Kiddom Texas: Matemáticas en Español – Grado 6

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

In unit 1, “Factores y múltiplos,” lesson 1, “Escribe expresiones donde las letras representan números,” is correlated to TEKS 6.7A. The lesson objectives are to explain orally how to create and solve an equation representing a situation with an unknown quantity and to write an expression with a variable to generalize relationships between quantities in a situation.

The scope and sequence includes a total of 13 units with associated ELPS and concepts identified for instruction. For example, in unit 3, lesson 1, “Figuras y sólidos,” the ELPS listed are 1.B and 2.C. The concepts include comparing (orally) the areas of figures that form a geometric pattern and understanding that the word area (orally and in writing) refers to how much of a plane a figure covers.

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 6 materials include suggested pacing to support three instructional calendars for 165, 180, and 210 days. The calendar document includes lessons and activities for a full year of instruction across three instructional calendars. For example, in the 165-day calendar, unit 1, “Factors and Multiples,” includes five lessons allotted seven days. In the 180-day calendar, unit 1 is allotted nine days. In the 210-day calendar, unit 1 is allotted 10 days.

Educators can access suggested pacing guides to review the standards addressed throughout the year and the time allocated to mathematical concepts, as well as suggested implementation calendars with options for varying numbers of instructional days. For example, unit 1, “Factores y múltiplos,” is allotted seven school days in weeks one and two. Unit 2 is allotted 13 school days in weeks three and four.

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.

In the Guía de recursos para maestros, the “Progressión coherente” section states, “Las narrativas de grado, unidad, lección y actividad describen decisiones sobre la organización de ideas matemáticas, conexiones con trabajos previos y futuros del grado, y el propósito de cada lección y actividad.” This guidance helps educators understand how mathematical concepts build over time.

In the Guía de recursos para maestros, the section titled “Enseñanza y aprendizaje basados en problemas” states that the approach of the program is to “brinda a los estudiantes acceso a las matemáticas a través de una progresión coherente.” This supports educators in understanding how mathematical ideas are intentionally sequenced to build conceptual understanding over time.

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

The materials include processes to support teachers in thoroughly understanding and preparing to teach each unit. The Guía de planificación provides an overview of key learning objectives and instructional sequencing. This guidance helps ensure instruction is aligned to learning goals while also supporting educator content knowledge. For example, in unit 6, the guide states, “En esta sección, los estudiantes exploran unidades de medida de longitud, volumen, peso y masa. Comienzan comparando unidades estándar como pulgadas y kilogramos, adquiriendo una sensación intuitiva de sus tamaños mediante actividades prácticas.”

The materials include protocols with corresponding guidance for lesson internalization by identifying the mathematical purpose, instructional sequence, student grouping, materials, educator monitoring expectations, and supports for learners in unit 13, lesson 7, activity 3. The activity guidance explains that students solve two proportional-allocation problems involving school advisors and connect the mathematics to prior work with distributing computers and assigning congressional representatives. Educators are guided to organize students in groups of two to four, provide calculators as needed, introduce the context, and monitor the strategies students use to assign advisors fairly. The lesson includes targeted supports for students with disabilities and multilingual learners, including breaking the task into manageable parts and using “Math Language Routine 2: Recopilar y mostrar” to capture and display mathematical language such as proporcional, tasas, and _____ por _____.

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

The materials do not 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 educators. 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.

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 educators. 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.

The materials include comprehensive unit overviews that provide the background content knowledge necessary to effectively teach the concepts in the unit. For example, in unit 4, lesson 3, “Métodos para multiplicar decimales,” the “Narrativa de la lección” explains how whole numbers and fractions connect to multiplying decimals. The materials state, “En grados anteriores, los estudiantes usaron el área de rectángulos para representar gráficamente y encontrar productos de números enteros y fracciones. Aquí hacen lo mismo para representar gráficamente y encontrar productos de decimales.”

The materials include comprehensive unit overviews that provide academic vocabulary necessary to teach the concepts in the unit. For example, in grade 6, unit 1, “Factores y múltiplos,” the “Narrativa de la unidad” provides a list of key terms with academic vocabulary necessary to teach the concepts in the unit. The section “Términos clave: descomponer” provides a definition of the word, stating, “Descomponer significa ‘separar’. Usamos la palabra descomponer para describir el acto de separar una figura para crear más de una nueva figura.” By explicitly identifying, defining, and providing examples of these key mathematical terms, the materials support students in developing and using the academic vocabulary needed to access the unit content.

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 grade 6 materials include family support resources in both English and Spanish for each unit to help families support student learning at home. For example, the grade 6 Materiales de apoyo familiar, unit 3, “Figuras y sólidos,” introduces key concepts students will learn and states, “Los estudiantes aprenderán que el área es la medida del espacio que cubre una figura y explorarán este concepto a través de actividades prácticas que implican componer y descomponer figuras.” This information helps families understand how students will develop foundational ideas about area through hands-on experiences.

The grade 6 materials include family support resources that emphasize skill development and application across units. For example, the grade 6 Materiales de apoyo familiar, unit 6, “Porcentajes, razones unitarias y conversiones,” provides explanations in both English and Spanish of upcoming learning goals and ways families can reinforce practice at home. The materials include the following information: “Los estudiantes aprenderán a comparar y convertir entre diferentes unidades de longitud, volumen, peso y masa usando razones.” This guidance supports families in understanding how students apply proportional reasoning to measurement conversions.

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).

The materials include comprehensive, structured, and detailed lesson plans that incorporate daily objectives, questions, tasks, materials, and instructional assessments required to meet the content standards of the lesson and that are aligned with the TEKS. In unit 1, lesson 1, activities 1–3, one of the cooldown questions states, “Marca la expresión que representa la cantidad de pulgadas que la planta creció esta semana. Explica cómo lo sabes.” This question aligns with TEKS 111.26.b.7.d, which focuses on generating equivalent expressions using properties of operations such as inverse, identity, commutative, associative, and distributive properties.

In unit 4, lesson 3, “Métodos para multiplicar decimales,” the lesson plan provides explicit guidance for teaching the objective, “Interpret different methods for calculating the product of decimals and evaluate (orally) their usefulness.” To support this objective, students are asked to analyze and compare solution strategies through questions such as, “Analiza los dos métodos, luego conversa estas preguntas con tu compañero. ¿Qué método te parece más razonable? ¿Por qué?” The materials guide educators in facilitating student discussions about the efficiency and reasonableness of different methods, as well as having students explain how Elena and Noah could each calculate 0.16 × 0.03 and determine whether both approaches produce the same result. This lesson aligns with TEKS 111.26(b)(3)(E), which requires students to fluently multiply and divide positive rational numbers while developing conceptual understanding and mathematical reasoning.

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.

The materials include a lesson overview that identifies the educator and student materials needed to effectively deliver instruction. For example, in unit 3, lesson 4, “Paralelogramos,” the “Materiales requeridos” section lists the necessary resources, including herramientas de geometría. In this lesson, students use unidades cuadradas as a geometry tool to solve the problem, “Halla el área de cada paralelogramo. Muestra tu razonamiento.”

The materials include a lesson overview listing the suggested timing for each lesson component. In unit 1, lesson 1, “Escribir expresiones donde las letras representan números,” the “Cronograma de la lección” specifies the time allotted for each activity and component of the lesson. The embedded table indicates that activity 1 takes five minutes, and activities 2 and 3 take 15 minutes.

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

In unit 3, lesson 3, “¡Locura de rectángulos!,” the lesson narrative provides guidance for using an optional activity as extended practice to deepen students’ understanding of fractions and greatest common factors. The materials explain that the activities build on one another and provide students opportunities to connect the greatest common divisor of two numbers to related fractions through repeated reasoning. The guidance states, “Las actividades de esta lección se construyen unas sobre otras, brindando a los estudiantes la oportunidad de expresar la relación entre el máximo común divisor de dos números y las fracciones relacionadas a través del razonamiento repetido. Por lo tanto, las actividades deben realizarse en orden.” The materials also provide pacing guidance by noting that completing all activities may require multiple class periods and that educators may determine how much time to dedicate based on instructional needs. The narrative explains that the lesson is an optional opportunity for students to go deeper into previously learned standards and make connections across mathematical domains.

In unit 11, lesson 6, “Educación vs. ingresos: traza tu camino profesional,” the materials provide guidance for using lesson activities as extended practice by connecting mathematical concepts to authentic career and financial planning contexts. In activity 3, “El impacto de una carrera elegida en las metas financieras de por vida,” students research a career of interest, analyze education and training requirements, calculate projected earnings over 40 years, and connect their findings to the financial education plan developed in the previous lesson. Educator guidance includes structured discussion prompts, monitoring suggestions, and the implementation of “Math Language Routine 8: Apoyos para la conversación” to support students in explaining their reasoning, reflecting on financial implications, and revising their plans. For example, educators are prompted to ask questions such as, “¿Qué pasa si esta carrera no nos permite alcanzar nuestras metas financieras?” and “¿Qué requisitos hay para la educación y formación vocacional que no conocías? ¿Cómo ajustarás tu plan?”

