Kiddom Texas: Matemáticas en Español – Grado 4

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

In the grade 4 materials, a scope and sequence outlines the specific order of the Texas Essential Knowledge and Skills (TEKS) and concepts taught throughout the course. For example, in unit 1, lesson 1, section A, “Valor posicional de números enteros,” the lesson aligns with TEKS 4.2B, and the objective is “Representa, lee y escribe números enteros de varios dígitos hasta 1,000,000, incluyendo la notación desarrollada.”

The materials include a scope and sequence that outlines the TEKS and concepts taught in the course and can be found in the document titled Grado 4 Alcance y Secuencia. The scope and sequence is organized by unit and lesson number and includes column titles such as “Número de unidad en Kiddom,” “Número de lección en Kiddom,” “TEKS,” and “Objetivos de aprendizaje.”

Evaluation for 1.1b
Materials include suggested pacing (pacing guide/calendar) to support effective implementation for various instructional calendars (e.g., varying numbers of instructional days - 165, 180, 210).

The grade 4 materials include suggested pacing to support three instructional calendars titled Calendario de 165 días, Calendario de 180 días, and Calendario de 210 días. Each calendar includes the number of lessons, evaluation days, additional flexible days, total days, and suggested ideas for flexible days to support effective implementation of the materials.

The materials include pacing through three instructional calendar options for 165, 180, and 210 days. The calendar options document includes lessons and activities for a full year of instruction. In the 165-day calendar, unit 1, “Factors and Multiples,” includes five lessons scheduled over seven days. In the 180-day calendar, unit 1, “Factors and Multiples,” includes five lessons scheduled over nine days. In the 210-day calendar, unit 1, “Factors and Multiples,” includes five lessons scheduled over 10 days.

Evaluation for 1.1c
Materials include an explanation for the rationale of unit order as well as how concepts to be learned connect throughout the course.

The TX Math Teacher Course Guide K–5 ES includes a rationale for unit order by providing explanations about why units are sequenced as they are and how each unit contributes to an overarching learning goal. Within the course guide, the section “Las narrativas cuentan la historia” provides explanations of how each unit is structured through a narrative that describes the mathematical work to be developed within that unit.

Additionally, within the TX Math Teacher Course Guide K–5 ES, the section titled “Principios del diseño del currículo de matemáticas de Kiddom” provides an explanation of the logic behind unit progression and the conceptual connections across units. For example, the materials state, “students are supported in making connections with prior knowledge and upcoming grade-level work, emphasizing the importance of teachers understanding the progressions within the materials.”

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

The materials include protocols with corresponding guidance for unit internalization in the document titled Guía de planificación a nivel de sección, which provides educators with support to understand and prepare to teach the unit by previewing unit objectives and vocabulary. For example, this section explains that in unit 1, students will deepen their understanding of multi-digit whole numbers, extending their knowledge to numbers up to 1,000,000,000.

The materials include protocols that guide educators in internalizing each lesson by reviewing the objective, key vocabulary, and success criteria. In the lesson narrative titled “Narrativa de la lección,” the materials explain how unit 1, lesson 1, extends understanding of place value to numbers within 1,000,000,000, and how the activities develop conceptual understanding of expanded notation and the relative value of digits. The lesson narrative supports educators in internalizing the objective “Representa, lee y escribe números enteros de varios dígitos dentro de 1,000,000,000, incluyendo en notación extendida.”

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

The grade 4 materials do not fully provide guidance for instructional leaders to support curriculum implementation and address instructional challenges. Although the materials include support for using the rostering system to organize student information and data, they do not include resources that help instructional leaders support effective classroom instruction. Additionally, the materials do not provide implementation-monitoring tools, leadership frameworks, or professional learning resources designed to build instructional leaders’ capacity to support educators in implementing the curriculum.

The materials include resources that support educators during implementation. However, the materials do not provide guidance for instructional leaders on how to use the curriculum to support the educator. The materials do not include professional learning resources or training modules designed to build instructional leaders’ capacity to support educators in implementing the curriculum with fidelity.

Evaluation for 1.2a
Materials include comprehensive unit overviews that provide the background content knowledge and academic vocabulary necessary to effectively teach the concepts in the unit.

In unit 4, lesson 3, “Fundamentos de las fracciones (parte 1),” under “Narrativa de la lección,” the materials include a comprehensive overview needed to teach fractions. For example, the narrative includes the purpose of the lesson, which is for students to understand and combine fractions with the same denominators using pictorial models and objects. Students explore the use of pictorial models to understand fractions, solve addition problems with the same denominators, and synthesize their learning through discussion and reflection.

In “Grado 4, Unidad 1: Valor posicional y patrones,” the unit overview includes the academic vocabulary necessary to effectively teach the concepts addressed in the unit. Examples of academic vocabulary include “múltiplo de un número — El resultado de multiplicar ese número por un número entero. Por ejemplo, 18 es un múltiplo de 3, porque es el resultado de multiplicar 3 por 6”; “recta — Un conjunto de puntos que están dispuestos en una línea recta y se extienden infinitamente en direcciones opuestas”; and “redondeo — Una forma formal de decir a qué número está más cerca un número dado. Por ejemplo, para 182, el número 180 es el múltiplo de diez más cercano y 200 es el múltiplo de cien más cercano.” The inclusion of these definitions and examples in the unit overview provides teachers with the academic vocabulary needed to effectively teach the concepts of place value and patterns.

Evaluation for 1.2b
Materials contain supports for families in both Spanish and English for each unit with suggestions on supporting the progress of their student.

The family support materials for grade 4 provide information in English and Spanish about unit objectives and ways families can support learning. For example, in unit 5, “Comparación multiplicativa y medidas,” the materials state, “Los estudiantes aprenden a comprender la multiplicación como una forma de comparar cantidades . . .” and explain that students apply this understanding to measurement conversions. The materials also provide family support suggestions, such as “Involucra a tu hijo en actividades del mundo real . . .” Families can engage in real-world math situations by comparing quantities while baking, converting measurements, using visual models, and encouraging comparative language during homework.

The materials provide families with information in both English and Spanish about unit goals and practical ways to support learning at home. For example, in unit 4, “Suma y resta de fracciones,” families receive information about what students are learning about fractions. The materials state, “Los estudiantes aprenderán cómo sumar y restar fracciones que tienen el mismo denominador.” The materials also offer specific suggestions for supporting student learning at home, such as discussing fractions while cooking, using everyday items like pizza slices or pieces of fruit to demonstrate fractional parts, encouraging students to draw models or use household objects to represent fraction problems, and having students explain their reasoning as they solve problems. These recommendations help families reinforce classroom learning through meaningful, real-world experiences and mathematical conversations at home.

Evaluation for 1.3a
Materials include comprehensive, structured, detailed lesson plans that include daily objectives, questions, tasks, materials, and instructional assessments required to meet the content and language standards of the lesson (aligned with the TEKS and the ELPS).

In unit 2, lesson 2, “Descubrimiento de los decimales,” the objective of the lesson about decimals is listed as “Comprende el valor de los dígitos en decimales hasta las centésimas usando notación desarrollada, reforzando destrezas de descomposición de números.” The objective aligns with TEKS 111.6.B.2.B. In this lesson, students will understand the value of digits in decimals through the hundredths place using expanded notation, reinforcing skills in decomposing numbers.

The materials include lesson plans with opportunities for students to engage with the content through tasks required to meet the language standards of the lesson. For example, in the “Launch” section of unit 2, lesson 2, activity 1, students build on their previous understanding of place value as they work together to express 78.35 in expanded notation. The activity encourages mathematical discourse and gives the educator insight into students’ understanding of decimal place value in relation to the TEKS.

Evaluation for 1.3b
Materials include a lesson overview listing the teacher and student materials necessary to effectively deliver the lesson, and the suggested timing for each lesson component.

In unit 3, lesson 4, “Mismo tamaño, tamaños relacionados,” students use visual representations to reason about fractions that are the same size and locate them on a number line. In addition to number lines, the materials needed to effectively deliver the lesson about equivalent fractions are fraction strips and tape diagrams. The educator materials state, “Los estudiantes razonaron sobre fracciones equivalentes, usando tiras de fracciones, diagramas de cinta y rectas numéricas.”

In grade 4, the materials include guidance and recommendations for the suggested timing of lesson components, such as inquiry and exploration, direct instruction, guided practice, collaborative work, and independent work. For example, in unit 3, lesson 4, activity 2, “Fracciones en las rectas numéricas,” the activity narrative indicates a duration of 15 minutes. The materials state, “El propósito de esta actividad es recordar a los estudiantes su trabajo en tercer grado utilizando rectas numéricas como una manera de razonar sobre fracciones. Primeras preguntas.”

Evaluation for 1.3c
Materials include guidance on the effective use of lesson materials for extended practice (e.g., homework, extension, enrichment).

