Math Journeys Texas, Grade 5

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 TEKS and ELPS are included in the scope and sequence document, which is organized into different sections called “Journeys” in the program.

The Journey Instructional Guide provides both vertical and horizontal rationales that explain how learning progresses across and within grade levels. For example, Journey A2 describes how students extend prior learning about numbers 1–20 to counting, comparing, ordering, and estimating numbers through 100. 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 is designed to start with an educator-led small group lesson, followed by logsheets for students to complete independently with educator feedback, and then optional additional practice in Atlas Math, an adaptive digital practice. 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.

Lesson plans include recommendations for adapting instruction to meet student needs through embedded supports, extensions, and alternative instructional approaches. For example, in Journey C4, station 8, the “Wrap-Up” section includes a support feature that suggests using three-digit dividends with the three smallest divisors and recommends base 10 blocks as manipulatives to support student understanding.

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

The TEKS Correlation Guide identifies recommended skill entry points based on diagnostic assessment results. Lessons within each journey are aligned to the specific TEKS content and process standards, allowing educators to connect instructional activities to assessed skills.

The placement test report sample for grade 5 provides the TEKS that were assessed on the diagnostic assessment and proposed grouping structures for students. The “Proposed Group Distribution” section of the report suggests grouping students based on assessment data and recommends an entry point for each group. The report also includes a page for each student, providing a snapshot of their current understanding and a suggested entry point. The materials do not include a sample grade 5 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 directing educators to understand how the journey builds on prior learning, review lesson objectives and mastery criteria, identify scaffolds and enrichment opportunities, gather materials, and analyze learning progressions before instruction.

The Program Guide includes three lesson preparation protocols: “Preparing for a Building Session,” “Preparing for a Consolidating Session,” and “Preparing for an Additional Practice Session.” These sections support lesson internalization by helping educators clarify objectives, anticipate misconceptions, organize materials, identify formative assessment opportunities, analyze student responses, and use progress-monitoring data to guide 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 support instructional leaders in implementing 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.”

Training resources include on-demand videos, such as “Let’s Add by Breaking Down,” that model instructional strategies. The Math Journeys Coaching Guide includes look-fors, predictable struggles, and ready-to-use rubrics during observations for instructional leaders. These materials are broken into four phases: “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 TEKS-aligned learning objectives, student materials, educator resources, and lesson components. For example, Journey C2, station 4 identifies the learning objectives, “establish connections between quotitive 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 provides 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 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 a 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 quotitive division and partitive division” and “develop understanding of the relationship between division and multiplication when one factor is unknown.” Adaptive activities are identified within the lesson plan through the “Activities Added to Individualized Practice After Session Completed” section. 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 Program Guide describes multiple assessment resources, including diagnostic and progress-monitoring tools that generate TEKS-aligned student performance data. The digital practice progress reports identify student performance by the TEKS to support instructional planning and ongoing monitoring. Additional TEKS-aligned assessments include formative logsheets, embedded lesson assessments, and end-of-journey assessments 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 family letter can be translated through the platform and is available in both Spanish and English.

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 families can 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, are intended to 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 student mastery of targeted skills and determine readiness for advancement within the instructional pathways.

The Placement Test Implementation Guide gives an even more in-depth description 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 the placement test is used 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 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 states, “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 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.

The end-of-journey test and formative assessments are available in both digital and print versions. For example, in grade 5, the D6 end-of-journey assessment includes an End-of-Journey Test: Instructions and Guide that provides a link to download printables, including the D6 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 toggle 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 specifying when calculators can be used. 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, supports include the ability to toggle the language between English and Spanish, but no other content or language supports. The program guide clearly explains the embedded accommodations, but it 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 5 provides results for 24 distinct assessment tasks mapped directly to TEKS 5.2–5.5 and mathematical process standards 5.1.A–5.1.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. An example of each question type, along with a visual and an explanation of each item type, is included in the guide. 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 that incorporate multiple interactive item types and varying levels of complexity. The logsheets for each lesson include text-entry, numeric-entry, equation-writing, number line, and problem-solving tasks that require students to apply mathematical understanding in different ways. For example, the logsheets in Journey D6 ask students to compare equations, solve real-world word problems, and reason about rational numbers. In Journey D6, station 7, students use the digits 1–3 and a decimal point to “form the greatest possible number and smallest possible number and add them.” Another task asks, “What is the total cost of his purchase? If I pay with a $50 bill, how much change should I get back?” This task requires students to apply decimal operations in a real-world context.

The Digital Assessment Item Bank provides educators with formative checkpoint questions that span multiple levels of cognitive complexity and include a variety of interactive item formats. For example, Journey Test D5 includes the open-response item, “Solve the multiplication: 3 × 2/8,” while Journey Test D4 includes the multiple-choice question, “In the morning John hiked 4 8/10 miles. In the afternoon he hiked 2 1/2 miles. Which equation can be used to determine the number of miles John hiked altogether?” These formative assessments include multiple interactive item types, such as single-select, multi-select, and text-entry responses, and provide students with opportunities to demonstrate understanding through tasks that progress from procedural fluency to multistep problem solving and mathematical reasoning.

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 5 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, along with a chart showing student performance by content strand and process standard. Additionally, guidance for interpreting student performance is included in the recommended starting journey, 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 5, the answer key for the D6 end-of-journey test provides the TEKS 5.3K for question 1 and provides the correct answer as 15.4 with the rationale, “3 represents 3 ones. Adding 3 ones to 12.4 increases the whole-number part by 3 and leaves 4 tenths unchanged, so the result is 15.4.” The answer key then outlines incorrect answers of 12.7 and 15.7 and a rationale which states, “Both answers change the tenths, but adding 3 changes only the whole-number part. 12.7 adds only 3 tenths, and 15.7 adds 3 ones and an extra 3 tenths.”

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 to help the educator use assessment results to plan purposeful tasks and activities aligned with the appropriate TEKS. For example, the report includes a chart listing the TEKS, the process standard, the percent correct for the class, and the percent 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 into a proposed group distribution 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 educator guidance and specific instructional activities to address student trends and misconceptions identified on assessments. In the assessment guidance matrix for test D4, the text connects specific performance trends under TEKS 5.3I and 5.3J to concrete instructional next steps under the “Additional Support” section. For example, when assessment trends show a student “interprets division by a unit fraction as making a smaller quantity,” the materials instruct the educator to respond by modeling “3 ÷ 1/8 and 4 ÷ 1/8 with fraction strips, number lines, or repeated filling contexts.” Similarly, to target the trend of treating fraction multiplication additively, the guidance directs the educator to use “fraction strips and area models to show why whole-number multiplication scales the numerator.”

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 5, in Journey D6, station 5, the materials include embedded guidance for educators to check for student understanding at key points during instruction. For example, when students are asked to represent money on an abacus and represent the amount, the lesson plans guide educators to “pay special attention to ensure they write the amounts as $23.25, instead of $23 and 25¢.” 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 E1, station 5, “Expressions with Multiple Operations I,” the goal is for students to generate the need to establish conventions about the order in which operations are solved and how this order is written. The lesson prompts educators to give students different scenarios, such as, “If Kai has 14 red flowers and 7 yellow flowers, and Jackson has the same number of each, how can we calculate the total number of flowers they have together?” Educators are guided to have students use a calculator to solve so they can understand that if you enter the operation 14 + 7 × 2 into the calculator, it does not give you the number you’re looking for (42) because it calculates the multiplication expression 7 × 2 before the addition expression.

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 the materials do not adjust in difficulty based on the students’ correct or incorrect responses. For example, in applet A148, “Prime Numbers Explorer,” when students create the incorrect prime number the activity prompts with a visual table of prime numbers. However, the next question in the program is not adjusted based on whether the student got the answer correct. The feedback is provided in the form of a vertical bar on the left side of the screen. A correct answer is marked with a green mark, while an incorrect answer is marked with a question mark.