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 6, the materials include a variety of unit-level formative assessments that use diverse task types and question formats. For example, the “Grado 6.2 Verifica tu preparación (A)” assessment includes multiple-select items, written responses, ordering tasks, drawing activities, and open-ended questions that require students to explain their reasoning and represent their mathematical thinking. These tasks assess a range of prerequisite skills, including fraction comparison, fraction operations, ordering rational numbers, and solving contextual problems. For example, question 2 combines a multiple-select item with a written explanation to assess students’ understanding of multiplication and division situations involving fractions.

In grade 6, assessments incorporate conceptual reasoning prompts, problem-solving tasks, and real-world application questions that elicit students’ prior knowledge and understanding. For example, the “Grado 6.6 Verifica tu preparación (A)” assessment in unit 6 includes the prompt, “Las cucharadas y litros son algunas unidades que se usan para medir el volumen. ¿Cuáles son otras unidades que se usan para medir el volumen?” This open-ended question requires students to draw on their prior knowledge of measurement units, helping educators assess readiness for upcoming learning related to volume and measurement.

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 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.”

In unit 9, lesson 4.1, “Calentamiento: conversación numérica,” is identified as a formative assessment designed to elicit student thinking and promote mathematical discourse. Additionally, the math course guide includes an “Assessment Guide” that defines the purpose of diagnostic assessments. The guide states, “Cada unidad comienza con Verifica tu preparación, una evaluación diagnóstica de los conceptos y destrezas previas a la unidad, así como algunos ítems que evalúan lo que los estudiantes ya saben sobre algunos contextos y conceptos clave abordados en la unidad.” This explanation identifies “Verifica tu preparación” as a diagnostic assessment intended to measure students’ prior knowledge and readiness for the upcoming unit.

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

The materials include educator guidance that supports the administration of instructional assessments. 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 (written), specifies that students complete the assessment individually, and describes its use within each unit, supporting consistent administration across classrooms.

The materials provide overviews of assessments, outline the time to administer each task, and provide step-by-step guidance for administering each component of the assessment, including information to support educators in understanding the assessments and ensuring consistency and standardized administration across examiners. For example, the “Guía de la evaluación” provides an overview of the various types of assessments educators will administer throughout each unit. The materials also provide educators with guidance on how formative assessments are used. The materials state, “Usa evaluación formativa para proporcionar retroalimentación o para planificar instrucción adicional.”

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

In grade 6, the materials include summative assessments that align with the objectives and learning goals of each unit. For example, one of the instructional goals in unit 6 is, “desarrollar y aplicar la comprensión de las razones unitarias para comparar, analizar y resolver problemas que involucren relaciones proporcionales.” The summative assessment for the unit directly aligns with this goal by requiring students to apply proportional reasoning in a real-world context. In “Grade 6.6 Evaluación final de la unidad (B),” question 3 asks, “Hay 9 niños en una clase que toman el autobús a la escuela, y hay 15 niños en total en la clase. ¿Qué porcentaje de los niños toma el autobús a la escuela?” This item assesses students’ ability to reason about ratios, proportions, and percentages, demonstrating alignment with the unit’s instructional focus.

The 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.

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

In grade 6, assessments incorporate procedural tasks, opportunities to apply multiple skills in problem-solving situations, and items that require conceptual reasoning and interpretation of mathematical representations. For example, in unit 12, “El proceso estadístico y los resúmenes numéricos de datos,” the “Grado 6.12 Evaluación final de la unidad (A)” includes TEKS-aligned items that assess students’ ability to analyze and interpret statistical data. One item asks, “Este diagrama de caja muestra información sobre la cantidad de mensajes de texto que algunos estudiantes enviaron en un día. ¿Cuál es el número mediano de mensajes de texto enviados por los estudiantes?” This task requires students to interpret a box plot and apply their understanding of measures of center, demonstrating a level of complexity beyond simple procedural computation.

In grade 6, the “Grado 6.7 Verifica tu preparación (A)” assessment includes TEKS-aligned items that evaluate prerequisite skills related to number sense, rational numbers, distance, and coordinate graphing. The assessment features tasks at varying levels of complexity, beginning with foundational concepts such as comparing fractions and decimals on a number line and advancing to the interpretation of distances in real-world situations and the identification of points on the coordinate plane. More complex items require students to apply mathematical reasoning, represent and label coordinates in contextual scenarios, and explain their thinking through written responses and visual representations. The use of multiple-select, fill-in-the-blank, constructed-response, and drawing tasks provides students with multiple ways to demonstrate understanding while assessing a range of cognitive demands.

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

The materials support educators in analyzing assessment results and determining students’ levels of understanding and proficiency. For example, in unit 3, “Figuras y sólidos,” the “Grado 6.3 Evaluación final de la unidad (B)” includes scoring guidance that helps educators evaluate the depth of students’ conceptual understanding. One assessment item asks, “Lin dibuja este diagrama de cinta para 4 ÷ 3/4. Lin dice que 4 ÷ 3/4 = 5 1/4 porque hay 5 grupos de 3/4 y queda 1/4. ¿Estás de acuerdo con Lin?” The scoring rubric includes leveled sample responses titled, “Respuesta mínima de Nivel 1,” “Respuesta de Nivel 2,” and “Respuesta de Nivel 3.” This scoring information provides criteria for distinguishing varying levels of student understanding and enables educators to interpret student responses accurately.

In grade 6, the assessment “Grado 6.2 Verifica tu preparación (A)” includes educator notes that help identify student misconceptions and instructional next steps. The assessment provides guidance for interpreting results by stating, “Si la mayoría de los estudiantes tiene dificultades con este ítem, planea apoyarlos en su comprensión de fracciones representando visualmente las fracciones en este problema . . .” This guidance helps educators analyze student responses, identify areas of misunderstanding, and determine appropriate instructional support.

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

In unit 6, “Verifica tu preparación (B),” the materials provide guidance for using included tasks and activities to respond to student trends in performance on assessments related to percents, unit rates, and conversions. The diagnostic assessment identifies prerequisite knowledge and connects assessment results to specific lessons and activities for targeted intervention. Educator guidance explains the concept measured by each assessment item, identifies common misconceptions, and recommends specific instructional responses based on patterns in student performance. For example, when most students demonstrate difficulty dividing to produce decimal quotients, the materials recommend revisiting practice problems from lessons 1–3, using lesson 4, activity 3 for additional contextual practice, and incorporating estimation warm-ups to help students evaluate the reasonableness of their answers. Similarly, guidance for students who struggle with fraction-to-decimal conversions directs educators to lesson 7, activity 1 to strengthen connections between fractions and decimals before applying these concepts to unit rates and scale.

Formative assessments, such as “Check Your Readiness” and cooldowns, include recommendations for targeted support based on student performance within the “Note for Evaluating Responses” and “Responding to Student Thinking” sections. These recommendations provide specific guidance for where and how to use included lessons and activities to support students’ development of essential course skills. For example, in grade 6, unit 2, “Verifica tu preparación (A),” the materials state, “Si la mayoría de los estudiantes tienen dificultades con este ejercicio, planifica usar la Lección 2 para apoyar a los estudiantes en la representación y razonamiento sobre situaciones de multiplicación y división. Después de compartir sus respuestas de la Actividad 1, pide a los estudiantes que piensen en una historia para esta expresión y que la comparen con las respuestas compartidas.” This guidance explicitly connects assessment results to specific follow-up instruction within the materials and directs educators to use included lessons and activities to address student needs.

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 2, lesson 1, “Tamaño del divisor y tamaño del cociente,” the materials include educator guidance for differentiated instruction by providing targeted scaffolds for students who struggle with comparing and ordering division expressions. The materials recommend breaking the task into smaller parts, beginning with fewer cards or simpler examples, and gradually increasing complexity as students build understanding. Educators are guided to prompt students to compare two quotients at a time and use previously learned patterns about the relationship between dividends, divisors, and quotients rather than relying solely on computation. Additional supports, including visible vocabulary references, collaborative grouping, and extension activities, help students progress toward grade-level proficiency while engaging with the same mathematical concepts as their peers.

In unit 6, lesson 4, “Comparemos velocidades y precios,” educators support students who are still developing understanding of percentages, unit rates, and conversions. The materials state, “Selecciona a estos estudiantes para que compartan su estrategia primero. Mantén estas representaciones visibles . . . Resalta el uso de la división para calcular el precio por lata y el uso del lenguaje ‘por 1’.” This guidance directs educators to have students share visual models such as tables and double number lines, keep these representations visible during discussion, and explicitly connect them to more efficient strategies for calculating unit rates.

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 6, the materials include pre-teaching and embedded supports for unfamiliar vocabulary. The guidance includes sentence frames aligned to unit 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 6, lesson 13, “Resolver problemas de porcentaje,” educators use sentence frames to support students in explaining percentage-based problem-solving. For example, the materials state, “Invita a los estudiantes a hablar sobre sus ideas con un compañero antes de escribirlas. Proporcionar estructuras de oraciones para ayudar a los estudiantes a explicar su estrategia. Por ejemplo: ‘Primero, yo _____ porque . . .’, ‘Observé _____ así que . . .’ y ‘¿Cómo lo conseguiste?’” The materials guide educators to have students discuss ideas with a partner before writing and to use sentence frames to support explanation of their strategies.

