Math Journeys Texas, Kindergarten

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 outlines the TEKS that correspond to each lesson, including both the content and process standards for educators. Additionally, the TEKS and ELPS are included in the scope and sequence document, which is organized into different sections called “Journeys.” The Journey Instructional Guide provides a detailed rationale for each journey, and section 2 provides a vertical rationale for how each journey builds on the previous to prepare students for the next concept. The A2 Journey Instructional Guide, which focuses on identifying numbers and counting from 1–100, extends the concept of counting, comparison, ordering, and estimation from the previous journey, which focused on numbers 1–20. The horizontal rationale is provided in section 3, which explains how the learning moves from concrete to representational to abstract models of number sense.

The materials in Math Journeys include an Alignment Guide that outlines the TEKS, ELPS, and concepts covered in the A1 Journey Instructional Guide. The materials state, “The guide shows the alignment among the TEKS, ELPS, lesson concepts, and the learning progression across A1.” Additionally, the materials include a rationale for learning paths across grade levels and within the same grade level. The Journey Instructional Guide states, “Lessons 1–4 build the oral counting sequence, numerical-quantity matching, and early-classification/comparison language.”

As the materials state, “the guide shows the alignment among the TEKS, ELPS, lesson concepts, and the learning progression across A1.” The materials also include a rationale for learning paths across grade levels and within the same grade level. For example, lessons 1–4 build the oral counting sequence, numerical-quantity matching, and early classification and comparison of numbers.

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

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

The materials in group 1 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. For example, the materials list 10 practices for effective math intervention in the program overview, such as, “identify each student’s starting point, set realistic and meaningful goals, create a safe environment where students dare to think, foster conceptual understanding, guide reasoning through modeling, questioning, and feedback.”

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

The materials in Math Journeys include a TEKS Correlation Guide with recommended skill entry points based on students’ diagnostic assessment results. The journeys and lessons are aligned to the TEKS content and process standards, and the guide helps educators implement instruction by connecting lessons and activities to specific standards within each journey. The TEKS Correlation Guide alone does not provide a recommended skill entry point.

The placement test report sample presents data on the TEKS assessed in the diagnostic assessment and proposes grouping structures for students. The “Proposed Group Distribution” section of the report recommends how educators should group students and identifies an entry point for each. The report also offers a snapshot of each student that shows their current understanding and suggests an entry point for instruction.

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

The materials in the Math Journeys Program Guide, “Lesson Internalization and Planning” section, include protocols with corresponding guidance for both unit and lesson internalization. The “Preparing for a Journey” section shows educators how journeys build on prior learning, as well as how to review lesson objectives and mastery criteria, identify scaffolds and enrichment opportunities, gather materials, and analyze learning progressions prior to instruction. The materials also include lesson internalization protocols through the sections “Preparing for a Building Session,” “Preparing for a Consolidating Session,” and “Preparing for an Additional Practice Session.” These sections guide educators to clarify objectives, anticipate student thinking and misconceptions, prepare materials, identify formative assessment opportunities, analyze student responses, and use progress-monitoring data to inform future instruction.

In the unit, Journeys A, “Counting and Numbers,” the learning brief allows educators to preview the key standards and objectives for the upcoming lessons. The learning brief states, “In this journey, students practice rote counting and identify numbers in the 1–10 range.”

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

The materials in Math Journeys include resources to help instructional leaders with the implementation of the program. In the Program Guide, the “Hands on Partnership” section provides examples of how a math success manager can partner with and provide guidance for instructional leaders. Examples include “facilitation of [professional learning community] sessions, student work and progress monitoring analysis, aligning instructional practices, and developing a long-term tier two math model.” Additionally, on-demand video resources in the training materials provide guidance on specific strategies, like “Let’s Add by Breaking Down.” These videos can help educators and leaders who may need more guidance on a specific strategy.

The grade K materials include guidance for instructional leaders to support educators in implementing the materials as designed. For example, the Math Journeys Coaching Guide states that it serves as a practical guide for principals, coaches, and instructional leaders who model, co‐teach, and observe cycles with educators as they implement Math Journeys. The guide outlines the four stages a classroom progresses through as the intervention takes hold, identifies the key indicators and predictable challenges at each stage, and concludes with ready‐to‐use rubrics for observation, self‐reflection, and lesson preparation.

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.

The materials in the Math Journeys Program Guide include detailed guidance for lesson implementation and a breakdown of the instructional components in each journey lesson, along with suggested time frames for each part. The materials outline components such as educator-led instruction, student exploration, discussion, and practice activities to support consistent implementation of the program as designed. Additionally, the materials include multiple assessment resources, such as diagnostic and progress-monitoring tools. The digital practice progress reports provide TEKS-aligned assessment data by identifying student performance for each of the TEKS to support instructional decision-making and monitor student growth.

In grade K Math Journeys, each lesson includes the ELPS, and the header of every question displays the TEKS and ELPS codes. This journey’s test covers 3.C, 4.C, and 4.F of the reading and writing domains. The A0 Lesson-Level ELPS Companion states, “Primary concept: Identify quantities and numerals from 1 to 3 through song, fingers, and body movement.” The primary TEKS contents are K.2.A, K.2.B, and K.2.F. The ELPS focus areas include listening (B, C) and speaking (A, B). The language objective states, “Students will identify and say the numbers 1, 2, and 3 while matching number words to finger and body representations in the song.” The language demand is, “Echo key number words, respond to oral prompts, use one more, and name first, second, and third in sequence.”

Additionally, the materials provide guidance on how to use the companion by using one short section per lesson to add a compact ELPS layer, without crowding the main guide. Each lesson note names the language objective, language demand, expected oral and written output, and a simple placement note for classroom use.

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 in Math Journeys include guidance for lesson planning through clearly organized lesson components within the program structure. To support effective pacing and implementation, the materials outline the instructional sequence of “build, consolidate, and practice,” along with suggested time frames for each part of the lesson. Additionally, the lesson framework is designed to help educators deliver instruction in a consistent and purposeful manner as they provide students with a structured learning routine. The organization of the lesson components supports educators in managing instructional time effectively and ensuring that each stage of learning is addressed throughout the lesson.

The Math Journeys materials give educators detailed lesson plans for each station, including a summary and learning objectives aligned to the TEKS. The cluster overviews list the ELPS language goals and each lesson’s wrap-up features TEKS-aligned checks for understanding. For example, in the Journey A1, lesson 1, wrap-up activity, students answer the question, “Some friends are sitting around the table, and another one arrives. How many are there in total?” This question checks whether students understand the concept of addition.

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 math journey provides a family letter in Spanish to help caregivers support learning at home. The letter suggests questions to ask, vocabulary to use, and specific activities caregivers can use at home to support learning. Educators can change the letter’s language to English using the language toggle in the top right corner.

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 clearly defines each assessment and explains its purpose. Educators use the diagnostic assessment at the start of instruction to identify students’ prior knowledge and decide which journey fits each student. Additionally, educators use formative assessments, such as warm-ups, logsheets, and embedded checks for understanding, to adjust instruction in real time. The materials describe summative assessments, like end-of-journey tests and cumulative progress reports, as tools educators use to evaluate student mastery of targeted skills and determine if students are ready to advance in the instructional pathway.

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

The materials in Math Journeys include guidance to ensure consistent and accurate administration of instructional assessments. The Placement Test Implementation Guide includes clear and detailed guidance for educators to administer the diagnostic assessment 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 notes 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.

Additionally, the End-of-Journey Test: Instructions and Guide shows educators how to lead the assessment and have students respond individually. The guide walks educators through how to check information before test day, administer the test, and review the results. The guide also specifies the approximate length, estimated testing time, materials needed (digital and external), number of assessment items, and examples of response types.

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 materials help educators administer the digital end-of-journey test by offering printable versions of all assessment components. The End-of-Journey Test: Instructions and Guide gives educators downloadable links to printable PDFs, which include the administration instructions, journey tests, and answer key. With these resources, educators can flexibly give assessments in classrooms with limited digital access or when they prefer paper-based testing.

Educators cannot enable or disable text-to-speech for individual students on the digital platform. Only students themselves can independently activate audio support at any time. Educators can change the language of instruction, but this is the only type of language support provided. Furthermore, the digital interface does not provide calculators, meaning educators must supply and manage calculators during assessments.

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 and questions. In the placement test report sample, the diagnostic assessment identifies student performance across multiple mathematical strands, then processes and connects results to specific instructional pathways within the Math Journeys progression. The diagnostic assessment incorporates three levels of Depth of Knowledge (DOK) to measure students’ understanding and reasoning skills. For example, one assessment item asks, “How many fruits are there?” which aligns with DOK 1 because it requires students to demonstrate factual knowledge and basic recall. Another item asks, “What is the result of this operation: 43 + 26?” which reflects DOK 2 because students must apply procedural knowledge and mathematical computation skills. In addition, the assessment includes higher-level items such as, “Which of these numbers will be in the 5th position of the following sequence?” This question aligns with DOK 3 because it requires reasoning, pattern recognition, and problem solving skills. Collectively, these assessment tasks demonstrate that the diagnostic assessment measures student understanding across more than two levels of cognitive complexity.