Linear interactive checks for understanding with feedback hints are given in the digital applet practice, but they are not adaptive as they do not dynamically alter lesson progression, scaffolding, or mathematical rigor based on student performance. For example, when logged in as a student completing the grade 5 digital milestone task, applet A137.2, “Identifying prime and composite numbers,” inputting incorrect answers triggers text-based definitions as hints along with green and red visual indicators to display right and wrong answers. However, the underlying digital environment does not change, numerical rigor is not dynamically scaffolded, and virtual visual tools are not deployed to actively resolve the user’s conceptual misunderstanding. Upon registering incorrect responses for every problem in the task, the platform requires the student to repeat the entire lesson and completely freezes progress, locking out access to any other core or interactive activities. Because the system restricts a student’s learning path through linear gatekeeping rather than modifying lesson pacing, content sequences, or complexity in real time, the materials fail to provide true adaptive personalization throughout their 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.

Embedded support boxes throughout the lesson plans scaffold instruction for students who have not yet reached proficiency. In Journey D6, station 7, the “Wrap-Up” section includes multiple targeted intervention strategies in clickable support boxes for educators. For example, educators are directed to “provide mixed dollar bills and $1 and cent coins so students can solve the task using manipulatives” as tactile supports to reinforce conceptual understanding. The lesson also incorporates computational supports by instructing educators to “provide a calculator so students can solve the calculations involved in the task. You can also place the digits that should go in the first or last position so they only need to complete the remaining blanks.” These embedded supports give educators immediate access to differentiated instructional strategies.

Additionally, the materials provide explicit educator guidance to scaffold grade-level instruction through an intentional progression from concrete to abstract. For example, in the Journey D5, station 13, warm-up, the guide instructs the educator to build prerequisite readiness using familiar interval contexts (30-minute and 15-minute blocks in three hours) without revealing abstract equations. In development activity 1.2, the scaffolding bridges this intuition to fractional units. The instructions direct the educator to project a fraction pieces widget and visually model placing one-fifth pieces below two whole units until the space is covered, to show why the answer is 10. To reinforce this scaffold under standard 1B, the educator is provided specific support questions to connect the model back to a multiplicative frame, including: “What math do you see in this juice-sharing situation?” and “How can you check that 10 cups is a reasonable answer?”

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 academic vocabulary development. The vocabulary section introduces key content vocabulary, English–Spanish cognates, false cognates, and multiple-meaning words. In addition, the Journey Language Banks embedded throughout the materials promote academic vocabulary development through routines, discourse, and writing. Within the D6 Journey Language Bank, the pre-teach/embedded language routine support bank provides three routines that identify when to use the routine, educator actions, and the lessons in which the routine is most effective. For example, routine 1, “Model-Pronounce-Define-Use-Check,” is recommended “before or at the first moment a new decimal term matters” in lessons D6.1, D6.5, and D6.13. The routine directs educators to say the term, define it, use it, and check for understanding. This guidance supports educators in developing academic vocabulary related to division problem solving.

Additionally, the materials provide embedded instructional steps to develop vocabulary with unfamiliar references. In grade 5, lesson D6.4, “Subtracting Decimals by Jumping,” the core educator materials provide in-the-moment language support to navigate situational text barriers. For example, the lesson integrates the Mathematical Language Routine (MLR) 6, “Three Reads,” protocol at the point of use, explicitly guiding the educator to remove numerical values during the first read and ask, “What is this situation about?” This mechanism actively guides educators to have a conversational focus on the real-world story scenario, such as buying juice and finding change, across tier-differentiated language proficiency tracks before mathematical computation begins on the number line. 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 D5, station 7, scaling up and scaling down are unfamiliar references. The materials suggest to “point to 0 on the Number Line, then to where 3 × 2 lands. ‘Scale up means the product lands farther from 0. Scale down means it lands closer to 0. Scale here is not a tool for weighing.’ Students point to a product and say scale up or scale down.”

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 include extension tabs with enrichment activities. For example, in the grade 5, Journey D6, station 2, “Wrap-Up” section, the extension tab states, “Ask students to verify that if the kangaroo is initially at point 3.50 and makes a jump of 3 and a jump of 0.50 in any direction, the six numbers it can reach are also six whole numbers.”

The Enrichment and Extension Guide provides guidance for which students should receive extension work and how to facilitate each extension logsheet, as well as the extension tips embedded in the lessons. For example, the Journey D5 Enrichment and Extension Guide provides tables for the five lessons within the journey that have extension tips embedded in the lessons. The table includes information on the intent of the task, what to listen for, and defines what counts as mastery. For example, in Journey D5, station 1, the table states the task is for students to “find another multiplication expression with the same result for each multiplication equation.” The table guides educators to look for “a different multiplication with the same value, typically by trading between the whole number and the numerator: twice as many copies of a piece half as large, or the whole product collapsed into copies of the unit fraction.” 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.

Students access digital assessments and applets through Atlas Math, which includes text-to-speech accommodations and digital help supports; however, educators cannot enable or disable these supports for individual students. According to the Program Guide, the available content and language supports are limited to changing the language between English and Spanish; no additional content or language supports can be activated or deactivated by educators. The Program Guide also 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 that will 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.

Opportunities for students to demonstrate understanding of the lesson topic are embedded throughout the lesson plans. For example, in Journey D6, station 5 includes guided discussion opportunities during the warm-up. The section also provides explicit guidance in a blue support box to assist educators in facilitating the discussion. In addition, students demonstrate understanding by using manipulatives to model decimals. The lesson incorporates two types of manipulatives, money and an abacus, to represent decimals. Activity 1 provides step-by-step guidance on introducing the abacus, a new instructional tool for students.

Additionally, the materials provide explicit educator guidance on offering student options and educator supports across multiple modalities. In the Teacher Guide, core lessons systematically embed process standard support boxes, such as standard 1G. In the grade 5, Journey D2, station 7, multiplicative frame lessons, the text explicitly guides educators to let students complete frames to choose between displaying visual factor-pair combinations or explaining and justifying their choices using precise mathematical language. The materials provide the required educator-facing scaffolding by scripting exact questions for the educator to facilitate these modalities, including: “Could you justify why 4 and 5 are the only possible factors that can be placed in the bottom cells for 20?” and “How can you explain, using factor and product language, which pairs work for 40 and why a pair such as 4 × 10 must be excluded?”

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.

The “Warm-Up” sections include explicit prompts and guidance that help educators anchor big ideas and activate prior knowledge. For example, in Journey D6, station 4, students solve subtraction word problems and the warm-up directs educators to display an applet in which students use a previously learned strategy to subtract on a number line by decomposing numbers. This activity requires students to draw on their prior understanding of how to subtract decimals.

Explicit educator-facing guidance and structural frameworks are embedded throughout the materials to anchor big ideas and outline learning progressions across mathematical concepts. In cluster overview D, “Rational Numbers,” the text describes the critical cognitive shift students make as they transition from discrete counting to continuous measurement of a single whole partitioned into equal parts. Multiple representations, including area models and number lines, are used to visualize fractional relationships and the placement of values between zero and one. Additionally, dedicated mathematical background text boxes outline specific learning milestones that guide educators through a progression of concepts, including deepening the representation, equivalence, and comparison of fractions; connecting decimal fractions to decimal numbers; and developing operational strategies for addition, subtraction, multiplication, and division across both fractional and decimal notation.

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

Varied instructional approaches, including hands-on exploration, whole-group and small-group discourse, and independent consolidation, are embedded throughout Math Journeys. Each lesson transitions from guided discovery, in which educators facilitate hands-on exploration, to formalization. For example, Journey D6, station 2, engages students in using a number line to explain their understanding of place value and participate in discourse during the warm-up. Students compare and contrast two number lines and support their reasoning by responding to prompts, including, “Could you explain your reasoning about why these Number Line representations are the same or different?” and “How can you use precise mathematical language to justify whether 4.183 is closer to the left end or the right end in each image?” These prompts and activities support students in formalizing their understanding of place value and decimal representation on a number line.

Each lesson plan provides educators with explicit directions and guidance to make each lesson effective and engaging. For example, Journey D6, station 2, begins with a warm-up using a whole-group discussion. The first development activity is whole-group instruction on how to place decimals on a number line. The second development activity is small-group practice where groups are asked to manipulate the placement of a decimal on a number line. The final section is the wrap-up, where students practice individually on logsheets. Each section of the lesson has 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.