In grade 6, the materials include pre-teaching and embedded supports for academic language in text. For example, in the beginning of unit 7, lesson 9, educators are guided to introduce the coordinate plane with four regions by explaining that, just as the number line was extended to include negative numbers, both axes of the coordinate plane can also extend to include negative coordinates. The guidance instructs educators to use the term cuadrante to describe the four regions of the coordinate plane and suggests supporting vocabulary development by explaining that the prefix cuad- means “four,” with examples such as cuádriceps and cuadrilátero. Students are then given 10 minutes to work followed by a group discussion. This guidance supports pre-teaching of key vocabulary and connects mathematical language to familiar linguistic patterns to support understanding of coordinate concepts.

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.

In unit 1, lesson 5, “Usar múltiplos comunes y factores comunes,” the materials provide educator guidance for differentiated instruction, enrichment, and extension activities for students who have demonstrated proficiency with grade-level content and skills. In “Actividad 5.4: Bingo de múltiplos y divisores,” the materials direct educators to monitor student responses, address misconceptions, and facilitate discussions that require students to justify their reasoning and explain how numbers meet specific criteria. The activity provides enrichment opportunities through higher-level questions that prompt students to analyze whether the game involves luck, strategy, or both, and to evaluate which numbers are most advantageous to include on a bingo board. Students who demonstrate proficiency extend their mathematical thinking by examining number properties, comparing strategies, and defending their conclusions using precise mathematical language. For example, students are asked, “Are some numbers better to have on a game board than others? Explain how you know,” which challenges them to analyze characteristics of composite numbers, factors, and multiples and justify their reasoning beyond basic identification skills.

In unit 6, lesson 7, “Más sobre la velocidad constante,” the materials provide educator guidance for differentiated instruction and include enrichment opportunities for students who demonstrate proficiency with grade-level content and skills. In activity 7.3, students determine Jada’s and her cousin’s individual speeds and extend their thinking by calculating their combined rate of 12.5 miles per hour as they move away from each other. Students who demonstrate proficiency are further challenged to compare different unit rates, including minutes per mile, miles per minute, and miles per hour, and justify which rate is most useful for answering questions in the given context. These extension opportunities encourage students to apply their understanding of unit rates in more complex situations and deepen their mathematical reasoning.

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 2, lesson 3, “Interpretemos situaciones de división,” the materials provide explicit educator guidance for modeling the relationship between multiplication, division, and visual representations. Educators are directed to monitor student work, select examples, and sequence diagrams from concrete to abstract representations to build understanding. In “Actividad 3.2: Mermeladas caseras,” there are discussion prompts that help educators explain the meaning of tape diagram components and connect them to multiplication equations. Educators are also guided to make division relationships explicit by connecting the diagrams and equations to division situations. For example, educators ask questions such as, “What does the ‘?’ represent?” and “How do the three parts of each multiplication equation relate to the diagram?” to model connections between visual representations and mathematical operations.

The materials include support for instruction through visuals that aid educator explanations. For example, in unit 6, lesson 3, “Convertir unidades,” educators use visual representations to support students’ understanding of unit conversion in real-world contexts. The materials state, “Muestra la imagen del enunciado de la tarea para que todos la vean . . . di a los estudiantes que es una señal de tránsito que podrían ver mientras conducen . . . Esta es una señal que podrían ver mientras conducen en Canadá u otro país que usa el sistema métrico.” This guidance directs educators to display a real-world image, help students interpret its meaning, and connect the example to the use of metric units in different countries.

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

In grade 6, the materials include educator guidance and recommendations for effective lesson delivery and facilitation using a variety of instructional approaches, including math stations to reinforce learning. For example, in unit 6, lesson 2, “Medir con unidades de diferentes tamaños,” educators use station rotations to help students practice measuring with different units. The materials state, “Estación 1: volumen de una caja . . . Estación 2: longitud . . . Estación 3: volumen de agua . . . Estación 4: pesos y masas de 2–3 objetos . . . Estación 5: volumen de sal . . .” This guidance directs educators to engage students in multiple measurement stations where they measure and compare quantities using both customary and metric units, including volume, length, and mass.

In grade 6, the materials use a variety of instructional strategies, including think-pair-share. For example, in unit 7, lesson 4, “Comparar velocidades y precios,” educators use partner discussions to help students generate and discuss mathematical questions about speed. The materials state, “Muestra la velocidad constante de Tyler, Kiran y Mai y pide a pares de estudiantes que escriban posibles preguntas matemáticas sobre la situación . . . Luego, invita a pares seleccionados a compartir sus preguntas con la clase . . . Finalmente, revela las preguntas reales en las que se espera que trabajen los estudiantes.” This guidance supports educators in having students work in pairs to generate questions about comparing speeds, share their ideas with the class, and transition into the lesson’s problem-solving task.

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.

In unit 1, lesson 5, “Actividad 5.2: Múltiplos y divisores,” the materials support multiple types of practice by engaging students in collaborative group work, guided whole-class discussion, and independent reasoning across a sequence of structured problems. Students first work in pairs to solve problems involving multiples, factors, least common multiple, and greatest common factor, then transition to independent classification and reflection on problem types. The materials include educator guidance for implementation, recommending 15 minutes of partner work followed by whole-class discussion to synthesize understanding. For example, in the “Fiesta” problem, students work with a partner to determine quantities of cups and plates using packages of eight and six, then explain whether the situation involves common multiples or common factors. The lesson concludes with a whole-class discussion where selected students share their reasoning and the class compares strategies and organizes responses in a shared table to consolidate understanding.

In grade 6, the materials support guided practice and include guidance for educators as well as recommended instructional structures such as whole-group work to support effective implementation. For example, in unit 7, lesson 3, “Comparar números positivos y negativos,” educators model how temperatures relate to positive and negative numbers and facilitate discussion about comparing signed numbers. The materials state, “Pide a los estudiantes que expliquen cómo podrían decir cuál de los dos números negativos es mayor en el problema 2. Después de recopilar 1 o 2 respuestas, muestra la siguiente imagen para que todos la vean.” This guidance directs educators to elicit student reasoning about comparing negative numbers, gather student responses, and use visual representations to support whole-class discussion.

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.

In grade 6, educators are provided with implementation guidance that supports the effective use of the materials with emergent bilingual (EB) students in bilingual/ESL settings while maintaining alignment with state standards. For example, in unit 7, lesson 1, “Números positivos y negativos,” the materials incorporate “Math Language Routine 6: Tres lecturas” and provide explicit educator directions for implementing the routine. The materials state, “Usa esta rutina para apoyar la comprensión de los estudiantes sobre los cambios de temperatura . . . . En la primera lectura . . . . En la segunda lectura . . . . En la tercera lectura . . . .” This guidance helps educators structure a three-read routine in which students first make sense of the context, then identify relevant quantities, and finally determine strategies for solving problems involving positive and negative numbers, supporting both mathematical understanding and language development for EB students in bilingual/ESL instructional settings.

In unit 7, lesson 2, “Puntos en la recta numérica,” the materials use “Math Language Routine 8: Apoyos para la conversación” to support academic discussion. The language routine states, “Para apoyar a todos los estudiantes a participar en la discusión al final de la clase, proporciona un marco de oración como _____. Estoy de acuerdo con _____ porque observo . . . Haz que los estudiantes compartan su respuesta con un compañero antes de la discusión grupal.” This guidance directs educators to provide sentence frames for academic discussion, structure partner sharing prior to whole-class discussion, and prompt students to justify their reasoning when interpreting values on a number line.

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 unit 4, lesson 1, “Usar decimales en un contexto de compras,” the materials include supports for building background knowledge through oral and written discourse. In activity 2, “Planificar una cena,” the materials provide guidance for leveraging students’ prior experiences with budget planning for a dinner party. For example, under “Desarrollo de conocimientos previos,” the materials state, “Qué supone la actividad: El estudiante necesita poder leer un folleto de supermercado—precio por unidad, precio por libra, ofertas por cantidad—y reconocer las categorías del menú (plato principal, guarnición, postre).” The materials require students to activate prior knowledge by discussing and writing about the total budget and cost per guest. Within the lesson, students develop background knowledge to make sense of decimals through the party planning scenario.


In unit 12, lesson 15, “Cuartiles y rango intercuartílico,” the materials include supports for cross-linguistic connections in oral and written discourse through learning routines so that students learn about statistical processes and numerical data summaries. In “Calentamiento: Observa y pregúntate: dos fiestas,” the materials provide “Conexiones translingüísticas.” The materials state, “Palabras que se transfieren: Casi todo lo demás, con los patrones -ción → -tion y -dad → -ty: distribución / distribution, variación / variation, desviación / deviation, variabilidad / variability. Añada absoluta / absolute, rango / range, datos / data, cuartil / quartile, intercuartílico / interquartile, diagrama / diagram.” In summary, the materials state, “Transferable words: Almost everything else, following the patterns -ción → -tion and -dad → -ty: distribución / distribution, variación / variation, desviación / deviation, variabilidad / variability. Add absoluta / absolute, rango / range, datos / data, cuartil / quartile, intercuartílico / interquartile, diagrama / diagram.”