Additionally, the materials include diagnostic assessments with interactive item type questions and tasks, including the placement test for grades K–2, which provides a variety of interactive question types designed to assess student understanding through multiple response formats. The assessment incorporates multiple item types such as multiple-choice, numerical-response, and number line placement questions, which allows students to demonstrate their mathematical thinking in different ways. For example, in questions 1 and 2, students select a multiple-choice answer. In questions 3–8, students generate numeric responses independently. For questions 13 and 14, students place responses accurately on a number line. The inclusion of these varied item types supports a more comprehensive evaluation of student learning by measuring conceptual understanding, procedural accuracy, and mathematical reasoning across multiple formats.

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 also include interactive formative assessment item types with varying complexity levels. For example, student logsheets and digital activities include multiple-choice questions, number entry, text entry, and number line activities. Educators use wrap-up assessments to check learning throughout each lesson. One assessment item asks students, “Which number do you hear?” which aligns with DOK 1 because students demonstrate factual knowledge and basic recall. Another item asks, “What number is shown?” which reflects DOK 2, as students apply procedural knowledge and mathematical computation skills to identify the number from visual images. The assessment also includes higher-level items, such as, “Which number is missing?” To answer a DOK 3 question, students reason, recognize patterns, and use problem solving skills. Students also use problem solving skills to determine the missing number in a sequence. Together, these assessment tasks show that the formative assessments measure student understanding across more than two levels of cognitive complexity.

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 provide educators with guidance on detailed scoring information for interpreting student performance data through the placement test report sample. The instructional assessment report displays each student’s score and identifies the corresponding performance category, including low, medium-low, medium-high, and high achievement levels. In addition to reporting performance outcomes, the sample report offers recommendations for instructional placement based on assessment results. This comprehensive approach supports educators in making informed and individualized instructional decisions aligned with student learning needs.

For example, the Journey Test A1 answer key gives educators detailed rationales for both correct and incorrect responses in the “Item-by-Item Guide” section. These rationales help educators understand student thinking and spot potential misconceptions. For instance, question 4 asks, “How many apples are there?” The rationale for the correct answer states, “There are 10 apples shown. Only the apples are counted; the pears and bananas are not part of the answer.” The rationales for incorrect answers of eight and nine explain, “These values are too small because they do not account for all ten apples in the picture.” For responses 11 and 12, the rationales clarify, “These values are too large for the number of apples shown and do not match the apple count of ten.” By explaining both correct and incorrect responses, the answer key helps educators analyze student errors, interpret student understanding, and make informed instructional decisions.

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 instructional materials provide guidance for responding to student performance trends through the placement test report. The “Placement in Journeys” section explains how assessment results can be used to support instructional decision-making and student placement. Specifically, the placement test report assists educators in grouping students for targeted instruction and determining appropriate placement within the Math Journeys program. Based on students’ assessment performance, the report generates suggested starting points for each student within the journey progression. This enables educators to differentiate instruction according to demonstrated levels of understanding and academic readiness.

The instructional materials in the End-of-Journey Test Answer Key and Formative Guide shows educators how to use activities and instructional supports in response to student performance. The guide details each assessment item, describes possible student misconceptions, and includes related skills and standards. The guide also includes “the student may…” statements and recommendations for additional support. Together, these components help educators analyze student responses and identify areas where students misunderstand concepts or lack skills. Additionally, the guide suggests targeted instructional support and interventions so educators can address student needs and promote academic growth based on assessment results.

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 instructional materials provide an educator dashboard that automatically compiles real-time student assessment data to help educators monitor progress and make instructional decisions. In the digital practice section, the general progress dashboard details weekly student performance by reporting hours spent on activities, the number of activities completed, overall grades, and performance data aligned with the TEKS standards for both the class and individual students. For example, the dashboard displays scores for each TEKS standard and uses a color-coded system—dark green, light green, orange, and red—to indicate varying levels of student performance and mastery. This feature enables educators to quickly identify strengths, areas of concern, and trends in student achievement so they can inform targeted instruction and intervention.

The instructional materials provide a digital platform that allows students to progress through the Math Journeys program based on their assigned placement levels. During activities, students receive immediate real-time feedback on their responses through visual indicators, such as a checkmark for correct answers and a question mark for incorrect answers. While this feature supports immediate response validation and engagement during practice activities, the program does not provide students with tools to independently monitor or track their overall academic progress and growth over time. As a result, students have limited opportunities to engage in self-monitoring or reflect on long-term performance trends within the program.

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 Math Journeys instructional materials prompt educators to foster student reasoning and mathematical discussion in the “Cardinal Counting (up to 100)” lesson plan, “Development: Activity 1” section. Educators ask students to explain how grouping snap cubes by tens makes counting more efficient, and how groups of ten, along with leftover cubes, represent and communicate a total number of snap cubes. In “Development: Activity 2,” the lesson incorporates frequent checks for understanding by directing students to hold up their response pads after prompts. These strategies promote mathematical communication and critical thinking and give students opportunities to verbally express their understanding.

Furthermore, the materials guide educators to conduct checks for understanding during lesson activities. For example, within Journey B3, “Consolidating Addition and Subtraction I,” station 5, “Addition: Jumping Without Bridging,” the “Wrap-Up” section includes prompts that help educators reinforce key concepts. Educators explain that an empty number line represents the interval between numbers, rather than a starting point, and students then identify reference tens before solving. Across lessons, the wrap-up sections include expandable tips, observation keys, and targeted look-fors to help educators interpret student thinking and determine whether learning goals have been met. These supports provide questioning strategies that assist educators in assessing, understanding, and informing instruction.

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

The digital Atlas Math Applet Library provides multiple opportunities for frequent checks for understanding throughout instructional activities. For example, in activity A1, “Words and Numerical Digits: Let’s Identify,” students listen to a number spoken orally and select the correct response from three answer choices. This activity provides immediate feedback based on student responses. If a student selects the correct answer, a green check mark appears and the next question is displayed. If a student selects an incorrect answer, the chosen response turns red and a question mark appears. The same question is then repeated orally with a reduced number of answer choices to support student understanding and provide another opportunity for success. If a student answers correctly after an initial incorrect response, the progress bar turns green but still includes a question mark to indicate that the first attempt was incorrect. If the student answers incorrectly a second time, the correct response is automatically identified, and the progress bar displays green stripes to indicate two incorrect attempts on the same question. These features provide educators with ongoing opportunities to monitor student understanding, identify misconceptions, and respond to student needs during instruction.

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 include educator guidance for scaffolded activities designed to support students who have not yet demonstrated proficiency with grade-level concepts. Each lesson within the program contains a representation page accompanied by either a support or extension logsheet that provides differentiated instructional opportunities based on student needs. The lesson materials include explicit directions for implementing the support logsheet with students who require additional assistance. For example, in lesson 14, “Acoustic Counting Forward,” the original logsheet displays six circles in sequential order, labeled one through six, with the four middle circles already fully colored. In comparison, the accompanying support logsheet provides circles labeled only one through five, which requires students to identify and complete the missing number independently. This scaffolded modification reduces the cognitive demand of the task while providing targeted support for counting sequence development and reinforcing students’ understanding of number order.

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.

The materials provide linguistic accommodations for each math journey, such as a Journey Language Bank. These items offer general language support strategies and routines, such as using sentence frames and discourse starters to help students use technical language during classroom discussions. The suggestions also offer discourse and writing prompts for specific lessons. Additionally, the materials provide educators with embedded supports for developing academic vocabulary within the lesson activities. These supports are integrated throughout various sections of the lessons to strengthen students’ conceptual understanding and mathematical language development. For example, in lesson 2, “Counting Forward to 10,” the “Development: Activity 1” section includes a video in which students listen to and sing along with “The Finger Song.” The song and accompanying visuals introduce students to counting vocabulary as they connect spoken number words to visual representations of quantity. This multimodal approach reinforces vocabulary acquisition and strengthens students’ foundational understanding of counting concepts.

In addition, these materials provide a Journey Language Bank and redirections for each journey, including routines and quick bridges for unfamiliar references. A lesson-by-lesson table names the unfamiliar references in each lesson, as well as the bridge educators use beforehand to pre-teach the text’s context. The A0 Journey Language Bank and redirections offer additional pre-teach routines, discourse moves, writing prompts, and redirection options that educators can use selectively. For example, in lesson A0.1, the unfamiliar reference may be the song “The Three Flowers.” In the quick bridge before the lesson, educators show three flowers and tell students, “This is a song about three flowers. One, then one more, then one more.” Students repeat flower and one more.

