respond to the reasoning of others. The materials provide prompts such as, “Could you explain how you decided to break down the jump of 5/6?” and “Do you see a pattern in how we can break jumps that cross units?”

Journey D3, station 3, “Comparing Fractions,” appropriately integrates the TEKS process standards. For example, process standards 1C and 1G feature at the top of the lesson plan. In activity 1.1 students compare the fractions displayed on the board, then hold up their screens to show their answers. Students then engage in a discussion about the choice they made and the strategy they used. Students engage in process standard 1G when they explain and justify the correct fraction comparison using precise mathematical language. For example, when determining that the fraction comparison statement B: 2/6 < 2/8 is false, students must display their reasoning about why 2/6 is actually greater than 2/8. Students must explain that when the numerators are the same a smaller denominator means larger parts, so the fraction is greater. In activity 2.3 students continue comparing fractions on their Magic Pads and justifying results with their partner. Students engage in process standard 1C when they select the technique that makes comparing 3/4 and 5/8 most efficient. Students may decide whether finding a common denominator or a common numerator is the better approach, noticing that since 4 is half of 8, finding that 3/4 is equivalent to 6/8 is a straightforward way to make the comparison.

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

The materials include a description of how the learning pathways incorporate and connect the TEKS process standards throughout. The Journey Instructional Guides include a table showing the lesson, concept covered, the TEKS, process standard, and why the lesson matters for each lesson in the journey. For example, the C4 Journey Instructional Guide explains that lesson 8 is a “useful pattern lesson for prediction, justification, and careful reading of groups of related equations” and lists process standard 1F, “Analyze mathematical relationships to connect and communicate mathematical ideas.” Educators can use the table to see the process standards incorporated throughout the pathway and how they connect to the rationale of the lesson.

Additionally, the cluster overview within the Journey Instructional Guide describes how the learning pathways incorporate and connect the process standards throughout. The D4 Journey Instructional Guide explains how the recurring actions of represent, rename, compare, justify, and interpret, connect the process standards across all D4 lessons into a coherent pathway. Journey D4, station 7, explicitly tags process standards 1A and 1F, and the guide links forward-jump language from this session to the common denominator and word problem work in Journey D4, station 10 and in Journey D4, station 13. The horizontal rationale explains that these process standards “deliberately revisit the same big ideas” across new representations throughout the journey.

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

The session header at the top of each lesson plan includes which process standards feature in the lessons. A drop-down box feature makes the process standard easy to locate and provides a clear explanation of both the process standard and how the lesson incorporates it. Journey D4, station 7, “Adding and Subtracting Fractions on the Number Line,” explicitly tags process standard 1A and 1F, signaling to the educator that students will apply mathematics to everyday contexts and analyze mathematical relationships to connect and communicate mathematical ideas. The Teacher Guide connects the 1A tag directly to tasks where students use the number line to model the growing of plants and the 1F tag to tasks where students explain how they used the number line to decompose fractions to make subtraction easier. This lesson overview allows educators to anticipate which process standards are active before instruction begins.

Journey D4, station 7, highlights that the lesson fully integrates process standard 4.1.F, “Analyze mathematical relationships to connect and communicate ideas.” The summary at the top of the screen states that the process standard will be incorporated when students “connect forward and backward jumps with addition and subtraction of fractions, respectively.” The Teacher Guide demonstrates how students analyze relationships for adding and subtracting on a number line directly from this tagged process standard during activity 2. In this activity, students use a number line to solve the problem of how much water remains in a cube. This lesson-level overview specifies which process standard is embedded and where it appears within the lesson.

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

Students have opportunities to think mathematically, persevere through solving problems, and make sense of mathematics throughout the Math Journeys lessons. For example, Journey C4, station 9, provides opportunities for students to think mathematically by solving division problems through deducing nearby facts. For example, activity 1.1 asks students to demonstrate 100 ÷ 6 using groups of 10, then deduce nearby facts like 99 ÷ 6 and 98 ÷ 6 without restarting from scratch, requiring sustained mathematical thinking. Additionally, students have the opportunity to persevere through solving problems in activity 1.2 by extending known facts such as 70 ÷ 7 = 10. Students progress to derive 72 ÷ 7 and 78 ÷ 7, adjusting when remainders exceed the divisor. Finally, in activity 2, students apply their understanding that 60 ÷ 3 and 18 ÷ 3 can combine to solve 78 ÷ 3 in a word problem context.