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 through the “Evaluación final de la unidad” that require students to demonstrate understanding aligned to the TEKS by representing and interpreting relationships between variables using multiple representations. The assessment includes a variety of practice opportunities through question types such as fill-in-the-blank, drawing, short-answer, and written response, requiring students to apply their understanding of tables, graphs, and equations in different contexts. For example, question 1, part d asks students, “Sea ‘S’ el número de tazas de agua con gas y ‘J’ el número de tazas de jugo de manzana. Escribe una ecuación que muestre cómo están relacionados ‘S’ y ‘J’.” Students write an equation that represents the relationship between independent and dependent quantities, demonstrating their ability to analyze and communicate connections among variables. This aligns with TEKS 111.26.b.6.B, which requires students to write equations representing relationships between independent and dependent quantities from a table.

The materials provide practice opportunities through unit 12, “El proceso estadístico y los resúmenes numéricos de datos,” in “Evaluación intermedia de la unidad (A)” that require students to demonstrate depth of understanding aligned to the TEKS through analysis, representation, and justification. The instructional assessment requires students to interpret data distributions, identify features such as center and variability, and apply conceptual understanding of statistical measures rather than recall. For example, one assessment task asks students, “Dibuja dos diagramas de puntos, cada uno con 7 o menos datos, de modo que: ambos diagramas de puntos muestren datos con aproximadamente la misma media . . . . ¿Cómo puedes saber, visualmente, que un diagrama de puntos muestra datos con una DMA mayor que otro?” Students create and compare dot plots with similar means but different variability, requiring them to analyze how data is distributed and justify their reasoning. These practice opportunities align with TEKS 111.26.b.12.A, which requires students to represent numerical data graphically using dot plots, stem-and-leaf plots, histograms, and box plots.

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

In unit 3, lesson 5, the questions and tasks progressively increase in rigor by requiring students to move from evaluating possible solutions to representing and justifying their mathematical reasoning. In the section “¿Estás listo para más?” students analyze a real-world percent mixture problem involving paint. Students are presented with the scenario: “Diego quiere pintar su habitación de morado. Compró un galón de pintura morada que es 30% pintura roja y 70% pintura azul. Diego quiere agregar más azul a la mezcla para que la mezcla de pintura sea 20% roja, 80% azul.” Students then determine the amount of blue paint Diego should add by testing possible solutions through the questions: “¿Cuánta pintura azul debe añadir Diego? Prueba las siguientes posibilidades: 0.2 galones, 0.3 galones, 0.4 galones, 0.5 galones. Escribe una ecuación en la que represente la cantidad de pintura que Diego debe agregar. Revisa que la cantidad de pintura que Diego debe añadir sea una solución a tu ecuación.” These tasks increase in complexity and require students to apply operations with rational numbers, connect percentages to equations, and justify their solutions. This lesson aligns with TEKS 111.26.b.3.E by requiring students to fluently multiply and divide positive rational numbers within a problem-solving context.

In unit 3, “Figuras y sólidos,” lesson 19, the questions and tasks increase in complexity by requiring students to analyze constraints, compare strategies, and justify their solutions. Students are asked to solve an open-ended packing problem: “Encuentra cuántas cajas de joyería caben en la caja. Explica o muestra cómo se pueden empacar las cajas de joyería en la caja de envío. Dibuja un boceto para mostrar tu pensamiento, si es necesario.” Students determine how many jewelry boxes fit inside a shipping container, explain their packing strategy, and use a sketch to represent their reasoning when appropriate. The task moves beyond finding an answer by requiring students to evaluate possible arrangements, make decisions about the most effective solution, and communicate their mathematical thinking. This lesson aligns with TEKS 111.26.b.1.G by requiring students to display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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

In unit 12, lesson 17, “Usemos diagramas de caja,” the materials demonstrate coherence by connecting key statistical concepts, including center, spread, and variability, within the same grade level. Students create and interpret box plots, compare distributions, and use numerical summaries to analyze data. The lesson narrative states, “En esta lección, los estudiantes crean diagramas de caja a partir de conjuntos de datos. Comparan e interpretan diagramas de caja para distribuciones con la misma mediana pero IQR muy diferentes, y utilizan los diagramas para responder preguntas.” Students identify the five-number summary and analyze how measures such as the median, interquartile range (IQR), and range describe data distributions. The materials guide educators to support students in making connections by asking them to examine “las medianas, los IQR y los rangos de los conjuntos de datos” when comparing box plots. Students also discuss how two datasets can have the same median but different IQRs, connecting these measures to the concepts of center and spread.

In unit 13, lesson 6, the materials demonstrate coherence by connecting previously learned data concepts and mathematical representations to new applications. The “Narrativa de la actividad” states, “todos los estándares relacionados ya se han abordado en unidades previas,” indicating that students apply concepts developed earlier in the course within a new context. Students build on prior understanding of data organization, frequency tables, and representations as they analyze election data and create relative frequency tables. For example, students use the task, “La gente tiende a estar más satisfecha con los resultados electorales si ganan sus opciones principales. Usa una tabla de frecuencias relativas para mostrar para cuántas personas, y qué porcentaje, la opción ganadora fue su primera opción, segunda opción, tercera opción o última opción.” This connects the mathematical ideas of organizing data, representing relationships, and interpreting quantities across contexts. By applying previously learned data concepts to analyze patterns and communicate relationships within a data set, the lesson reinforces connections among mathematical representations and concepts aligned with the TEKS 111.26.b.1.E.

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.

In unit 9, lesson 3, “Ecuaciones y desigualdades,” the materials demonstrate coherence across units by connecting mathematical language and concepts from previous grades to current learning and future algebraic reasoning. Students build on prior understanding of operations and equality as they develop a deeper understanding of equations, variables, and equivalence. The lesson narrative explains that students may recognize that equations such as 4 + x = 6 and 6 = 4 + x represent the same relationship, but the materials emphasize the importance of explicitly developing students’ understanding that the equal sign represents a balanced relationship. The materials state, “Cuando escribimos una ecuación, significa lo mismo si se intercambian los dos lados.” Students use balance models and equations to connect visual representations with algebraic structures, reinforcing the idea that both sides of an equation must remain equivalent. For example, students solve equations and “Utiliza los móviles para explicar qué significa cada solución,” connecting prior arithmetic reasoning to solving one-variable equations.

In unit 12, lesson 5, the “Narrativa de la lección” demonstrates coherence across units by connecting current learning to future mathematical content and language. The materials state, “En esta lección, los estudiantes continúan usando diagramas de puntos para desarrollar su comprensión del centro y la dispersión . . . . En lecciones futuras, harán sus descripciones y análisis más precisos, a medida que aprendan sobre medidas de centro y dispersión.” Students build on prior experiences interpreting dot plots while developing mathematical language related to center and spread. The lesson also prepares students for future work by introducing informal descriptions that will later be extended to formal statistical measures, such as the mean, median, and measures of variability. This progression connects current statistical reasoning to future learning through increasingly precise mathematical concepts and language aligned to the TEKS.

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..

In unit 8, lesson 5, the “Narrativa de la lección” explains that the purpose of the lesson is for students to strengthen their ability to interpret and solve problems involving data presented as decimals. The materials state, “Cada etapa está cuidadosamente diseñada para construir sobre su comprensión de los decimales y la interpretación de datos, preparándolos para resolver problemas de un solo paso utilizando datos decimales.” This demonstrates coherence at the lesson level by intentionally connecting students’ prior understanding of decimals and data interpretation to new problem-solving tasks. Each activity builds on previously learned concepts while preparing students to apply those skills in increasingly complex data contexts.

In unit 13, lesson 2, “Si nuestra clase fuera el mundo,” the materials demonstrate coherence at the lesson level by connecting students’ prior knowledge of ratios, percentages, and proportional reasoning to new applications involving real-world data. Students build on previously learned concepts as they apply scaling and proportional relationships to model global population data. For example, students are asked, “Supongamos que tu clase representa a todas las personas del mundo. Elige varias características sobre la población mundial que hayas investigado. Encuentra el número de estudiantes en tu clase que tendrían las mismas características. Crea una exposición visible que incluya un diagrama que represente esta información.” Students use proportional reasoning to determine how many classmates represent specific population characteristics and communicate their results using diagrams. By applying previously learned ratio and percentage concepts in a real-world context, the lesson extends students’ understanding of proportional relationships and prepares them for more advanced work with functions and proportional reasoning.

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

In unit 10, lesson 4, “Tablas, ecuaciones y gráficos, ¡qué combinación!,” the materials provide spaced retrieval opportunities by requiring students to revisit and apply skills from earlier units, including equations, ratios, tables, and patterns, in new algebraic contexts. For example, students analyze the scenario, “Diego llegó a un acuerdo con sus padres . . . . ¿Qué representa cada variable en la ecuación? ¿Cuál es la variable independiente? ¿Cuál es la variable dependiente? Explica cómo lo sabes.” This task prompts students to retrieve prior learning about equations and variables while identifying and justifying the independent and dependent quantities in a real-world situation. Students also revisit proportional reasoning in question 1: “Tyler compró 4 onzas de semillas de vegetales por $10 . . . . Completa la tabla . . . . Escribe una ecuación que muestre la relación entre el peso de las semillas en onzas y el costo en dólares.” By applying previously learned concepts of ratios, tables, and equations across multiple tasks, the lesson reinforces prior learning through spaced retrieval and supports long-term retention of algebraic concepts.