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

The materials provide an Enrichment and Extension Guide, which states, “This is a practical guide for educators to determine who receives extension work in Journey A0 (Identify Numbers and Counting (1-10), how to facilitate it, and how to assess student responses. It complements the lesson plans and every referenced task already exists in the Classroom Manager.” The guide also gives educators directions on how to use it, including specific grade-level extensions. For example, if “the student is ahead within the journey’s content” educators “decide who qualifies using the signals in section 2; facilitate and assess with section 3 (Extension logsheets) and section 4 (Extension tips), during the session and at wrap-up.” For above-content extension the guide states, “The student is ahead of the journey itself. The pattern of signals across sessions and tests (section 2, third signal) triggers the early End of Journey Test. If the student scores above 75%, they advance to the next journey (section 5).”

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

The digital materials do not include the ability for educators to enable or disable text-to-speech and content and language support. Additionally, digital materials do not include digital calculators.

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 materials let students demonstrate their understanding of mathematical concepts in different ways. Math Journeys lesson plans intentionally use the concrete, pictorial, abstract (CPA) instructional sequence to give students multiple opportunities to show what they know. For example, in development activity 1, students use their bodies to count down from five. Students then group themselves in fives to act out the song by standing on chairs and pretending to be bottles on the wall. Before starting, the educator explains that each group must agree on the order in which they will fall. Educators can have all groups sing together or take turns. The lesson plan offers explicit questions and tips to support students’ ability to perform and express mathematical concepts.

Additionally, these materials let students show their understanding of mathematical concepts in multiple ways. In Journey A0, lesson 7, “Cardinality 0–20,” students sort objects in the Atlas Math digital applet, explain their sorting verbally, draw pictorial models like graphs, and analyze those graphs. Each lesson guides educators with suggested questions, a process standards section that details relevant examples, and different options for administering the activity in the classroom.

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

The materials provide explicit prompts and guidance for educators to build on students’ prior knowledge. For example, in the warm-up at the start of each session, students activate prior knowledge to anchor big ideas. In lesson A0.9, length is introduced through a connection with cardinal counting. The development then continues to connect measurement and number sense through different activities.

Furthermore, the materials provide explicit prompts and guidance that help educators build on students’ prior knowledge and connect key mathematical ideas. For example, Journey B1, lesson 5, “Quantity 0–6,” activates students’ prior knowledge through prompts about geometric shapes. Students answer questions such as, “What geometric shape is Byte on?” “What geometric shape is around him?” and “Is there another type of area marked out? Do you think we can find it?” The lesson also helps educators anchor big ideas and highlight mathematical patterns and relationships through games, discussion, and multiple means of representation. After students play the game, the class uses guided questions to analyze scores together, such as “How much did you score with each pebble? In which area did the pebbles land?” “What is each person’s total score?” and “Was it possible or impossible to score 0 points? And 7 points?” Throughout the lesson, students represent their thinking in multiple ways to deepen understanding. For example, students use symbol and number cards to create equations to represent their scores. Students also draw pictures to model the game and connect their drawings to the corresponding equation.

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

The lesson plans in these materials guide educators to use varied instructional approaches, including collaboration, structured conversation, and direct instruction, to facilitate lessons. Educators typically lead the warm-up with whole group discussion. During development, educators combine whole-group and small-group work, providing multiple opportunities for hands-on exploration and student discourse. The wrap-up assigns individual work for consolidation, followed by a group debrief. Each session transitions smoothly from guided discovery, where educators facilitate hands-on exploration, to formalization. The warm-up explicitly guides educators to ask the class, “How many fingers do you see?” and “Which numbers would you add?”

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

The materials explicitly state how both the conceptual and procedural emphasis of the TEKS are addressed throughout the lessons. In lesson 3, “Mathematical Situations,” students develop conceptual understanding as they analyze a picture that represents students sitting on a bench and standing beside the bench. Students then use themselves as physical representations to model the situation, such as demonstrating four students on the bench and one student off the bench, to strengthen their understanding of concrete mathematical relationships. After building this conceptual understanding, students transition to procedural application as they write number sentences that represent the situation, and correctly use mathematical symbols for addition, subtraction, and equality. This progression from concrete modeling to symbolic representation supports students’ ability to connect conceptual understanding with procedural skills, while they also develop an understanding of joining and separating quantities.

These materials guide educators to implement multitiered intervention by providing a structured framework for grouping students by proficiency. The placement test report sample helps educators make instructional decisions by identifying entry points for targeted small-group instruction based on placement assessment results. The report gives educators detailed data to determine instructional needs, organize flexible groups, and differentiate support by individual understanding. Placement results also help educators assign students to the appropriate Math Journeys program pathway to ensure that instruction matches each student’s learning needs and readiness.

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

The grade K materials in Math Journeys include enrichment and extension methods that support a variety of student engagement opportunities throughout the lessons. Support and extension sections embedded within instructional activities provide differentiated supports that allow educators to extend or scaffold learning based on individual student readiness and performance. The lessons also include explicit educator guidance for effective implementation through questioning prompts, manipulatives, number lines, partner discussions, visual models, and independent practice opportunities. For example, students use multiple strategies to solve problems, calculate distances on number lines, compare solution methods, and justify their mathematical reasoning through whole-group, partner, and individual activities. These scaffolded supports and enrichment opportunities equip educators with the tools needed to differentiate instruction for students while maintaining meaningful engagement with grade-level mathematical concepts and skills.

The materials include multiple forms of engagement designed to support enrichment and extension opportunities for students through interactive, collaborative, and hands-on learning experiences. For example, in lesson 6, “The Number Sequence,” students participate in a group discussion during the warm-up activity in which they analyze a character’s actions using accompanying digital media that incorporates a number line representation. This discussion-based activity encourages students to build conceptual understanding through verbal reasoning and shared observations. In the first development activity, students further engage with mathematical concepts through a kinesthetic learning experience using a giant floor number line. Students stand on a selected number and determine the number that is one step forward or backward, which reinforces the concepts of one more and one less through physical movement and active participation. Additionally, the materials support enrichment through embedded extension tips within the lesson plans. For example, in the same lesson, educators extend student learning by “playing a similar game with number cards, either on the floor or on a table to skip count by twos or fives.” This extension activity promotes continued exploration of number patterns while allowing educators to adapt instruction through additional hands-on and collaborative practice opportunities.

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

Lesson overviews within Math Journeys include a section with specific prompts and guidance for educators to use with students during instruction. Educators are instructed to pause at key moments to check for understanding, address misconceptions, and give immediate, meaningful feedback in real time. This support equips educators to deliver responsive instruction and keep students on track toward lesson objectives. Lesson plans also encourage rich discourse and conversation, with explicit prompts for educators to check understanding and offer formative assessment during the lesson. For example, in development activity 2, educators prompt students to represent inequalities on the board using < or >. Students then create inequalities, write them on their pads, and hold them up to check answers.

Additionally, the materials guide educators to provide timely and responsive feedback through the Journey Language Banks. These banks identify likely student responses or misconceptions, explain what that indicates about understanding, and suggest educator redirections. For example, in lesson A1.3, the materials highlight a common misconception in which a student states the answer but cannot explain the meaning of minus one. The language bank notes that the student may recite the song without understanding the change in quantity. To address this, the materials instruct educators to rebuild the verse with five visible objects, remove one, and ask “What changed?” The educator then revoices the concept by saying “One fell, so there is one less,” to reinforce subtraction as a change in quantity.

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.

Each journey includes a Lesson-Level ELPS Companion that provides a table with the ELPS focus, language objective, language demand, expected oral and written output, and suggested use for each lesson. Furthermore, some lessons provide embedded linguistic accommodations for all levels of the ELPS. For example, in Journey C2, station 2, the lesson includes supports for reading word problems at each proficiency level.

The materials provide a digital Teacher Guide with embedded linguistic accommodations for all current levels of the ELPS. In Journey A8, “Rote Counting,” activity 2, the bilingual tip includes a suggestion for each of the five proficiency levels. For example, the lesson guides the educator to prompt pre-production students with, “Say a number: the student steps to it and points, no talking.” Additionally, the suggestion for advanced students states, “Stepping forward adds 1 and stepping backward subtracts 1, every time.”

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 offer text-to-speech, visual manipulatives, and a support button to help with content, but do not provide linguistic accommodations. For example, in applet 95, “Numbering and Calculation,” when students select the wrong equation, the help button prompts, “Listen to the number and select the correct response.” Students can use this prompt to find the number they hear, but the materials lack linguistic accommodations for students who do not understand due to language barriers.

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

The materials guide educators to implement bilingual/ESL programs effectively using the Texas Bilingual/ESL Implementation Guide. The guide has two sections: part one, “Reusable Core for All Math Journeys,” and part two, “Journey Addendum.” Part one gives foundational guidance on the structure and implementation of Math Journeys and notes that mathematical content changes from one journey to the next. The guide also describes key features of the journeys and explains their importance for bilingual and ESL instruction. For example, the guide highlights the frequent use of visible models and hands-on materials, such as objects, drawings, number paths, ten frames, graphs, cubes, cards, and movement-based activities, as critical features.

In the Texas Bilingual/ESL Implementation Guide, part two, “Journey Addendum,” provides educators with guidance specific to each individual journey by outlining the math focus, primary language demands, and model-specific implementation notes. For example, when the math focus is identifying quantities and numerals one through three, the guide identifies the primary language demands as echoing number words and using terms such as one more, first, second, and third.

