Math Journeys Texas, Grade 3

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 that identifies the TEKS addressed within each lesson and includes both the content and process standards. The scope and sequence further organizes the standards across the program’s instructional “Journeys.” This organization allows educators to identify the TEKS and ELPS associated with each unit of study.

The Journey Instructional Guide provides a rationale for the learning progression within and across and grade levels. For example, the Journey A2 materials extend students’ understanding of number concepts from identifying numbers and counting within 20 to counting, comparing, ordering, and estimating within 100. In section 3, the guide explains how instruction progresses from concrete experiences to representational and abstract models. The guide supports both the vertical and horizontal alignment of mathematical concepts.

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 Program Guide outlines recommendations for implementing the materials through a three-part instructional framework of “build, consolidate, and practice.” Lessons begin with educator-led small-group instruction and progress to independent practice using logsheets. There are opportunities within the independent practice for educator feedback. Additional practice is available through Atlas Math, the program’s adaptive digital component. In the Program Guide, the “Lesson Internalization and Planning” section includes suggestions for planning instruction based on learner variability, multiple representations, extensions, and just-in-time supports.

The lesson plans include integrated recommendations for adapting the instruction based on student needs. The “Wrap-Up” section provides intervention supports, extensions, visual models, and alternative explanations to use during instruction. For example, in Journey C4, station 8, the materials include a linked support that suggests introducing division with three-digit dividends using smaller divisors. Additionally, the materials recommend using money or base 10 blocks to model the task with concrete representations before moving to abstract procedures.

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

The TEKS Correlation Guide connects the lessons and activities within each journey to the specific TEKS content and process standards. Educators can use the guide alongside diagnostic assessment results to identify standards-based instructional pathways and determine which lessons align to targeted student needs.

The placement test report sample displays the TEKS assessed through the diagnostic assessment and recommends student groupings based on performance data. The “Proposed Group Distribution” section of the report includes suggested instructional groupings and entry points for each group. Individual reports provide a snapshot of a student’s current understanding and include recommended starting points for instruction based on those assessment results.

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

In the Program Guide, the “Lesson Internalization and Planning” section provides protocols for both unit and lesson preparation and planning. The “Preparing for a Journey” section guides educators to examine how each journey builds on prior learning. The section further reviews lesson objectives and mastery expectations, identifies opportunities for scaffolding and enrichment, gathers instructional materials, and analyzes learning progressions before instruction begins.

Three sections support educators in implementing the program’s instructional components: “Preparing for a Building Session,” “Preparing for a Consolidating Session,” and “Preparing for an Additional Practice Session.” These sections instruct educators in how to clarify lesson outcomes, anticipate misconceptions, prepare materials, identify formative assessment opportunities, analyze student responses, and use progress-monitoring data to inform their subsequent instruction.

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

In the Program Guide, the “Hands on Partnership” section outlines the ways instructional leaders can support educators in implementing the Math Journeys content. Educators have the opportunity to collaborate with other educators through professional learning community support. The section also outlines additional ways leaders can provide support, such as through analysis of student work and progress-monitoring data, alignment of instructional practices, and development of long-term intervention systems. These examples demonstrate how campus leaders can support educators in implementing the program as designed.

The training materials include on-demand videos that model the instructional strategies used throughout the program. For example, the video, “Let’s Add by Breaking Down,” demonstrates an approach educators can use during instruction. These resources support both educators and campus leaders in understanding specific strategies.

The Math Journeys Coaching Guide includes look-fors, predictable struggles, and ready-to-use rubrics for instructional leaders during observations. These materials break down into four implementing phases including “Getting Math Journeys Going,” “Shifting the Instruction,” “Adapting to the Classroom,” and “Assessing Learning.” For example, the look-fors clearly outline that in the “Shifting the Instruction” phase, “conclusions and formalizations come after the exploration not before.” The materials provide very specific 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 include the TEKS-aligned learning objectives, student materials, educator resources, and lesson components. For example, the materials in Journey C2, station 4 identify 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.” The Program Guide includes a suggested timeframe for the various lesson components in the “Inside a Journey” section. Additionally, the C2 Lesson-Level ELPS Companion provides the ELPS alignment. A table contains 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 provides multiple assessment resources including diagnostic and progress-monitoring tools that support instructional decision-making. The digital practice progress reports identify how students perform in each of the TEKS, while formative assessments such as logsheets, embedded lesson assessments, and end-of-journey assessments provide an additional measurement of student understanding. The answer key provides the TEKS and ELPS alignment for each question.

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

The materials include adaptive lesson plans that provide the TEKS-aligned learning objectives, student materials, educator resources, and lesson components. For example, Journey C2, station 4 states that the objective is to “establish connections between quotitive division and partitive division” and “develop understanding of the relationship between division and multiplication when one factor is unknown.” The lesson plan identifies adaptive activities in the “Activities Added to Individualized Practice After Session Completed” section. In the Program Guide, the “Inside a Journey” section includes suggested instructional 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 includes multiple assessment resources, such as diagnostic and progress-monitoring tools. The digital practice progress reports provide TEKS-aligned assessment data by identifying and connecting student performance to each of the TEKS standards. This data supports instructional decision-making and monitors student growth. Additionally, there are formative assessments called logsheets incorporated into the lesson plans. The end-of-journey assessments align to the TEKS and can be found in the answer key.

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

Each journey includes a family letter providing information about student learning. The letter contains 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 to support communication with families.

The 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 the various instructional models and strategies. In addition, the “How Can We Help from Home?” section provides games and activities that 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 directions for the activity and a visual representation.

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 clearly define the diagnostic assessment as a tool used at the start of instruction to identify prior knowledge and to determine in which journey pathway to place students. Formative assessments, such as warm-ups, logsheets, and integrated checks for understanding, inform real-time instructional adjustments. Finally, the summative assessments include the end-of-journey tests and cumulative progress reports. These assessments provide opportunities to evaluate student mastery of targeted skills and determine readiness for advancement within the instructional pathways.

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

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

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

The end-of-journey and formative assessments are available in both digital and print versions. For example, in grade 3, the C2 end-of-journey assessment includes a document that provides a link to download printables including the C2 end-of-journey assessment, answer key, and instructions guide.

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

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

The materials include diagnostic assessments with TEKS-aligned tasks or questions. Additionally, a sample diagnostic assessment and placement report for grade 3 provides results for 24 distinct assessment tasks mapped directly to TEKS 3.2–3.5 and mathematical process standards 3.1.A–3.1G.

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

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

The materials include a variety of formative assessments with TEKS-aligned tasks including interactive item types with varying complexity levels in the logsheets for each lesson. For example, the logsheets in Journey C2 include text entry, numeric entry, and completing visual representations. The logsheets also require students to “write the equation that helps you find the answer,” “complete these multiplicative frames,” and represent situations with a drawing and multiple equations.

A Digital Assessment Item Bank includes questions from the digital formative checkpoints that educators can view. The questions span more than two levels of complexity and progress from basic arithmetic recall to multistep fractional word problems and comparative justification tasks. For example, Journey Test B4 asks an open-response question, “What is the result of this operation? 734 – 286.” In contrast, Journey Test D5 asks a multiple choice question, “Which situation can be represented by this equation: 4 divided by 1/8?” Furthermore, these checkpoints feature more than two unique interactive item formats, incorporating standard single-select buttons, multi-select criteria boxes, and interactive text-entry fields containing custom stacked fraction lines.

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

The materials in Math Journeys include a placement test report sample for grade 3 providing detailed scoring information and guidance for interpreting student performance. For example, the “Results: General Overview” section shows a colorful graphic comparison of low, medium-low, medium-high, and high percentages of students in the program, the region, and the class. The report includes student-by-student scoring information with a chart that shows student performance by content strand and process standard. Additionally, the recommended starting journey includes guidance for interpreting student performance along with suggested instructional focus points for each student.

Each end-of-journey test includes a downloadable answer key that includes rationales for correct and incorrect responses. For open-ended item types the answer key also provides rationale for common incorrect answers. In grade 3, the answer key for the end-of-journey test for C2 provides the TEKS 3.4.F for question 1 and provides the rationale that “36 is correct because 9 x 4 represents 4 groups of 9, and 4 groups of 9 equal 36.” The answer key then provides an incorrect answer of 13 and the rationale, “13 is incorrect because it is the sum of 9 and 4, not the product.”