In unit 13, lesson 3, the “Narrativa de la lección” explains how the lesson is intentionally sequenced to build conceptual understanding. The materials state that students “exploran la relación entre el máximo común divisor de dos números y las fracciones relacionadas mediante una representación geométrica.” The narrative also emphasizes that the activities are designed to build on one another, stating, “Las actividades de esta lección se construyen unas sobre otras, brindando a los estudiantes la oportunidad de expresar la relación entre el máximo común divisor de dos números y las fracciones relacionadas a través del razonamiento repetido. Por lo tanto, las actividades deben realizarse en orden.” This guidance shows that each activity extends students’ understanding by connecting geometric representations, greatest common factor, and fraction concepts through a purposeful progression of learning.

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

In unit 3, “Figuras y sólidos,” lesson 2, the materials provide interleaved practice opportunities by requiring students to apply previously learned concepts of area, decomposition, rearranging, and equivalence within the same lesson. For example, students are asked, “La diagonal de un rectángulo se muestra. Descompón el rectángulo a lo largo de la diagonal y recompón las dos piezas para formar una figura diferente. ¿Cómo se compara el área de esta nueva figura con el área del rectángulo original? Explica cómo lo sabes.” This task requires students to connect geometric reasoning with their understanding that area is preserved when shapes are decomposed and rearranged. Students also engage in interleaved practice through an open-ended prompt: “Un estudiante dijo: ‘No podemos encontrar el área de la región sombreada porque la figura tiene muchas medidas diferentes . . .’” Students evaluate the statement and apply prior knowledge of decomposition and area strategies to justify their reasoning rather than relying on a single procedure. These interleaved practice opportunities reinforce connections among multiple mathematical concepts and align with TEKS 111.26.b.8.B, which requires students to model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes.

In unit 11, “Verifica tu preparación (A),” the materials provide interleaved practice opportunities by requiring students to apply previously learned concepts related to decimal operations within financial literacy contexts. For example, question 1 asks students to “Usa la división larga para encontrar el valor de 78.9 ÷ 2.” The lesson narrative explains, “El contenido de este problema se encuentra por primera vez en la Lección 3: Balancear un registro de cheques: Depósitos, retiros y transferencias. Este problema permitirá a los estudiantes practicar habilidades relacionadas con valores decimales, especialmente los relacionados con el dinero, y cómo realizar operaciones básicas con decimales hasta la centésima más cercana.” This guidance requires students to revisit previously learned decimal computation while connecting it to financial literacy concepts introduced in lesson 3. By integrating decimal operations with real-world applications involving money and check registers, the assessment provides interleaved practice that reinforces connections among mathematical concepts rather than isolating a single skill.

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 analyze real-world situations by interpreting tables, expressions, and equations that represent relationships involving unknown quantities. The lesson requires students to evaluate patterns in numerical data and connect repeated calculations to algebraic representations, such as using the expression (0.5c) to represent money earned from selling lemonade or (59 + d) to represent a person’s height relative to Elena. For example, students interpret the table showing lemonade sales and determine that raising $127.50 corresponds to selling 255 cups by analyzing the relationship between the quantities and applying the expression to a specific situation. Tasks in the materials allow opportunities for students to compare strategies, use tape diagrams, and discuss how expressions and equations represent the same relationships.

In unit 12, “Evaluación intermedia de la unidad,” students interpret, analyze, and evaluate models and representations for statistical concepts and numerical data summaries. In the mid-unit assessment, students analyze a histogram representing survey data about time spent on homework and respond to questions that require determining which statements must be true. For example, the materials describe the tasks and questions, “Los estudiantes respondieron a una encuesta. La encuesta pidió a los estudiantes que informaran la cantidad de tiempo que dedicaron a hacer la tarea durante una semana, redondeado a la hora más cercana. Este histograma muestra los datos. ¿Cuál de estas afirmaciones debe ser verdadera?” Students interpret the distribution, identify features such as center and variability, and evaluate statements based on the graphical representation.

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

In unit 1, lesson 2, questions and tasks require students to create conceptual models of algebraic relationships by matching equivalent expressions and justifying their reasoning using the properties of operations. Students analyze and represent equivalence by pairing expressions such as (5(a + b)) with (5a + 5b) and (2x + 3) with (3 + 2x), demonstrating how different symbolic forms can represent the same mathematical situation. For example, students use the distributive property to explain that (5(a + b)) and (5a + 5b) represent the same relationship, creating a model that connects symbolic expressions to mathematical structure. Through partner discussions and comparisons of multiple representations, students refine and communicate their models while developing understanding that algebraic expressions can be rewritten in equivalent forms.

In unit 9, lesson 3, questions and tasks require students to construct and interpret hanger diagrams to represent equations and relationships. The materials include tasks in which students match equations to hanger diagrams and use the diagrams to determine unknown values. For example, students are presented with four hanger diagrams and are asked to match an equation to each hanger. They analyze relationships and find correspondences between the two representations. Then students use the diagrams and equations to find the unknown value in each diagram. This value represents the solution to the equation. Students use visual representations to model equations and relationships between quantities.

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

In unit 2, lesson 9, students apply their understanding of fractions, division, and equal groups to determine the total capacity of a water dispenser using partial information presented in a real-world context. Students interpret visual representations, written descriptions, and contextual clues to determine whether the unknown represents the size of a group, the number of groups, or the whole amount. For example, students use the information that 3/4 liter fills 2/5 of the dispenser to write multiplication and division equations, draw tape diagrams, and reason that the dispenser holds 1 7/8 liters. Students compare and discuss multiple solution strategies, including diagrams, equations, and verbal reasoning, to justify solutions in a measurement context.

In unit 3, lesson 19, “Encajar cajas dentro de cajas,” questions and tasks provide opportunities for students to apply conceptual understanding to new problem situations and contexts involving geometric shapes and solids. In activity 2, “los estudiantes ahora calculan el costo de enviar cajas de joyería en cada una de las cajas de tarifa fija del USPS. Cada miembro del grupo seleccionará uno de los 4 tamaños de cajas de envío de tarifa fija, decidirá la mejor orientación para las cajas de joyería dentro de la caja de envío y calculará el costo de enviar 270 cajas.” Students select a box size, determine placement of items, and calculate shipping costs based on constraints. Educators observe group strategies and encourage the use of drawings or paper models when needed.

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

In unit 1, lesson 4, students repeatedly identify multiples and common multiples within a real-world context involving prize distribution at a bakery. The task develops fluency as students list multiples of 5, 9, and 12, recognize common multiples, and determine which customers receive prizes. For example, students determine that customer 36 receives both a blueberry muffin and a slice of carrot cake because 36 is a common multiple of 9 and 12. Through repeated practice identifying multiples and common multiples and explaining their reasoning in context, students build accuracy and efficiency when solving number theory problems.

In unit 3, lesson 2, “Hallar el área descomponiendo y reorganizando,” the materials include a set of cumulative exercises that provide repeated opportunities for students to apply procedures for finding area through decomposition and rearrangement. Within these cumulative tasks, students apply area concepts using a variety of geometric figures and contexts. Students build fluency by consistently decomposing shapes, rearranging pieces, comparing areas, and determining missing dimensions, reinforcing efficient strategies for finding area. The materials state, “Un estudiante dijo: No podemos encontrar el área de la región sombreada porque la figura tiene muchas medidas diferentes, en lugar de solo una longitud y un ancho que pudiéramos multiplicar. Explica por qué la afirmación del estudiante sobre el área es incorrecta.” In this task, students explain why a complex shaded figure can still have its area determined by decomposing it into rectangles or by enclosing it within a larger rectangle and subtracting the unshaded regions.

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 1, lesson 5, students analyze a variety of real-world scenarios and determine whether each situation requires finding common factors, the greatest common factor, common multiples, or the least common multiple. Students evaluate the efficiency of different solution methods by first identifying the appropriate mathematical concept and then selecting strategies such as listing multiples, identifying factor pairs, drawing diagrams, or using equations to solve problems. For example, students determine that finding the largest possible banner size from 72-inch and 90-inch pieces of fabric requires using the greatest common factor, while coordinating flashing lights requires finding the least common multiple, and justify why one procedure is more appropriate than another.

In grade 6, the materials provide opportunities for students to practice the application of efficient, flexible, and accurate mathematical procedures. In unit 2, lesson 8, students apply fraction division procedures by representing and solving a real-world problem involving part-to-whole relationships. Students demonstrate flexibility by writing multiplication and division equations, drawing tape diagrams, and using these representations to determine the total cost of the camping trip. For example, students reason that if $240 represents 2/5 of the total cost, then each 1/5 is worth $120, leading them to determine that the total cost is $600. Through the use of equations, visual models, and contextual reasoning, students strengthen procedural fluency while applying efficient and accurate strategies for solving fraction division problems.