In unit 7, lesson 20, “Todo tipo de atributos,” the materials provide guidance for using an optional extension activity to extend students’ application of previously learned concepts. The lesson guidance explains that “El propósito de esta lección opcional es que los estudiantes utilicen lo que saben sobre simetría y rectas paralelas y perpendiculares para completar o dibujar figuras con esas características en un plano.” The materials further explain that the lesson is optional because it allows students to synthesize the classification and categorization work completed throughout the unit. The provided plans support students in applying their understanding of parallel lines, perpendicular lines, measurements, and figure attributes while completing or creating figures. This guidance supports educators in using the optional lesson as extended practice to reinforce and deepen students’ understanding of geometric attributes.

In unit 7, “Problemas de práctica de la sección A,” the materials provide additional opportunities for students to apply geometric concepts through continued practice. Students draw, identify, and explain geometric concepts such as lines, rays, line segments, parallel lines, and angles in a variety of contexts. For example, students are asked to “Dibuja 4 rectas diferentes que pasen por puntos en el plano. Al menos dos de las rectas deben cruzar otra recta. Marca al menos 3 segmentos diferentes en tu dibujo.” The practice problems allow students to apply geometric vocabulary and concepts beyond individual lesson activities. Answer keys and sample student responses provide educators with information to evaluate student work and identify areas for additional support.

Evaluation for 2.1a
Materials include a variety of instructional assessments at the unit and lesson level (including diagnostic, formative, and summative) that vary in types of tasks and questions.

In grade 4, the materials include a variety of instructional assessments at the unit level, including a formative assessment that varies in types of questions. For example, the “Punto de control de la sección A” includes varying types of questions such as text-entry, drawing a number line, and drag-and-drop. One of the questions in the assessment asks students to “Estima la ubicación de los siguientes números en la recta numérica: 0.43, 0.3, 0.09, 1.2, 1.02.” Students draw on a canvas to enter their response.

In the Guía de recursos para maestros, each unit includes a section titled “Evaluaciones” that includes an evaluation for the whole unit to be used as a summative assessment. For example, the final assessment for unit 2, in which students learn about decimals, includes varying question types such as fill-in-the-blank (question 1), short-answer (questions 2 and 3), and ordering (question 5).

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

The materials define the purpose of each assessment type and explain how the results should be used. Formative assessments, such as “Enfriamientos” and “Section Checks,” are intended to monitor student understanding and inform instructional decisions. Summative assessments, such as unit assessments and the end-of-course assessment, are designed to measure students’ mastery of grade-level standards and learning objectives. The assessment guide provides specific guidance for interpreting student performance and planning next steps for instruction. For example, the materials state that the “Enfriamiento” (cooldown) is “a brief check to determine whether students understand the main concepts of the lesson” and should be used “as a formative assessment to plan future instruction.”

The materials include the purpose and importance of assessments in supporting educators with lesson planning, delivering instruction, monitoring student progress, and making data-driven instructional decisions throughout the school year. For example, the Guía de recursos para maestros highlights the availability of assessment guidance at the lesson, section, and unit levels. It explains that each lesson includes an “Enfriamiento” (cooldown) in the form of an exit ticket. It further defines how this activity serves as a brief check to determine if students understand the main concepts of the lesson and how to use the formative assessment to plan future instruction.

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

The materials include educator guidance to ensure consistent administration of instructional assessments, including the allotted timing of each test. For example, the “Enfriamiento” for unit 7, lesson 17, includes the notation “5 minutos” at the top of the page, indicating that students should spend five minutes completing the task. This guidance helps establish uniform administration conditions across classrooms.

The Descripción del curso and “Guía de la evaluación” provide information about the assessment types used throughout the course and explain how and when they should be administered. For example, the materials state, “Cada unidad incluye una Evaluación final escrita que los estudiantes completan individualmente para evaluar su progreso hacia los objetivos de aprendizaje de la unidad y los estándares del nivel de grado para el curso. Los ítems están alineados con los estándares relevantes del nivel de grado.” This guidance identifies the assessment format as written, specifies that students complete the assessment individually, and describes its use within each unit, supporting consistent administration across classrooms.

Evaluation for 2.1d
Diagnostic, formative, and summative assessments are aligned to the TEKS and objectives of the course, unit, or lesson.

In grade 4, diagnostic, formative, and summative assessments are aligned to the TEKS and the learning objectives identified throughout the scope and sequence. Each lesson lists the specific TEKS and corresponding learning goals, ensuring that assessments measure the intended knowledge and skills. Formative assessments, such as “Enfriamientos” and “Section Checks,” monitor student progress toward lesson and section objectives, while unit and end-of-course assessments evaluate mastery of TEKS-aligned standards. This alignment provides a coherent system for measuring student learning from individual lessons through the entire course.

In grade 4, the materials include summative assessments that are aligned to the TEKS and objectives of the unit. The materials identify alignment with the summative assessment and the TEKS for the course by including the TEKS number directly next to assessment items. For example, unit 8, “Datos y finanzas personales: evaluación y recursos de fin de curso” has the TEKS next to each item in this summative assessment. One question states, “Selecciona el número donde el valor del 6 sea 1,000 veces el valor del 6 en 463.” The corresponding TEKS is 111.6.b.2.A.

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

The materials include formative and summative assessments with multiple-choice, text-entry, and open-response items aligned to the TEKS. In unit 7, “Geometría,” the “Evaluación final de la unidad” includes item types that require different levels of student thinking and response. For example, multiple-choice items primarily assess students’ ability to identify or recognize geometric attributes, while text-entry items require students to generate a response independently. Open-response items demand a higher level of complexity because students must apply their understanding, justify their reasoning, and communicate mathematical thinking. For example, the item “¿La figura es un rombo? ¿Es un rectángulo? Explica cómo lo sabes” aligns to TEKS 111.6.b.6.D and requires students not only to classify the figure but also to explain the attributes that support their conclusion.

In grade 4, the summative assessment for unit 7, “Evaluación final de la unidad 7,” includes TEKS-aligned items that measure students’ understanding of angle measurement and the use of a protractor at varying levels of complexity. The assessment begins with tasks that assess foundational procedural skills, such as determining angle measures from protractor diagrams and applying subtraction to calculate the correct angle. The level of rigor then increases as students are required to independently construct and upload evidence of a 135-degree angle using a protractor. The assessment also demonstrates a progression in mathematical complexity across item types. For example, question 1 asks students to identify right triangles, requiring basic recognition of geometric attributes, while question 6 requires students to apply their understanding of perimeter, symmetry, and the properties of quadrilaterals to solve multistep problems and justify their reasoning, reflecting a higher level of mathematical thinking and problem-solving.

Evaluation for 2.2a
Instructional assessments and scoring information provide guidance for interpreting student performance.

The materials help educators identify students’ strengths, weaknesses, gaps in understanding, and common misconceptions based on assessment results. For example, in unit 2, “Decimales,” the “Punto de control de la sección B” includes educator guidance for interpreting student responses. Students solve the word problem, “Elena compró un cuaderno de dibujo por $7.95 y un paquete de lápices por $3.40. ¿Cuánto gastó en total?” The accompanying scoring guidance identifies a likely misconception by stating, “Conceptos erróneos: algunos estudiantes pueden desalinearse los decimales u olvidarse de llevar el 1 del lugar de las décimas, lo que da como resultado un total incorrecto.” This guidance helps educators interpret incorrect responses by identifying the underlying misunderstanding, allowing them to determine students’ learning needs and plan targeted instruction.

The materials include a summative assessment for unit 7, “Geometría,” titled “Evaluación final de la unidad 7,” and educator support materials that outline how each assessment item should be scored. For example, the section “Note for Evaluating Responses” provides a sample answer key and scoring guidance to help ensure consistent evaluation of student responses and mastery of the assessed concepts. Question 6 asks students whether they agree that Elena can make a square frame if each side measures 37 cm in length and the perimeter is 146 cm. The “Note for Evaluating Responses” gives the correct answer as “No” and provides the rationale. The materials state, “Ejemplo de respuesta: Un cuadrado tiene 4 lados de la misma longitud, por lo que el perímetro del cuadrado de Elena sería 4 × 37 cm y eso es 148, que es demasiado.”

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

The materials do not provide guidance for the use of included tasks and activities to respond to student trends in performance on assessments. The materials include tasks and activities for responding to trends in performance. However, the materials do not include guidance on the use of these tasks and activities. The materials do not include guidance that supports the educator in using assessment results to purposefully plan tasks and activities for the appropriate grade level.

The materials encourage educators to use assessment information to inform future instruction, but they do not provide specific guidance for responding to student trends in performance on assessments. For example, the Guía de recursos para maestros states that the “Enfriamiento” activity is given at the end of the lesson as “un breve chequeo para determinar si los estudiantes comprenden los conceptos principales de la lección” and instructs educators to “Utilízala como una evaluación formativa para planificar la instrucción futura.” However, the materials do not explain how to analyze patterns or trends across student responses, identify groups of students with similar needs, or recommend specific follow-up tasks or instructional activities 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 materials provide educators with tools to monitor student progress and growth over time. Educators can access multiple reporting features, including the Assignment View, Scoreboard, Scorecard, Question View, and Standards View, which offer data on both individual student performance and class-wide achievement across assignments, assessment items, and standards. For example, the resource “What Do My Reports Mean?” provides explicit guidance for interpreting and using key reporting tools, including the Class Grade Average line graph, Mastery Groups pie chart, Class Standard Mastery tracker, and Student Standard Progress reports. These features allow educators to analyze changes in student performance over time, monitor mastery of specific standards, group students according to performance levels, and track individual growth toward learning goals.