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

The materials in the Math Journeys program for grade K include embedded guidance to support emergent bilingual (EB) students as they develop academic vocabulary, increase comprehension, build background knowledge, and make cross-linguistic connections through oral and written discourse. The materials include academic vocabulary, estimation discussions, collaborative conversations, and written representation activities that support students’ language development during math instruction. For example, in the lesson “Estimating and Comparing,” students use vocabulary such as meter, estimation, less than, and equal as they estimate quantities, count the elements in a set, and compare quantities. The “Emergent Bilingual Students” section provides structured oral language prompts and opportunities for students to use comparison language during classroom discussions as they revise, clarify, and explain their mathematical thinking.

The grade K Math Journeys materials include embedded guidance that supports EB students as they develop academic vocabulary, increase comprehension, build background knowledge, and make cross-linguistic connections through both oral and written discourse. Each lesson includes a “Key Vocabulary” section that identifies cognates and false cognates to strengthen language development. For example, in lesson 8, “Estimating and Comparing,” the listed cognates include estimate/estimar and compare/comparar. Students further develop academic vocabulary during the development activities and wrap-up as they discuss the attributes of shapes and explain, both orally and in writing, their estimates of how many people or objects will fit within a given space. The lesson also supports comprehension through a progression of learning experiences in which students first discuss and record estimates for larger areas before they apply the same concepts to smaller areas using various manipulatives. These activities build background knowledge and provide opportunities for guided practice, which enables students to apply their understanding independently during the wrap-up.

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.

The grade K Math Journeys materials provide practice opportunities and instructional assessments throughout the learning pathway that require students to demonstrate depth of understanding aligned to the TEKS. For example, the lesson “Decomposing Ten” gives students practice opportunities to show a deep understanding of TEKS K.2.I. Students find multiple combinations of toy vehicles that total 10 wheels to solve an open-ended problem, and use ten frames, manipulatives, and math sentences to reason additively and explore several solution pathways. In activity 2, educators “encourage pairs to find and represent other solutions. They can draw sketches or diagrams on blank paper, use manipulatives or ten frames, among other methods.” As the activity progresses, students work in pairs and use ten frames to verify that the sum of the wheels is 10. Examples of this include two tricycles and one car, or representing the expression 2 + 4 + 4 with 1 scooter and 2 cars. Students identify multiple combinations, model their thinking, and communicate their mathematical reasoning. Through these practice opportunities and instructional assessments, students demonstrate depth of understanding aligned to the TEKS throughout the learning pathway.

The Math Journeys grade K materials provide practice opportunities and assessments in which students demonstrate depth of understanding aligned to the TEKS. For example, Journey B1, lesson 13, provides both practice opportunities and an instructional assessment that require students to go beyond simple recall of number pairs. Rather, students use math sentences to perform decompositions of nine, such as 4 + 5 = 9 and 9 - 4 = 5, in order to demonstrate the additive relationship and inverse operations required by TEKS K.3.A. These assessments demonstrate depth of understanding by requiring students to represent relationships symbolically rather than relying on surface-level fact recall.

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.

The grade K Math Journeys materials include questions and tasks that increase in rigor and complexity, and lead to grade-level and above grade-level proficiency in the mathematics TEKS. In Journey B1, “The Number 10 (Part 2),” students build on prior learning from earlier sessions in which they decomposed numbers and represented quantities in multiple ways. In activity 1, students work collaboratively to identify pairs of numbers that compose 10, organize the pairs systematically, and analyze relationships among the addends to determine whether all possible combinations have been found. The educator asks students questions such as, “What happens to the other number when one addend goes up by 1?” and “Could you find a way to make sure you have all the pairs?” These questions require students to move beyond finding answers and engage in mathematical reasoning about completeness, patterns, and relationships among numbers. Students justify their reasoning and explain how organizing the pairs helps verify their solutions. In activity 2, students apply their understanding of number pairs that sum to 10 within a game context that requires them to recall, identify, and use complementary pairs fluently. During the activity, students use drawings, numbers, and equations to independently represent decompositions of 10. The materials include an extension activity that encourages students to move from pictorial representations to more abstract and symbolic representations, including recording equations such as 3 + 7 = 10 and 0 + 10 = 10. Across the learning pathway, students progress from composing and decomposing numbers to analyzing patterns, justifying mathematical relationships, and representing solutions symbolically. This increases the rigor and complexity of the tasks and supports both grade-level and above grade-level proficiency in the mathematics TEKS.

The grade K materials present questions and tasks that gradually increase in rigor and complexity to help students achieve and exceed grade-level proficiency in the mathematics TEKS. In a lesson titled “The Number 8 (Part 1),” students use manipulatives, drawings, and equations to decompose the number eight into two or more addends. Educators ask questions such as, “If one tower uses 5 cubes, how many cubes does the other tower need? How do you know?” and “Do you think there might be another way to make two different towers that still use all 8 cubes?” To solve the equation, students explore multiple combinations, explain their reasoning, and compare different representations. Additionally, these questions help students deepen their understanding of number relationships and part-part-whole concepts. The challenge of determining combinations of more than two addends allows students to find every possible way to decompose eight, represent their thinking in different ways, and justify how their combinations total eight. The materials also encourage students to communicate their mathematical thinking and connect concrete, pictorial, and symbolic representations. These experiences build number sense, strengthen problem-solving skills, and improve mathematical communication to support both grade-level mastery and opportunities for advanced proficiency.

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

Within the grade K Math Journeys materials, students connect patterns, big ideas, and relationships to demonstrate coherence across concepts horizontally within the grade level. Lesson 15, “Cardinal Counting,” connects acoustic counting to cardinal counting by explaining that students must recite number sequences before they can determine how many objects are in a set. In activity 2, students compare sets of objects without counting to determine which set has more elements and which color has more snap cubes. Students use two-hand strategies to link cardinality to comparison, including by removing one cube of each color at a time or lining up cubes in parallel rows to see which set has more. After students compare sets, the educator introduces the symbols >, <, and =, so students can represent these relationships symbolically. The materials explicitly tag process standard 1C, as students select a technique to determine which color has more snap cubes. This process links concrete comparison methods to the formal notation students learn in the same session. Educators ask, “What method did you choose to figure out which color has more snap cubes? Why?” and, “What would work better for your handfuls: lining up the snap cubes in rows or removing one from each color?” Across the lesson, students connect counting, cardinality, comparison, and symbolic representations to demonstrate coherence across mathematical concepts within the grade level.

The grade K materials demonstrate coherence across concepts horizontally within the grade level by intentionally developing students’ understanding of number decomposition as a foundational concept for addition and subtraction. In lesson 11, “Introducing the Additive Part-Part-Whole Frame,” students work with numbers between two and 10 and use an additive part-part-whole frame to represent multiple decompositions of a number. In activity 1, students decompose numbers less than 10 into an addition operation of two numbers. Educators guide students to decompose five in a different way and check if all possible pairs that make five have been found. Through these experiences, students learn that a whole can be composed of different combinations of parts, which strengthens their understanding of number relationships and flexibility with numbers. The lesson summary reinforces these concepts as students model and discuss various decompositions to help them recognize the connection between parts and wholes. This understanding serves as a critical bridge to later learning within the grade level, where students apply decomposition strategies to addition and subtraction within 20. By connecting number composition and decomposition to operational thinking, the materials support a coherent progression of mathematical concepts and deepen students’ conceptual understanding of addition and subtraction.

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 grade K Math Journeys materials demonstrate coherence vertically across concepts and grade bands by connecting patterns, big ideas, and relationships. The materials provide a horizontal layout that builds on previous scopes within the grade level. For example, cluster overview A explicitly documents vertical coherence across grade bands by outlining a “Learning Progression” section. This section traces how prior knowledge (recognizing numbers one through three, basic size comparison) builds through cluster A (counting to 100, number lines, place value) into the next cluster (transitioning to strategic calculation using number composition for addition and subtraction beyond ten). The “Mathematical Background” section further demonstrates vertical coherence by showing how the bead string introduced in A2 evolves across subsequent journeys, from concrete 20-bead models to the two-color number line to the empty number line. This explicitly connects grades K–6 concepts through a documented progression of representations and big ideas, such as place value, cardinality, and number decomposition. The overview explains how these representations evolve across later journeys to help students connect cardinality, counting, number decomposition, and place value concepts over time. Through the documented progression of concepts, representations, and mathematical relationships, the materials demonstrate vertical coherence across concepts and grade bands.