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 5, lesson 19, “Un problema de Fermi,” students generate, solve, and present open-ended estimation problems that require them to select and justify appropriate mathematical strategies. Students evaluate the efficiency and flexibility of different approaches by deciding whether tools such as double number lines, tables, measurements, or proportional reasoning are most effective for solving their chosen problem. Students investigate how long it takes to make sandwiches for everyone in their town, decompose the problem into manageable subquestions, estimate quantities, and determine whether their procedures produce reasonable and accurate results. Activity 3 states, “Haz una lluvia de ideas con al menos cinco problemas de Fermi que quieras investigar y resolver. Si te quedas atascado, considera comenzar con ‘¿Cuánto costaría . . . ?’ o ‘¿Cuánto tiempo tomaría . . . ?’” Through comparing classmates’ visual displays and discussing similarities and differences in solution methods, students critique and refine mathematical processes and select efficient, flexible, and accurate strategies for solving complex real-world problems.

In unit 12, lesson 1, students evaluate procedures, processes, and solutions through discussion and analysis of multiple responses to real-world scenarios involving data collection. In activity 1, students share and compare reasoning, and the class discusses agreement or disagreement across responses. For example, the materials state, “Haz una encuesta entre los estudiantes sobre sus respuestas a cada escenario presentado en la primera pregunta y pide a algunos estudiantes que expliquen su razonamiento para cada uno. Después de que cada estudiante comparta su explicación, pregunta a la clase si están de acuerdo o en desacuerdo y por qué.” Students evaluate and discuss different reasoning approaches when interpreting categorical data and justifying conclusions.

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

In unit 2, lesson 6, educators observe the different strategies students use, including drawing tape diagrams, rewriting whole numbers or mixed numbers as fractions, writing multiplication or division equations, and checking solutions with multiplication. The activity supports students in moving from visual reasoning to more efficient abstract reasoning by prompting them to use tools, compare strategies with peers, and reflect on multiple ways to understand a division expression. For example, students solve how many 3/8-inch books form a six-inch stack by representing the situation with a tape diagram or by writing a division equation, then verify the solution using multiplication. Educators use geometry tools, partner discussions, reflection prompts, and the “Math Language Routine 1: Cada vez más fuerte y más claro” routine to guide students as they refine explanations and move toward more precise and efficient fraction division strategies.

In grade 6, the materials contain embedded supports for educators to guide students toward increasingly efficient approaches to solving mathematical problems. In unit 5, lesson 3, educators begin with concrete demonstrations and taste tests before guiding students toward more abstract reasoning with equivalent ratios and proportions. The lesson provides explicit educator prompts, such as asking students how to make mixture A taste like mixture B and encouraging them to compare ratios and explain why doubling both ingredients maintains the same flavor. Educators also guide students from informal reasoning about taste and diagrams to more efficient strategies using proportional relationships and symbolic representations, including proportions to determine equivalent mixtures. Embedded discussion questions, sentence frames, and recommendations for sequencing student responses support educators in scaffolding learning from concrete models and visual representations to generalized reasoning about equivalent ratios and proportional relationships.

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

In grade 6, the materials explicitly state how the conceptual and procedural emphases of the TEKS are addressed in unit 1, “Evaluación final de la unidad (A).” The assessment balances conceptual understanding and procedural fluency by requiring students to explain mathematical reasoning, identify misconceptions, and accurately apply procedures across multiple content strands. For example, students determine common factors and common multiples, rewrite algebraic expressions using properties of operations, represent multiplicative relationships using tables and equations, and graph polygons on the coordinate plane. The accompanying educator guidance identifies common student misconceptions, explains what specific errors may reveal about students’ conceptual understanding, and provides recommendations for instructional follow-up, such as revisiting lessons on factors and multiples with manipulatives when needed. By integrating explanation, representation, problem-solving, and procedural accuracy along with performance descriptors and instructional next steps, the materials explicitly address the balance between conceptual understanding and procedural proficiency.

In unit 5, lesson 16, “Ecuaciones en la vida real,” the materials explicitly describe how conceptual understanding and procedural fluency are addressed in teaching mathematical reasoning. Conceptually, students interpret relationships between quantities and understand how equations represent real-world situations. Procedurally, students construct equations, organize data in tables, and analyze representations to model situations. For example, the materials state, “En esta lección, los estudiantes participan en actividades que desarrollan su comprensión de cómo las situaciones de la vida real pueden representar gráficamente utilizando relaciones aditivas y multiplicativas. Los estudiantes usarán ecuaciones en la forma y = ax o y = x + a para representar escenarios de la vida real que implican suma o multiplicación secuencial. El objetivo es conectar conceptos algebraicos abstractos con experiencias y aplicaciones concretas, utilizando su conocimiento previo de la lección anterior para ampliar su comprensión.” The lesson narrative and learning goals describe how students connect conceptual understanding with procedural application.

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 3, “Recetas,” questions and tasks include the use of concrete models and manipulatives, pictorial representations (figures and drawings), and abstract representations of mathematical reasoning. Students analyze visual diagrams that represent ingredient amounts in drink mixtures and use these models to compare ratios and determine equivalent relationships between mixtures. In activity 2, students use the diagrams to interpret and complete ratio statements about different mixtures. For example, “Aquí están los diagramas que representan tres mezclas de polvo para bebida y agua: ¿Cómo compararía el sabor de la mezcla A con el sabor de la mezcla B? Utiliza los diagramas para completar cada enunciado: La mezcla B usa _____ tazas de agua y _____ cucharaditas de mezcla para bebida. La razón de tazas de agua a cucharaditas de mezcla para bebidas en la mezcla B es _____.” Students use these representations to move from concrete mixture contexts and visual models to numerical ratio expressions that describe equivalent relationships.

In unit 12, lesson 3, activity 2, students use visual displays to represent and communicate mathematical relationships using equations, tables, graphs, and written explanations of constants and slopes, connecting multiple representations of the same relationship. Students also use a number line as a visual model to represent estimates and compare values. The materials state, “Haz que cada estudiante coloque su calcomanía en el lugar correcto sobre la recta numérica. Si más de una persona hizo la misma estimación, la segunda persona debe colocar su punto más arriba en la pizarra en la misma posición horizontal.” Students use the number line representation to organize and compare estimates, supporting connections between concrete placement of values and underlying numerical relationships.

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 5, lesson 4, “Mezclas de colores,” questions and tasks include supports for students in connecting, creating, defining, and explaining concrete and representational models to abstract concepts of ratios. In activity 2, “En esta actividad, los estudiantes mezclan diferentes cantidades de lotes de una receta de color para obtener un tono determinado de verde. Observan cómo múltiples lotes de la misma receta producen el mismo tono de verde que un solo lote, lo que sugiere que las razones de azul a amarillo en las dos situaciones son equivalentes. También crean una razón que no es equivalente para producir una mezcla que es un tono diferente de verde.” Students use real-world mixture contexts, tables, and ratio reasoning to explain equivalence and non-equivalence across representations.

In unit 13, lesson 4, activity 2, students use graphical representations and coordinate plane tools to construct images by plotting ordered pairs and adjusting their coordinates to represent a desired design. This activity supports students in connecting spatial relationships, coordinate values, and transformations within the coordinate plane as they refine their representations. For example, the materials state, “Los estudiantes usan tecnología de representación gráfica para crear una imagen de su propio diseño. Mientras determinan los pares ordenados necesarios para crear su imagen, tienen oportunidades de pensar sobre las distancias en el plano de coordenadas, los signos en los pares ordenados, las reflexiones a través de un eje y la distancia desde el cero. Los estudiantes hacen uso de la estructura mientras ajustan sus pares ordenados hasta que la imagen se ve exactamente como quieren.” Students use structure and repeated adjustment of ordered pairs to connect geometric representations with symbolic coordinate values and refine their models.

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

In grade 6, unit 3, lesson 1, activity 2, “¿Más rojo, verde o azul?” the materials provide opportunities for students to develop academic mathematical language through the language development routine “Recopilar y mostrar.” The routine states, “En esta rutina, el maestro circula y escucha a los estudiantes mientras hablan, escribiendo las palabras, frases o dibujos que los estudiantes produzcan. El lenguaje recolectado se muestra visualmente para que toda la clase lo use durante la lección y la unidad. Generalmente, la exposición contiene diferentes ejemplos de estudiantes usando funciones del lenguaje disciplinario, como interpretar, justificar o comparar. El propósito de esta rutina es capturar una variedad de palabras y frases usadas por los estudiantes en una exposición a la que los estudiantes puedan referirse, ampliar o conectar durante discusiones futuras, y para aumentar la comprensión de los estudiantes sobre el lenguaje usado en las conversaciones matemáticas.” The routine supports students in using visual displays of mathematical language and student-generated examples to develop and communicate mathematical ideas.

In unit 3, lesson 1, “Cubrir el plano,” the materials provide opportunities for students to develop academic mathematical language using visuals, manipulatives, and language-development strategies while learning about geometric shapes and solids. In activity 2, students use visual representations and manipulatives to compare the areas covered by different tiling patterns. The materials state, “Esta actividad pide a los estudiantes comparar las cantidades del plano cubiertas por dos patrones de mosaicos, con el objetivo de apoyar dos ideas clave de la unidad: Si dos figuras pueden colocarse una sobre la otra de modo que coincidan exactamente, entonces tienen la misma área. Una región puede descomponerse y reorganizarse sin cambiar su área.” The lesson incorporates “Math Language Routine 2: Collect and Display,” in which student language is captured, organized, and refined to support precise mathematical communication. Students describe geometric relationships using informal language that is developed into more formal mathematical vocabulary.