The materials provide tools that enable students to monitor their own progress and growth. Students have access to reports that display class grade averages, standards mastery, assignment performance, and progress trends over weekly or monthly periods. For example, students can view Class Standard Mastery reports that present color-coded mastery levels, allowing them to track their progress over time. Students can also select specific standards to access related assignments, grades, and educator feedback. These features support students in actively tracking their learning, reflecting on their performance, and understanding their progress across standards and assignments.

Evaluation for 3.1a
Materials include teacher guidance for differentiated instruction, activities, and paired (scaffolded) lessons for students who have not yet reached proficiency on grade-level content and skills.

In unit 4, lesson 5, the materials provide guidance for supporting students who are developing understanding of how to use number lines to add fractions. For example, the materials state, “Los estudiantes pueden no establecer una conexión entre las fracciones en los problemas y las rectas numéricas porque estas últimas no muestran etiquetas fraccionarias. Considera pedir a los estudiantes que etiqueten la recta numérica para mostrar las fracciones o preguntar, ¿Dónde crees que se etiquetaría en esta recta numérica?” This guidance supports educators in helping students make explicit connections between fraction representations and number lines by having students add or interpret labels on an unlabeled number line, strengthening their understanding of fractional placement and representation.

In unit 4, lesson 14, the materials provide support for students who are developing understanding of how to subtract fractions. For example, the materials state, “Los estudiantes pueden no estar seguros de qué cantidad es desconocida en cada problema o cómo está relacionada con lo que se da. Considera pregunta ¿Qué sabemos sobre la pulsera?, ¿Qué no sabemos pero queremos saber?, y ¿Qué necesitamos hacer para averiguarlo?” This guidance supports educators in helping students identify known and unknown information in word problems and use questioning strategies to determine what is being asked and how to solve for the unknown quantity.

Evaluation for 3.1b
Materials include pre-teaching or embedded supports for unfamiliar vocabulary and references in text (e.g., figurative language, idioms, academic language).

In grade 4, the materials provide sentence frames aligned to each unit’s problem-solving tasks, along with sample educator scripting that models how to introduce and use the frames to support academic language during math discussions. In unit 4, lesson 1, “Problemas con grupos iguales de fracciones,” educators are guided to use sentence frames as a problem-solving technique when working with equivalent fractions. If students do not yet understand that factors in expressions can be interpreted in multiple ways or that different expressions can represent the same quantity, educators are encouraged to use visual diagrams to help students represent and explain their thinking by illustrating different groupings of the same quantity and prompting them with questions such as “¿Dónde ves _____ grupos de _____?” This support helps students identify and articulate equivalent relationships among quantities.

In grade 4, the materials provide suggested manipulatives and visual representations, including fraction bars, place-value charts, and number lines, and guide educators to use these tools alongside academic vocabulary to build students’ understanding of abstract mathematical concepts. For example, in unit 4, lesson 2, “Fracciones como sumas,” educators are directed to use measuring cups to develop students’ understanding of fractions while reinforcing academic language. The guidance prompts educators to observe how students represent quantities in multiple ways, such as repeated fills of a measuring cup, products of a unit fraction and a whole number, and sums of unit and non-unit fractions. Educators are encouraged to highlight and name the different strategies students use with manipulatives and explicitly connect those strategies to academic vocabulary and mathematical representations.

Evaluation for 3.1c
Materials include teacher guidance for differentiated instruction, enrichment, and extension activities for students who have demonstrated proficiency in grade-level content and skill.

The materials provide enrichment opportunities such as project-based learning activities, research projects, and creative tasks that require students to synthesize and extend their understanding of content. In unit 4, lesson 5, “Suma de fracciones,” educators are guided to provide enrichment for students who have demonstrated proficiency in adding fractions. For example, the activity offers students additional practice using number lines to decompose fractions into sums of other fractions and record these decompositions as equations.

In unit 4, lesson 16, “Historias con fracciones,” educators provide open-ended tasks that require students to apply their understanding of adding and subtracting fractions in real-world scenarios, such as designing a budget for a community project. For example, the purpose of the activity is for students to create and solve problems involving fraction addition and subtraction by first writing stories that match a given value or equation along with specific constraints. The guidance also includes conversation supports, such as sentence frames to facilitate small-group discussion: “_____ y _____ son iguales/similares porque . . .” or “_____ y _____ son diferentes porque . . .”

Evaluation for 3.2a
Materials include explicit (direct) prompts and guidance to support the teacher in modeling and explaining the concept(s) to be learned.

In unit 4, lesson 7, “Frenesí de fracciones de referencia,” educators are guided in helping students analyze how fractions relate to benchmark fractions through the prompt, “¿Cómo se relaciona cada nueva fracción con las fracciones de referencia? ¿Es mayor que, menor que o está entre dos fracciones de referencia?” This prompt encourages students to determine whether a fraction is greater than, less than, or between two reference fractions.

In unit 4, lesson 1, “Números decimales,” educators are guided in modeling decimal concepts by having students represent decimals with concrete models and then connect those representations to decimal notation, fraction notation, and diagrams. The guidance states, “Aquí se hace explícita la idea de que dos decimales pueden ser equivalentes, al igual que dos fracciones pueden ser equivalentes. Cuando los estudiantes conectan cantidades en forma de palabras, forma decimal y forma fraccionaria, razonan de manera abstracta y cuantitativa,” emphasizing how multiple representations support students’ understanding of equivalence and strengthen their abstract and quantitative reasoning.

Evaluation for 3.2b
Materials include teacher guidance and recommendations for effective lesson delivery and facilitation using a variety of instructional approaches.

In unit 5, lesson 1, “Veces más que,” educators use manipulatives and student-generated representations to develop students’ understanding of multiplicative comparison and measurement concepts. For example, the materials state, “El propósito de esta actividad es que los estudiantes construyan sobre lo que saben acerca del lenguaje de ‘dos veces’ y ‘dos veces más’ para representar situaciones de comparación. En esta actividad, se anima a los estudiantes a representar la situación de una manera que tenga sentido para ellos, aunque las representaciones discretas (cubos o dibujos) son el enfoque de la síntesis de la actividad. Las representaciones utilizadas en esta actividad sirven como base para los diagramas de cinta más abstractos que se usarán más adelante en la sección.” Through this activity, educators build students’ understanding of multiplicative comparison language by having them model situations with concrete representations such as cubes or drawings. Students use these hands-on models to make sense of comparison relationships and prepare for the more abstract tape diagrams introduced later in the unit.

In grade 4, the materials include educator guidance and recommendations for effective lesson delivery and facilitation using a variety of instructional approaches, including hands-on exploration with math manipulatives. In unit 1, lesson 1, the materials provide structured guidance for collaborative, independent, and whole-class instruction. In activity 1, “Clasificación de tarjetas: números grandes,” educators begin with partner discussions, facilitate a card-sorting task, and guide students to use place-value language while working with expanded notation before returning to partner discussion. The lesson includes questioning strategies that prompt students to describe multiplicative relationships and use place-value charts or base 10 blocks to represent concepts concretely. Educators also collect and display student thinking, address common misconceptions, and lead a synthesis discussion that connects place value to multiplication and division.

Evaluation for 3.2c
Materials support multiple types of practice (e.g., guided, independent, collaborative) and include guidance for teachers and recommended structures (e.g., whole group, small group, individual) to support effective implementation.

The materials support multiple types of practice through a combination of guided instruction, independent work, and collaborative learning opportunities, with educator guidance and recommended structures to support effective implementation. In unit 2, lesson 6, “Compara y clasifica decimales,” students engage with decimals through guided questioning and modeling with base 10 manipulatives, independently compare and order decimals, and discuss and justify their reasoning with a partner. The lesson provides educator prompts, monitoring suggestions, and discussion questions to support implementation in partner, small-group, and whole-class settings while reinforcing conceptual understanding of place value and decimal comparisons.

Similarly, in unit 4, lesson 5, “Suma de fracciones,” the lesson follows an instructional sequence that integrates multiple practice structures. During the “Inicio,” students work with a partner while the educator prompts them to analyze addition expressions with questions such as “¿Qué notas sobre los números?” During the activity, students work independently for 5–7 minutes and then discuss their responses with a partner. In the “Síntesis de la actividad,” the educator facilitates a whole-group discussion in which students share solutions and analyze relationships among fractions and sums using prompts such as “¿Qué observas sobre los números en cada suma? ¿Cómo se relacionan con los números en las fracciones que se están sumando?” These instructional routines provide clear guidance for educators to implement guided, independent, collaborative, and whole-group learning opportunities throughout the lesson.