The materials in Math Journeys include multitiered intervention methods for various types of practice and structures with educator guidance to support effective implementation. The lessons include multiple types of practice, such as collaborative partner work and whole-group discussions embedded throughout the lesson activities. In Journey D6, station 5, the materials include opportunities for students to develop understanding of decimal numbers through the context of money independently in the logsheets as a whole group, and in small groups through the development activities. The lesson provides opportunities for students to relate bills and coins to their place value on an abacus in a small group. Embedded blue boxes support the educator in guiding students. For example, a blue box embedded during the small-group activity states, “Since this is the first time students are working with this abacus, represent the amount gradually: first the hundreds, then the tens, the ones, the tenths, and finally the hundredths.” These embedded supports help educators with effective implementation of the lesson.

The materials provide small-group, collaborative, and individual intervention methods across various types of practice and instructional structures to support struggling learners. For example, in Journey D5, station 11, the fraction of a set lesson, the core content is explicitly split into segmented learning environments to deliver targeted, small-group remediation alongside universal whole-class and independent settings. Within the small-group structure, the curriculum alters the type of student practice by providing diverse tactile, visual, and symbolic scaffolds. Students use physical circular pieces, dry erase boards, and digital lesson slides to simultaneously model problems using objects, drawings, and equations while completing multiplicative frames. The small-group segment incorporates a dedicated inquiry-based “what do you notice” slide prompt to guide students to discover the inverse relationship between operations. Following a whole-class true-and-false routine anchored in process standard 1F, the instructional approach transitions to the independent practice phase, which utilizes a specialized, scaffolded support version of the standard student logbook to offer further multitiered practice variations for struggling learners. It concludes with an independent “what do you notice” metacognitive writing prompt. Furthermore, the materials provide educator-facing implementation guidance by embedding specific educator scripts and explicit guiding prompts, such as “What patterns do you notice when you divide by different natural numbers?” and “What patterns do you notice when you divide by different natural numbers?” This actively scaffolds student understanding and facilitates the true-and-false discourse.

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 5, Journey D6, station 3, the materials guide the educator to extend the practice of adding and subtracting decimals, stating “add the challenge of finding the result that is as close as possible to 3.” The wrap-up also provides a question where a receipt is provided and students add up the items to find the total. An extension tab states, “Ask how much the total purchase would have been if, instead of buying fresh onions, Dad had bought a 1 kg package of chopped frozen onions for $1.40.” These tabs provide enrichment and extension guidance. Additionally, most lessons provide an extension version of differentiated logsheets.

Additionally, the materials provide extension logsheets, which are activities that can be used for enrichment with some students while others are working on the traditional logsheet. These extension logsheets include more puzzle-like activities that require productive struggle and discovery from the students. For example, in Journey D6, station 7, the extension logsheet asks students to complete number sentences about rounding decimals by filling in given digits. They are asked to find all possible combinations that make that statement true.

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

Embedded throughout Math Journeys are prompts and guidance that support educators in providing timely feedback during lesson delivery. Specific prompts are incorporated into the lessons, enabling educators to pause at key moments to check for understanding, address misconceptions, and provide immediate, meaningful feedback. For example, in Journey D6, station 10, students solve word problems involving the multiplication of decimals. After presenting the word problem, the materials direct educators to ask, “How does the equation 4 × 1.30 = 5.20 solve the problem of how much money the friends collected?” and “Why does using multiplication make sense when each of the 4 friends contributes the same $1.30?” The lesson also provides guidance for responding to students who use different strategies by stating, “Based on the Warm-Up, acknowledge and validate the different strategies students use to repeatedly add the same decimal number.” These embedded supports equip educators with the tools needed to deliver responsive instruction and keep students progressing toward the lesson objectives.

The materials include lesson-specific misconceptions and how to respond with feedback and correction in the Journey Language Bank, section 4, anticipated responses/misconceptions/redirection bank. For each misconception, the educator is provided with what it may mean and suggested redirection. For example, in Journey D6, station 1, and Journey D6, station 2, a common misconception is that students may say that “3.74 is greater than 3.8 because 74 is greater than 8.” The materials suggest this is because “the student is reading the digits like whole numbers instead of comparing place values” and instructs the educator to have the students “rewrite 3.8 as 3.80” or “compare the tenths first.”

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 D2, station 8, development activity 1, the lesson embeds emergent bilingual (EB) student supports. Students list multiples of 9 and 12 and label the common multiples. The EB student supports state to keep both lists visible as students circle numbers that appear on both lists. The supports also provide 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 “circle or point to a number they find in both lists,” while a student at the advanced level may use the stem, “Common multiples of ___ and ___ repeat every ___, so the next is ___.”

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 materials include text-to-speech, visuals of manipulatives, and a support button to help with the content, but do not provide linguistic accommodations. For example, in applet 289, “Addition and Subtraction of Fractions,” when students miss finding a common multiple, the help button provides fraction models to show equivalent fractions. Students can use this to help find an equivalent fraction and solve the addition problem, but the materials do not provide linguistic accommodations if a student does not understand the language.

In the Program Guide, the “Overview” section explains that the educator should provide accommodations during the “Build” and “Consolidate” portions of the lesson so that academic vocabulary is learned before the student reaches the adaptive portion of the lesson. Therefore, the linguistic accommodations are not 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, each journey includes a Lesson-Level ELPS Companion guide. 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 D6, station 2, “Jumping with Decimals on the Number Line,” the Lesson-Level ELPS companion recommends that educators “use this section to extend the teacher guide support with the language objective, the expected oral/written output, and a short precision prompt during facilitation.” The guide also states the language objective as, “Students will describe decimal positions on a number line and justify forward or backward jumps using relational language.” 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 on how to integrate each state-approved model in the classroom. Across the elementary grade band, the curriculum provides a dedicated, standalone Texas ESL/Bilingual Implementation Guide designed to support educators in navigating the distinct structural and pedagogical demands of 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 directly to these state-approved frameworks, rather than offering generic, one-size-fits-all directives, the materials ensure that educators are fully equipped 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.

Embedded guidance throughout the lessons supports EB students in developing academic vocabulary, increasing comprehension, building background knowledge, and making cross-linguistic connections through both oral and written discourse. In Journey D6, station 1, development activity 2 incorporates MLR3, “Critique, Correct, Clarify,” as a linguistic support. The routine directs educators to present an incorrect statement, such as “2.5 rounds to 2 because 5 is closer to 0 than to 10,” and have students point out and correct the statement with “2.5 rounds to 3 because we round up when the number is equally distant from both.” Educators then provide a correct statement and ask students to clarify why it is correct. This embedded linguistic guidance supports EB students in acquiring both language and content for success.

Additionally, the materials include embedded guidance to support EB students in developing academic vocabulary and building background knowledge through oral and written discourse. For example, Journey D2, station 10 integrates an active “Two Truths and One Lie” framework within the daily warm-up routine to activate students’ prior mathematical concepts before core content delivery. For written discourse, this lesson launch prompts students to use their personal pads to physically write down which projected statement is false, anchoring their background choices textually. To ensure academic rigor, the text layout embeds formal math vocabulary structures such as, prime number, directly into the written choices. Simultaneously, for oral discourse, the warm-up script directs the educator to cue students to display their written answers and immediately facilitate a brief group discussion regarding the responses. This dual-layer warm-up structure successfully ensures that students actively read, write, and verbally debate academic vocabulary terms while bridging their prior mathematical understanding to the current lesson focus.

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 includes 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 D4, station 10, “Adding Fractions II,” students engage in progressively more complex tasks. In activity 1.1, students begin by reasoning how to add 3/10 and 7/100 and recall that they need common denominators. Next, in activity 1.2, students use fraction pieces to create equivalent fractions in order to have common denominators or find the least common multiple to add fractions. Finally, students represent adding fractions, use their operational knowledge of adding fractions, and express results in simplest form across multiple fraction pairs in their formative assessment logsheet.

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 D5, station 11, “Dividing Fractions by Whole Numbers I,” the materials require grade 5 students to demonstrate a depth of understanding aligned to TEKS 5.3.L. The materials provide practice opportunities that begin with using models and real-world context to connect dividing a fraction by a whole number to multiplying by the reciprocal. As the lesson progresses, the depth of practice opportunities increase as students are asked to repeat this observation with different whole numbers by writing abstract equations. The materials dive to an even deeper understanding in activity 2, when students justify whether this newly discovered pattern would apply to any natural number. Each station or lesson includes a formative assessment known as the logsheet. The logsheet for station 11 requires students to divide fractions by a whole number at three different levels of understanding: abstract equations, real-world problems, and in writing. The logsheet used as an instructional assessment demonstrates the depth of understanding because it requires students to apply their understanding of dividing fractions by a whole number to a new situation, rather than replicate a practiced example.