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

The placement test report sample provides information that supports the educator in utilizing the results from assessments to support the purposeful planning of tasks and activities for the appropriate TEKS. For example, the report includes a chart that lists the TEKS, the process standard, the percentage of correct responses for the class, and the percentage of correct responses for students using the program. Educators can use the class trends to prioritize the necessary TEKS. Additionally, the “Placement in Journeys” section groups students in the class into 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 the student trends and misconceptions identified on the assessments. In the assessment guidance matrix for the C2 test, the text connects specific performance trends in multiplication and division to concrete instructional next steps under the “Additional Support” section. For example, to address trends in which students confuse multiplication with addition (such as adding factors instead of multiplying), the materials guide educators to use concrete tools, such as array models or equal-groups counters, to visually demonstrate scaling.

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 in the digital practice portion. The report allows educators to track student progress and growth over time, as well as overall performance by topic and student expectation. For example, the report provides information that includes the total number of 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 3, in Journey C2, station 7, the materials include embedded guidance for educators to check for student understanding at key points during instruction. In activity 1, students explore 5 x 7. The educator then asks students to draw a rectangle with 7 columns and 5 rows and find the total. The embedded guidance states, “The goal is for students to quickly deduce that the answer is the same as before because they have represented the same rectangle but rotated. This opens the door to talk about the commutative property of multiplication.” This check for understanding allows educators to correct any misconceptions before continuing with the remainder of the lesson.

Additionally, the guidance materials for educators include scripted questions and PowerPoint presentations. For example, in Journey C2, station 5, “Connecting Multiplication and Division I,” the goal is for students to understand that they can use division to determine the number of equal groups they can make from a quantity or the number of items they can place in each group. The lesson begins by asking students if the following statement is true or false: “12 is a result in the 4 times table.” Furthermore, the script prompts educators to ask, “What operation did you use to find the answer?” to ensure students understand the connection between multiplication and division.

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

The materials provide frequent checks for understanding during digital practice in the Atlas Math Applet Library, but do not adjust the level of difficulty based on students’ responses. For example, in applet A95, “Multiplication,” if students select an incorrect answer, the activity prompts, “How many groups of 5 can you make?” Students receive immediate feedback through a progress indicator that displays a green check mark for correct responses and a question mark for incorrect responses. However, the next question does not change based on student performance, and the activity continues with the same sequence of questions.

The digital applets do not provide adaptive checks for understanding within an activity. For example, in applet D101, “Area of Rectangles,” incorrect responses result in feedback indicating whether the answer is correct or incorrect, but the activity continues with the same types of questions. The level of rigor does not adjust, nor does the applet provide additional scaffolds or changing instructional supports based on student performance. The materials do not provide hints, virtual manipulatives, text-to-speech support, or alternative item types that respond to student misconceptions during the activity. As a result, the materials provide frequent checks for understanding but do not demonstrate adaptive checks that adjust instruction in real time based on individual student responses.

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.

The materials contain embedded, explicit educator support throughout the lesson plans to scaffold instruction for students who have not yet reached proficiency. In the lesson plan for Journey C2, station 3, the “Wrap-Up” section outlines multiple targeted intervention strategies in clickable support boxes for the educator. For example, the teacher is directed to “Ask students to write the number of rows and columns that make up the set of dots” to reinforce procedural and conceptual understanding. Additionally, the materials incorporate tactical modifications, instructing the educator to “Provide students with colored tiles to represent the rectangular arrangement associated with each multiplicative frame.” These embedded components ensure educators have immediate access to differentiated strategies.

Additionally, the materials provide explicit educator guidance and structured lesson activities to scaffold instruction for students working toward grade-level proficiency. In grade 3, the Journey C2, station 2 activity on multiplicative models and comparison phrases outlines a step-by-step scaffolding routine that uses concrete manipulatives to build prerequisite understanding of the grade-level objective. The instructions guide educators to first construct a single tower of snap cubes and then create an identical second tower. The guidance then directs educators to stack the towers to demonstrate the concept of “twice as many.” To support the transition from concrete representations to abstract multiplication, the materials provide specific prompts for educators, such as “Could you explain how your towers show twice as many cubes?” and “Do you see how we can write this tower model as a multiplication equation?” This progression and guided questioning supports students who have not yet reached proficiency in deriving a multiplicative equation from a concrete model.

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.

Math Journeys provides language-development supports through the “Key Vocabulary” section at the beginning of each lesson and in the Journey Language Bank within each journey. These resources support academic vocabulary development by identifying key content vocabulary, English–Spanish cognates, false cognates, and multiple-meaning words. The embedded Journey Language Bank also includes routines that support listening, discourse, and writing.
For example, in the C2 Journey Language Bank, the pre-teach/embedded language routine support bank provides four routines with sample educator actions and student responses to use across lessons. Educators implement routine 1, “Model-Pronounce-Define-Use-Check,” before or when introducing a keyword in model-based lessons C2.1, C2.3, and C2.4. This routine directs educators to say the term, define it, use it in context, and check for understanding, thereby supporting students’ development of the academic vocabulary identified in the lessons.

The C4 Journey Language Bank includes support for unfamiliar references in the text through routine 4, “The Reads for Multiplicative Stories.” In this routine, educators preview numbers during the first read and ask, “What is this story about?” to help students clarify unfamiliar real-world contexts before computation begins. Additionally, in the Journey Language Bank, the “Unfamiliar References” section includes a “Quick Bridge Before the Lesson” section that guides educators on supporting unfamiliar references. In Journey C1, station 3, tower and cubes may be unfamiliar. The materials suggest “Build one tower of 5 cubes and hold it up. ‘This is a tower. It is made of 5 cubes. Every tower today has the same number of cubes.’ Students then build one tower and say tower and cubes.”

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.

Math Journeys provides enrichment opportunities through extension activities embedded within extension tabs. For example, in the grade 3, Journey C2, station 6, “Wrap-Up” section, the extension tab directs educators to “Ask students to draw different multiplicative frames that have the numbers 20, 30, 40, etc. at the top.”

The Enrichment and Extension Guide provides guidance for which students receive extension work and how to facilitate each extension logsheet. Additionally, there are extension tips embedded in the lessons. For example, the Journey C1 Enrichment and Extension Guide provides tables for those nine lessons within the journey with extension logsheets. The table includes information on the intent of the task, what to listen for, and what counts as mastery. For example, in Journey C1, station 1, the table explains that the intent of the extension is for students to notice what happens to the number of rods and ones when you half or double a number. The table guides educators to listen for “Students predicting from the ones digit whether a double will regroup (5 or more ones means a new rod), and noticing that halving forces an exchange exactly when the tens digit is odd.” This guide further 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.

Math Journeys includes digital assessments and Atlas Math, which provide text-to-speech accommodations and digital support. However, the materials do not allow educators to enable or disable supports for individual students. The Program Guide explains what content and language supports are available, including the ability to switch between English and Spanish. However, educators cannot enable or disable any additional content or language supports. Moreover, the Program Guide states, “Calculators are not provided within the digital interface; instead, they are provided and managed by educators during test administration.”

The lessons in the Atlas Math digital applets include a speaker button for text-to-speech and a help button that provides 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.

The lesson plans provide opportunities for students to demonstrate understanding of the lesson topic through multiple instructional experiences. For example, in Journey C2, station 6, the “Warm-Up” section engages students in guided discussion and provides explicit educator guidance within a blue support box to facilitate the discussion. Students also demonstrate understanding by working with manipulatives in small groups. The lesson walks educators through how to help students translate their manipulative models into multiplication and division equations. The guidance prompts educators to offer guiding questions such as, “How many groups have been formed?” and “How many colored tiles are in each group?” Students further demonstrate understanding by creating pictorial representations. Additionally, activity 2 asks students to draw multiplicative frames to represent multiplication and division.

Math Journeys provides explicit educator guidance for offering students options and support across multiple modalities. In the Teacher Guide, core lessons systematically embed process standard support boxes, including standard I.C. For example, in the grade 3, Journey D1, station 13, fraction-comparison lessons, the text explicitly directs educators to allow students to choose among physical fraction pieces, a unified visual model, or numerical notation. The materials further support implementation by providing scripted questions to facilitate these modalities, including “Do you see how we could use fraction pieces to help us compare these?” and “Could you explain which way of showing these fractions makes them easier to compare?”

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.

Math Journeys provides a “Warm-Up” section with explicit prompts and guidance that help educators anchor big ideas and activate prior knowledge. For example, in Journey C2, station 5, students connect division to multiplication as the warm-up activity displays six numbers and asks students to identify numbers from the six times table. This activity requires students to activate their prior knowledge of multiplication before making connections to division.