Evaluation for 3.3a
Materials include teacher guidance on providing linguistic accommodations for various levels of language proficiency [as defined by the English Language Proficiency Standards (ELPS)], which are designed to engage students in using increasingly more academic language.

This guidance is not applicable because the Spanish program does not require guidance on providing linguistic accommodations.

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

The materials provide instructional strategies that support emergent bilingual (EB) students in developing academic language while engaging with grade-level content. In unit 4, lesson 2, “Fracciones como suma,” the materials incorporate “Math Language Routine 7: Compara y conecta” to promote structured mathematical discourse. For example, the materials direct educators to lead a discussion supporting synthesis of concepts: “Dirige un debate en el que se comparen, contrasten y conecten las diferentes representaciones. Pregunta: ¿Cómo muestran estas diferentes representaciones la misma información?, ¿Qué tienen en común cada una de estas representaciones? y ¿En qué se diferencian?” This routine guides educators in facilitating discussions that help students compare, contrast, and connect multiple representations while strengthening their mathematical understanding and academic language development.

In unit 2, lesson 1, “Números decimales,” the materials provide implementation guidance that supports educators in state-approved bilingual and ESL programs through the integration of “Math Language Routine 2: Collect and Display,” a strategy designed to strengthen academic language development. Educators are directed to intentionally collect, display, and revisit key mathematical vocabulary from previous activities, encouraging EB students to use and build upon shared language resources throughout the lesson. The lesson also incorporates multiple language modalities by having students connect concrete models, fraction notation, decimal notation, and verbal descriptions of quantities, supporting comprehension and language acquisition simultaneously. Additionally, structured partner discussions, educator prompts, and opportunities to explain equivalent decimals such as 0.6 and 0.60 provide meaningful practice in mathematical discourse while reinforcing both content understanding and language proficiency.

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

In Grade 4, Unit 4, Lesson 5, “Fracciones equivalentes,” the materials include embedded guidance to support emergent bilingual (EB) students in developing academic vocabulary, increasing comprehension, building background knowledge, and engaging in mathematical discourse. The lesson incorporates visual representations, including diagrams and number lines, to support students’ understanding of fractions. The materials state, “En esta actividad, verás diagramas y rectas numéricas que representan fracciones” and “Tómate 5 minutos para trabajar de forma independiente en el primer problema. Después, comparte tu pensamiento con tu compañero.” These supports provide opportunities for students to interpret mathematical representations, connect visual models to fraction concepts, and communicate their mathematical thinking through partner discussion.

In “Grado 4, Unidad 7, Lección 18: Simetría en las figuras,” the materials provide embedded guidance to support emergent bilingual students in developing academic vocabulary and making cross-linguistic connections. In “Calentamiento: Observa y pregúntate: Ver doble,” the strategy “Conexiones translingüísticas: Palabras que se transfieren” identifies Spanish-English cognates and vocabulary related to geometric concepts, including simetría / symmetry, geometría / geometry, línea / line, vertical / vertical, central / central, diseño / design, reflejo / reflection, and atributo / attribute, as well as hexágono / hexagon, rombo / rhombus, triángulo / triangle, and trapecio / trapezoid. This guidance helps students recognize connections between the two languages and develop the academic vocabulary needed to understand and discuss geometric concepts.

Evaluation for 3.3d
If designed for dual language immersion (DLI) programs, materials include resources that outline opportunities to address metalinguistic transfer from English to the partner language.

This guidance is not applicable because the program is not designed for dual language immersion (DLI) programs.

Evaluation for 4.1a
Practice opportunities over the course of a lesson and/or unit (including instructional assessments) require students to demonstrate depth of understanding aligned to the TEKS.

The materials provide practice opportunities that require students to demonstrate depth of understanding aligned to the TEKS by applying place value, rounding, number comparisons, and algebraic reasoning across multiple contexts. Students move beyond procedural application by explaining their reasoning, justifying conclusions, and using mathematical representations to support their thinking. For example, in unit 1, section C, practice problems require students to analyze scenarios, construct numerical comparisons, and represent real-world situations using expressions and equations. Question 2 asks students to analyze the altitudes of three airplanes that round to 30,000, 31,000, and 32,000 feet, requiring students to apply understanding of rounding and place-value relationships. These practice opportunities provide students with repeated opportunities to demonstrate conceptual understanding, mathematical reasoning, and application of the grade-level TEKS.

The materials provide practice opportunities, including instructional assessments, that require students to demonstrate depth of understanding aligned to the TEKS by applying decimal concepts related to place value, comparisons, addition, and subtraction. In unit 2, the “Evaluación final de la unidad 2” states that the purpose of the assessment is to demonstrate student understanding of decimals by solving problems that involve multiple mathematical concepts rather than only identifying decimal values. For example, one instructional assessment question asks, “¿Cómo se escribe el número 5,003.71 en notación desarrollada?” requiring students to represent a decimal number using expanded notation. This aligns with TEKS 111.6.b.2.B, which requires students to represent the value of digits in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals. These practice opportunities allow students to apply place-value understanding, analyze numerical relationships, and demonstrate conceptual understanding of decimal representation.

Evaluation for 4.1b
Questions and tasks progressively increase in rigor and complexity, leading to grade-level proficiency in the mathematics TEKS.

In unit 2, lesson 9, “Decimales y dólares,” the practice opportunities throughout the lesson require students to demonstrate a deep understanding of decimals by connecting hundredths, place value, fractions, and monetary values in meaningful contexts aligned to the TEKS. Rather than simply identifying decimals, students represent and interpret decimal amounts using play money, applying their knowledge of how dimes and pennies relate to tenths and hundredths of a dollar. For example, students model $0.37 using three dimes and seven pennies and explain how the digits represent 3 tenths and 7 hundredths of one dollar. These tasks promote conceptual understanding by requiring students to justify representations, make connections among multiple mathematical ideas, and apply their learning to real-world financial literacy situations.

In unit 6, lesson 3, “Multiplicación y división de números de varios dígitos,” students progress from identifying factor pairs to analyzing mathematical relationships and justifying whether numbers are prime or composite based on evidence. Students begin by finding the number of rectangles that can be created for a given area, connecting rectangular models to factor pairs. They then analyze and discuss their strategies in groups and respond to questions such as “¿Cuáles son todos los números primos en nuestra lista? ¿Cómo sabemos que son primos?” and “¿Cuáles son todos los números compuestos en nuestro conjunto? ¿Cómo sabemos que no son primos?” These questions require students to move beyond identifying factors to explaining patterns, analyzing relationships, and defending mathematical conclusions.

Evaluation for 4.2a
Materials demonstrate coherence across units by explicitly connecting patterns, big ideas, and relationships between mathematical concepts.

The Grado 4 Alcance y Secuencia provides guidance on students’ repeated application of place-value reasoning, equivalence, multiplicative relationships, visual representations, and problem-solving strategies across whole numbers, decimals, fractions, geometry, measurement, and data analysis. These connections reinforce consistent mathematical structures and relationships across units while supporting alignment to the grade 4 TEKS. For example, unit 1, “Valor posicional y patrones,” connects to the mathematical concept of “Construyamos y descompongamos números usando bloques de base diez, enfatizando el valor cuantitativo del valor posicional.” Students build and decompose numbers using base 10 blocks while analyzing the value represented by each digit. This learning aligns with TEKS 111.6.b.2.B, which requires students to represent the value of digits in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals.

The course overview and scope and sequence demonstrate coherence across units by connecting mathematical concepts and relationships throughout the year. Concepts introduced in unit 1, such as place value, comparing numbers, and patterns, are revisited and extended in unit 2 through decimal place value, decimal comparisons, and decimal operations, helping students build on prior understanding. Similarly, unit 3 develops fraction equivalence and comparison, which provide the conceptual foundation for the work with fraction addition and subtraction in unit 4, reinforcing relationships between representations and operations. Across the curriculum, students apply patterns, reasoning, and mathematical processes in new contexts, demonstrating a coherent progression toward mastery of the grade 4 TEKS.

Evaluation for 4.2b
Materials demonstrate coherence across units by connecting the content and language learned in previous courses/grade levels and what will be learned in future courses/grade levels to the content to be learned in the current course/grade level.

The materials demonstrate connections across grade levels by building on students’ prior understanding of fractions, place value, and representations while extending learning to new decimal concepts. In unit 2, lesson 4, “Decimales equivalentes,” students apply previous knowledge of equivalent fractions, tenths, and hundredths to understand why different decimal representations can represent the same value. Students connect earlier learning about equivalent quantities to decimal relationships, recognizing that decimals such as 0.2 and 0.20 are equivalent because they represent the same amount. The lesson uses familiar representations, including hundred grids, number lines, and equations, to support mathematical language and reasoning from previous grades while increasing precision with decimal concepts. This learning prepares students for future work with comparing, ordering, and operating with decimals by reinforcing the relationship between place value and equivalent representations.