Furthermore, the materials demonstrate vertical coherence across concepts and grade bands, as outlined in the Journey Instructional Guide. This guide provides a clear developmental rationale for how mathematical concepts, patterns, big ideas, and relationships progress across clusters to build students’ understanding over time. For example, in the A0 cluster, students establish a foundational bridge from readiness skills to formal school mathematics as they connect number words to quantities, introduce numeral recognition, develop the counting sequence, and make concepts such as one more and one less visible through songs, movement, ordering activities, and hands-on comparisons. The guide explains that the A1 cluster builds on these foundational understandings by extending number concepts to 1–20 and strengthening students’ ability to compare quantities, count backward, and develop an understanding of early teen numbers. The progression continues in the A2 cluster, where students expand their understanding of numbers to 100, use number lines to represent and locate numbers, and develop place value concepts through concrete and pictorial models, including bead strings, money, and abacuses. This intentional sequence of learning demonstrates how concepts are revisited and expanded across clusters to ensure that students build on prior knowledge while developing increasingly sophisticated mathematical understandings.

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.

This guidance statement is not applicable to the program because kindergarten does not have a preceding course with prior knowledge to connect to the current grade level and future grade levels.

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

The grade K Math Journeys materials provide opportunities for students to develop and apply mathematical knowledge and skills through distributed practice over time. For example, in lesson 15 students revisit and apply number decomposition relationships they previously developed throughout earlier sessions in Journey B1. During activity 1, students use previously learned decompositions to identify, organize, and justify all pairs of numbers that sum to 10. Educators ask, “How many pins were knocked down?” and “How many remained standing?” To answer these questions, students retrieve previously learned number relationships and apply them to determine whether all possible combinations have been found. In activity 2, students continue applying these number relationships as they play a card game in which they identify complementary pairs that total 10. The distributed practice extends to subsequent lessons, where students use these same decomposition relationships to solve new mathematical situations and verify solutions using ten frames. Through repeated opportunities to retrieve, apply, and extend previously learned concepts across multiple sessions, the materials support distributed practice over time.

Another opportunity for retrieval practice is provided in Journey B1.9, “The Number 8 (Part 1),” where students revisit and apply strategies and concepts introduced in earlier lessons throughout section B1. Students count and build towers with snap cubes, find number pairs with number cards, create pictorial representations, write equations, and use other familiar methods to represent and decompose the number eight. In activity 1, students work in groups to determine how many combinations can be used to make eight, and then use cards or equations to represent their solutions. These activities draw upon strategies previously used to compose and decompose the numbers five through seven in earlier sessions, which enables students to retrieve and apply prior learning while extending their understanding of number relationships. Through these repeated opportunities, students strengthen their mathematical understanding and fluency across the learning pathway.

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

The grade K Math Journeys materials provide interleaved practice opportunities with previously learned skills and concepts across learning pathways, in which students actively apply skills and concepts in varied contexts. For example, the “Mathematical Background” section explains that cluster B sessions mix previously learned strategies rather than isolating them. This section details several strategies from Journey B2, such as bridging through 10, combining jumps in different directions, and optimizing jumps of 10, which students use to solve similar problems. Students draw on skills from earlier sessions and combine them in new ways. The learning progression column highlights this approach by describing how students shift from simple counting to flexible, strategic arithmetic, emphasizing that cluster B intentionally interleaves strategies. Rather than practice a single strategy in isolation, students draw on skills developed across multiple sessions and learning experiences to select and apply different approaches to solve problems efficiently. The learning progression highlights a shift from simple counting to strategic and flexible computation, in which students connect previously learned concepts and procedures across the pathway. Through opportunities to revisit, apply, and integrate multiple strategies developed over time, the materials provide interleaved practice with previously learned skills and concepts across learning pathways.

Additionally, the grade K materials interleave practice with previously learned skills as students regularly switch between different types of problems within a single lesson. For example, in lesson 4, “Counting Natural Objects,” students use counting skills from A0 and A1.1–A1.3 to count sorted natural objects. In activity 3 students apply additive reasoning from A0.13 and A0.14 to immediately solve addition and subtraction word problems with those same quantities. This structure avoids isolated practice because students alternate between counting and problem-solving skills, which strengthens their flexibility. The Teacher Guide unites sorting, counting sets, and solving word problems under shared learning goals, and the vocabulary supports both addition and subtraction. By integrating multiple concepts into a single activity sequence, the materials encourage students to think flexibly and connect mathematical ideas across contexts.

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

The materials provide opportunities for students to interpret, analyze, and evaluate mathematical models. For example, in lesson 3, “Mathematical Situations,” students look at pictures of children getting on and off of a raised surface and interpret what is happening. Educators ask questions such as, “What do you see?” “What happened?” “How many are on the ground?” and “How many are there in total?” In activity 2 students analyze the additive part-part-whole frame, which the Teacher Guide describes as a “more abstract lens” on the same situation, connecting the concrete picture to a symbolic representation. Students then evaluate which math sentence (e.g., 5 - 2 = 3) correctly represents each scenario as they interpret the visual context, analyze the frame, and evaluate the accuracy of the written response.

In the lesson titled “The Number 8 (Part 1),” students develop understanding of TEKS K.2.I as they interpret, analyze, and evaluate mathematical models and representations. In activity 1, students interpret cube tower models to determine how two groups of cubes can be combined to make a total of eight while also meeting the condition that the towers are different heights. Educators ask questions such as, “Do you think there is more than one solution?” and “Can you find any other solutions?” To answer these questions, students analyze multiple representations and identify possible combinations. Students then evaluate whether each model satisfies the criteria for a “magical combination,” and explain why combinations such as 4 + 4 do not qualify. In activity 2, students continue evaluating representations as they classify combinations as “magical” or “non-magical” based on the established mathematical rules, which demonstrates their understanding of the concept through reasoning and justification.

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

In grade K Math Journeys students create concrete and pictorial models of mathematical situations. For example, students use snap cubes to physically build towers of different heights that total eight. In Journey B1.9, “The Number 8 (Part 1),” students use eight cardboard boxes as verification tools and place them side by side to compare combinations. In the wrap-up the educator hands out a representation page and asks each student to “make a representation on the blank page: Can you explain what we did today?” This gives students a direct opportunity to create pictorial representations of the mathematical situation they explored with concrete materials.

The lesson “Mathematical Situations” serves as a key point in the progression toward mastery of TEKS K.2.I, K.3.A, K.3.B, and K.3.C. To begin, students use bears and a soft building block to make concrete representations of addition and subtraction situations, such as placing five bears on the block and removing two to model 5 - 2 = 3. Students then use an additive part-part-whole frame, described as a “more abstract lens,” to represent the same mathematical situation and record the corresponding equation. This progression from concrete manipulatives to pictorial and symbolic representations helps students connect mathematical actions to mathematical notation. Process standard 1E is explicitly embedded in the lesson because students create and use representations to organize, model, and communicate their mathematical thinking.

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

In Journey B, “Additive Thinking,” students apply counting strategies to identify quantities in sets of objects. For example, in one lesson, students apply their conceptual understanding of decomposition to a novel puzzle context in which they are told that toy cars have four wheels, scooters have two, and tricycles have three. The materials state, “A family buys several toys, and between everything they buy, there are 10 wheels in total. Which toys could they have bought?” Students must transfer their understanding of part–whole additive structure to this unfamiliar multi-addend problem. To do so, they select tools from the table, such as ten frames, colored tiles, snap cubes, or drawings, to test combinations and record math sentences such as 2 + 4 + 4 = 10. Activity 2 extends this further as educators ask, “Could you write a math sentence to show how many wheels there are? Which tool—ten frames, cubes, or a drawing—helped you solve the problem, and why did you choose it?” Students continue flexible application of the same conceptual understanding across new combinations, and when they find a solution, they also write down the operation. Additionally, educators ask, “What types of toys did they buy?” “How many of each type did they buy?” and “In total, how many toys did the family buy?” Students share and compare solutions with classmates, which demonstrates the application of conceptual understanding in a new mathematical context.

The grade K Math Journeys materials include questions and tasks that provide opportunities for students to apply conceptual understanding to new problem situations and contexts. For example, in Journey B1.14, “The Number 10 (Part 1),” activity 1, students apply their understanding of decomposition in a new context beyond the cubes, number cards, and counting activities used in previous lessons. Students work in groups to explore and test different arrangements of 10 bottles or cans to use as bowling pins, then determine which arrangement allows them to knock down the most pins. As they organize the bottles, students apply previously learned concepts in a hands-on, real-world situation to create groups that compose 10. Students also mark a throwing line 10 feet away, which provides an additional opportunity to count to 10 in a meaningful context. In activity 2, students use ten frame representations to record the pins that were knocked down and count the total. To complete these tasks, students transfer and apply their conceptual understanding of counting, composing numbers, and representing quantities to new problem situations and real-world contexts.

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

The materials include tasks designed to develop the automaticity and fluency students need for grade-level math. For example, in Journey A, activity 1, students stand in a circle, each holding a random 21 through 100 number card, and must instantly name the number before and after their neighbor’s card. The activity rotates through the entire deck, which requires rapid automatic recall for any number in the range. Activity 2 raises the demand; students alternate two chosen body movements while skip-counting backward from 100 to 10 by tens, then count backward from 100 to one by ones. These are kinesthetic routines that ingrain the full descending sequence as automatic. Additionally, in activity 3 small groups of students arrange sets of four to five random cards from least to greatest, then from greatest to least, and then exchange cards and repeat to drill number-order fluency at pace. The session’s stated learning objective explicitly targets the full range of counting forward and backward from any number between one and 100. Through repeated counting, sequencing, and ordering activities, students develop automaticity and fluency with number patterns and counting skills that support future mathematical learning.