The materials provide explicit educator-facing guidance and structural frameworks that anchor big ideas and outline learning progressions across mathematical concepts. In the cluster overview C, “Multiplicative Thinking,” the text describes the critical cognitive shift students make as they transition from additive thinking to multiplicative scaling. The materials systematically connect these big ideas through multiple representations and explicitly direct the use of rectangular arrays and fixed-interval jumps on a number line to visualize multiplication as iterated addition. Additionally, the embedded mathematical background text boxes identify specific learning milestones and guide educators through a progression of concepts. This includes a structured timeline for introducing multiplication tables, connecting inverse operations through multiple strategies, and transitioning students from visual models to the abstract standard algorithm.

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

Math Journeys offers varied instructional approaches, including hands-on exploration, whole-group and small-group discourse, and independent consolidation. Each lesson is structured to transition from guided discovery, during which educators facilitate hands-on exploration, to formalization. For example, in grade 3, Journey C2, station 6 asks students to explore the relationship between multiplication and division through hands-on discovery. Students sort colored tiles into groups and explain the relationship using prompts such as, “Could you explain how the tiles and circles show both multiplication and division?” and “How does your arrangement of tiles connect to the equations we wrote?” These prompts, along with the activity, help formalize the relationship between multiplication and division.

Each lesson plan provides educators with explicit directions and guidance for implementing effective and engaging instruction. For example, Journey C2, station 6 begins with a “Warm-Up” section that utilizes whole-group discussion. The first development activity transitions to small-group instruction in which students use manipulatives to model division. The second development activity returns to whole-group instruction and uses a multiplicative frame to represent division equations. The lesson concludes with the wrap-up during which students practice independently on logsheets. Each lesson component includes explicit guidance, step-by-step directions, and educator prompts that support implementation.

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

Math Journeys incorporates multitiered intervention methods through varied practice structures and embedded educator guidance that support effective implementation. Lessons include multiple opportunities for practice, including collaborative partner work and whole-group discussions integrated throughout the lesson activities. For example, in grade 3, Journey C2, station 7 provides opportunities for students to independently practice multiplication and division with an area model using the logsheets. Students then work collaboratively during the development activities. Students work in pairs to build an area model, determine the area, and finally represent the model using both multiplication and division. Embedded blue support boxes guide educators in facilitating these experiences. For example, a blue box during partner work states, “The goal is for students to avoid counting square by square and instead use their multiplication knowledge to find the answer (35) by multiplying 5 × 7.” These embedded supports equip educators with guidance for implementing the lesson effectively.

The materials incorporate various intervention methods, including small-group, collaborative, and independent activities, to support learners across varied instructional structures. For example, Journey C2, station 6 segments core instruction into targeted learning environments that provide small-group support alongside independent settings. Within the small-group structure, students engage with hands-on manipulatives, including colored tiles and specialized multiplicative frames, to support understanding. The materials further scaffold instruction by providing educator-facing scripts and exact guiding questions, such as, “How does your arrangement of tiles connect to the equations we wrote?” These prompts support educators in actively facilitating and scaffolding student understanding as learners work through the examples.

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

Clickable extension tabs embedded throughout the lessons provide opportunities for enrichment and extension. For example, in grade 3, Journey C2, station 6 guides educators in extending students’ understanding of the connections between multiplication and division by asking students to draw different multiplicative frames with the numbers 20, 30, 40, and so forth at the top. These extension tabs provide enrichment and extension guidance within the wrap-up. Additionally, most lessons include an extension version of differentiated logsheets to further support enrichment opportunities.

Furthermore, the materials in Journey C2, station 9 include enrichment and extension methods that support various forms of engagement. Students engage in productive practice of multiplication facts on a hundreds chart and the lesson plans include explicit implementation prompts such as, “Ask students to predict the patterns they would color on the 100s chart for the 10, 11, and 12 times tables.” This extension pushes students to identify the pattern on the hundreds chart before they have learned this level of multiplicative thinking.

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

The materials in Math Journeys include prompts and guidance to support educators in providing timely feedback during lesson delivery. The lessons provide educators with specific prompts and guidance to use with students during instruction. These embedded prompts direct educators to pause at key moments during the lesson to check for student understanding, address misconceptions, and provide immediate and meaningful feedback to students in real time. For example, in Journey C2, station 2, students practice the concept of multiplicative comparisons with snap cubes. After building their first comparison, the prompt in the lesson plan states, “Ask: Which tower has more snap cubes? How many snap cubes does it have?” Additionally, the guidance provides prompts for how students should respond, such as “Students should explain that the new tower has 8 more snap cubes than before (additive comparison), or that the new tower has twice as many snap cubes as the original (multiplicative comparison). If students don’t use the word double, ask what term describes when one quantity is 2 times another.” This support equips educators with the tools needed to deliver responsive instruction and keep students on track toward meeting the lesson objectives.

In the Journey Language Bank, section 4, the anticipated responses/misconceptions/redirection bank includes lesson-specific misconceptions and corresponding feedback and correction strategies. For each misconception, the materials provide educators with an explanation of what the misconception may indicate and suggested redirection strategies. For example, in Journey C2, station 2, a common misconception occurs when “a student says the taller tower has ‘8 more’ but does not name it as ‘twice as many.’” The materials explain that this response indicates “the student sees additive comparison but not multiplicative comparison.” Educators should next compare the original and new towers side by side and ask, “How many copies of the first tower do you see?” Educators should then restate the response using double or twice as many.

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 C2, station 4, development activity 1, the lesson embeds emergent bilingual (EB) student supports. In this activity, students receive 34 tiles that represent cows and must separate them into five stables. The EB student supports 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 pre-production level students build five groups of six tiles while pointing to the leftovers. The high intermediate level students may state, “First I made groups of six, then I counted four cows left over.”

Additionally, each journey includes a Lesson-Level ELPS Companion that provides a table identifying the ELPS focus, language objective, language demand, expected oral output, expected written output, and suggested lesson implementation. 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 for content, but do not provide linguistic accommodations. For example, in applet 95, “Multiplication: Let’s Select Multiplications,” when students select the wrong equation, the help button asks, “How many times do you have to add 4 to obtain this result?” Students can use this prompt to help them find an equivalent fraction and use digital manipulatives. However, the materials do not provide linguistic accommodations if a student does not understand due to language barriers.

In the Program Guide, the “Overview” section explains the lesson structure. The educator should provide accommodations during the “Build” and “Consolidate” portions of the lesson to ensure students learn the academic vocabulary, 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.

The Lesson-Level ELPS Companion embedded within each journey supports educators in implementing instruction for EB students. The materials state that educators can use the guide to “add a compact ELPS layer without crowding the main guide.” The document provides a language objective, language demand, expected oral and written outputs, and suggested implementation guidance. For example, in Journey C2, station 2, “Times Table Comparison Model,” the Lesson-Level ELPS Companion recommends that educators “contrast additive language with multiplicative language every time students compare towers. When students say more, prompt them to decide whether they mean more cubes or twice as many cubes.” The guide also identifies the expected oral output. “The new tower has ___ more cubes, and it is ___ times as tall as the original tower.” This guidance supports educators in effectively implementing the materials with EB students.

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

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

Embedded lesson guidance supports EB students in developing academic vocabulary, increasing comprehension, building background knowledge, and making cross-linguistic connections through both oral and written discourse. In grade 3, Journey C2, station 13, development activity 2 incorporates linguistic support through the Mathematical Language Routine (MLR) 5, “Co-Craft Questions.” The activity asks students to identify reversed products in the three times table, such as 12 and 21. The routine then invites students to generate mathematical questions about the pattern, including “Which times tables have a pair of reversed products?” “Why does the nine times table have so many?” and “Does the eight times table have any?” This embedded linguistic guidance supports EB students in acquiring both the language and content knowledge necessary for success.

Additionally, the materials include embedded guidance to support EB students in building background knowledge and making cross-linguistic connections through oral and written discourse. For example, Journey C2, station 12 successfully activates students’ prior mathematical concepts across both language media during the daily warm-up routine. For written discourse, this fact-review launch prompts students to use their dry erase boards to write out operational equations and map numerical patterns, anchoring their prior knowledge in writing before core content delivery. For oral discourse, the activity integrates MLR5, “Co-Craft Questions,” which provides explicit educator-facing facilitation scripts to demonstrate a multiplication model and directly invite students to generate mathematical questions aloud. To ensure comprehensive language support, this oral routine explicitly notes that educators can bridge to native-language cognates during the group discussion, thereby establishing an active oral cross-linguistic connection through student-generated questions.