The materials include a progression document titled “Progresión coherente” within the TX Math Teacher Course Guide K–5 ES that describes how mathematical ideas are organized, how learning connects across grade levels, and how students build understanding through increasingly complex representations and applications. The document states, “Las narrativas de grado, unidad, lección y actividad describen decisiones sobre la organización de las ideas matemáticas, conexiones con trabajos previos y próximos de grado, y el propósito de cada lección y actividad.” For example, in grade 3, students develop foundational multiplication understanding by representing equal groups with drawings. In grade 4, students extend this understanding by interpreting representations of multiplicative comparison. In unit 5, lesson 2, “Interpretar representaciones de comparación multiplicativa,” students analyze and interpret representations of discrete objects, such as connecting cubes, and linear diagrams in which each unit is visible. The lesson explains that these representations prepare students for more abstract linear diagrams used in future learning. This work aligns with TEKS 111.6.b.5.A, which requires students to represent multistep problems involving the four operations using strip diagrams and equations with a letter representing the unknown quantity. The progression connects prior multiplication concepts to current multiplicative comparison strategies while preparing students for future algebraic reasoning.

Evaluation for 4.2c
Materials demonstrate coherence at the lesson level by connecting students' prior knowledge of concepts and procedures from the current and prior grade level(s) to new mathematical knowledge and skills..

The materials demonstrate coherence at the lesson level by connecting students’ prior understanding of fractions, equivalent fractions, and operations with tenths and hundredths to new learning about decimal notation. For example, in unit 2, lesson 1, “Números decimales,” students build on previous work with fractions by using fraction concepts to interpret tenths and hundredths as decimals and connect fraction notation, word form, and decimal form. The lesson uses cuadrículas de cuadrados unitarios as visual representations to connect familiar fraction models to new decimal concepts while reinforcing place-value understanding. Students represent tenths and hundredths by shading unit-square grids and then write the corresponding decimal values. For example, students connect their prior understanding of fractions such as 1/10 and 1/100 to decimal representations such as 0.1 and 0.01 by matching shaded models to fraction and decimal notation. This activity builds on earlier work with denominators of 10 and 100 and extends students’ understanding by connecting visual models, fraction language, and decimal place value.

The materials demonstrate coherence at the lesson level by connecting students’ prior understanding of place value, multiplication strategies, and representations to new mathematical skills. For example, in unit 6, lesson 9, “Multiplicar 2 números de dos cifras,” the materials explain that students build on previous learning with place-value strategies, stating that “Anteriormente, los estudiantes usaron el razonamiento del valor posicional para descomponer un factor en una expresión de multiplicación para multiplicar números de hasta cuatro dígitos por números de un dígito.” In this lesson, students extend these ideas to multiply two two-digit numbers and reflect on why it is useful to decompose both factors by place value. As students analyze connections between expressions and diagrams, they recognize that partial products involving single-digit factors or multiples of 10 can be found mentally. The materials also identify rectangular area models as a tool to connect visual representations and multiplication expressions, helping students understand how decomposing factors supports solving more complex multiplication problems. This progression connects prior multiplication concepts and procedures to new learning while developing students’ understanding of two-digit multiplication strategies.

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

In unit 3, lesson 5, students engage in spaced retrieval by revisiting prior learning about fraction equivalence and comparison from earlier lessons and units. Students apply previously learned strategies, such as using benchmark fractions, fraction strips, common numerators and denominators, and reasoning about relative size, to determine whether fraction statements are true or false. This repeated application provides opportunities for students to retrieve foundational fraction concepts while strengthening understanding of equivalent fractions. The routine also revisits familiar representations, including diagrams and visual models, allowing students to apply prior knowledge in new but related contexts. For example, in the warm-up, Part B requires students to determine that 3/4 = 6/8. Students retrieve their understanding of equivalent fractions by recognizing that each fourth can be partitioned into two equal eighths, showing that both fractions represent the same amount. Students then use this prior knowledge to explain why the statement is true, reinforcing fraction equivalence strategies across lessons and units.

For example, in unit 5, lesson 3, “Resuelve problemas de comparación multiplicativa,” students engage in spaced retrieval by revisiting and applying prior understanding of comparison, equal groups, multiplication, and measurement concepts in a new context. Students build on previous work with additive comparison from grade 3, such as determining how many more or less, and extend this understanding to multiplicative comparison situations. In the “Two Cities” task, students apply their prior knowledge of equal groups to interpret one city’s data as a unit and compare quantities multiplicatively. Students also retrieve measurement skills by identifying the quantity being compared and determining the appropriate unit, such as miles or residents, within the context of the problem. By applying previously learned concepts to new comparison situations, students reinforce understanding while extending their mathematical reasoning.

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

In unit 1, “Valor posicional y patrones,” practice problems in section A provide interleaved practice opportunities by combining previously learned place-value concepts with multiple numerical skills within the same practice set. Students apply understanding of place value and expanded notation while also practicing writing numbers in words, adding four-digit numbers, and subtracting four-digit numbers. For example, the task “Escribe cada número en notación desarrollada: 101,496,345” requires students to analyze the value of each digit and represent the number using expanded form. Across the practice problems, students switch between different concepts and procedures rather than focusing on a single skill, requiring them to apply previously learned strategies and determine an appropriate method for solving each problem.

In unit 4, section A, the practice problems provide interleaved practice opportunities by combining multiple previously learned fraction concepts within a single practice set, requiring students to apply and connect skills from different lessons and units. Students revisit fraction addition with like denominators, equivalence, decomposition, number-line representations, and comparison to benchmark fractions while solving problems in varied contexts, including measurement and sharing situations. For example, students solve 3/8 + 2/8 using a visual model, represent 3/4 as different sums of fractions, and determine whether a fraction sum is closer to 1/2 or 1 using benchmark fractions. By requiring students to switch between different fraction concepts and representations, the practice set supports students in applying prior learning across problem types rather than practicing a single skill in isolation.

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

In unit 1, lesson 1, students interpret, analyze, and evaluate mathematical models as they use base 10 blocks and diagrams to represent large numbers in multiple ways. Students examine how changing the value of the smallest cube affects the value of all other blocks and use these relationships to represent numbers in the ten-thousands, hundred-thousands, and millions. For example, students analyze Lin’s strategy of assigning a value of 100 to the small cube and represent 415,700 using four large cubes, one large square, five long rectangles, and seven small cubes while explaining the value each block represents. Students compare representations, identify equivalent groupings, and justify their choices using labels and drawings.

In unit 8, lesson 8, “Comprender los gastos: una mirada a los costos fijos y variables,” students interpret, analyze, and evaluate a monthly budget of $3,500 by classifying expenses as fixed or variable. Students respond to questions such as “¿Es la renta un gasto fijo o variable? ¿Qué observas que te hace pensar eso?” and “¿Qué otros gastos pueden ser gastos fijos? ¿Cómo lo sabes? ¿Por qué estos otros gastos son gastos variables?” Students use evidence from the budget table to justify their classifications and explain relationships among expenses. Students also sort expenses in a role-play activity, identify patterns, and write a rule describing the difference between fixed and variable expenses before sharing and discussing their reasoning.

Evaluation for 5.1b
Questions and tasks require students to create models to represent mathematical situations.

In unit 1, lesson 3, students use place-value blocks and place-value charts to create concrete and visual models representing the value of each digit in whole numbers. Students work in small groups to build numbers and represent place-value relationships using manipulatives and charts. For example, when representing 4,567 in this task, students use place-value blocks to show four thousands, five hundreds, six tens, and seven ones and label each part of their model to match the value of each digit.

In grade 4, unit 7, lesson 9, activity 2, students create models by drawing a circle and partitioning it to represent angles. Students are prompted to draw a circle and divide it to approximate degrees, as described in the activity: “En esta actividad, los estudiantes intentarán dibujar un círculo, partirlo para aproximar los grados y entender el concepto de un grado como 1/360 de un círculo.” Students then use their drawings to represent angle relationships within the circle. To close the activity, educators facilitate comparison of student models with the prompt, “Comparte y compara los círculos dibujados por los estudiantes.”

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

In unit 1, lesson 3, students apply their understanding of place value and expanded notation to read, write, and represent numbers in the millions and billions. Students extend previously learned place-value concepts to a new context by using place-value charts to identify patterns across periods and represent increasingly larger numbers. For example, students represent “Ocho mil setecientos sesenta y seis millones ochocientos treinta y seis mil ochenta y seis” as 8,766,836,086 in standard form using a place-value chart. By applying familiar place-value relationships to numbers in the millions and billions, students transfer their conceptual understanding to a new mathematical context.

In unit 5, lesson 4, “Resuelve problemas de comparación multiplicativa con números grandes,” students apply their understanding of multiplicative comparison to new problem situations using strip diagrams, equations, estimation, and reasoning. Students interpret relationships between quantities, analyze missing information, and connect strip diagrams to equations to solve unfamiliar problems. For example, students are given a strip diagram showing the pages Lin and Kiran read during a reading competition and respond to prompts and questions such as “Escribe una ecuación de multiplicación que compare las páginas leídas por Lin y Kiran,” and “¿Cuántas páginas en total leyó Kiran?”