The grade K materials include tasks intentionally designed to develop student automaticity and fluency with foundational number relationships. For example, in lesson 15, “The Number 10 (Part 2),” students work in small groups and use zero through 10 number cards to identify all possible pairs that make 10. In activity 1 students analyze number relationships and build fluency with combinations of 10. Students then extend their understanding by playing the game “Lucky Escape,” where they use cards dealt to them while holding any remaining cards and create pairs that add to 10. Students take turns offering a remaining card to a peer, who determines whether the card can be combined with one already in hand to make 10. Once students have shared their pairs of numbers, educators ask, “Do you think we have all the possible pairs? What could we do to find out? Could ordering the pairs of numbers help?” Students then order the pairs systematically—zero and ten, one and nine, two and eight, and so on—so they can see that as one addend increases by one, the other decreases by one. Students use that relationship to confirm when they have found all possible pairs. Play continues until only one card remains. Throughout these activities, students repeatedly practice composing 10, which strengthens automaticity, fluency, and strategic problem-solving skills.

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

The learning pathway in lesson 15, “The Number 10 (Part 2),” provides efficient, flexible, and accurate mathematical procedures as it guides students from conceptual understanding to application. For example, in activity 1 students practice flexible and accurate procedures as they find all pairs that sum to 10 through student-selected strategies. Students can order pairs systematically, use snap cubes, or apply known facts. To guide students toward a systematic approach that ensures accuracy and completeness, the educator asks, “Could ordering the pairs of numbers help?” The session then transitions into the lucky escape card game where students accurately apply decomposition facts in a dynamic context and correct pairing under time pressure. The materials provide repeated, increasingly fluent practice of the same procedure across two connected activities within the learning pathway.

The learning pathways provide opportunities for students to practice the application of efficient, flexible, and accurate mathematical procedures. During the activity portion of the lesson, students use snap cubes to model a specific quantity and then record the corresponding mathematical expression. As students solve, they share their strategies with peers and engage in discussions about which approaches are the most efficient to find a solution. Through these activities, students practice flexible counting strategies as they develop accuracy and procedural fluency. Students then extend their learning by applying these procedures to addition expressions within 20, which reinforces conceptual understanding and strengthens foundational skills that support future mathematical learning.

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.

In the lesson “Number 7 (Part 2),” activity 2, students learn to evaluate the efficiency, flexibility, and accuracy of mathematical representations throughout the learning pathways. Students find all number card pairs that add to seven and evaluate their work by answering educator questions such as, “Do you think you have found them all?” “What could you do to check?” and “Would ordering them help?” This directly engages students in evaluating strategies for completeness and accuracy. Additionally, students evaluate whether two arrangements represent the same decomposition and then remove duplicate pairs if found. On the wrap-up representation page, students answer, “What do you know about pairs that add to seven? What have you learned today?” which prompts a final evaluation and reflection on their own mathematical reasoning and representations.

Within the grade K Math Journeys materials, students evaluate mathematical representations, models, strategies, and solutions for efficiency, flexibility, and accuracy. In lesson 3, “Addition: Counting On I,” students use snap cubes to solve addition problems with specific quantities, such as 5 + 3. The educator asks, “How many snap cubes are there in total?” Students then check their answers as a class and explain the strategies they used to solve the problem. The materials guide the educator in how to help students identify the most efficient strategy and encourage them to apply that strategy in future work. Additionally, the lesson supports students in exploring more efficient counting methods to promote accuracy as students develop flexibility in understanding and learn how to apply multiple problem-solving strategies.

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

In the Math Journeys grade K lesson, “Counting with the Bead String I,” students choose more efficient methods for solving math problems. The lesson highlights process standard 1C, which emphasizes efficiency. Students use this standard when they select how to place a spoken number on the bead string. Instead of counting each bead from one, students use number sense and reference points at 5, 10, and 15 to reach numbers like 6, 9, 14, or 19 more quickly. For example, students might start at 15 and count forward to 19, or start at 10 and count back to 9. The educator identifies the less efficient method of counting every bead from one and provides concrete examples of the better approach of jumping from a reference point. Educators use step-by-step instructions to help students choose the more efficient strategy, and the wrap-up reinforces this guidance during the activity. By encouraging students to use the color pattern for counting, this activity helps them move away from one-by-one counting and adopt reference-point-based strategies.

The grade K materials contain guidance to support students as they learn to select increasingly efficient approaches to solve mathematics problems. For example, in lesson 4, “Cardinal Counting,” students are given between 11 and 19 snap cubes and count them by first creating a group of 10 and then placing the remaining cubes to the side. Students organize quantities into groups of 10, which emphasizes place value concepts while supporting cardinal counting of teen numbers through the representation of one group of 10 and additional ones. Educators observe whether students use the group of 10 as a strategy or count each cube individually. This supports students learning to recognize place value relationships and select more efficient strategies for solving mathematical tasks.

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

The grade K learning pathway explains how concepts are addressed both conceptually and procedurally. For example, in the lesson “Counting and Numbers,” the “Mathematical Background” section explicitly distinguishes the conceptual and procedural emphases across the B cluster. The conceptual emphasis includes students learning the number 10 and the base ten number system, which supports the development of additive thinking. Working with decomposing 10 helps students recognize number combinations flexibly. The procedural emphasis is addressed through an explicit progression of strategies: bridging through 10, combining directional jumps, optimizing jumps of 10, and the standard algorithm. The sections name each strategy and how it develops computational fluency across the B learning pathway.

The materials explicitly state how both the conceptual and procedural emphasis of the TEKS are addressed throughout the lessons. In lesson 3, “Mathematical Situations,” students develop conceptual understanding as they analyze a picture that represents students sitting on a bench and standing beside the bench. Students then use themselves as physical representations to model the situation, such as demonstrating four students on the bench and one student off the bench to strengthen their understanding of concrete mathematical relationships. After they build this conceptual understanding, students transition to procedural application as they write number sentences that represent the situation and correctly use mathematical symbols for addition, subtraction, and equality. This progression from concrete modeling to symbolic representation helps students connect conceptual understanding with procedural skills while developing an understanding of joining and separating quantities.

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.

Throughout the grade K Math Journeys pathway, students use concrete objects, pictorial models, and abstract strategies across lessons. For example, the lesson “Decomposing 10” includes the word problem, “A family buys several toys; between everything they buy, there are 10 wheels in total. Which toys could they have bought?” This problem guides students through the full concrete-representational-abstract (CRA) progression within a single addition-within-ten task, which directly addresses TEKS K.3.C. For the concrete phase, students manipulate snap cubes, colored tiles, or plastic caps to test different toy combinations. Students use as many pieces as there are wheels on each toy, and different colors for each vehicle type. For the representational phase, students use ten frames as pictorial verification models and draw pictures to bridge from hands-on objects to visual representations of addition. In the abstract phase, the educator asks, “Could you write a math sentence to show how many wheels there are?” During the wrap-up student pairs record their solution as a number sentence, such as 2 + 4 + 4 = 10, with process standard 1C tagged to the activity.

The materials provide opportunities for students to use concrete, pictorial, and abstract representations as they develop an understanding of subtraction with regrouping. For example, in Journey B3, lesson 12, activity 1, students manipulate physical base ten rods and units on a tray to represent 75 and subtract 47. When the ones place does not contain enough units, students physically exchange one rod for 10 units to experience regrouping as a value-preserving action rather than a procedural rule. In activity 2, students use $10 and $1 bills to model the same regrouping process. Although still connected to a real-world context, the bills serve as visual representations of tens and ones to help students see the relationship between exchanging one ten for ten ones in a different format. This allows students to apply the same place-value reasoning across representations. Students then move to an abstract representation through the decomposition strategy widget and column diagram, where regrouping is recorded symbolically using numbers and place-value notation. Students represent the exchange, subtraction process, and final solution without relying on manipulatives to demonstrate the abstract reasoning required by TEKS 2.4.B. By progressing from physical base ten blocks, to visual money models, to symbolic recording in a column diagram, the lesson helps students connect concrete experiences, pictorial representations, and abstract procedures while maintaining a focus on the underlying place-value concepts.

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

In the materials students use concrete and pictorial models to understand algorithms. For example, the “Mathematical Background” section explicitly describes a CRA progression. The section states that students use base ten blocks “as a scaffold to support additive operations and gradually approach the standard algorithm.” The section provides a step-by-step example in which students first represent 45 and 34 with physical base ten blocks (concrete), then decompose each number into tens (rods) and ones (crosses) on paper (representational), before they group the totals and record an intermediate step. Finally, to complete the addition, students combine the recorded tens and ones (abstract) and make explicit the supports for connecting, creating, and explaining each transition from concrete to abstract.