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. Lessons are scaffolded to build conceptual understanding and support student progression toward mastery. Journey D2, station 9, “Introducing Prime Numbers,” builds from exploring arrays for prime numbers, to connecting the exploration to the definition of prime numbers and divisibility rules, to build proficiency for TEKS 5.4.A. In activity 1, students begin at DOK 1 by forming rectangles with 13 squares, then rise to DOK 2 as students explain why only one rectangle exists, connecting that constraint to the definition of prime numbers. Then in activity 2, students “find all the prime numbers in the range 1–100” and engage in process standard 1F by explaining patterns in eliminating nonprime numbers and analyzing relationships between nonprime numbers and their factors. The activities build a deep conceptual understanding of prime numbers.

Students engage with questions and tasks, including enrichment and extension materials, that increase in rigor and complexity, helping them reach and exceed grade-level proficiency in the mathematics TEKS. Journey D2, station 9, “Introducing Prime Numbers,” includes a differentiated extension logsheet to further challenge students. The logsheet requires students to identify prime numbers on a hundreds chart, while the extension logsheet requires students to “find the 8 numbers that have 3 different prime factors among their factors.” This extension pushes the DOK beyond identifying which provides 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 5, cluster D, “Rational Numbers,” links factoring numbers and fraction equivalence into a coherent horizontal concept to add fractions with unlike denominators. In Journey D4, station 10, “Adding Fractions II,” students begin the lesson with a warm-up using fraction widgets to add fractions with common denominators and find equivalence through simplifying or seeing the equivalence with the fraction widgets. Then, students discuss why they cannot add 3/10 and 7/100 by combining the numerator, naming the big idea that the denominator tells the size of the equal parts. Next, students use fraction widgets to find common denominators for 1/4 and 1/3 and then add the fractions using the common denominator of twelfths. This lesson directly builds on multiples to find common denominators and fraction equivalence introduced in Journey D2.

The lesson plans within a journey demonstrate a coherence across concepts horizontally by connecting patterns, big ideas, and relationships. For example, Journey D5 builds understanding of multiplicative thinking with fractions over 18 lessons. Lesson 9, “Completing Multiplicative Frames with Fractions,” is in the middle of the journey and provides multiple examples of horizontal coherence of concepts. Activity 3 asks students to take their conceptual understanding of multiplying fractions from lesson 8, where they used area models to visualize multiplication of fractions, and begin to multiply fractions abstractly as they complete multiplicative frames. Students use their ability to multiply fractions abstractly in lesson 11, where they connect dividing by a whole number to multiplying by the reciprocal. Students also use the same visual of multiplicative frames in lesson 16, when they begin abstractly dividing fractions, thus connecting 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 how the next cluster will build upon the current cluster. For instance, cluster overview D, “Rational Numbers,” includes a learning progression that explains that “students have worked in the realm of whole numbers,” and how they have developed a “foundational understanding of area and factors that will be essential for visualizing fractions” in the previous cluster. The document then outlines that in this cluster “students begin to realize that a single whole can be divided into equal parts” and are introduced to the relationship between fractions, decimals, and percentages. The progression finally explains that the next cluster moves into abstract systems and magnitudes, including integers and powers. This progression shows the connecting patterns, big ideas, and relationships.

Additionally, the D4 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 22 content standards across a specific cluster, including five grade 3 TEKS, seven grade 4 TEKS, nine grade 5 TEKS, and one grade 6 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 three 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 will learn across the different levels. This pictorial progression traces student expectations as they learn and practice addition and subtraction of fractions with the same denominator using multiple strategies, and ultimately develop strategies to add and subtract fractions with different denominators, focusing on finding representations with the same denominator. 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 prior knowledge and connects to concepts in the current lesson. For example, in Journey D4, station 11, “Subtracting Fractions II,” educators conduct a warm-up with fraction widgets and subtracting fractions with the same denominator. The Teacher Guide explicitly provides support to connect learning to prior knowledge stating, “Students should connect separating pieces with subtraction. As we did with addition, reinforce this idea to keep students from being tempted to subtract numerators and denominators.” Then, activity 1.1 and 1.2 use fraction widgets to help students visually and conceptually understand that a common denominator is needed, connecting the new skill of subtracting with unlike denominators to previously learned concepts of adding fractions with unlike denominators and fraction equivalence. Additionally, the Teacher Guide notes to “extend the strategy of finding a common denominator to fractions greater than one when they are written as a fraction. Later, this will help us add and subtract mixed numbers.” This coherence in concept of adding and subtracting fractions with unlike denominators and the use of visual fraction widgets helps students solidify their conceptual knowledge of needing common denominators.

In grade 5 the materials demonstrate coherence vertically across concepts and grades within each journey. For example, Journey D5 takes the multiplicative relationship and the multiplicative frame representation learned in grade 3 and practiced in grade 4 and extends the learning to fractions and decimals. Specifically, Journey D5, station 9, activity 1, has students complete a multiplicative frame for multiplying fractions using an area model that they had learned in the previous lesson. The multiplicative frame is a fundamental representation throughout the entire Math Journeys program. Multiplicative frames are also used to multiply and divide decimals throughout Journey D6.

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. Journey D6, station 1 asks students to identify locations of a whole number on a number line through the Atlas Math digital applet. This activity requires students to retrieve whole-number place value and number line skills from previous journeys before bridging to decimal concepts. Furthermore, activity 1.1 asks students to locate decimals on a number line by giving two lengths, resurfacing prior learning about measuring length. Finally, activities 1.2 and 1.3 layer locate decimals on a number line and compare numbers with money and weight measurement concepts, requiring students to recall their prior knowledge of hundredths, thousands, measurement, and value of numbers.

The components in the lessons provide space retrieval opportunities for both concepts and skills throughout the program. For example, in Journey D4, station 14, the wrap-up requires students to apply estimation skills to assess the reasonableness of a real-world problem. This concept was taught previously in the journey during the wrap-up from station 13. This concept is taught and reinforced with short activities over several days. Journey D4, station 14 is also an example of spaced retrieval opportunities for the skill of placing fractions on a number line. This skill is first taught in Journey D4, station 6, used again during Journey D4, station 7, and during the Journey D4, station 14, warm-up.

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 for each cluster that outlines the progression of concepts and skills that are connected across the journeys within the cluster. For example, the cluster plan in cluster overview D, “Rational Numbers,” outlines how students are first introduced to numbers that represent part of a whole through fraction representation in various models including area, length, and set models in Journey D1. Next, the materials bridge the concepts and strategies around multiplicative thinking and divisibility learned in Journey C with divisibility for comparing and operating with fractions in Journey D2. Finally, students recall and build on previous concepts learned in additive and multiplicative thinking to add, subtract, multiply, and divide fractions in Journeys D4 and D5. This cluster plan in the overview outlines how the concepts are connected throughout the journeys or learning pathways.

In grade 5, the materials provide routine interleaved practice opportunities by mixing geometry and number theory concepts across pathways and lessons. In the Journey E1, station 4, “Applying Operations,” wrap-up, students complete two distinct sections to practice flexible strategy selection within a single assignment. The first section requires students to observe prisms to complete a table detailing total volume, visible cubes, and hidden cubes. The subsequent section immediately alters the mathematical constraint, forcing students to shift their strategy to a number theory context. In this task, students must utilize only prime numbers for the prism dimensions to equal a specified volume. This rapid alternation between visual spatial counting and prime number classification requires dynamic decision-making on every problem.

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 10, activity 1.1 requires students to interpret, analyze, and evaluate combining 3/10 and 7/100 using fraction pieces. The activity requires students to reason about the size of the pieces and explain that two fractions “cannot be added by simply combining the numerators.” Students are required to analyze how tenths and hundredths represent different-sized pieces visually and symbolically and thus find the same sized pieces or a common denominator by changing the tenths into hundredths. The lesson plans provide guidance through sample responses and questions, such as, “What does the denominator tell us about the size of each part?” and “Can you explain why we can’t just add the numerators when we add three-tenths and seven-hundredths?”