Additionally, Journey D4, station 5, introduces students to the addition and subtraction of mixed numbers. Students think mathematically throughout the warm-up as they identify what fraction fills the missing space. Activity 1 guides students through making sense of an additive part-part-whole frame and how it represents both addition and subtraction. Students must also persevere through activity 2 as they take the concepts of fraction addition and subtraction and apply them to mixed numbers.

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

The educator-led lessons support students in understanding, explaining, and justifying the multiple ways to solve problems and complete tasks. Grade 4, Journey D3, station 3, “Comparing Fractions,” requires students to justify fraction comparisons in an activity, “Two Truths and One Lie,” where students explain which fraction comparison is a lie and why. This activity requires students to analyze fraction comparisons using the common-denominator and common-numerator strategies. The next activity requires students to compare fractions such as 3/4 and 5/8 using multiple strategies. Additionally, the lesson tags process standard 1C, requiring students to compare the efficiency of both strategies. The prompt asks, “Could you find a strategy to create equivalent fractions that makes comparing 3/4 and 5/8 easier?” and “Why might converting 3/4 to 6/8 be a more efficient technique than finding a common numerator for this problem?”

Throughout the journeys, students engage in understanding, explaining, and justifying multiple approaches to solving problems and completing tasks. In Journey D3, station 9, “Practicing Fractions and Decimals,” activity 1, students explore two valid solution paths for the same flag problem. Students start at the green flag and jump forward 7/10 or add 7/10 + 7/10 directly and express the result as a mixed number. By writing their chosen strategies in logbooks and sharing them, students explain and compare different methods. During the class discussion, the educator validates both approaches by showing that both methods lead to the same result, helping students justify why different solution paths are equivalent.

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/or educators.

Various lessons and activities throughout the pathways provide multiple opportunities for students to demonstrate their understanding of mathematics through hands-on activities, discussion, and writing with their peers and educators. For example, Journey D4, station 7, activity 1.2, asks students to solve two fraction addition equations independently in their log books and then explicitly states that the educator should “guide a class discussion to share strategies.” Throughout the lesson, students use a number line to model real-world problems involving adding and subtracting fractions in a plant growth context. The lesson plans provide multiple prompts for students to discuss their strategies with their peers and educators, such as “Could you explain how you decided to break down the jump of 5/6?” and “Do you see a pattern in how we can break jumps that cross units?”

In grade 4, Journey D2, the station 5 lesson on factors and multiples provides explicit, structured opportunities for students to make sense of mathematics by simultaneously doing, writing about, and discussing mathematical concepts with peers and educators. In the warm-up students actively do math through an auditory and kinesthetic whole-group “Choral Count” routine. Arranged in a circle, students count aloud sequentially up to 50. However, they must perform an agreed-upon physical movement instead of speaking any number that contains the digit 4 or is a multiple of 4. Following this routine, the lesson transitions to a conceptual discussion where the educator models the inverse relationship between factors and multiples using a familial analogy. This collaborative exploration continues into the activities, where students work together to do math and discuss math by constructing physical chains using dry-erase boards and number cards. Students collaborate to evaluate and arrange their boards so that the numbers form a continuous chain based on their relationships to factors and multiples. This cooperative task applies process standard 1F, requiring students to use precise mathematical language to analyze and communicate these numerical relationships. To guide this student discourse, the lesson provides the educator with two explicit prompts: “Could you explain how the numbers in your chain are related to each other?” and “In the chain 12–4–20, what factor-and-multiple relationship does the 4 have with each of its neighbors?” Finally, during the wrap-up two differentiated logsheets promote student accountability by requiring students to independently formulate written responses to analytical questions such as, “What link have you found between the three answers and the factors of 12?” By seamlessly weaving physical modeling, collaborative discourse, and targeted writing exercises into a single lesson, the materials comprehensively ensure that students employ multiple communication pathways to deepen their conceptual understanding.