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 C2, station 5, “Connecting Multiplication and Division 1,” students engage in progressively more complex tasks. In activity 1, students begin by evaluating true/false statements about multiplicative relationships and then create multiplicative frames connecting known facts to related division statements. In activity 2, students solve word problems and model with manipulatives in the multiplicative frames. Finally, students create division equations and solve real-world word problems 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 D1, station 12, “Comparing Fractions,” the materials require grade 3 students to demonstrate a depth of understanding aligned to TEKS 3.3.H. The practice opportunities begin with using models to compare fractions with like denominators. The beginning examples provide students with a visual fraction model and require the students to identify the two fractions and then compare them with the correct comparison symbol. As the lesson progresses, the depth of practice opportunities increase. Students view a set of comparison statements with like denominators and must verbally explain their observations about comparing fractions with like denominators. The materials dive to an even deeper understanding providing practice opportunities in real-world context. The lesson then repeats this progression of practice opportunities that builds a deeper understanding by comparing fractions with like numerators. Each station or lesson includes a formative assessment known as the logsheet. The logsheet for station 12 also requires students to demonstrate a depth of understanding by comparing fractions with like numerators or like denominators when given a fraction model, constructing their own model to compare fractions, and explaining their reasoning in writing.

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 intentionally designed to increase in rigor and complexity, guiding students toward grade-level proficiency in the mathematics TEKS. Scaffolded lessons build conceptual understanding and support student progression toward mastery. The grade 3, Journey C1, station 6, “Introducing the Four Times Table,” builds from exploring products of two to products of four by doubling the products to build proficiency for TEKS 3.4.F. In activity 1, students explore this idea with visuals of sets of objects and analyze the pattern through listing multiplication equations in order. Explicit instructions guide educators to “Help students notice that skip counting by 4s is like doubling the results of skip counting by 2s, or moving from one result to the next by making forward jumps of 4.” Then in activity 2, students continue to practice the concept of doubling through building arrays that show the multiple means of scaffolding for conceptual understanding, to guide students toward grade-level proficiency.

Students encounter questions and tasks, including enrichment and extension materials, that increase in rigor and complexity, guiding them toward grade-level and above grade-level proficiency in the mathematics TEKS. For example, in Journey C3, station 2, “Multiplication Beyond Basic Facts I,” students start with 10-column partial products and progress to full area model diagrams for problems like 18 × 7. Students build grade-level TEKS 3.4.G proficiency as they work with increasingly larger factors. In the wrap-up, students find the number of hours in a week, applying multi-digit multiplication to real-world contexts that go beyond basic fact recall. Furthermore, a dedicated extension logsheet provides students with additional above grade-level written resources to further challenge and extend their learning.

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 3, cluster C, “Multiplicative Thinking,” links equal-groups thinking with multiplication, and division with array reasoning and inverse-operation language into a coherent horizontal concept. In Journey C2, station 7, “Connecting Multiplication and Division III,” students draw rectangles with rows and columns connecting the area model to a multiplication equation and multiplicative frame. Then, students connect the multiplicative frame to the four multiplication and division equations. This lesson directly builds on the equal-groups thinking and array building initially introduced in Journey C1, and inverse-operation language that was introduced in previous lessons in Journey C2.

In grade 3, the lesson plans within a journey demonstrate a coherence across concepts horizontally by connecting patterns, big ideas, and relationships. For example, Journey D1 builds understanding of fractions and fraction representation across 16 lessons. Lesson 9, “Equivalent Fractions, Area Model,” is in the middle of the journey and provides multiple examples of horizontal coherence of concepts. The warm-up activity, “Which One Doesn’t Belong?” refers back to what students learned in lesson 1 about recognizing halves and lesson 3 where they began using fraction notation. Activity 1 introduces equivalent fractions by requiring students to identify halves using area models, while also referring back to lessons 1 and 3. Activity 2 extends activity 1 from identifying halves to identifying equivalent fractions of other benchmark fractions using area models. Students use the equivalent fractions they identified in area models as a foundation for lessons 10 and 11 where they use other models to represent equivalent fractions. Lesson 9 concepts repeat in lesson 12 where students use area models to compare 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 describing how the next cluster will build upon the current cluster. For instance, cluster overview C, “Multiplicative Thinking” includes a learning progression that explains “students have moved away from counting by ones to thinking in groups,” and “established the ‘part–whole’ logic where students understood a total quantity can be broken down and rearranged” in the previous cluster. The document then outlines that in this cluster “Students begin to see that adding repeatedly is slow, so they transition to working with equal groups.” The progression finally explains that the next cluster takes the rectangular array and skip-counting strategies from this cluster and “applies them to partitioning in sharing.” This progression shows the connecting patterns, big ideas, and relationships.

Additionally, the C3 Learning Brief demonstrates vertical coherence across multiple grade levels to support a wide continuum of mathematical learning. This document outlines a unified mathematical progression by mapping a comprehensive sequence of 16 content standards across a specific pathway, including seven grade 3 TEKS, seven grade 4 TEKS, and two grade 5 TEKS. To address varying student entry points and potential learning gaps, the document includes a diagnostic section that explicitly differentiates between students who have achieved mastery of the prerequisite mathematical concepts, and students who require active reinforcement of foundational skills. The overview features visual models and instructional text that map out a persistent vertical strategy for multi-digit multiplication. This progression traces student expectations from initial basic fact fluency, advances through intermediate conceptual models, and culminates in abstract standard algorithm proficiency. By embedding this comprehensive vertical scope at the beginning of the pathway within the broader cluster, the materials ensure that multidimensional calculation skills build upon a continuous, cross-grade framework 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 activity that activates prior knowledge and connects to concepts covered in the current lesson. For example, in Journey C1, station 6, “Introducing the Four Times Table,” educators conduct a warm-up through the Atlas Math digital applet where students recall doubles. This recall is useful for the current lesson of constructing the four times table. Then, activity 1 guides students to understand that 3 x 4 = 12 because the double of 3 x 2 is 12. This connects prior skip-counting procedures to a new multiplicative structure. Additionally, the Teacher Guide notes that students already know the product of 2 x 4, which equals 4 x 2. This provides a preview of how this doubling relationship will extend into the eight times table in Journey C1, station 7. This coherence in the concept of doubling to find new products helps students solidify their knowledge of equal-groups thinking.

In grade 3, the materials demonstrate coherence vertically across concepts and grades within each journey. For example, Journey C2 reinforces grade 2 concepts of arrays and transitions into multiplication. This introduces big ideas and patterns that will be used for multi-digit multiplication in grades 4 and 5. Specifically, in lesson 3, activity 1.1 begins by projecting an array and asking students “how many rows, columns, and dots they see in total.” After students identify the factors and products on the array, the educator introduces the multiplicative frame to visually represent the relationship between factors and products. The multiplicative frame is a fundamental representation throughout the entire Math Journeys program. In grade 4, Journey C4, station 10 uses multiplicative frames to model the relationship between multiplication and division. Journey D5, station 9 additionally uses multiplicative frames in grade 5 to assist students in applying multiplicative thinking to fractions.

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

The Math Journeys materials offer spaced retrieval opportunities by revisiting previously
learned concepts across different learning pathways. In grade 3, Journey C2, station 5 asks students to “select the results corresponding to the indicated times table,” requiring students to retrieve skip-counting skills built in the previous journey. Furthermore, activity 1 asks students to analyze true/false statements about multiplication equations and complete multiplicative frames previously introduced in Journey C2, station 3. Finally, the wrap-up includes a word problem for students to apply division reasoning first established in Journey C2, station 4.

The components in the lessons provide space retrieval opportunities for both concepts and skills throughout the program. For example, in Journey D1, station 12, the warm-up activity activates prior knowledge of the concept that the size of a whole must be equal in order for fractions to be compared. This concept was taught previously in Journey D1, station 9. This concept is also used again in the wrap-up in station 12. This concept is taught and reinforced with short activities over several days. Journey D1, station 12 also provides spaced retrieval opportunities, revisiting the skill of writing in fraction notation given an area model of a fraction. This skill is first taught in Journey D1, station 3, and used again during the warm-up activity in Journey D1, station 5. Students revisit the skill again in Journey D1, station 9, activity 1 and again during Journey D1, station 12, activity 1.

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

In grade 3, the materials provide routine interleaved practice opportunities by mixing distinct operations and geometric attributes across pathways and lessons. In Journey E1, station 3, “Mixed Word Problems II,” students use number cards to evaluate inequality equations involving mixed multiplication and addition. During the first developmental activity of the same lesson, students solve a multistep word problem requiring double multiplication and addition to determine a total. The materials then alter the situation to force a subtractive solution path instead. In the wrap-up students navigate scrambled multistep word problems to practice flexible strategy selection. One task requires a combination of division and addition operations. The subsequent task immediately switches to a geometric context where students calculate total vertices across multiple squares and triangles. This constant alternation between operations and geometric attributes prevents predictable pattern-following.