Throughout the lessons, students actively interpret, analyze, and evaluate models and representations of mathematical concepts and situations. For example, in Journey D6, station 11, students estimate the area of a projected rectangle using whole square centimeters, then analyze why both the area model and the standard algorithm yield results with the same digits. Students determine which unit of measure, millimeters or centimeters, most clearly shows decimal placement, making explicit connections between 26 × 42 = 1,092 mm² and 10.92 cm². The educator guides students to observe and explain the structural relationship between these two computed forms, deepening their understanding of the connection between models and algorithms.

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

Journey D4, station 10, “Adding Fractions II,” includes opportunities for students to manipulate fraction pieces to add fractions with unlike denominators. The warm-up opens with a review activity asking students to add fractions with common denominators using fraction pieces. The students connect joining the pieces together to adding, and then connect this concrete model to the equation. Then, in activity 1.1 students use fraction pieces showing how fourths can be converted to eighths to find a common denominator when combining 1/4 and 3/8. Finally, the logsheets require students to draw a fractions representation of the fractions they are combining. For example, the students shade 1/4 and 1/2 of the fraction model and write the matching equation as 1/4 + 2/4 = 3/4. The lesson plans include questions to support connecting the representations to the mathematical situation, prompting, “What does the denominator tell us about the size of each part?” and “Can you explain why we can’t just add the numerators when we add three-tenths and seven-hundredths?”

The progression of lessons provide ample opportunities for students to create their own models and representations. In Journey D4, station 11, students use fraction bars as a concrete model to represent subtraction of fractions with different denominators. In activity 1.1 students build 7/10 with 1/10 fraction bars and 1/5. Then students build an equivalent fraction for 1/5 using 1/10 bars to determine the difference for 7/10 and 1/5. In the logbook students apply what they have learned about subtracting fractions with different denominators by shading in and crossing off parts of a pictorial representation.

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

In Journey D3, station 7, students continue to build on their conceptual understanding of decimal or fractional place value. Students are asked to measure objects that are between 50 centimeters and three meters. Then the educator leads a discussion about one of the lengths recorded in meters, such as 1.25 meters, and applies students’ decimal place-value understanding from money contexts to measurement, which is a new context. This lesson bridges the conceptual understanding of place value to real-world measurement. The lesson plans explain that in the previous money context, 1.25 can be decomposed into 1 and 2/10 and 5/100, but in the context of measurement it can be decomposed into 1 meter and 25 centimeters. Additionally, the lesson includes a drop down support to guide the process standard 1A, “Apply mathematics to problems,” with questions such as, “What does 1.25m tell you about the actual length of the object you measured?” and “How can the same decimal be interpreted differently depending on the context (money or measurement)?” 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 Journey D4, station 14, “Estimation in Fractions,” activity 2.2, students must apply the conceptual understanding of the benchmark fractions zero, one-half, and one to the new, real-world problem situation of determining whether half of a five gallon tank is enough water for Mei’s garden. Again, in activity 2.4, students are given the opportunity to apply the conceptual understanding of benchmark fractions to a new, real-world situation in the context of hiking when reasoning about how far Jackson still needs to travel.

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

The warm-ups and wrap-ups embedded in particular lessons provide tasks and practice that are designed to build student fluency and automaticity in performing math operations. For example, the Journey D4, station 12, “Productive Practice: Fractions,” wrap-up, tasks students to repeatedly create and solve addition and subtraction equations using number cards across four tasks. The lesson plans provide guidance, stating, “An efficient strategy is to keep the denominators the same for both addition equations and only change the numerators.” This repeated practice reinforces fluency. Additionally, the remaining tasks in the wrap-up require students to find equations with the greatest result, smallest result, and matching results. This added layer sustains engagement while building accurate, automatic computation.

The lessons incorporate regular routines and practices to build student automaticity and fluency with grade-level calculation skills. In the Journey D6, station 11, warm-up, the educator writes the expressions 1.40 x 10, 1.40 x 100, and 1.40 x 1,000 on the board to establish fact automaticity. The curriculum encourages students to calculate these numerical products mentally from memory. Individuals utilize instant, rule-based place value shifts to determine the totals effortlessly without executing long multiplication algorithms. This quick mental drill reinforces rapid numerical retrieval to build long-term fact automaticity. During the developmental activity, the materials transition to an estimation and area modeling task to develop strategic calculation fluency. Students analyze a 3.4-by-8 area model and estimate the geometric area using whole centimeter units while ignoring the decimal values. The educator reasons that the final product must be greater than 24 square centimeters. Students then calculate the exact area using the area model or the standard algorithm for 34 x 8. After determining a product of 272, students evaluate their previous whole-number estimates to accurately place the decimal point after the seven. This multistep process develops computational fluency by forcing individuals to cross-check algorithmic calculations using logical estimation strategies.

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

In grade 5, the Journey D5, station 5, “Multiplying Fractions by Whole Numbers III,” warm-up, requires students to look at a visual of fraction pieces that represents three groups of 2/9 and asks them to solve and find all equivalent multiplication expressions. This task requires students to use flexible thinking to create multiple expressions. Additionally, the lesson plan provides explicit guidance for educators to help students practice the application of efficient and accurate procedures. The lesson plan states, “If a student suggests repeated addition 2/9 + 2/9 + 2/9 = 6/9, acknowledge their contribution, but ask them to find a more efficient strategy, guiding them toward multiplication.” The lesson plan also provides guidance on a specific strategy stating, “Use this opportunity to highlight that when multiplying a whole number (for example, 3) by a fraction (for example, 2/9), multiply the numerators and keep the denominator the same, because this calculates 3 groups of 2 pieces of 1/9 (3 × 2 × 1/9).” The specific guidance leads to 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 many opportunities to practice them. In Journey D4, station 12, students use two strategies to add and subtract fractions with unlike denominators. One strategy is using fraction circles and the other is calculating a common denominator. These strategies are flexible because they can be used with any unlike denominators, efficient because students can choose which strategy to use for each problem, and accurate as students will arrive at the correct sum or difference no matter which strategy they choose.

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 throughout learning pathways. For example, Journey D5, station 5, “Multiplying Fractions by Whole Numbers III,” provides students with two representations of 1/3 × 6 and asks them to compare the representations and strategies for similarities. The lesson plans provide guidance on the similarities and differences stating to accept statements, such as, “Both representations there are 6 circles and only 2/6 are colored” and “In both, something is divided into 3 equal parts and only 1 part is colored.” The guidance shares a difference as, “In the left one, what you divide into 3 equal parts is the set of 6 circles, and in the right one, each of the circles.” These statements guide students to flexibly evaluate the representations for accuracy. Additionally, activity 2 guides students through a similar situation with both “eight times 3/4 of an hour” and “3/4 of 8 hours.” Throughout the lesson students evaluate and observe that both representations and strategies are valid while educators guide the conversation to discuss the efficiency and accuracy of the representations and strategies.

Throughout Journey D5, station 13, students actively evaluate mathematical representations, models, strategies, and solutions for efficiency, flexibility, and accuracy. In activity 1.2 students compare two solution paths for 2 ÷ 1/5, repeated grouping with the fraction pieces widget and recognizing the multiplication pattern, and determine which method is more efficient. In activity 1.3 students extend this comparison to 2 ÷ 1/3, checking the accuracy of their results against the multiplicative frame. Finally, in activity 2 students assess whether multiplying by the denominator is always a valid strategy, evaluating the accuracy of this generalized approach.

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

In Journey D5, station 5, students warm up by providing a matching equation for an image of three groups of 2/9 projected with fraction pieces. The guidance explicitly states that if students suggest repeated additions with 2/9 + 2/9 + 2/9 = 6/9, educators should “acknowledge their contribution, but ask them to find a more efficient strategy, guiding them toward multiplication.” In activity 1 students multiply to solve 1/3 × 6. Students compare the strategies of taking 1/3 of 6 or 6 groups of 1/3 and discuss how both strategies provide the same product. This work around comparing strategies helps students consider which strategy may be more efficient for them and how efficient strategy may change based on the problem complexity.