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

In unit 2, students build automaticity and fluency with decimal concepts through the section A practice problems, which require repeated practice representing, comparing, and interpreting decimals using expanded notation, visual models, number lines, and place-value reasoning. Students write decimals in expanded notation, match expanded forms to decimal numbers, identify decimals represented on hundredths grids, compare decimals, and locate decimals on number lines. For example, students write 1.37 as (1 × 1) + (3 × 0.1) + (7 × 0.01), match expanded forms to decimal numbers, and identify 0.53 from a shaded hundredths grid. This repeated practice across multiple decimal representations builds fluency with decimal place value and supports grade-level work with decimals.

In unit 6, “Multiplicación y división de números de varios dígitos,” section B practice problems, students build automaticity and fluency with multi-digit multiplication through repeated practice solving multiplication problems using grade-level strategies and algorithms. For example, students solve problems such as “Hay 4,218 estudiantes en el distrito escolar A. El distrito escolar B tiene 3 veces más estudiantes que el distrito escolar A. ¿Cuántos estudiantes hay en el distrito escolar B?” Students apply multiplication strategies across multiple practice problems, using consistent procedures to solve real-world situations. This repeated, focused practice strengthens procedural fluency with multi-digit multiplication and supports grade-level computation.

Evaluation for 5.2b
Materials provide opportunities for students to practice the application of efficient, flexible, and accurate mathematical procedures within the lesson and/or throughout a unit.

In unit 2, lesson 12, students are provided practice opportunities to use estimation to support efficient and accurate subtraction of decimals. Students round decimals to the nearest whole number, estimate differences, and compare their estimates to their exact answers to determine whether their solutions are reasonable. For example, students round 23.56 to 24 and 4.96 to 5, estimate the difference as 24 − 5 = 19, and then solve the subtraction problem exactly. Through these practice opportunities, students discuss their estimation strategies with a partner and use their estimates to verify the accuracy of their computations.

In unit 6, lesson 18, “Diferentes formas de resolver problemas,” students evaluate multiple solution strategies for the same multiplication problem. Students analyze five educator-presented strategies and respond to prompts such as “¿Qué estrategia se acerca más a la tuya?” and “Con un compañero, toma turnos para explicar cómo tu estrategia se parece al póster que elegiste.” Students then discuss an alternative strategy with a partner and apply it to solve 14 × 35. By comparing, explaining, and applying different solution methods, students evaluate procedures for flexibility and efficiency within the lesson.

Evaluation for 5.2c
Materials provide opportunities for students to evaluate procedures, processes, and solutions for efficiency, flexibility, and accuracy within the lesson and throughout a unit.

In unit 1, lesson 6, students evaluate procedures and strategies for representing and locating numbers on number lines using place-value reasoning. Students compare approaches such as identifying midpoint benchmarks, using interval patterns, and reasoning about relative position to determine an efficient way to place numbers. For example, when locating 347 on a number line, students identify that 350 is halfway between 300 and 400 and use this benchmark to accurately place 347 slightly to the left of the midpoint rather than labeling every interval individually. Through discussions with peers and comparisons across number lines representing 347, 3,470, 34,700, and 347,000, students recognize that although place values increased by factors of ten, the numbers maintained the same relative position, reinforcing flexible and accurate place-value reasoning.

In unit 3, lesson 6, “Fracciones en las rectas numéricas,” students evaluate mathematical representations by comparing equivalent fractions on number lines and fraction strip models. Students analyze relationships between denominators, justify why fractions are equivalent, and determine accurate ways to represent the same value. For example, students respond to prompts such as “Explica a tu compañero por qué la marca en el extremo derecho puede etiquetarse con fracciones con diferentes números” and “Explica a tu compañero por qué las fracciones que escribiste son equivalentes.” By comparing representations and justifying their reasoning, students evaluate the accuracy and flexibility of different approaches for representing equivalent fractions.

Evaluation for 5.2d
Materials contain embedded supports for teachers to guide students toward increasingly efficient approaches.

In unit 1, lesson 8, the materials include embedded educator supports that guide students toward increasingly efficient approaches for comparing and ordering multi-digit numbers using place-value reasoning. Educator prompts encourage students to explain their reasoning, compare strategies with peers, and analyze relationships among large numbers rather than relying solely on digit-by-digit comparisons. For example, when determining that Andre’s score must be greater than 980,000,032 but less than 1,000,000,000 to place second, educators are guided to discuss why many possible nine-digit numbers fit within this range, helping students refine their understanding of numerical magnitude and intervals. Through purposeful questioning and discussion, the materials guide students toward more efficient place-value strategies for comparing and ordering large numbers.

In unit 6, lesson 10, “Multiplicación y división de números de varios dígitos,” the materials include embedded educator supports that guide students toward increasingly efficient multiplication strategies. During the lesson synthesis, educators prompt students to compare the standard algorithm and the partial products method by asking, “¿En qué se parecen estos dos métodos? ¿En qué se diferencian?” Educator guidance highlights that both methods multiply tens and ones, but the standard algorithm combines those steps into a more organized process using regrouping instead of writing each partial product separately. These embedded discussions help students recognize when a more efficient strategy can be used while reinforcing their understanding of multiplication procedures.

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

In the TX Math Teacher Course Guide K–5 ES section “Enseñanza y aprendizaje basados en problemas,” the materials explain how conceptual understanding is developed through problem-based learning while procedural fluency is built as students apply and refine strategies over time. The guide states, “Los estudiantes aprenden matemáticas haciendo matemáticas, en lugar de ver a alguien más hacer matemáticas o que les digan qué hacer. ‘Hacer matemáticas’ significa aprender conceptos y procedimientos matemáticos mientras se participa en las prácticas matemáticas . . .” including interpreting problems, reasoning abstractly and quantitatively, modeling with mathematics, and using tools strategically. By engaging in these mathematical practices, students develop both conceptual understanding and procedural fluency through active problem-solving rather than passive instruction.

The course guide explains how the conceptual and procedural emphasis of the TEKS is addressed through a coherent lesson structure consisting of four phases: “Calentamiento,” “Actividades de instrucción,” “Síntesis de la lección,” and “Enfriamiento.” Warm-ups such as “Observa y pregúntate,” “Conversación numérica,” and “¿Qué sabes sobre _____?” prompt students to explain their thinking, make connections among representations, and justify solutions. Instructional activities introduce new concepts, representations, and mathematical language while providing practice opportunities for mental computation and numerical reasoning. The course guide also explains that lessons conclude with a synthesis and independent cooldown to check student understanding of both concepts and procedures and to inform instructional adjustments.

Evaluation for 5.3b
Questions and tasks include the use of concrete models and manipulatives, pictorial representations (figures/drawings), and abstract representations, as required by the TEKS.

In unit 5, lesson 18, “Problemas sobre perímetro y área,” the materials include tasks that require students to use concrete models, pictorial representations, and abstract representations to explore perimeter, area, and measurement conversions as required by the TEKS. Students use inch tiles and measurement tools to construct and measure figures, providing a concrete model of the concepts. Students then use grid paper, visual diagrams, and drawings to represent their thinking and analyze relationships among measurements. In activity 2, students apply these representations to a real-world task involving the design of a class banner, determining how to use or partition poster paper to meet design requirements and converting between feet and inches. Students connect their models to abstract representations by writing equations, performing calculations, and explaining their reasoning to determine area, perimeter, and unit conversions.

In unit 1, lesson 8, students use abstract representations by comparing and ordering multi-digit numbers written in standard numerical form and justifying their reasoning using place-value concepts. Students also use pictorial representations, including drawings, diagrams, and place-value charts, to model and explain their comparison strategies, particularly as they revise and strengthen their explanations during “Math Language Routine 1: Cada vez más fuerte y más claro.” For example, students compare 54,000,000 and 587,000,000 by representing the values according to their place-value positions and explaining that 587,000,000 is greater because it is in the hundreds of millions, while 54,000,000 is only in the tens of millions. The lesson also provides opportunities for students to use concrete place-value representations as needed, helping them connect concrete models, pictorial representations, and abstract numerical reasoning when comparing large numbers.

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 unit 2, lesson 1, students use base 10 manipulatives, hundred grids, and shaded diagrams to build conceptual understanding of tenths, hundredths, fractions, and decimal notation. The lesson supports students in connecting concrete models, such as flats, rods, and unit cubes, with pictorial representations. The lesson also integrates abstract decimal and fraction notation through educator questioning and structured discussions. For example, students model “uno con treinta y tres centésimas” using one flat, three rods, and three unit cubes, representing the quantity on two hundred grids. Students then connect this model to the abstract forms 1.33 and 1 33/100. Embedded supports, including visual-spatial scaffolds, two-column representation charts, and prompts, assist students in reading decimals such as “one and thirty-three hundredths.”