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 D1, station 9, “Warm-Up” activity, “Which One Doesn’t Belong?” requires students to interpret, analyze, and evaluate four area model fraction representations by justifying which representation does not belong. One sample response states that figures A, B, and C represent one-half of the area or an equivalent form because the shaded part equals the unshaded part, while figure D has four shaded squares out of a nine square unit figure and therefore does not represent one-half. The lesson plans provide guidance through sample responses and questions for educators to ask, such as “Could you explain another reason why this shape might not belong?” and “How can you justify, using the number of shaded and total squares, that the 9-square figure does not represent one-half like the others?”

Questions and other tasks immerse students in interpreting, analyzing, and evaluating models and representations of mathematical concepts and situations. For example, in Journey C2, station 3, activity 1.1, students interpret a dot array by identifying rows, columns, and the total, then actively map these quantities to the three cells of the multiplicative frame. Then in activity 1.3 students evaluate incomplete frames, determine the missing number, and explain whether a given value correctly completes the relationship.

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

The Math Journeys materials include questions and tasks that provide opportunities for students to create concrete models and pictorial representations to represent mathematical situations throughout the lessons. For example, Journey C1, station 8, “Connecting Counting and Multiplication,” activity 1.1, includes opportunities for students to build equal-groups models. Students work in pairs to draw six large circles and place four colored tiles in each one. The activity then connects the concrete equal-groups model to representing multiplication. In activity 2 students represent multiplication through creating arrays with the tiles. Finally, the logsheets require students to draw the array to create pictorial representations of the given multiplication statement. The lesson plans include questions to connect the representations to the mathematical situation prompting, “Could you explain how the number of circles and tiles in each circle relate to your multiplication equation?” and “Do you understand what each number in your equation represents in the tiles arrangement?”

The progression of lessons provides students with ample opportunities to create their own models and representations. In grade 3 students use circles to repeatedly represent an area model for fractions. In Journey D1, station 3, students model fractions with cookies and use a printable to cut out so that they can physically manipulate the fractions. These paper cookies are a concrete model that students refer back to as they move towards area model representations. In the logbook students partition and shade their own pictorial representations of the fractions they discovered earlier with the concrete models.

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

The materials include questions and tasks that provide opportunities for students to apply conceptual understanding to new problem situations and contexts. For example, in Journey C1, station 14, activity 1.1, students begin with the conceptual understanding of building an equal-groups model to represent multiplication. Students then progress to using the equal-groups model to solve real-world word problems in activity 2. Students respond to questions requiring them to “draw 8 rectangles” to represent cars and “4 lines” to represent the number of people in each car. Students then must solve the problem of “how many passengers fit in eight cars.” This lesson bridges the conceptual understanding of multiplication and the concrete models, to using the concrete models to solve word problems. Additionally, the lesson includes a drop down support to guide the process standard 1A, “Apply mathematics to problems.” Guiding questions include “Could you explain how your drawing shows the real situation with the cars and passengers?” and “Do you see how your equation represents the number of eggs in all the egg cartons?” 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 D1, station 3, “Representing Fractions: Area Model,” activity 2.1, students apply the conceptual understanding of needing common denominators or the same sized piece when distributing unit fractions in the new, real-world situation of sharing cookies equally among three children. The wrap-up asks students to distribute different quantities and fractions of foods equally. This activity provides students with additional opportunities to apply their conceptual understanding that any remaining unit fractions that are not sufficient to give each of three students a portion represents a remainder that no one gets.

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

The Atlas Math digital applet practices are embedded in the lesson warm-ups or used as extra practice for students and provide tasks designed to build students’ fluency and automaticity in performing math operations. For example, Journey C1, station 8 uses applet A95, pack 9, in the warm-up, and asks students to rapidly identify which multiplication equation matches a displayed product. This applet builds fact fluency and automatic retrieval across the previously learned multiplication facts. Additionally, fluency and automaticity build through repeated practice of the skills embedded with the conceptual understanding in the lessons. For example, Journey C1, station 8, activity 1.1, asks pairs of students to place colored tiles into circles and write the matching equation, such as drawing six circles and placing four tiles inside each circle then writing the equation 6 x 4 = 24. This repeated practice while changing group sizes builds fluency and automaticity while reinforcing the conceptual understanding of multiplication.

The lessons incorporate routines and practices to build student automaticity and fluency with grade-level multiplication skills. In the Journey C1, station 12, developmental activity, the educator displays an array consisting of two columns and six rows. The curriculum instructs students to determine the total product without counting individual dots one by one. To facilitate strategic fluency, the educator guidance outlines two flexible calculation methods. Students either double the value of six using known fact components or skip count by twos to efficiently solve the problem. These exercises develop computational fluency through flexible strategy selection rather than rigid procedures. In the wrap-up, the materials transition to a dedicated routine designed to measure and build fact automaticity. Students evaluate a multiplication grid containing various basic facts from previously studied times tables. The task requires students to shade studied products green and instantly record the products they successfully recall from memory in each box. Students must then evaluate numerical relationships to fill in the remaining products for the unshaded squares. This memory-based recall activity directly reinforces instant, effortless fact retrieval to establish long-term automaticity.

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

The materials provide efficient, flexible, and accurate mathematical procedures throughout the learning pathways. In grade 3, Journey B5, station 14, “Comparing Addition and Subtraction Strategies,” activity 1 requires pairs of students to solve 3,455 + 1,280 using both decomposition and jumping along the number line strategy, and then switch. The use of both strategies side-by-side directly practices flexibility in strategies. The lesson plans provide prompts for students to “discuss which strategy they find more efficient for solving the given addition operation.” This discussion guides students to solving efficiently and accurately. Finally, in the wrap-up activity students practice adding and subtracting with the various strategies, reinforcing efficiency, flexibility, and accuracy.

The lessons build in a variety of efficient, flexible, and accurate mathematical procedures and provide students with many opportunities to practice them. In grade 3, Journey C4, station 12, students solve division problems using mental math, a column diagram, or division by decomposition. Students can choose which strategy to use and then compare results with another student who used a different strategy. This lesson is an example of students practicing efficient, flexible, and accurate procedures. The students have the flexibility to choose which strategy is most efficient and they compare two different strategies for accuracy.

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

Students evaluate the efficiency, flexibility, and accuracy of mathematical representations and models throughout learning pathways. For example, in grade 3, Journey B5, station 14, “Comparing Addition and Subtraction Strategies,” students solve 3,455 + 1,280 using the decomposition and jumping along the number line strategy. After solving, students exchange work and then verify accuracy and identify errors. Finally, students discuss “which strategy they find more efficient for solving the given addition operation,” evaluating flexibility and efficiency. The lesson repeats the same steps for subtraction, requiring students to justify efficiency and confirm that both approaches yield identical, accurate results, regardless of the strategy. This task requires students to evaluate mathematical strategies and solutions for accuracy, flexibility, and efficiency.

Throughout Journey D1, station 13, students evaluate mathematical representations, models, strategies, and solutions for efficiency, flexibility, and accuracy. In activity 1.2, students compare two fractions presented in different models and signal which is greater. Students then discuss strategies such as representing both fractions in the same model or converting to fraction notation to check for shared numerators or denominators. In activity 2 students work in groups to find fractions greater than and less than 2/6 using fraction circles. Students evaluate which representations make comparisons most efficient and select the tool or technique that best supports their reasoning. Finally, students must verify their accuracy by checking that their approach yields a defensible and correct ordering.

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

Educators guide students to select increasingly efficient approaches to solve math problems. In Journey B5, station 15, students explore multiple strategies to subtract 1,000 - 754 including jumping backwards on a number line, jumping forward from 754, and decomposing with place value blocks. The lesson plan guides educators to discuss which strategy is more efficient. Additionally, the lesson plan provides a compensation or equivalent subtraction strategy by directing educators to ask, “What if you came to the store with $1 less, but what you bought also cost $1 less?” Students have the remainder of the lesson and the wrap-up to continue exploring the various strategies and decide which one may be most efficient based on the number characteristics. As students work toward increasing efficiency, educators can use the provided guidance such as, “Students are free to choose which operation to use with each strategy. Students who understand the benefits of each strategy will likely choose strategy D for the most challenging subtraction (6,000 - 2,345).”