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

In grade 6, unit 2, lesson 5, the materials include embedded educator guidance that scaffolds students’ development and use of academic mathematical vocabulary within mathematical contexts. As students transition from concrete manipulatives to tape diagrams, they use mathematical language to describe division situations involving fractions. The embedded “Math Language Routine 2: Collect and Display” guidance directs educators to gather and display student-generated mathematical language, including phrases such as “equal parts,” “same size,” and “group of _____,” creating a shared language resource for students to reference during the lesson. Educator prompts, including questions such as “How are the values in the equation represented in the tape diagram?” and “What do the blue and white parts represent?” support students in connecting vocabulary, visual representations, and symbolic notation. Additional supports, including sentence frames, visual models, exemplar responses, and opportunities to verify solutions using multiplication, provide structures for students to use mathematical language while explaining their reasoning.

In unit 8, lesson 2, “La propiedad distributiva,” the materials include embedded educator guidance to scaffold and support students’ development and use of academic mathematical vocabulary in the context of teaching expressions. In activity 1, “Calentamiento: Conversación numérica: formas de multiplicar,” the lesson incorporates “Math Language Routine 8: Apoyos para la conversación” by providing sentence frames to support students as they explain their strategies. The materials state, “Muestre marcos de oración para apoyar a los estudiantes cuando expliquen su estrategia.” The provided sentence frames include “Primero, _____ porque . . .” and “Noté _____, así que . . .” The materials also recommend that some students rehearse their responses with a partner before sharing with the class. These embedded supports provide students with structures for explaining their reasoning and using mathematical language during discussions.

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, the materials include embedded educator guidance to support the application of appropriate mathematical language, including vocabulary, syntax, and discourse, through opportunities for students to explain and justify their reasoning. In “Verifica tu preparación (A),” students use mathematical language as they identify fractions and decimals on a number line, simplify algebraic expressions by combining like terms, and explain how they determined the factors of a number. The task states, “Haz una lista de todos los factores del número 20. Explica cómo encontraste tus respuestas.” The educator guidance includes exemplar responses, such as explaining that “factors of 20 are numbers that divide evenly into 20,” which models mathematical vocabulary and syntax for student responses. The materials also provide diagnostic guidance and instructional follow-up recommendations to support educators in responding to student thinking and facilitating mathematical discussions.

In unit 6, lesson 8, “Resolvamos problemas de tasas,” the materials include embedded educator guidance to support mathematical conversations using appropriate vocabulary, syntax, and discourse while learning about percentages, unit rates, and conversions. In activity 2, students compare unit rates and explain whether a comparison represents a fair offer using mathematical reasoning. The materials state, “Encontrar y comparar las razones unitarias tanto del paquete original como del nuevo paquete. Si la razón unitaria es la misma, la oferta es justa. Si la razón unitaria es menor, el dependiente está ofreciendo un descuento. Si la razón unitaria es mayor, el dependiente no está siendo justo.” The activity includes discussion opportunities in which students explain reasoning, compare strategies, and justify decisions using mathematical language related to unit rates and value comparisons.

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

In unit 10, lesson 3, “Más relaciones,” the materials integrate the TEKS process standards through opportunities for students to represent, analyze, and communicate mathematical relationships using multiple representations. In activity 2, students represent relationships using tables, graphs, and equations, communicate their thinking orally and in writing, and apply problem-solving strategies to determine relationships between quantities. The materials state, “En esta actividad, los estudiantes consideran la relación entre la longitud y el ancho de diferentes rectángulos con la misma área dada y se les pide que comparen estrategias para encontrar varias longitudes y anchuras. Interpretan cómo el gráfico muestra lo que sucede con el ancho cuando la longitud cambia y qué significan los puntos representados en el gráfico en el contexto del problema.” Students analyze relationships among variables, interpret representations, compare strategies, and justify conclusions, demonstrating the integration of the TEKS mathematical process standards within the lesson.

In grade 6, unit 11, lesson 6, the materials integrate the TEKS mathematical process standards through opportunities for students to analyze financial situations, represent transactions, and justify decisions using mathematical reasoning. Students use a check register to record deposits, withdrawals, and transfers, calculate balances, and make financial decisions based on a provided budget. Educator guidance supports students in explaining how they determined balances and evaluating whether financial choices align with the given scenarios. Through modeling financial situations, solving problems, and communicating reasoning, students apply the TEKS mathematical process standards within the context of personal financial literacy.

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

In unit 1, lesson 5, the materials provide a description of how the TEKS mathematical process standards are incorporated and connected throughout the course by embedding problem-solving, mathematical communication, reasoning, and strategic tool use into daily instruction. The materials describe opportunities across lessons for students to analyze mathematical relationships, justify solutions, and communicate their thinking using mathematical language. For example, in activity 4, “Bingo de múltiplos y divisores,” students apply their understanding of factors and multiples while explaining their reasoning, identifying patterns, and justifying their selections. The activity includes embedded supports such as “Math Language Routine 8: Apoyos para la conversación,” sentence frames, and educator-facilitated discussions to support students in communicating mathematical ideas and responding to the reasoning of others. These materials demonstrate how the TEKS mathematical process standards are incorporated and connected throughout the course through recurring opportunities for problem-solving, reasoning, representation, and communication.

In grade 6, Guía de recursos para maestros, the materials provide a description of how mathematical processes are incorporated and connected throughout the course through classroom structures that support mathematical discourse, problem-solving, and the communication of mathematical ideas. The materials state, “Los materiales curriculares incluyen estructuras en el aula que apoyan a los estudiantes a tomar riesgos, participar en el discurso matemático, luchar productivamente con los problemas y participar de maneras que hagan visibles sus ideas. Es a través de estas estructuras en el aula que los maestros tienen oportunidades diarias para aprender y aprovechar los entendimientos y experiencias de los estudiantes, y para posicionar a cada estudiante como un aprendiz capaz de matemáticas.” These structures provide opportunities for students to engage in mathematical reasoning, explain their thinking, discuss strategies, and make their mathematical ideas visible throughout the course.

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

In grade 6, Guía de planificación por sección, the materials describe how the TEKS mathematical process standards are incorporated and connected throughout the unit by engaging students in problem-solving, representation, reasoning, communication, and justification across a progression of learning experiences. In the equations section, students develop and interpret tape diagrams and balanced hanger diagrams to model equations, connect visual models to algebraic symbols, and justify solutions by substituting values and explaining their reasoning using mathematical vocabulary. As students progress, they use multiple representations, including diagrams, equations, and real-world contexts, to communicate mathematical ideas and make connections among representations. In the inequalities portion of the lesson, students translate verbal descriptions into symbolic inequalities, graph solution sets on number lines, analyze constraints in contextual situations, and evaluate solutions.

In unit 12, “El proceso estadístico y los resúmenes numéricos de datos,” the Guía de planificación por sección provides a description of how the TEKS mathematical process standards are incorporated and connected throughout the unit on statistical processes and data summaries. The guide describes opportunities for students to represent mathematical relationships, justify reasoning, analyze constraints, communicate mathematical ideas, model real-world situations, and evaluate solutions using multiple representations. In section A, the materials state, “En esta sección, los estudiantes aprenden los fundamentos de la indagación estadística formulando preguntas de encuesta y observando respuestas. Comienzan distinguiendo entre datos numéricos y categóricos y comprendiendo la importancia de las unidades en la recolección de datos. Los estudiantes practican la clasificación de datos y reconocen cómo la estructura de una pregunta influye en el tipo de respuestas.” These descriptions demonstrate how the TEKS mathematical process standards are incorporated and connected throughout the unit progression through opportunities for problem-solving, reasoning, communication, and representation.

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

The Guía de recursos para maestros provides an overview of the TEKS mathematical process standards incorporated into lessons through instructional routines that support problem-solving, mathematical communication, reasoning, and representation. The guide explains how instructional routines are used across lessons to structure mathematical thinking and discourse. The materials state, “Las rutinas de instrucción (IR) ofrecen oportunidades para que todos los estudiantes participen y contribuyan a conversaciones matemáticas. Las IR son invitacionales, promueven el discurso y son predecibles por naturaleza. A lo largo del currículo, las rutinas se introducen de manera intencionada para construir una comprensión colectiva de su estructura. Se seleccionan para actividades basándose en su alineación con los objetivos de aprendizaje de la unidad, la lección o la actividad. Aunque cada rutina tiene un propósito específico diferente, todas comparten el propósito general de apoyar a los estudiantes para acceder a las matemáticas, y todas requieren que los estudiantes piensen y se comuniquen matemáticamente.” The guide describes how these routines are incorporated at the course, lesson, and activity levels to provide opportunities for students to engage in mathematical reasoning, communication, and representation.

In grade 6, the materials provide an overview of the TEKS mathematical process standards within each lesson by engaging students in problem-solving, mathematical communication, representation, and reasoning through structured instructional routines. In unit 2, lesson 8, students interpret a division expression, create a real-world context, and generate a question that matches the equation using multiple representations. The lesson includes opportunities for students to exchange scenarios with partners, explain how their contexts align with the division expression, and justify solutions during whole-class discussions. Embedded educator guidance prompts educators to ask clarifying questions, support connections between division and multiplication relationships, and facilitate peer analysis of mathematical situations. Through these lesson activities and discussion structures, students engage in reasoning, modeling, representation, and communication of mathematical ideas.