In unit 7, lesson 2, “Puntos, rectas, rayos y segmentos,” students connect concrete, representational, and abstract understandings of geometric concepts by using line segments and rays to draw familiar two-dimensional shapes, letters, and numbers. As part of the activity, the materials state: “El propósito de esta actividad es que los estudiantes utilicen segmentos de recta y rayos para dibujar figuras bidimensionales familiares, letras y números. Dibujar sobre papel cuadriculado ayuda a reforzar la idea de que los segmentos tienen un punto final definido en ambos extremos y a distinguir los rayos de los segmentos. Al describir y comparar figuras, los estudiantes usan el vocabulario de la actividad anterior.” Through these tasks, students use grid paper to distinguish between rays and segments while describing and comparing their drawings using geometric vocabulary, supporting connections between visual models and formal geometric language.

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

In unit 1, lesson 1, “Calentamiento: ¿Cuántos ves?,” students use base 10 block representations to describe and analyze numbers greater than 1,000, reinforcing place-value concepts through visual models. The lesson incorporates language development strategies such as think time, partner discussions, and educator questions, including “¿Cuántos ves?” and “¿Cómo los ves?,” to encourage students to explain their reasoning using precise mathematical vocabulary. For example, students describe the representation of 1,325 as “1 mil, 3 centenas, 2 decenas y 5 unidades” and discuss relationships among the blocks, such as recognizing that a large cube represents 1,000 unit cubes and a flat represents 100 units.

In unit 7, lesson 5, “¿Qué es un ángulo?,” the materials provide opportunities for students to develop academic mathematical language using visuals, manipulatives, and other language development strategies to learn about geometry. In activity 3, students develop academic mathematical language through visuals, partner discourse, and geometric representations. Students identify and draw angles in their environment, including in text, graphics, and physical surroundings, and reinforce the idea of an angle as a figure formed by two rays that share an endpoint. Students use and explain vocabulary such as ángulo, rayo, vértice, recta, and punto extremo while describing figures, identifying geometric relationships, and communicating mathematical ideas.

Evaluation for 5.4b
Materials include embedded teacher guidance to scaffold and support students' development and use of academic mathematical vocabulary in context.

In unit 2, lesson 13, educators introduce a real-world budgeting scenario in which students use decimal place-value charts to select picnic items and calculate total costs. The lesson provides educator prompts, such as “¿Cómo podemos asegurarnos de mantenernos dentro de nuestro presupuesto?,” and “¿Cómo puede ayudarte ajustar el valor posicional a mantenerte dentro del presupuesto?,” to encourage students to discuss and apply vocabulary related to decimals, place value, budgeting, and precision. Students work collaboratively to explain their calculations, compare totals, and discuss how place value supports accurate addition of decimals.

In unit 5, lesson 10, “Problemas de medición de varios pasos,” educators receive guidance to scaffold and support students’ development and use of academic mathematical vocabulary related to measurement and multiplicative comparisons. In activity 1, the materials guide educators in supporting students as they extend their use of measurement vocabulary. For example, “En esta actividad, los estudiantes aplican su conocimiento de centímetros, metros y kilómetros, realizan conversiones de unidades y razonan multiplicativamente para comparar y ordenar distancias. Los estudiantes tienen la oportunidad de decidir qué unidad usar para hacer comparaciones (es decir, si convertir todas las distancias a metros, a centímetros o a kilómetros).” The materials guide educators to use “Math Language Routine 1: Stronger and Clearer Each Time,” in which students articulate their reasoning using precise mathematical language about unit relationships and multistep problem-solving.

Evaluation for 5.4c
Materials include embedded teacher guidance to support the application of appropriate mathematical language to include vocabulary, syntax, and discourse to include guidance to support mathematical conversations that provide opportunities for students to hear, refine, and use math language with peers and develop their math language toolkit over time as well as guide teachers to support student responses using exemplar responses to questions and tasks.

In unit 1, lesson 2, students work collaboratively with place-value blocks and charts to identify, label, and discuss the value of each digit in large numbers. The lesson provides educator prompts such as “¿Cuántos cientos hacen mil?” and “¿Qué dígito nos muestra cuántas decenas tenemos?” to support students in using place-value vocabulary, including miles, cientos, decenas, and unidades. Educators are guided to facilitate mathematical conversations in which students explain their thinking, discuss strategies, and refine their understanding with peers. The materials also include exemplar responses, such as “identificar que, en 4.567, el dígito 4 representa 4 unidades de millar o 4.000,” to support educators in guiding student responses.

In unit 6, lesson 12, “Situaciones que involucran grupos de igual tamaño” provides educator guidance to support students’ use of mathematical vocabulary, syntax, and discourse related to multiplication and division. In activity 1, the materials guide educators to display questions students wrote for the equation 84 ÷ 7 = ? and ask, “¿Qué representa el 84 en ambos problemas?” Educators are also prompted to invite students to share their strategies and highlight strategies that show equal-sized groups and reasoning that connects multiplication and division. Students use division vocabulary as they formulate mathematical questions, share strategies with peers, and discuss relationships between multiplication and division.

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

In unit 1, lesson 5, “Valores por diez,” the materials integrate the TEKS mathematical process standards as students explore place-value relationships and patterns. In activity 2, students engage in problem-solving by analyzing place-value relationships, constructing equations, and interpreting mathematical patterns. Students communicate mathematical reasoning through partner discussions, written explanations, and multiple representations, including equations and contextual models. For example, “En esta actividad, los estudiantes interpretan las relaciones entre los valores del mismo dígito en diferentes números y escriben ecuaciones de multiplicación y división para representar estas relaciones. A medida que completan y analizan la tabla, los estudiantes reconocen que el valor del dígito en una fila es diez veces mayor que el valor del dígito en la fila inferior. Los estudiantes trabajan para articular estas relaciones con precisión, usando palabras y ecuaciones y reciben retroalimentación de sus compañeros sobre las ecuaciones que están escribiendo. Durante la síntesis, los estudiantes conversan sobre por qué una ecuación de multiplicación o de división puede usarse para representar la misma relación.” Students use representations, explain relationships, and discuss strategies to communicate mathematical ideas.


In unit 6, lesson 17, students apply their understanding of multiplication, division, factors, and multiples to solve contextual problems involving remainders and unknown factors. Students use multiple strategies, including partial products, partial quotients, and decomposition of dividends into familiar multiples, while explaining their reasoning orally and in writing. Students interpret division situations by considering the number of groups or the size of groups and connect division situations to multiplication equations. Through mathematical discussions, “Math Language Routine 8,” written explanations, and opportunities to compare strategies, students are able to use representations to justify their reasoning and communicate mathematical ideas.

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

In the TX Math Teacher Course Guide K–5 ES section titled “Enseñanza y aprendizaje basados en problemas,” the materials provide a description of how the TEKS mathematical process standards are incorporated and connected throughout the course. For example, the course guide states, “En K–5, modelar con matemáticas es resolver problemas. Es la resolución de problemas lo que ofrece oportunidades para que los estudiantes observen, se pregunten, estimen, planteen problemas, creen representaciones, evalúen la razonabilidad y hagan revisiones continuamente, según sea necesario. En los primeros grados, estas oportunidades involucran diversas destrezas precursoras de modelado que apoyan a los estudiantes a resolver problemas de manera flexible . . . Además de las destrezas precursoras y las etapas de modelado que aparecen a lo largo de las lecciones, cada unidad culmina con una lección que aborda explícitamente estas destrezas y etapas de modelado mientras integra el trabajo matemático de la unidad.” This description explains how modeling, problem-solving, representations, and mathematical reasoning are incorporated throughout the course.

In unit 1, lesson 3, students extend their understanding of place value by reading, writing, and analyzing numbers through the billions using place-value charts and collaborative discussions. The lesson connects prior learning with larger numbers and uses multiple representations to support mathematical understanding. Educator guidance directs educators to address misconceptions related to zeros and place-value periods, facilitate mathematical discourse through targeted questions, and encourage students to explain their reasoning. The lesson demonstrates how the TEKS mathematical process standards are incorporated through representations, communication, and reasoning within the unit.

Evaluation for 5.5c
Materials include a description for each unit of how TEKS process standards are incorporated and connected throughout the unit.

The Grado 4 Alcance y Secuencia demonstrates recurring opportunities for students to engage in mathematical communication, justification, modeling, problem-solving, representation analysis, and interpretation across instructional pathways. The lesson learning targets illustrate how the TEKS mathematical process standards are integrated throughout the course. For example, “Decimales sección A: valor posicional decimal” describes how students interpret tenths and hundredths in decimal notation using unit square grids. Another example is “Suma y resta de fracciones sección A: sumar fracciones,” where students recognize that a fraction can be decomposed into a sum of fractions with the same denominator and write equations to represent fraction decomposition.

In unit 2, Guía de planificación a nivel de sección, the narratives for sections A and B describe how students engage in reasoning, problem-solving, and mathematical communication throughout the unit. In section A, students use hundred grids, unit squares, and number lines to connect fraction and decimal notation, compare and order decimals, and justify equivalencies such as 0.10 = 0.1. In section B, students apply place-value reasoning, estimation, and standard algorithms to add, subtract, and round decimals while evaluating the reasonableness of their solutions. The section narratives describe how students use representations, analyze mathematical relationships, and communicate their reasoning across the unit.