Throughout the pathways, the materials deliver explicit educator guidance to support students in selecting increasingly efficient mathematical approaches. For example, in the Journey C3, station 2, developmental activity, the educator uses the Atlas Math digital applet to display a vegetable garden array. Because students struggle to execute an exact count due to large column dimensions, the educator instructs individuals to decompose the multiplication problem of 18 x 7 into expanded form components. Students compute 10 x 7 and 8 x 7 before combining the partial products. The educator reinforces this scaling by utilizing a box area model to demonstrate 18 x 4, airing a digital video to connect basic facts to larger products under process standard 1F. To facilitate this efficiency check, the educator asks two embedded prompts, “Do you notice a pattern in how we can break apart larger numbers to make multiplication easier?” and “Could you use what you learned about breaking apart 18 × 4 to help you solve 35 × 6?” During the wrap-up the materials provide targeted remediation guidance to actively discourage inefficient procedures. Students evaluate grid area models for the expressions 17 x 4 and 23 x 5. Although individuals can physically count the small grid squares to determine the answer, the curriculum explicitly establishes the expectation that students decompose the two-digit factors instead. The guidance materials instruct the educator to actively correct any students counting by ones, prompting them to utilize the more efficient partial product strategies. Students solidify this efficiency progression by utilizing the abstract box method to solve a contextual two-digit by single-digit word problem.

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 C2, station 3, “Introducing the Multiplicative Frame,” explicitly states the conceptual emphasis of TEKS 3.4D by connecting arrays and dot counting to the multiplicative frame. This concept shows that multiplying consists of counting the total number of objects arranged in groups or arrays when the number of rows and columns is known. Activity 1.3 explicitly addresses this procedural emphasis, asking students to “write the number that completes each multiplicative frame,” practicing missing-factor computation. 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 D1, station 3, the learning objectives explicitly state the emphasis on both concepts and procedures for TEKS 3.3.A and 3.3.C. Learning objective A explains the concept of fractions being created when objects are divided equally. It explicitly states, “when a shape is divided into three or six equal parts by area, each part represents one-third or one-sixth, respectively.” Learning objective B explains the procedure of writing a fraction in different forms as it explicitly states that students will “express fractions in different ways: in words (four-sixths) and using fraction notation (4/6).”

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. In grade 3, Journey C2, station 3, “Introducing the Multiplicative Frame,” activity 1.1 provides opportunities for students to count dots arranged in rectangular arrays using colored tiles as concrete models, then record the totals in the multiplicative frame. Then, activity 1.2 directs students to “draw the corresponding multiplicative frame,” providing the opportunity to create pictorial representations of rows, columns, and products. Finally, activity 1.3 requires students to write the missing number completing each frame, moving to abstract symbolic notation of multiplication and division relationships. This lesson provides opportunities for students to use concrete models, pictorial representations, and abstract models as required by TEKS 3.4D.

The lessons actively engage students in using concrete models, pictorial representations, and abstract models as required by the TEKS. Journey D1, station 3, “Representing Fractions: Area Model,” requires students to work in groups to cut and distribute cookie printouts, using manipulatives as concrete fraction models aligned with TEKS 3.3.A. Students then draw pictorial representations on the printout, shading portions to show each child’s share as a fraction. For the wrap-up, students write fraction notations with the numerator and denominator, completing the progression to abstract symbolic representation.

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

The lesson plans provide comprehensive support for students in connecting, creating, defining, and explaining concrete models as they transition to abstract concepts. In Journey D1, station 3, “Representing Fractions: Area Model,” students explore fractions with concrete models by cutting and distributing four paper cookies evenly among three people. This activity allows students to either cut each of the paper model cookies into thirds and distribute four-thirds or distribute one whole cookie to each student first and partition the remaining cooking into thirds, allowing each student to receive one and one-third cookies. Activity 1 of the same lesson connects how much cookie each student receives to the word form of fractions as well as the abstract fraction notations. The resources provide a visual along with the fraction in words and in fraction notation. Additionally, the lesson plans provide guidance by stating, “Be sure to connect the three representations: the graphic, the verbal, and the numerical (fraction notation).”

Educators use the lesson plans to guide students through Journey B5, station 9, using the explicit, structured opportunities for students to connect, create, define, and explain both concrete and representational models to abstract concepts, as required by the grade 3 TEKS. In the second segment of activity 2 the materials support the creation and explicit definition of concrete models by directing each student to physically manipulate base 10 blocks to independently represent multi-digit values of 2,517 and 3,164. This task requires students to actively define the mathematical place value assigned to each physical manipulative—thousands, hundreds, tens, and ones—on their tray to construct the quantities. To ensure students explain and connect these physical manipulations to abstract mathematical principles, the materials incorporate process standard 1F, which prompts students to analyze and define the structural place value relationships that make regrouping mathematically viable. The lesson plan uses two explicit student-facing prompts to drive the conceptual explanation and definition of regrouping mechanics: “Do you understand why we can exchange 10 rods for 1 flat?” and “Could you explain how the value stays the same when we regroup?” Furthermore, the lesson explicitly targets mathematical communication for EB students using MLR3, “Critique, Correct, Clarify.” This specialized framework requires students to define and write out their mathematical justifications and corrections using the structural sentence stems: “The regrouping step was missing because ___ ones make 1 ten and ___ ones are left” and “This number should be ___ because we need to regroup ___ tens as 1 hundred.” In the third segment of activity 2 the materials systematically transition from concrete tools to the creation of representational models. Students must independently generate a pictorial model by sketching out base 10 block symbols for 2,635 and 582 to determine the total sum, defining what each drawn symbol represents in the combined value. The lesson culminates by connecting these concrete and pictorial experiences directly to abstract notations. The tasks guide students to align and evaluate their expressions using an expanded algorithm, forcing a direct cognitive bridge between the visual, physical, and symbolic algorithmic processes.

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

The materials in Math Journeys provide consistent opportunities for students to develop academic mathematical language through the use of visuals, manipulatives, and language goals. The materials provide a cluster overview for each cluster and a “Cluster Plan” section that lists the language goals and academic vocabulary words. For example, cluster overview C, “Multiplicative Thinking,” includes language goals for reading, speaking, listening, or writing along with academic vocabulary words for C1–C4. The speaking language goal for C1 states, “students will describe a multiplication scenario as a specific number of equal groups” and provide the example “I have 3 groups of 4.” The listening goal reads, “students will identify the total when given oral descriptions of rows and columns in an array.” A separate box lists the academic vocabulary words equal groups, array, row, column, skip count, factor, product, and repeated addition. 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 help students develop academic mathematical language by using visuals, manipulatives, and targeted language strategies. In Journey C2, station 3, “Introducing the Multiplicative Frame,” students view dot arrays and connect the visual patterns of rows and columns to the terms factor and product which they record in the multiplicative frame. Students then draw their own arrays and write the corresponding frames, bridging hands-on models to symbolic math notation. The guidance for EB students encourages educators to keep the words row, column, factor, product, and total visible so students can reference each term while describing their work. Educators revoice student answers using precise mathematical language, such as, “So you’re saying that multiplying 2 by 7 gives a product of 14?”

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 to support routines. The Journey Language Bank additionally provides discourse and writing supports to scaffold, support, and extend students’ use of academic mathematical vocabulary when communicating with peers and educators. In grade 3, the C2 Journey Language Bank provides embedded supports to scaffold, support, and extend students’ use of academic vocabulary for moving between models, equations, and pattern language with multiplication and division. In Journey C2, stations 8–13, the pre-teach/embedded language routine support bank states to “prompt students to notice, compare, and connect patterns using posted words such as row, column, diagonal, adjacent, product, and multiple.” Additionally the bank requires students to restate “informal observations with precise mathematical language.” Furthermore, the discourse-and-writing bank provides partner talk, quick write, and “revise your explanation” prompts, aligning with the suggested supports for Journey C2, station 8. The prompts provide a partner talk stem that states, “The ___ times table makes a ___ pattern on the chart because ___.” Students should then “color one chart and write two pattern observations.” These oral and writing communication prompts increase students’ use of academic mathematical vocabulary.

In grade 3 the educator guidance in Journey C2, station 7 outlines steps to scaffold academic language through grid models and MLR2. The instructions for activity 1 direct the educator to have student pairs draw a rectangle with five columns and seven rows on grid paper to calculate the total area. After writing the equation 5 × 7 = 35 on the board, the educator prompts students to draw a rotated rectangle featuring seven columns and five rows. To extend mathematical thinking, the educator notes highlight this layout as an opportunity to discuss the commutative property. After writing the corresponding equation 7 × 5 = 35, the educator summarizes both geometric representations using a multiplicative frame. To transition to inverse operations in activity 2, the educator guides students to determine a missing factor by drawing a rectangle with three columns that occupies 21 total squares. Using a multiplicative frame, the educator models how to represent four related multiplication and division equations. To support EB students, MLR2 instructs educators to collect and display student thinking by grouping visual descriptions with their corresponding mathematical equations. Furthermore, instructions guide educators to revoice student contributions using rows, columns, factors, product, and divide to build a classroom reference tool. During the wrap-up a support drop-down menu outlines specific scaffolding strategies, directing the educator to introduce physical snap cubes so struggling learners can build concrete models.