Throughout the pathways, the materials deliver explicit educator guidance to support students in selecting increasingly efficient mathematical approaches. In the Journey D5, station 1, warm-up, the educator displays two unique arrangements of circular fraction pieces that both equal 6/12. The educator asks individuals to generate arguments supporting how the visual images represent both identical and different structural properties. In task 2 the curriculum prompts students to write addition equations to express each fractional layout. Students identify that different expressions can decompose the same underlying value. In task 3 the educator guides students to evaluate a more streamlined numerical representation for the second arrangement. The educator introduces multiplication as a direct alternative to repeated addition by replacing 3 addends of 2/12 with a single multiplication expression. The educator reinforces this scaling by demonstrating that 6 iterations of 1/12 equal 6 × 1/12. During the developmental activity the educator continues to guide students to connect multiplicative thinking to repeated addition. This systematic instruction helps students select the more efficient multiplicative approach.

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 D6, station 11, “Multiplying Decimals: Decimal Factors,” explicitly states the conceptual emphasis of TEKS 5.3D, multiplying a decimal number by a whole number to the concept of the area of a rectangle. When multiplication of a decimal by a whole number is tied to the concept of area, students are able to visualize the size of the product. The lesson plans guide the conceptual understanding by providing a visual and having students first estimate the area by only using whole numbers, and then breaking down the steps of multiplying the decimal. The procedural emphasis is equally explicit, directing students to “calculate the exact area using either the Area Model diagram or the standard algorithm.” The lesson plans address both the conceptual and procedural emphases of the TEKS in the summary, learning objectives, and activity sequence.

The lessons include guidance to help educators address the conceptual and procedural emphasis of the TEKS. In Journey D4, station 10, the learning objectives explicitly state the emphasis on both concepts and procedures for TEKS 5.3H. Learning objective A provides a conceptual emphasis that “adding two fractions with the same denominator can be interpreted as the result of joining two pieces of a group when the pieces are the same size.” Learning objective B provides a procedural emphasis on “how to find a common denominator when we want to express two fractions with different denominators.”

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. Journey D5, station I, “Multiplying Fractions by Whole Numbers I,” starts with opportunities for students to analyze pictorial representations in the warm-up. Two images of fraction pieces are projected and students are asked to “find arguments to support both that the two images are the same and that they are different.” Students should recognize that the pictorial models show the same total, but one image can be represented with addition, while the second image can be represented with multiplication. Additionally, activity 1 of the same lesson requires students to use fraction pieces to build a concrete model and write the matching multiplication expression. Students can group four 1/12 pieces into two equal groups of 2/12 and write a matching multiplication expression of 2 × 2/12 with the goal of connecting it to multiplicative thinking and the abstract process of calculating. The lesson concludes with students solving for “how much Mei runs in total in 2 weeks, considering the number of quarter hours you run in 1 week.” This one lesson provides opportunities for students to use concrete models, pictorial representations, and abstract models as required by TEKS 5.3I.

The lessons guide students through concrete, pictorial, and abstract modeling, fully aligning with the TEKS. The warm-up in Journey D4, station 10, requires students to use the fraction pieces widgets as a concrete model to combine same-denominator pieces, such as 2/9 and 4/9. In activity 1.1 students shift to pictorial modeling by drawing fraction bars and number lines to represent unlike denominators, such as the addition of 3/10 and 7/100. Finally, students record equivalent fractions and their sums as abstract equations, completing the progression from concrete to abstract understanding.

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.

In Journey D4, station 10, “Adding Fractions II,” students explore adding fractions with fraction pieces as concrete models. In the warm-up students use the concrete fraction pieces to physically combine same-denominator pieces and connect the concrete action to the equation 2/9 + 4/9 = 6/9. Students also practice simplifying with finding equivalent fractions using the concrete models. Activity 1.2 asks students to represent 1/4 and 1/3 with fraction bars, then create an equivalent-fraction diagram before writing the abstract addition equation. The Teacher Guide links the representational model to the abstract algorithm by prompting educators to ask, “How do the pieces help you understand why we find a common denominator?”

Educators use the lesson plans to guide students through Journey D6, station 6, with explicit, structured opportunities for students to connect, create, define, and explain both concrete and representational models to abstract concepts as required by the grade 5 TEKS. In the warm-up the materials support the explicit definition of concrete and representational models by presenting students with a pictorial representation of three $1 bills and two dimes. Without assistance, students must independently analyze and write out the total abstract value of the collection, requiring them to accurately define and assign the correct place value to each specific bill and coin to collectively determine that the total value represents 3 and 2/10. In the first segment of activity 1 students must explain and connect these currency models to abstract mathematical principles. To support this, the materials incorporate process standard 1F, prompting the class to analyze the structural relationship between the money model and the standard decimal place value system. This conceptual explanation and definition of positional value is driven by two explicit student-facing prompts, “How are the relationships between dollars, dimes, and pennies similar to the relationships between hundreds, tens, and ones?” and “Could you explain why we need to regroup when the number of cents in the subtrahend is greater than in the minuend?” The materials systematically transition to the creation and handling of representational models during the subtraction tasks and the wrap-up. When solving subtraction problems, the materials force students to actively manipulate the pictorial model by crossing out the exact subtrahend value rather than just viewing a completed static image. To definitively prove their conceptual understanding of subtraction mechanics, students are required to physically cross out a subtrahend and explicitly draw how whole bills are regrouped into 10 separate dimes. The lesson culminates by connecting these visual and physical experiences directly to abstract notations. The tasks guide students to use the pictorial model to add the bills and coins to determine the answer to decimal addition through decomposition, forcing a direct cognitive bridge between visual monetary sketches, regrouping drawings, and abstract decimal algorithms.

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

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 D, “Rational Numbers,” includes language goals for reading, speaking, listening, or writing, along with academic vocabulary words for D1–D6. The reading language goal for D4 states, “Students will analyze word problems to determine if they need to ‘join’ or ‘separate’ fractional parts.” The speaking goal reads, “Students will describe the requirement of having a common denominator before performing operations.” Then the academic vocabulary words like denominators, unlike denominators, common denominator, least common multiple (LCM), sum and difference are listed in a separate box. 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 use visuals, manipulatives, and targeted language strategies to help students develop academic mathematical language. In Journey D2, station 9, activity 1, students represent all possible rectangles for 13 squares on grid paper, anchoring the terms factor and prime number to a visual arrangement with concrete manipulatives. The guide instructs educators to connect each rectangle to factor pairs before formally naming prime numbers, tying vocabulary to the physical model. Video 30 on the “Sieve of Eratosthenes” reinforces the same terms by visually eliminating numbers across a hundreds chart.

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. The D6 Journey Language Bank provides embedded supports to scaffold, support, and extend students’ use of academic vocabulary for decimal reasoning. The pre-teach/embedded language routine support bank in Journey D6, stations 1, 5, and 13, states for the educator to “say the term aloud, point to the model, define it in student-friendly language, use it immediately in the task, then check with a quick question or matching prompt” when introducing a new decimal term such as nearest, tenth, hundredth, or expanded form. Additionally, the discourse-and-writing bank provides partner talk, quick write, and “revise your explanation” prompts, aligning with the suggested supports for Journey D6, station 1. The prompts provide a partner talk stem that states, “___ is between ___ and ___. It is closer to ___ because ___.” Students are then prompted to “write one decimal in words and say which benchmark value it is closest to.” These oral and writing communication prompts increase students’ use of academic mathematical vocabulary.

In grade 5 the embedded educator guidance in Journey D4, station 10, systematically scaffolds, supports, and extends students’ use of academic mathematical vocabulary in context during peer-to-peer and whole-class communication. Throughout the warm-up and activities, targeted support drop-down menus and tiered sentence frames assist language acquisition as students model adding fractions with like and unlike denominators. For instance, when adding 1/4 and 3/8, instructions direct the educator to provide physical fraction pieces to help students explicitly see and communicate how a fourth splits into two equal parts so both addends are expressed as eighths. The lesson plans embed tiered sentence frames to scaffold mathematical communication for EB students. The guidance explicitly extends vocabulary use by instructing the educator to encourage students to move to the next proficiency level by increasing the complexity of their language using targeted frames: “To add these fractions, I need to find a common denominator of ___. Then I convert ___ to ___ by ___” and “When I add fractions with denominators of ___ and ___, I find a common denominator by ___, which gives me ___, because ___.” The wrap-up requires students to independently use and extend their academic vocabulary by sketching out pictorial models to represent their mathematical thinking. To ensure students are analyzing structural mathematical relationships through process standard 1F, the educator utilizes structured questions to evaluate student discourse. For fractions with like denominators, educators are directed to ask, “Why do you think the figure is divided into 4 parts?” For fractions with unlike denominators, educators ask, “Why do you think you can’t do the addition on a single figure?” These questions push students to mathematically justify and explain with academic vocabulary.