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

In the TX Math Teacher Course Guide K–5 ES section titled “Qué hay en una lección de matemáticas de Kiddom,” the materials provide educators with an overview of how the TEKS mathematical process standards are incorporated into lessons. The lesson structure includes opportunities for students to explain reasoning, analyze relationships, and communicate mathematical ideas. For example, in the instructional routines section, the materials state, “Conversación numérica - La secuencia de problemas en una conversación numérica anima a los estudiantes a buscar estructura y utilizar razonamientos repetidos para evaluar expresiones y desarrollar fluidez computacional. A medida que los estudiantes comparten sus estrategias, conexiones y construyen sobre las ideas de los demás, desarrollan comprensión conceptual.” These routines are embedded within lessons and describe how students engage in mathematical processes during instruction.

In unit 1, lesson 4, “Enfriamiento: el valor de los dígitos,” students write numbers in expanded notation and explain relationships between digit values, incorporating the TEKS mathematical process standards through representation, reasoning, and mathematical communication. Students use symbolic representations to decompose numbers and analyze how the value of a digit changes based on its position, connecting mathematical ideas through place-value relationships. The lesson provides opportunities for students to communicate mathematical ideas and justify their reasoning through written explanations. Educator guidance recommends small-group discussions before the cooldown and provides examples of expected responses, supporting students as they explain strategies, refine their reasoning, and communicate mathematical thinking through the TEKS mathematical process standards.

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

In unit 6, lesson 19, activity 1, “Dos verdades y una mentira,” the materials provide opportunities for students to think mathematically and persevere through problem-solving by analyzing multiple problem situations and determining whether statements are true or false. Students solve each situation using strategies, collaborate with peers, and explain their mathematical reasoning. The materials provide guidance for supporting productive struggle through a structured problem-solving process: “Desarrollar el esfuerzo y la perseverancia. Divide esta tarea en partes más manejables. Por ejemplo, invita a los estudiantes a realizar los siguientes pasos: selecciona una situación para comenzar, lee el problema en voz alta y visualiza la situación, representa la situación con un dibujo matemático o un diagrama de cinta, escribe una ecuación para representar el problema, resuelve o estima la solución de la ecuación. Revisa con los estudiantes para proporcionar retroalimentación y motivación después de cada parte.”

In unit 8, lesson 8, “Comprender los gastos: una mirada a los costos fijos y variables,” the materials provide opportunities for students to make sense of mathematics by applying concepts of fixed and variable expenses to real-world financial situations. In activity 2, students collaboratively create a budget, predict future expenses, and allocate funds based on given information. The materials state, “En esta actividad, los estudiantes aplicarán su comprensión de gastos fijos y variables para crear un presupuesto para un escenario dado. Predecirán gastos futuros basándose en gastos anteriores y asignarán el dinero en consecuencia.” The activity requires students to reason through financial scenarios, develop representations, and apply mathematical understanding to solve problems.

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

In unit 1, lesson 13, activity 2, “Propiedades mágicas de la multiplicación,” students apply the properties of multiplication to identify strategies for simplifying calculations using multiples of 10 and 100. Students work collaboratively to discuss and justify their approaches using mathematical language. The educator guidance states: “En esta actividad, los estudiantes utilizarán las propiedades distributiva, asociativa y conmutativa de la multiplicación para resolver problemas que implican multiplicar números por 10 y 100. Trabajarán con un compañero para conversar sobre los problemas y resolverlos, utilizando las rutinas de lenguaje matemático para apoyar su aprendizaje.” The materials also prompt students to explain how strategies can simplify multiplication by asking, “¿Cómo puede descomponer números simplificar el proceso de multiplicación? Pregunta cómo cada propiedad podría hacer que el cálculo sea más fácil.”


In unit 7, lesson 17, “Formas de observar los cuadriláteros,” the materials support students in representing, explaining, and justifying mathematical ideas by comparing attributes, examples, and non-examples of quadrilaterals. In activity 3, students work with a partner to identify and communicate different ways to describe geometric attributes. The materials state, “En esta actividad opcional, los estudiantes trabajan con un compañero para practicar nombrar y buscar ciertos atributos en cuadriláteros. Cada compañero tiene la oportunidad de seleccionar un atributo particular que podría tener un cuadrilátero y hallar ejemplos y no ejemplos. Su compañero debe deducir el atributo que eligió basándose en los ejemplos y no ejemplos.” The activity requires students to explain their reasoning and use evidence from examples and non-examples to justify their conclusions.

Evaluation for 6.1c
Materials are designed to require students to make sense of mathematics through multiple opportunities for students to do, write about, and discuss math with peers and teachers.

In grade 4, unit 1, lesson 2, activity 1, “Representar milésimas en un plano,” the materials are designed to require students to make sense of mathematics through multiple opportunities to do, write about, and discuss mathematical ideas with peers and educators. Students represent thousandths by shading decimal grids and writing corresponding decimal numbers. Students also create their own representations of decimals such as 0.328, demonstrating their understanding through drawings and written responses. Throughout the lesson, educator prompts such as “How does the diagram represent the number you wrote?” and “How does the diagram represent each digit in the number?” require students to explain the connection between visual representations and decimal place value. Partner discussions provide opportunities for students to communicate mathematical ideas using precise vocabulary, including tenths, hundredths, thousandths, and decimals. Students engage in hands-on modeling, written responses, and mathematical discourse as they represent, explain, and discuss their understanding of decimal place value.

In grade 4, unit 5, lesson 17, “Exploración del área del rectángulo,” the materials provide opportunities for students to make sense of mathematics through activities that require students to explore, represent, write about, and discuss mathematical ideas. Students investigate the attributes of rectangles using classroom objects, rulers, and grid paper before representing their thinking by counting square units and applying the area formula. Students record observations and solutions to connect visual models with mathematical procedures for finding area. Throughout the lesson, students participate in partner and whole-class discussions to explain relationships between rectangle attributes and area calculations and compare solution methods. Reflection questions, such as “¿Qué parte de la lección de hoy te ayudó más a entender el área de un rectángulo?” and “¿Qué tan cómodo te sientes calculando el área usando tanto el conteo como el método de la fórmula?” prompt students to communicate their reasoning and reflect on their mathematical understanding.

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

In grade 4, unit 2, lesson 12, warm-up, “Conversación numérica: redondear números,” the materials support educators in guiding students to share and reflect on their problem-solving approaches through a structured number talk. The lesson provides educator guidance to allow silent think time, invite students to share their reasoning, record multiple strategies, and keep student work visible during discussions. Students explain their mental math strategies for finding missing addends, such as determining 645 + 355 = 1,000, and justify how place value and benchmark numbers support their reasoning. Educator prompts such as “¿Qué tienen en común todas las ecuaciones?” and “¿Es 1,000 el múltiplo de 100 más cercano a 645?” support educators in facilitating student explanations, comparisons of strategies, and justification of mathematical thinking.

In unit 8, lesson 13, “Explorando el papel de las instituciones financieras,” the materials support educators in guiding students to share and reflect on problem-solving approaches related to data and personal finances. In activity 2, “Navegando la responsabilidad financiera: las implicaciones de pedir dinero prestado,” educator guidance includes prompts that encourage students to explain and justify their reasoning when analyzing financial decisions. The materials state, “Invita a los estudiantes a pensar en cómo diferentes tasas de interés o duraciones de los préstamos podrían cambiar la cantidad a pagar. Pídeles que consideren el impacto a largo plazo de manejar los préstamos de manera responsable.” The activity provides opportunities for educators to facilitate partner discussions, group sharing, and reflection as students explain their approaches, evaluate outcomes, and justify their conclusions.

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

In grade 4, unit 1, lesson 2, “Expandiendo números,” the materials include prompts and guidance to support educators in providing explanatory feedback based on student responses and anticipated misconceptions about place value. The lesson includes reflection questions such as “¿Demostraron los estudiantes comprensión del valor posicional de los dígitos en números grandes?,” “¿Fueron los estudiantes capaces de convertir números enteros en notación desarrollada con éxito?,” and “¿Cómo puedo apoyar más a los estudiantes que tienen dificultades para reconocer la importancia de la posición de cada dígito?,” to help educators analyze student understanding and adjust instruction. Students use place-value cubes, charts, and number cards to represent whole numbers up to 1,000,000,000 and convert them into expanded notation. When students demonstrate misconceptions about digit value or place-value position, the materials guide educators to use models and questioning strategies to help students connect representations to the value of each digit.

In unit 7, lesson 16, “Descubrimiento del tipo de triángulo,” the materials include prompts and guidance to support educators in providing explanatory feedback based on student responses and anticipated misconceptions about classifying triangles. In activity 2, students work with a partner to classify triangles by their angles and explain their reasoning. The materials state, “En esta actividad, los estudiantes trabajarán con un compañero para clasificar triángulos en categorías según sus ángulos y explicar su razonamiento.” Educator guidance includes questioning strategies, visual supports, and discussion prompts to address misunderstandings about triangle classification and support students in clarifying and extending their mathematical reasoning.