In the materials students connect concrete experiences, representational models, and abstract mathematical notation. For example, in Journey B1, lesson 3, students begin with a concrete model and act out addition and subtraction situations with bears and a soft building block. Educators prompt students to “show this story using your bears,” which encourages them to create a physical representation of the mathematical situation and connect actions, such as joining or removing objects, to the underlying mathematical concepts. Students then transition to a representational model as they record the same situation in an additive part-part-whole frame. The Teacher Guide explicitly describes this as a “more abstract lens” on the concrete situation, which helps students understand that the visual representation communicates the same mathematical relationships shown with the manipulatives. Students learn to define and explain how the parts and whole represented in the diagram correspond to the objects they manipulated. Finally, students move to an abstract representation as they write a matching math sentence using +, -, and = symbols. Educators instruct students to “write down what happened using numbers and the minus sign,” which explicitly connects the visual model and the concrete action to symbolic notation. Students use equations to explain the situation and demonstrate that they can represent the same mathematical idea abstractly. Process standard 1E further supports these transitions by emphasizing the creation and use of representations to organize and communicate mathematical situations. The lesson does not treat the representations as isolated activities. Instead, it intentionally helps students connect, explain, and translate among concrete objects, visual models, and symbolic equations.

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

The cluster overview in the materials outlines clear language goals for each journey and focuses on building academic mathematical vocabulary through multiple strategies. In Journey B1 students use complete sentences and key terms—such as add, plus, subtract, minus, sum, difference, total, part-part-whole, and equation—to verbally describe how to join and separate sets. In Journey B2 students use comparative language—like double, near double, make a 10, count on, and count back—to compare and justify mental strategies. Students develop this vocabulary through manipulatives, visual models, and oral discussions throughout both journeys.

In the session “Counting and Numbers” students use manipulatives and language supports to develop academic mathematical language. Students build doubles with snap cube towers to connect the terms double and equal groups to concrete models before they write equations such as 2 + 2 = 4. To support students’ use of mathematical language during discussions, the educator includes sentence frames such as, “I made a double of,” “___ makes ___,” and “two groups of ___ is ___ because ___ plus ___ equals ___.” The guide also highlights the cognate double/doble and prompts students to connect the vocabulary to their cube models and drawings. This strengthens student understanding through visual and hands-on representations.

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.

In Journey B1, the cluster overview A, “Counting and Numbers,” includes a language bank with session-specific scaffolds for building academic vocabulary. The pair-finding language ladder, routine 2, prompts educators to display a simple sequence, from naming to explaining. Students hold cards or cubes as they speak and then read one equation aloud together as a way to practice vocabulary in a hands-on activity. The B1.4 discourse entry provides partner-talk frames at two levels: “___ and ___ make six” for naming, and “___ plus ___ equals six” for explaining. Educators are guided to “read the equation aloud and point to each quantity as you say it,” which helps students move from naming to explaining in context. For B1.13, the bank offers frames for both operations: “___ plus ___ equals nine” and “nine minus ___ equals ___”. The bank also includes a prompt that helps students use academic vocabulary to describe the relationship between addition and subtraction.

The materials include embedded guidance for educators to scaffold, support, and extend students’ academic mathematical vocabulary throughout the lesson activities. Students engage in structured mathematical conversations using precise mathematical language to explain and justify how their pair of cards combine to make six. To deepen mathematical discourse, educators ask questions such as, “Could you tell me exactly how many dots you need to add to your card to make 6?” and “Do you use the words plus and equals when explaining how you made 6?” To answer these questions, students use academic vocabulary as they articulate mathematical thinking clearly and accurately. The materials further support vocabulary development with educator-facilitated small group discussions that focus on exploring multiple ways to decompose six into two addends. These structured interactions create meaningful opportunities for students to communicate mathematical ideas with both peers and educators, which strengthens their understanding of mathematical concepts as they develop confidence and precision in academic mathematical language.

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

In the lesson “The Number 9 (Part 2),” students use mathematical language and academic vocabulary during discussions, including the EB student supports. In routine 2, “Collect and Display,” educators record students’ exact words in a visible space, then write the symbolic form beside them. For example, the educator writes “four plus five makes nine” next to 4 + 5 = 9. Students find the related subtraction sentence and add both to the display to create an addition sentence. Educators then underline the words addition and subtraction to highlight these academic terms in classroom conversations. The lesson explicitly tags process standard 1D, connecting spoken language to written equations and concrete models.

In Journey B, “Additive Thinking,” the materials include embedded guidance to support students’ application of academic mathematical language during classroom discourse. Educators ask, “Can you explain why you entered the nest of six?” after which students use symbol and number cards to read math sentences and explain their reasoning. Educators use sentence frames, such as “I need,” “ and ___ make six,” and “___ plus ___ equals six,” to help students apply appropriate mathematical vocabulary as they communicate their thinking.

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

In the lesson “The Number 5 (Part 2),” educators use clear prompts to lead conversations about math topics. In Journey B, “Additive Thinking,” the EB tip offers sentence frames that progress from naming to justifying. The sentence frames include, “This card has ___ and this card has ___,” “___ plus ___ makes five,” and “I paired these cards because ___ plus ___ makes five.” Educators keep the frames visible as they speak and students point to their cards as they speak. After student pairs use numerals and the plus sign to record their answers on the board, educators ask them to read an expression aloud. This structures peer interactions so students hear and refine mathematical language as they use academic terms like plus and makes. The wrap-up deepens this exchange as groups present their recorded pairs to the class.

In the lesson “The Number 6,” educators lead mathematical conversations that help students hear, refine, and use mathematical language with peers. In the “Nest of 6” activity students move around the classroom to find a classmate whose quantity card combines with theirs to make six. As they work, students use mathematical language to explain combinations. To support peer discussions, educators use sentence frames such as “I need ___,” “___ and ___ make six,” and “___ plus ___ equals six.” Students explain why their cards belong in the nest and further refine their mathematical language as they communicate with classmates.

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 materials include embedded guidance to anticipate a variety of student answers, including
exemplar responses to questions. In Journey B, “Additive Thinking,” the educator outlines specific scenarios, such as “If there are 2 students standing on chairs and 2 more get up, write 2 + 2 = 4; if there are 3 students standing on chairs and 2 get down, write 3 − 2 = 1.” The scenarios give educators models of the expected form of response for different situations. A support section anticipates that some students will struggle to construct the full equation and provides scripted redirecting questions for educators. The questions include, “How many students are standing on chairs?” “How many got down?” “How can you represent the students who got down in the math sentence?” and “How many are left now?” These questions guide students back to the correct mathematical representation without simply providing the answer.

The lesson “The Number 10” anticipates a variety of student answers and provides educator support for inaccurate responses. During the lucky escape card game, the Teacher Guide includes exemplar responses, such as “Sonya paired a six and a four. Does it add to 10?” and “Andrew has a 10. What does he need? A zero, but he does not have one, so he cannot make the pair.” These responses help educators recognize both correct and incorrect student reasoning. When students make incorrect pairs, educators revisit the visual aid from activity 1 that displays number pairs that make 10, which provides a clear strategy to redirect misconceptions and reinforce correct thinking.

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

The materials embed the mathematical process standards from the TEKS in every lesson. In “Quantity 0–6,” the lesson clearly labels process standard 1B in activity 1. To complete this task, students use a problem-solving model as they play the square and angle game. Students then analyze the scoring rules: two points for landing in the angle, one point for landing in the large square, and zero points for landing outside. Students plan where to aim their three pebbles and evaluate whether their strategy earns the most points. The educator supports this process with targeted questions, such as, “How can you get the most points in this game?” “What is your plan for throwing the pebbles?” and “Does your plan really give you the highest possible score? How do you know?”

The materials appropriately integrate the TEKS process standards by explicitly identifying the applicable standards at the beginning of each lesson and embedding opportunities for students to actively engage with those standards throughout instruction. For example, in lesson 14, “The Number 10,” the TEKS process standards 1.B and 1.G are clearly identified alongside the content standards to guide instruction. The lesson addresses TEKS 1.B as students apply a problem solving model to analyze and determine possible solutions for a pin-arrangement task. Students think critically, make decisions about their approach, and use mathematical reasoning to come up with the solution. The lesson further supports TEKS 1.G during the wrap-up as students utilize precise mathematical language to communicate their mathematical thinking and explain and justify their representations to the class. This intentional integration of process standards throughout the lesson promotes the development of critical thinking, problem solving skills, and mathematical communication as students deepen their conceptual understanding of mathematics.

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

The grade K Math Journeys materials include a description of how the TEKS process standards are incorporated and connected throughout the learning pathways. In the B Journey Instructional Guide, educators review a vertical justification that explains how students develop additive thinking across the journey as they engage in problem solving, reasoning, mathematical communication, representations, and connections. The guide also includes a lesson-by-lesson alignment table that identifies where specific TEKS process standards are embedded throughout the pathway. The guide demonstrates how these standards are incorporated across multiple sessions, rather than taught in isolation. In addition, the guide describes how process standards support the progression of content learning and provides examples of students using mathematical representations, discussing strategies, solving problems, and justifying their reasoning.