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

The Math Journeys lessons include educator guidance to support students in explaining their thinking, as well as generating and justifying mathematical arguments. In Journey D4, station 4, “Subtracting Fractions I,” the warm-up shows a picture of a pizza that two students are eating. One student is eating a circular slice and the other is eating a rectangular slice. The materials explicitly ask “if, with this information, they can determine what fraction of pizza each person has left.” This task requires students to analyze how much pizza is left and to generate and justify their response. The lesson plan provides prompts to support the task by asking students, “Could you explain why we can’t compare the pieces just by looking at the fractions?” and “Does the size of the whole pizza matter when comparing fractions?” Additionally, the materials support educators in guiding students to reflect on their problem-solving approaches through explanations, arguments, and justifications. Students subtract fractions using fraction pieces in activity 1. The materials include prompts that help students reflect on their learning, such as “Could you explain what happens to the denominator every time we subtract fractions?” and “Do you notice any pattern that helps us solve fraction subtraction?”

The Teacher Guide actively supports educators in helping students share and reflect on their problem-solving approaches through explanations, arguments, and justifications. In Journey D3, station 10, students participate in a structured “Two Truths and One Lie” activity using dry-erase boards. Students form mathematical stances, evaluate fractional expressions, and defend which comparison statement is false. The materials prompt students to justify correct comparisons using precise mathematical language and concrete fraction-tile models, thereby activating process standard 1G. Educators lead class-wide proof checks with targeted prompts, such as “Could you explain your reasoning about why statement B is false using what we know about denominators?” and “How can you justify, using precise mathematical language, why 2/6 is greater than 2/8 when both fractions have the same numerator?”

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

Educators guide students through provided explanatory feedback, targeted guidance, and prompts that address student responses and anticipated misconceptions. For example, grade 4, Journey D3, station 6, introduces students to decimals in the context of money. The Teacher Guide explicitly anticipates that students may struggle with expressing coins as a fraction of a dollar and states, “If students have difficulty expressing pennies and dimes as fractions of a dollar, ask them to look at the array representation.” Additionally, the materials provide slides with the coins on the array and provide multiple support prompts such as, “How many pennies can we put in each row? (10) And in each column? (10)” and “What fraction does 1 penny represent? (1/100)” Additionally, the guide scripts explanatory feedback for the language errors station, “although it’s not common in the context of money, in mathematics the fraction 1/10 is written as 0.1 in decimal form and is read as one tenth.” The anticipated misconception, along with the scripted prompts, equips educators to provide explanatory feedback.

The materials provide comprehensive prompts and guidance to support educators in delivering responsive explanatory feedback based on specific student actions and anticipated misconceptions. In the Journey D4, station 1, warm-up, the lesson anticipates a common misconception in which students incorrectly identify unequal partition fields as standard fractional parts. The materials provide a targeted prompt based on this anticipated misconception. The script directs the educator to ask, “Are all parts the same size?” to guide students to self-correct and to understand that fractional units must represent equal areas. In activity 1 the materials provide separate pedagogical guidance based on anticipated misconceptions. The guidance instructs the educator to look at student layouts and evaluate if pairs fail to consider real-world parameters, explicitly noting that non-adjacent or scattered cell selections suggest a breakdown in connecting mathematics to real-world contextual situations. The materials transition to providing explicit prompts based on student responses. The educator leverages process standard 1F by asking two prompts: “Could you explain how you know that these two fractions represent the same number?” and “How does the relationship between fifths and tenths on the grid help you find other equivalent fractions?” Finally, the materials provide separate guidance based on student responses within the support callout. The guidance advises the educator to alter their delivery and introduce physical fraction bars as a concrete manipulative intervention if a student’s response demonstrates that they are struggling to remember or identify fractions equivalent to one-half. By separating structural real-world layout warnings, targeted partitioning checks, and explicit manipulative interventions, the materials fully equip educators to deliver responsive feedback that targets the root cause of student errors.