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

The lesson plans in Math Journeys include embedded guidance to support students in
applying appropriate mathematical language and academic vocabulary during discourse. Journey C2, station 3, “Introducing the Multiplicative Frame,” includes guiding questions to prompt students to reason around the multiplicative relationship using precise mathematical vocabulary. For example, the lesson plan shows an array and asks students “how many rows, columns, and dots” they see in total. The embedded language supports prompts include, “What is the total?” and “Why did you place this number here?” The embedded guidance also directs educators to “revoice responses with precise language” and provides a sample statement such as, “so you’re saying that multiplying two by seven gives a product of 14?” The guidance also recommends partners restating the sentence using the same vocabulary which supports students in applying appropriate mathematical language and academic vocabulary.

Journey B5, station 15, includes deeply embedded communication and discourse structures to prompt students to actively apply precise mathematical language and academic vocabulary. During the warm-up the educator prompts the class to analyze 10 related subtraction problems to determine which calculations are most efficient to solve mentally. The instructions guide the educator in monitoring and validating appropriate mathematical language, explicitly providing two detailed, anticipated student reasoning responses. This expectation for student justification carries into the developmental activity, where students apply mathematics to real-world financial contexts. Before executing calculations, the educator facilitates contextual discourse using two intentional prompts: “Could you explain the problem in your own words and what you need to find out?” and “What does each number in the problem—$1,000 and $754—represent in the piggy-bank situation?” This forces students to interpret and mathematically explain the exact structural meaning of each quantity within the story scenario. Authentic student discourse and peer-to-peer language refinement serve as the core vehicle for the lesson’s concluding tasks. After the educator models multiple student-suggested subtraction strategies—including open number lines, decomposition with base 10 blocks, and compensation adjustments to the minuend and subtrahend—students independently apply these strategies in the wrap-up tasks. To evaluate their strategic selections based on problem complexity, the materials implement MLR3, “Critique, Correct, and Clarify.” This routine mandates that students exchange their written mathematical solutions, critically evaluate whether the technical language and visual work support the chosen strategy, and actively revise their own mathematical explanations based on peer feedback before sharing an improved solution with the collective class.

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

The materials include prompts and guidance in the lessons for educators to facilitate math-related conversations. For example, in Journey D1, station 12, “Comparing Fractions,” the warm-up asks students to explain how two students “could have traveled different fractions of the distance if they started from the same place and are now at the same point along the way.” The lesson plans include guidance explaining that since the two fractions, 2/3 and 1/3, are not equivalent fractions, the only possible explanation is that their end points differ. Additionally, the embedded prompts assist educators in helping students use and refine their math language. The prompts include, “Could you explain why we can’t compare these fractions directly?” and “How can you justify that Ethan’s home must be closer to school, using the fractions 2/3 and 1/3?” The lesson plan explicitly states, “This discussion provides an opportunity to observe whether students listen to their peers’ arguments and evaluate their reasoning.”

Additionally, Journey C1, station 5, provides opportunities for students to practice using new multiplication vocabulary as they explore multiples of five and 10. In activity 1 the materials provide guidance for facilitating these student responses, stating “for each student response, use the phrase ‘___ times 5 equals ___’”. This activity provides opportunities for students to hear their peers verbalize a multiplication fact, and opportunities to practice using the vocabulary themselves. Activity 2 also requires students to explain how they could determine the value of sets of five without time to count each item. The drop-down below step 2 provides guiding questions for the educator to help students refine and use mathematical language.

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.

The lessons include embedded guidance to help educators anticipate a variety of student responses, including exemplar answers. Journey D1, station 1, “Foundations with Fractions: 1/2,” anticipates that students may claim two tangram parts are not halves because they differ in shape. The lesson plans incorporate support and guidance stating, “If students struggle to recognize that the 2 parts represent the same amount, use the tangram pieces to overlap them and verify using manipulatives.” Additionally, the materials provide prompts such as “Could you explain why these two parts are equal even though they look different?” and “How can you justify that the blue parallelogram is one-half of the rectangle using precise mathematical language?” ​These supports help students articulate their fractional understanding of halves using precise academic language and vocabulary.

The lesson plans provide guidance that helps educators anticipate a variety of student answers, provide exemplar responses to questions and tasks, and support or redirect inaccurate student responses. For example, in Journey C2, station 3, “Introducing the Multiplicative Frame,” activity 1.3 anticipates that students might place a factor in the top cell instead of the total and offers the exemplar “The missing number is 3, because 3 times 5 or 5 times 3 equals 15.” The guide prompts educators to redirect inaccurate responses by asking students to name rows, columns, and the total out loud before re-entering numbers. This ensures that students connect the frame’s structure to the array, rather than applying it procedurally.

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

The Math Journeys materials appropriately integrate the TEKS process standards throughout instruction. For example, Journey C1, station 14, “Times Tables: Equal Groups Model,” activity 2, explicitly tags process standard 1A, “Apply mathematics to problems.” The lesson asks students to create equal-groups models with concrete tiles and then applies the models to real life situations, such as eight cars with four passengers each. The materials explain, “Students engage in process standard 1A when they apply multiplication to problems arising in everyday life, such as fitting passengers into cars or packing eggs into cartons. They translate the real-world details—eight cars with four passengers each, or five egg cartons holding six eggs each—into multiplication equations that match the situation they have drawn.” Students should be able to explain their reasoning and respond to the reasoning of others. The materials provide prompts such as, “Could you explain how your drawing shows the real situation with the cars and passengers?” and “Do you see how your equation represents the number of eggs in all the egg cartons?” to guide students in their reasoning.

Journey D1, station 3, “Representing Fractions: Area Model,” appropriately integrates the TEKS process standards into the lesson content. For example, the top of the lesson plan prominently features process standard 1A. It states, “1A - Apply mathematics to problems arising in everyday life.” In activity 1.3 students receive five cookies and must share them equally among two students. Students engage in process standard 1A by sharing five cookies equally between two friends and deciding what to do with the leftover cookie. Students connect fraction notation directly to a familiar sharing situation. Educators provide guidance to support this work with questions such as, “Could you explain how your drawing shows the way you shared the cookies?” and “How does the fraction help you show that each friend gets the same amount of cookie?”

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

The materials include a description of how the learning pathways incorporate and connect the TEKS process standards throughout. The Journey Instructional Guides include a table showing the lesson, concept covered, the TEKS, process standard, and why the lesson matters for each lesson in the journey. For example, the C2 Journey Instructional Guide explains that lesson 2 provides a “strong opportunity for comparison and precise justification” and lists process standard 1D, “Communicate mathematical ideas.” Educators can use the table to see how the pathway connects to each of the process standards, and how it connects to the rationale of the lesson.

Additionally, the cluster overview in the Journey Instructional Guide describes how the materials incorporate and connect the process standards throughout the learning pathways. Cluster overview D and the D1 Journey Instructional Guide explain how the materials incorporate process standard 1A in sharing contexts. Additionally, they connect it across area, set, and length model sessions throughout the D1 learning pathway. Journey D1, station 3, explicitly tags process standard 1A, linking cookie sharing situations to fraction notation, and the guide notes that this language reoccurs in Journey D1, station 4, and in Journey D1, station 5. The D1 scope and sequence shows that 1A, 1E, and 1G each appear in multiple lessons, demonstrating these cross-session connections.

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 the lessons incorporate. 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 featured in the lesson. Journey C1, station 14, “Times Tables: Equal Groups Model,” explicitly tags process standard 1A, signaling to the educator that students will apply mathematics to everyday contexts such as counting passengers and eggs. The Teacher Guide then connects this tag directly to tasks where students write multiplication equations matching real-world equal groups drawings. This lesson overview allows educators to anticipate which process standards are active before instruction begins.

Journey D1, station 3, identifies that the lesson incorporates the process standard 3.1.A. The summary at the top of the screen states that the process standard will feature in the lesson by using “fraction notation and the area model to represent fractions in real-world sharing situations.” The Teacher Guide demonstrates how students develop numerator and denominator language directly from this tagged process standard during the cookie sharing activity. This lesson-level overview identifies the embedded process standard and tracks its inclusion throughout the lesson.