These materials include a clear description of how the TEKS process standards are intentionally incorporated and connected throughout the learning pathways by explicitly identifying the standards within each lesson and providing guidance for educators on how to implement them during instruction. In cluster B1, “Numbers Decomposition and Addition and Subtraction through Counting (1–10),” the process standards are systematically embedded across the sequence of lessons to support the development of mathematical reasoning and conceptual understanding. For example, lesson 12 addresses TEKS 1.A as students apply mathematics to solve problems that arise in everyday situations. Lesson 13 integrates TEKS 1.D as students use multiple representations to communicate the same mathematical idea. The lesson also incorporates TEKS 1.G as students use precise mathematical language to display, explain, and justify their thinking. Lesson 14 embeds TEKS 1.B to continue the progression. In this lesson, students use a problem-solving model to work through mathematical tasks, alongside TEKS 1.G, which supports mathematical discourse through explanation and justification of student reasoning. Lesson 15 addresses TEKS 1.F when students analyze mathematical relationships and make connections between concepts before they consolidate their learning during the lesson 16 review. This intentional progression demonstrates how the process standards are connected across lessons to build students’ critical thinking, problem-solving abilities, mathematical communication, and deeper conceptual understanding over the course of the learning pathway.

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

In grade K, Journey B1.5, the lesson “Quantity 0–6” outlines the TEKS process standards within each session. A dedicated sidebar explicitly labels process standard 1B with the heading, “Use a problem-solving model,” at the point of instruction. The box explains how students apply the standard to analyze the scoring rules: two points for landing in the angle, one point for landing in the large square, and zero points for landing outside. Students plan where to aim their three pebbles and evaluate whether their strategy earns the most points. Educators ask questions such as, “How can you get the most points in this game?” “What is your plan for throwing the pebbles?” and “Does your plan really give you the highest score possible? How do you know?” By identifying the process standard and describing how it is applied during instruction, the materials provide an overview of the TEKS process standards incorporated into the lesson.

The materials clearly show how each lesson embeds the TEKS process standards by identifying the relevant standards and guiding educators on how to use them in instruction. In Journey A0.9, the lesson “The Cube Measurer” highlights the TEKS process standards 1C and 1E within the lesson structure to create purposeful math learning. For standard 1C, students choose and use an eight-cube measuring tool to solve measurement problems and decide the best way to complete each task. This lesson addresses standard 1E as students organize, record, and share measured lengths with representations that show their understanding of measurement. These instructional opportunities let students select mathematical tools, apply strategies, and represent their thinking in meaningful ways. The materials help educators consistently build mathematical reasoning, problem solving, and communication skills into daily lessons by explicitly identifying process standards and providing educator guidance.

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

In grade K Math Journeys, lesson B1.9 presents opportunities for students to think mathematically in the magical towers challenge. For this task, students find all pairs that decompose eight into two towers of different heights, which is a constraint with exactly three solutions (seven and one, six and two, and five and three) that students discover through sustained exploration. Students then extend the problem to combinations of three or more towers. Rather than direct students toward solutions, educators ask, “Do you think there is more than one solution? Can you find any other solutions?” The 1.F box states, “Do you think there might be another way to make two different towers that still use all eight cubes?” Students analyze relationships among tower heights, record combinations that sum to eight, discuss their reasoning with peers, and create representations of their solutions, thereby supporting mathematical thinking, perseverance, and sense making.

In Journey B1, lesson 17, students solve a 10-wheel puzzle as they find combinations of toy cars (four wheels), scooters (two wheels), and tricycles (three wheels) that total exactly 10 wheels. Students explore many possible solutions, reason through their choices, test ideas, and revise their thinking. The lesson encourages strategic problem solving by offering manipulatives such as ten frames, colored tiles, linking cubes, pencils, and blank paper. Student pairs choose the tools that best support their thinking, knowing that different groups may use different approaches. After they solve the problem, students explain their reasoning, describe the toys and quantities they chose, and reflect on which tools or representations helped them. These discussions focus on mathematical reasoning and justification, not just finding the right answer.

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

In grade K Math Journeys, the lesson “The Number 8 (Part 1)” teaches students that they can solve problems in many ways. Students tackle open-ended tasks and find several valid ways to solve them. For example, in the warm-up students use eight cardboard boxes to build two magical towers and find all possible combinations, such as seven and one, six and two, and five and three.To support this work, educators ask questions like, “If one tower uses five cubes, how many cubes does the other tower need?” and “Do you think there is another way to use all eight cubes to build two different towers?” In the wrap-up students show their solutions in different ways. These activities help students see that they can use multiple strategies to solve a problem.

The materials help students see that they can solve problems in different ways. In Journey B1, lesson 8, students use number cards to find every pair that adds up to seven. Students record each solution, organize their results, and check if they found every possible combination. Educators ask questions like, “Do you think you have found them all?” and “Would ordering them help?” to prompt students to justify their answers and explain their thinking. Students compare the math sentences they create, such as 3 + 4 = 7 and 6 + 1 = 7, and notice that different pairs can make the same total. These activities teach students that they can use many valid strategies and representations to solve a problem, and that they should explain and justify their choices.

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.

In lesson B1.7, “The Number 7 (Part 1),” students make sense of math by doing, writing, and discussing it with peers and the educator. In activity 1 students play a target practice game where they throw seven hoops over a cone and record their scores on a group chart. This hands-on activity builds students’ understanding of how to break apart the number seven. Educators visit groups and ask, “How many hoops are on the cone?” “How many are around it?” “Are there more hoops on the cone or around it?” and “Are there the same number of hoops on the cone as around it?” These questions guide students as they decompose seven in different ways. During the wrap-up students use drawings, numbers, symbols, arrows, or words on the representation page to explain their thinking. The session starts with a class goal, “identify pairs of numbers that add to seven,” and ends with a discussion where students compare group score charts and talk through their solutions. To complete the task, students explain and compare their representations and strategies with the whole class.

The materials help students make sense of math by doing, writing, and discussing it with peers and educators. For example, in Journey B1.2 students form small groups, choose quantity cards from zero to six, find every pair that adds up to five, and write each combination on blank paper. In activity 3 students share their findings during a class discussion. To encourage clear math communication and attention to structure, educators ask questions like, “Could you tell me exactly how many dots are on each card you chose?” and “Do you remember how we write the plus sign when we want to show adding?” Students then write on a representation page to answer, “What have we done today?” Throughout these materials, students use drawings, numbers, math sentences, quantities, symbols, arrows, and words to show that they can express math ideas in many connected ways.

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 materials help educators guide students to explain their thinking and justify their math strategies. For example, in the lesson “The Number 7 (Part 1)” students throw hoops, record their scores, and discuss their results. Educators visit each group and ask, “How many hoops are on the cone?” “How many are around it?” “Are there more hoops on the cone or around it?” and “Are there the same number of hoops on the cone as around it?” In response, students explain how they broke apart seven. In the wrap-up educators pick a student to show and explain their work to the class. To help everyone understand the solution and inspire their classmates, educators highlight numbers, signs, or operations in student work. This student sharing serves as both an explanation and a model for peers.

Furthermore, the materials help educators guide students to share and reflect on their problem solving. For example, in Journey B1, lesson 8, activity 2, students find every pair of cards that adds to seven. To encourage students to review their strategies and check for completeness, educators prompt students with questions like, “Do you think you have found them all?” “What could you do to check?” and “Would ordering them help?” The educator leads a class discussion where students explain how they ordered their pairs and what they learned, which shifts the focus from individual work to group reflection. In the wrap-up students present and explain their solution to the class, giving everyone a chance to communicate and justify their thinking.

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

In grade K Math Journeys, lesson B1.8, “The Number 7 (Part 2),” educators use prompts to give students clear feedback and address misconceptions. Students use connecting cubes to show their scores and write equations that equal seven. As students throw hoops and record their scores, educators visit each group and ask, “How many hoops are on the cone?” “How many are around it?” “Are there more hoops on the cone or around it?” and “Are there the same number of hoops on the cone as around it?” These questions prompt students to explain how they broke apart the number seven. In the wrap-up educators choose a student to show and explain their work to the class. Throughout instruction, educators point out numbers, signs, or operations in student representations and use them to help the class understand and inspire others. These prompts help educators respond to student thinking with feedback and address misconceptions by asking questions, not just giving answers.

The materials guide educators to give clear, explanatory feedback based on how students respond and what misconceptions they might have. In Journey B1, lesson 4, educators ask, “Can you explain why you entered the nest of six?” and expect answers like, “We entered six because it equals 3 + 3.” Students then turn their oral explanations into math sentences and show their reasoning in writing. For EB students the materials provide sentence frames that help them explain their thinking at different language levels, from “I need ___” for beginning-level students to “I have ___, and I know that if I add ___, it equals six because ___ and ___ together make six” for advanced-level students. In activity 2, step 5, educators lead a class check to see if each pair can “enter the nest.” This step gives educators a chance to correct misunderstandings and explain why a pair’s combination does or does not make six.