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

Journey D4, station 10, “Adding Fractions II,” includes guiding questions to prompt students to justify needing a common denominator using precise mathematical vocabulary. For example, the lesson plan asks students “how to add fractions when the two fractions we combine do not have the same denominator.” Then the guiding questions read, “What does the denominator tell us about the size of each part?” and “Can you explain why we can’t just add the numerators when we add 3/10 and 7/100?”

Student application of precise academic vocabulary and mathematical language is structurally embedded within the classroom’s problem-solving and discourse routines. During the Journey C4, station 15, warm-up, students are challenged to analyze mathematical constraints and formulate an organized plan to arrange three rolled digits—two, four, and five—into unique division operations featuring a two-digit dividend divided by a one-digit divisor. To facilitate this structural analysis and guide student reasoning, the educator uses two intentional verbal prompts: “Could you explain your strategy for finding all possible combinations of these three numbers in a division problem?” and “How can you check that you’ve found all the possible divisions?” These prompts require students to verbally justify their organizational methods and explain the mathematical placement of each digit within their custom-built division expressions. To support mathematical communication and expand peer-to-peer discourse, the lesson implements MLR7, “Compare and Connect.” After constructing their division operations, students are required to actively analyze and read the grouped equations to describe structural similarities and differences across the various numerical arrangements. The educator guidance embeds two explicit questions to evaluate and direct this student-led discourse: “What do you notice about the divisions that give a remainder of 4?” and “How are the equations with remainder 0 different from those with remainder 4?” This structured interaction forces students to apply precise place value and algorithmic language, such as dividend, divisor, quotient, and remainder, as they compare their mathematical plans and connect different representations with their peers.

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, Journey D4, station 10, “Adding Fractions II,” activity 1, asks students “how to add fractions when the 2 fractions we combine do not have the same denominator” and asks them to solve 3/10 + 7/100. The embedded support explains the goal is for students to recall from last year that the size of the pieces must be the same and that tenths can be represented as hundredths. Additionally, the embedded prompts assist educators in helping students refine and use math language. The prompts include, “What does the denominator tell us about the size of each part?” and “Can you explain why we can’t just add the numerators when we add 3/10 and 7/100?” This task provides an opportunity for students to listen to their peers’ arguments and evaluate their reasoning.

Journey D5, station 9, provides opportunities for students to hear, refine, and use vocabulary related to fraction operations. The warm-up activity, “Two Truths and a Lie,” provides the students with three fraction equations and the students must determine the equation that is incorrect. The educator is then directed to “guide a discussion to validate the answers together.” This gives the students an opportunity to hear and use math language with their peers. The next step provides three more equations and provides another opportunity for students to refine their use of math language after the first discussion.

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 D4, station 10, “Adding Fractions II,” anticipates that students may simply add numerators and denominators. Therefore the Teacher Guide redirects by prompting, “Read the operation aloud as three-tenths and seven-hundredths to show that the result should be close to 4 tenths.” An exemplar student response and guidance about the size of the denominators are embedded in the materials, stating, “Using precise mathematical language, students should explain that the two fractions name parts of different sizes—tenths versus hundredths—and that one must be rewritten as an equivalent fraction with a common denominator before they can add.” 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. In Journey D2, station 9, the Teacher Guide expects that students might think a 13x1 rectangle differs from a 1x13 rectangle and guides educators to clarify that rotating the figure does not change its geometric properties. The guide also expects that students might incorrectly classify one as prime and explicitly explains that “prime numbers must have exactly two distinct factors,” while one has only a single factor. When students claim a composite number like 57 is prime, the guide instructs educators to ask for its factors and reveal the 3 × 19 relationship.

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, process standard 1A, “apply mathematics to problems,” is explicitly tagged in Journey D4, station 14, “Estimation in Fractions,” activity 2. The lesson has students imagine they are gardeners and solve for fraction word problems. The materials explain, “Students engage in process standard 1A when they apply mathematics to problems arising in everyday life, such as deciding whether half of a 5-gallon water tank provides enough water for Mei’s garden or estimating how far Jackson still needs to hike after walking four and five-eighths miles of a seven and one-half mile trail. In each case, students connect fraction estimation to a practical decision—determining whether a resource is sufficient or how much distance remains.” Students are also expected to explain their reasoning and respond to the reasoning of others. The materials provide prompts to guide students in their reasoning, such as, “Could you explain how your mathematical solution relates to the real-world context?” and “When might you use estimation versus exact calculations when working with fractions in everyday situations?”

The TEKS process standards are integrated appropriately into Journey D5, Station 11, “Dividing Fractions by Whole Numbers l.” For example, process standard 1F is featured at the top of the lesson plan. In activity 2.2 students are told, “Dividing by a natural number is the same as multiplying by a fraction with numerator one and that natural number as the denominator.” Students are then asked to reflect on whether a statement is true or false. With this question, students should recognize that although they have only looked at the equivalence of dividing by 3, 2, and 4 and multiplying by 1/3, 1/2, and 1/4, respectively, this holds true for all natural numbers. Students address process standard 1F when they analyze the mathematical relationship between dividing by a natural number and multiplying by a unit fraction. After examining multiple cases—such as recognizing that dividing by three is the same as multiplying by 1/3, and that dividing by two gives the same result as multiplying by 1/2—students connect both operations and use that relationship to evaluate whether the generalization holds for all natural numbers.

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

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 D6 Journey Instructional Guide explains that lesson 1 “builds the place-value language students will reuse in every later decimal lesson” and lists process standards 1G and 1E since students are creating and using decimal representations and justifying mathematical ideas. Educators can use the table to see which process standards are incorporated throughout the pathway and how it is connected to the rationale of the lesson.

Additionally, the cluster overview alongside the Journey Instructional Guide describes how the materials incorporate and connect process standards throughout the learning pathways. The D4 Journey Instructional Guide describes how process standards 1F and 1G are incorporated throughout the learning pathway, explaining that 1F connects common-denominator reasoning to equivalence work from Journey D2 and Journey D3, while 1G supports precise justification of fraction operations. The guide explicitly connects these process standards from Journey D4, station 1, through Journey D4, station 16, showing how same-denominator reasoning, unlike-denominator strategies, and estimation are linked by recurring standards. This rationale demonstrates how process standards form a coherent thread across the entire additive fractions 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 are incorporated in the lessons. Additionally, a drop-down box embedded in the lesson makes the process standard easy to locate and provides a clear explanation of the process standard and how it is incorporated in the lesson. Journey D4, station 10, “Adding Fractions II,” explicitly tags process standard 1F and 1G, signaling to the educator that students will analyze mathematical relationships to connect and communicate mathematical ideas and use precise mathematical language to display, explain, and justify mathematical ideas and arguments. The Teacher Guide then connects the 1F tag directly to tasks where students analyze the relationship between the visual fraction pieces and the numerical equation, and the 1G tag to tasks where students explain and justify why the size of the fractional part matters when adding and subtracting fractions. This lesson overview allows educators to anticipate which process standards are active before instruction begins.

Journey D5, station 11, identifies that process standard 5.1.F, “Analyze mathematical relationships to connect and communicate ideas,” is embedded in this lesson. The summary at the top of the screen states that the process standard will be incorporated when students “discover that dividing 1/a by b is the same as multiplying 1/a × 1/b.” The Teacher Guide demonstrates how students analyze relationships for dividing by a whole number and multiplying by its reciprocal directly from this tagged process standard during a peer discussion in activity 2. 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.

Journey D4, station 14, provides opportunities for students to think mathematically through estimating where fractions are located on a number line using benchmark reference points like zero, one-half, and one. The lesson plans guide educators to prompt students with questions, such as, “Is the fraction 3/8 closer to 0 or to 1?” These prompts require students to think mathematically and make sense of mathematics. Activity 1C explicitly instructs students to “select estimation and number sense as techniques” rather than computing exactly, sustaining perseverance when exact answers are unavailable. Activity 2 presents a real-world water tank scenario where students use reasonableness to see if there is enough water to water the plants in a school garden, requiring students to make sense of fraction magnitude in an everyday context.