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

Students have opportunities to think mathematically, persevere through solving problems, and make sense of mathematics throughout the Math Journeys lessons. For example, Journey B4, station 13, provides opportunities for students to think mathematically by estimating the closest sum or difference without calculating the exact solution. In activity 2, the educator presents the problem, “Alex has 378 points in a video game. Martha has 161 points fewer than Alex. How many points does Martha have?” Students receive answer choices and guidance to eliminate 417 because Martha must have less than 378. They can also eliminate 317 because the difference between 317 and 378 is less than 100. The lesson then continues with additional problems without answer choices, prompting students to persevere and explain their reasoning. The guide states, “The following word problems do not include multiple choice options. Ask students to use estimation to solve them without performing exact calculations.”

Additionally, Journey D1, station 9, asks students to think mathematically in the warm-up as they analyze four partially shaded shapes and must determine which one does not belong. Students can reason using fractions, geometry, or area to justify which shape does not belong. Students persevere in problem solving during activity 1 as they determine different ways to represent 1/2 using a given set of fraction circles. The guidance in the drop-down under step four suggests that educators ask, “Do you notice a pattern in how the numerators and denominators are related?” which helps students construct meaning from their own discoveries.

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. In grade 3, Journey B5, station 14, ”Comparing Addition and Subtraction Strategies,” assigns each partner a different approach—decomposition, or a number line—to solve 3,455 + 1,280. Partners then compare their work to verify that both methods produce the same answer while discussing which they find most efficient for solving the given addition operation. The next activity repeats the process for subtracting 3,455 - 1,280, requiring students to explain and justify their preferred approach. Additionally, the lesson aligns with process standard 1C by requiring students to compare the efficiency of both methods and respond to questions such as “Could you explain why you think the Number Line strategy works better for this problem?” and “Do you think there might be problems where decomposition would be more efficient?”

Throughout each journey, students engage in understanding, explaining, and justifying multiple approaches to solving problems and completing tasks. In Journey C2, station 6, “Connecting Multiplication and Division II,” activity 1, students work in groups to distribute 35 colored tiles into five circles and record all the equations their arrangement produces. This activity helps students see that one model can represent both 5 × 7 = 35 and 35 ÷ 5 = 7. The guide instructs educators to write all suggested equations on the board and ask which multiplicative frame connects them. Students then create word problems for a new frame, explaining and justifying how one set of three numbers can support four distinct operations.

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. Students participate in hands-on activities, discussion, and writing with their peers and educators. For example, Journey C2, station 2, requires students to work in pairs to physically build snap cube towers, count tiles, and write equations to make sense of doubling through a hands-on activity. The materials guide educators to “ask students how many snap cubes are in the tall tower, how they know this, and what equation represents this.” Throughout the lesson students build towers, compare the heights to the original tower, and discuss whether it is “double” or “triple” the size. Additionally, students write equations requiring them to demonstrate mathematics through hands-on activities, discussion, and writing with their peers and educators.

In grade 3, Journey C1, station 8, “Connecting Counting and Multiplication,” provides explicit, structured opportunities for students to make sense of mathematics by simultaneously doing, writing about, and discussing mathematical concepts with peers and educators. In activity 1 students collaborate in pairs to actively do math by constructing concrete equal groups using circles and physical counters. This hands-on modeling is directly tied to peer interaction and writing about math, as one partner physically manipulates the groups while the other partner records the corresponding multiplication equation. This collaborative workflow integrates process standard 1E, requiring students to use representations to organize and record a multiplication idea. The lesson directs the educator to facilitate this conceptual modeling using two explicit prompts: “Could you explain how the number of circles and tiles in each circle relate to your multiplication equation?” and “Do you understand what each number in your equation represents in the tiles arrangement?” This multimodal approach continues into activity 2, where student pairs use counters to construct arrays. To ensure all learners can actively discuss math, the materials embed MLR 7, “Compare and Contrast,” to support EB students. The educator facilitates a guided academic discussion about the structural relationships between equal groups and arrays, and students consolidate their thinking by completing the written sentence frame, “___ groups of ___ and ___ rows of ___ both equal ___.” Finally, during the lesson wrap-up students maintain accountability for their written mathematical reasoning through differentiated logsheets. On the extension version of the logsheet students complete mathematical computations and independently write responses to three open-ended questions, such as “Which color has the most squares in each grid? Which has the fewest? What is the difference in the number of squares between the color with the most and the color with the fewest in each case?” By seamlessly weaving physical modeling, collaborative discourse, and targeted writing exercises into a single lesson, the materials comprehensively ensure that students employ multiple communication pathways to deepen their conceptual understanding.

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

The Math Journeys lessons include educator guidance to support students in explaining their thinking, as well as generating and justifying mathematical arguments. In Journey D1, station 12, “Comparing Fractions,” the warm-up provides the example of two students who walked different fractional distances of 2/3 and 1/3 with the same starting point. The materials explicitly state that educators should “ask students how they could have traveled different fractions of the distance if they started from the same place and are now at the same point along the way.” The lesson plans embed questions that require students to justify their mathematical arguments, stating “Could you explain why we can’t compare these fractions directly?” and “How can you justify that Ethan’s home must be closer to school, using the fractions 2/3 and 1/3?” Additionally, the materials support educators in guiding students to reflect on their problem-solving approaches through explanations, arguments, and justifications. Throughout the lesson, students compare fractions by analyzing the number of pieces when the denominators are the same and the size of each piece when the numerators are the same. In activity 2 students solve the problem, “Mei has drunk 3/8​ of a pint of juice. If Charles has drunk more than Mei, what fraction with a numerator of 3 might he have drunk?” The materials include prompts that help students reflect on their problem solving, such as “Do you notice a shortcut that helps you compare fractions with the same denominator?” and “Could you find a rule that helps you compare fractions quickly when they have the same numerator?”

The Teacher Guide actively supports educators in helping students share and reflect on their problem-solving approaches through explanations, arguments, and justifications. In Journey B4, station 13, “Addition and Subtraction Word Problems II”, activity 1 provides guidance for process standard 1B, prompting educators to ask, “How does knowing that 33 + 10 = 43 help you decide whether 133 + 210 is closer to 300 or 400?” Then, students share and defend their estimation strategies and problem-solving approach. The embedded EB student routine guides students to state whether the unknown should be greater or less than 378, then eliminate answer choices, prompting them to justify their operation choices together. After solving the problem, the Teacher Guide prompts students to explain their reasoning and reflect on the efficiency of their chosen strategy.

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

Educators guide students through provided explanatory feedback, targeted guidance, and prompts that address student responses and anticipated misconceptions. For example, in grade 3, Journey C1, station 14, students place tiles in circles to represent multiplication in the equal groups context. The Teacher Guide explicitly anticipates the most common misconception of students writing the equation with reversed factor order, such as eight groups of five equals 40, rather than five groups of eight equals 40. The support directs educators to “pay attention to both incorrect final results and cases where the order of factors does not match the situation presented.” The accompanying process standard 1A reinforces this with the scripted follow-up prompts: “Could you explain how your drawing shows the real situation with the cars and passengers?” and “Do you see how your equation represents the number of eggs in all the egg cartons?” These prompts guide educators to return students to the contextual meaning of the factor positions. The anticipated misconception, along with the scripted prompts, equips educators to provide targeted feedback that reinforces students’ mathematical understanding.

The materials provide comprehensive prompts and guidance to support educators in delivering responsive explanatory feedback based on specific student actions and anticipated misconceptions. In the Journey B5, station 14, warm-up, the lesson anticipates that students might struggle to mentally compute operational shifts between pairs of expressions. The materials provide a targeted prompt based on this anticipated misconception. Instructions guide the educator to ask which subtraction in each pair would be easier to calculate mentally if neither had been solved, prompting the students to notice that they can use the result of one subtraction to responsively find the answer to the other. Within that same warm-up, the materials provide separate guidance based on student responses. If student responses demonstrate a foundational difficulty in understanding what the subtraction operations mean, educators should alter their delivery and create contextual word problems using a number line or manipulatives. Moving into activity 1, the materials transition to providing explicit prompts based on student responses. The educator leverages process standard 1A by asking two prompts: “Could you explain the problem in your own words and what you need to find out?” and “What does each number in the problem—$1,000 and $754—represent in the piggy-bank situation?” The materials provide separate pedagogical guidance based on anticipated misconceptions regarding subtraction strategies. The guidance explicitly cautions the educator that a multi-digit subtraction like $1,000 - $754 can be uniquely challenging when using a base 10 decomposition strategy. By combining targeted calculation questions, contextual word problems, and guidance for decomposition strategies, the materials equip educators to deliver responsive feedback that addresses students’ mathematical misconceptions.