Additionally, in Journey D2, station 9, students draw rectangles on grid paper with a given area as a way to investigate prime numbers. Students think mathematically in activity 1 as they discover that the first example of 13 only has one possible rectangle, 13x1, and that even if the dimensions are reversed, 1x13, they are considered the same because the product is the same. Students persevere with problem solving in activity 1 as they search for other numbers that have only one set of factors. Activity 2 guides students in making sense of mathematics as they watch the video “Sieve of Eratosthenes” and list all the prime numbers between 1 and 100.

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. Journey D4, station 12, “Productive Practice: Fractions,” asks students to create two addition equations using fraction cards, write the sum as a mixed number, and compare the sums in order to circle the greatest sum. This activity requires students to find “the two with the greatest result” and then “the two with the smallest result.” The lesson plans directly state that “students are not expected to try all possible addition equations, but rather to investigate if there is a way to improve the results they obtained in the previous task. For example, if they used a fraction less than the unit, they can switch the numerator and denominator to increase one of the addends.” The lesson plans support students in explaining and justifying the multiple strategies to solve and complete the problems. Additionally, the lesson tags process standard 1B, prompting, “What plan do you have to find the fraction combinations that will give you the greatest possible sum?” and “Could you explain how you’re checking if your solution really gives the greatest possible result?” The tasks and guidance in the lesson plans require students to understand, explain, and justify their process.

Throughout the journeys, students engage in understanding, explaining, and justifying multiple approaches to solving problems and completing tasks. In Journey D5, station 5, “Multiplying Fractions by Whole Numbers III,” activity 2, students solve two word problems with identical numbers in different contexts, using either repeated addition or the commutative property. After solving, students explain how both methods of finding 1/3 of six and six groups of 1/3 produce the same product and justify which representation fits each situation. As students complete the tasks, educators display both interpretations on the board, validating that different approaches are legitimate solution paths.

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 D5, station 1, explicitly states, “This is a good session to use visibly random pairs.” The lesson begins with opportunities for small groups of students to use fraction circle pieces to physically build all decompositions of 4/12, recording each as an addition equation and doing math together with manipulatives. The groups then write the multiplication expression connecting repeated addition to n x 1/b in their logbooks. In the next activity students discuss how to solve 1/3 × 6. The materials prompt, “What does 1/3 × 6 mean in terms of equal groups?” and “Could you explain how finding 1/3 of 6 relates to division?” These prompts give students opportunities to demonstrate their understanding of multiplying fractions by a whole number through hands-on activities and discussion.

In Journey C4, the station 14 lesson on division and remainders provides explicit, structured opportunities for students to make sense of mathematics by simultaneously doing, writing about, and discussing mathematical concepts with peers and educators. During the warm-up the educator actively engages students in doing math and writing about math by rolling three dice to generate digits and requiring students to independently write all possible division arrangements, a two-digit number divided by a one-digit number, on individual dry-erase boards. This initial activity leverages process standard 1B, prompting students to analyze problem constraints, formulate a plan, determine a complete set of combinations, and evaluate missing arrangements. To facilitate this rigorous mathematical reasoning, the lesson provides the educator with two explicit prompts: “Could you explain your strategy for finding all possible combinations of these three numbers in a division problem?” and “How can you check that you’ve found all the possible divisions?” Following a targeted mathematical discourse about these arrangements, students independently solve the division expressions, shifting their focus to analyzing the resulting remainders. This multimodal approach seamlessly transfers into partner work during activity 2, where students work in pairs to play the “Remainder Race” game. Students collaboratively do math by taking turns rolling three dice, strategically selecting which digits will form the two-digit dividend and the one-digit divisor, calculating the quotient, and moving their game piece forward based on the value of the remainder. The game design directly requires strategic mathematical thinking. For example, if a student rolls a two, four, and five, that yields six possible expressions; the student must mathematically deduce that calculating 24 ÷ 5 is optimal because it yields the largest possible remainder. After the game concludes, the educator reinforces student accountability and peer interaction by leading a structured, whole-class discussion focused on the mathematical strategies and reasoning students discovered throughout the game. 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. Journey D4, station 14, “Estimation in Fractions,” shows students a false statement, 1/5 + 1/10 = 7/10, and prompts the educator to “ask students how they can tell that the addition equation is incorrect without calculating it.” The materials explicitly provide support for how students should recognize that it is incorrect: both fractions are less than 1/4, so they should use estimation and benchmark reasoning to understand that the sum should be less than 1/2. The tasks of estimating sums throughout activity 1 without calculating require students to use relational thinking and explain their justifications. Additionally, the materials support educators in guiding students to reflect on their problem-solving approaches through explanations, arguments, and justifications. In activity 2 the materials provide problem-solving situations where students do not need exact answers. The problems ask students to reason about whether there is enough water if there is half of a five-gallon tank and whether students need to do an exact calculation to see how much is left to walk. In both situations, the materials provide guidance on how students can use the nearest benchmark fraction to estimate the solutions. The lesson plans then prompt, “When might you use estimation versus exact calculations when working with fractions in everyday situations?”

The Teacher Guide actively supports educators in helping students share and reflect on their problem-solving approaches through explanations, arguments, and justifications. Journey D5, station 13, “Dividing Whole Numbers by Fractions,” begins with a real-world warm-up involving time intervals and monetary values, prompting students to justify their thinking and deduce how fractional parts scale to whole units. In the developmental activity students use rectangular fraction pieces to propose open-ended strategies for solving a division word problem before learning a formal procedure. Educators connect the expression 2 ÷ 1/5 = 10 to a multiplicative model, guiding students to reflect on the inverse relationship between multiplication and division. Additionally, the materials reinforce structural proof by activating process standard 1B. Students systematically evaluate their solutions and formally justify their answers by executing the inverse operation within a structured multiplicative frame, guided by prompts such as, “How can you check that 10 cups is a reasonable answer for 2 ÷ 1/5?” A graphic further illustrates the reciprocal operation in dividing whole numbers by unit fractions, and educators ask students to analyze numerical shifts and identify operational patterns.

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

In Journey D4, station 10, students add fractions with different denominators. The Teacher Guide explicitly anticipates that students may struggle with remembering the new equivalent fraction and the process, and states, “Students should suggest rewriting the 2 starting fractions so both have the same denominator.” The materials also recommend to “place special emphasis on the idea that we can transform a fraction by multiplying the numerator and denominator by the same number because doing so is equivalent to multiplying by one.” Additionally, the materials provide visuals of the fraction pieces, and suggest “If any student shows difficulty understanding how we go from fourths or thirds to twelfths, provide the fraction pieces to help them see that if we split the fourths into 3 equal parts and the thirds into 4, we get 2 fractions with denominator 12, which we can add directly.” The anticipated misconception, along with the suggested support, equips educators to provide explanatory feedback.

The materials provide structured prompts and guidance to support educators in delivering targeted, explanatory feedback that directly addresses student responses and anticipated structural misconceptions. In the Journey D5, station 12, fraction division warm-up, students are tasked with independently solving a sequence of operations on blank paper. The lesson plan instructs the educator to say, “Share four-fifths into two equal parts,” to directly prompt and guide students in translating abstract expressions into manipulative actions with fraction pieces. As the activity progresses, it is intentionally designed around a predictable misconception in which the dividend’s numerator is not evenly divisible by the whole-number divisor. To address this, the “Teacher Moves” and “Conclusions” sections provide embedded guidance instructing educators to use fraction pieces to demonstrate how scaling up a fraction creates an equivalent numerator, enabling equal sharing. The materials support the delivery of this feedback with targeted prompts based on anticipated misconceptions: “Can we transform the fraction in part c into an equivalent fraction that is easier to share?” and “When looking for that equivalent fraction, which numerator would make it possible to share the pieces?” Finally, the materials offer additional reactive guidance based on student responses in the support callout, instructing educators to pivot to a concrete sharing context, such as sharing cookies among people, if a student demonstrates difficulty understanding the core meaning of division. By embedding strategic equivalence prompts, script-driven partitioning directives, and contextual real-world pivots, the materials enable educators to systematically unpack student misunderstandings and provide responsive, conceptually grounded feedback.