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

In unit 2, lesson 11, “Usar un algoritmo para dividir fracciones,” the materials provide opportunities for students to think mathematically, persevere through solving problems, and make sense of mathematics when working with positive rational numbers. In activity 2, “Dividir una fracción entre una fracción,” students investigate why the algorithm for dividing fractions works rather than simply applying a memorized procedure. Students use tape diagrams, numerical examples, and partner discussions to identify patterns, compare representations, and develop a generalization for dividing any number by a fraction. The activity supports mathematical sense-making by requiring students to connect visual models to symbolic expressions, analyze Elena’s strategy, and explain why multiplying by the denominator and dividing by the numerator produces the same result as multiplying by the reciprocal. The “Math Language Routine 8: Apoyos para la conversación” and sentence frames such as “Sé que hay _____ porque . . .” and “Dibujé el diagrama así porque . . .” support students in explaining their reasoning, justifying conclusions, and refining their understanding through mathematical discourse. Students investigate, represent, discuss, and generalize mathematical relationships as they develop conceptual understanding of fraction division.

In unit 5, lesson 19, “Un problema de Fermi,” the materials provide opportunities for students to think mathematically, persevere through solving problems, and make sense of mathematics through open-ended problem-solving. In activity 2, “¿Quién fue Fermi?” students engage in tasks without a prescribed solution path and determine what information is needed, make assumptions, and select strategies to analyze a complex problem. The materials state, “En esta actividad, se introduce a los estudiantes al tipo de pensamiento útil para los problemas de Fermi. El propósito de esta actividad no es llegar a una respuesta, sino más bien ver diferentes maneras de descomponer un problema de Fermi en preguntas más pequeñas que se puedan medir, estimar o calcular.” Students refine their thinking as they break problems into smaller questions, estimate and calculate possible solutions, and discuss their reasoning with others. The open-ended structure of the task provides opportunities for students to engage in productive struggle and apply mathematical reasoning.

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 5, “Usar múltiplos comunes y factores comunes,” the materials support students in understanding, explaining, and justifying that there can be multiple ways to represent and solve problems involving factors and multiples. In activity 2, “Múltiplos y divisores,” students solve real-world problems involving factors, multiples, greatest common factors, and least common multiples, including determining arrangements for party supplies, tile sizes, first aid kits, and lockers with multiple stickers. Students analyze each situation to determine which mathematical relationship applies and explain why a particular strategy is appropriate. The materials state, “Los estudiantes hablar sobre las estrategias que usaron para determinar si están respondiendo problemas sobre el mínimo común múltiplo o el máximo común divisor. Anota frases comunes que escuches que usen como ‘más pequeño,’ ‘común,’ ‘más grande,’ ‘aparece dos veces’ . . . Escribe las palabras de los estudiantes en una exposición visual y sigue usándola como referencia durante la lección. Esto ayudará a los estudiantes a usar el lenguaje matemático mientras comprenden cómo los problemas se relacionan con factores comunes y múltiplos comunes.” These supports require students to explain, compare, and justify different strategies across mathematical contexts.

In unit 6, lesson 8, “Resolvamos problemas de tasas,” the materials support students in understanding, explaining, and justifying that there can be multiple ways to represent and solve problems involving percentages, unit rates, and conversions. In activity 2, “Clasificación de tarjetas: ¿Es una oferta?” students compare prices and determine whether offers represent a good deal by using strategies such as scaling, division, and proportional reasoning. Students explain and defend their reasoning during discussions and compare different approaches for determining unit rates. The materials state, “Encontrar y comparar las razones unitarias tanto del paquete original como del nuevo paquete. Si la razón unitaria es la misma, la oferta es justa. Si la razón unitaria es menor, el dependiente está ofreciendo un descuento. Si la razón unitaria es mayor, el dependiente no está siendo justo.” These supports provide opportunities for students to represent, solve, explain, and compare multiple valid approaches to the same mathematical task.

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 unit 1, lesson 4, “Múltiplos comunes,” the materials provide opportunities for students to make sense of mathematics through multiple opportunities to do, write about, and discuss mathematical ideas with peers and educators when learning about factors and multiples. In activity 3, “Premios en el día de la gran inauguración,” students analyze a real-world scenario involving common multiples and least common multiples, record their reasoning in written responses, and use strategies such as skip counting, listing multiples, or marking multiples with different colors or symbols to determine which customers receive one or more prizes. The materials state, “En la primera lectura, los estudiantes leen el problema con el objetivo de comprender la situación. En la segunda lectura, pide a los estudiantes que describan las cantidades en la historia. En la tercera lectura, pide a los estudiantes que dibujen representaciones de la situación y luego que propongan posibles estrategias matemáticas para responder las preguntas.” Throughout the activity, students explain and compare their reasoning with partners, while educators facilitate discussions in which strategies are shared, sequenced, and connected using mathematical language such as common multiple and least common multiple.

In grade 6, unit 10, lesson 4, activity 1, “Calentamiento: empareja ecuaciones y tablas,” the materials require students to make sense of mathematics through multiple opportunities to do, write about, and discuss mathematical relationships. Students analyze equations and tables, identify matching representations, record their matches, and justify their choices by reasoning about dependent and independent variables and the structure of each relationship. The “Túrnense” instructional routine supports mathematical discourse by requiring one partner to explain the reasoning behind each match while the other partner listens, asks questions, and collaborates to reach an agreement before switching roles. Educator prompts such as “¿Qué características de la ecuación o tabla te ayudaron a reducir las posibles coincidencias?, ¿Hubo alguna coincidencia en la que tú y tu compañero no estuvieron de acuerdo?” and “¿Cómo trabajaron para llegar a un acuerdo?” encourage students to compare and refine their mathematical thinking.

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 6, unit 6, lesson 8, activity 2, “Clasificación de tarjetas: ¿Es una oferta?,” the materials support educators in guiding students to share and reflect on their problem-solving approaches, including explanations, arguments, and justifications. Students first solve pricing scenarios independently, then explain their reasoning to a partner, compare alternative solution strategies, and revise their decisions based on peer discussion before reaching a consensus. Educator guidance directs educators to select students who used different strategies, such as comparing unit rates, scaling equivalent ratios, or using proportional tables, and facilitate a discussion highlighting the similarities and differences among the approaches. Through the “Math Language Routine 2: Recolectar y mostrar” routine, educators record and connect students’ mathematical language and strategies. The materials state, “Escucha y observa cómo los estudiantes razonan sobre las ofertas y toman sus decisiones (oferta o no oferta), y toma nota de las diferentes estrategias que usan para comparar razones unitarias . . . . Continúa agregando a la exposición y refiérete a ella durante la conversación con toda la clase, haciendo conexiones explícitas entre el lenguaje y las estrategias usadas (es decir, si los estudiantes comparan la tasa unitaria original con la nueva cantidad o viceversa). Esto ayudará a los estudiantes a comprender cómo calcular y comparar razones unitarias mientras aumentan la metaconciencia del lenguaje.”

In grade 6, unit 12, lesson 17, activity 3, “Aviones de papel,” the materials support educators in guiding students to share and reflect on their problem-solving approaches through explanations, arguments, and justifications while interpreting box plots. Students compare box plot representations, explain similarities and differences, and justify their interpretations using evidence from the data. Educator guidance prompts students to reflect on their reasoning and make connections between multiple representations. The materials state, “Usa esta rutina para apoyar a los estudiantes mientras interpretan las representaciones de los diagramas de caja de los datos. Pregúntales a los estudiantes, ‘¿Qué es lo mismo y qué es diferente entre los diagramas de caja?’ Invita a los estudiantes a usar el lenguaje de mediana, RIC, distribución, centro, dispersión mientras conversan las comparaciones. Pídeles a los estudiantes que hagan conexiones en los diagramas de caja con los valores de la tabla resumen de 5 números.” These supports guide students to explain, compare, and justify their mathematical thinking while reflecting on multiple interpretations of the data.

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 6, unit 1, “Evaluación final de la unidad (A),” the materials include prompts and guidance to support educators in providing explanatory feedback based on student responses and anticipated misconceptions about factors and multiples. The assessment guidance directs educators to analyze student performance and adjust instruction when misconceptions are identified. The materials state, “Si a la mayoría de los estudiantes les cuesta trabajo con este ítem, planifica dedicar más tiempo a la Actividad 2 de la Lección 16 y proporciona materiales manipulativos para ayudar a los estudiantes a representar gráficamente el problema y entender el contexto. Los maestros tendrán oportunidades a lo largo de las Lecciones 16 y 17 para ayudar a los estudiantes a comprender factores y múltiplos comunes mediante contextos. Estas lecciones están diseñadas para utilizar situaciones contextuales y desarrollar estrategias y definiciones para factor común, máximo común divisor, múltiplo común y mínimo común múltiplo.” This guidance supports educators in identifying misconceptions, providing concept-focused explanations using visual representations and contextual examples, and planning follow-up instruction to strengthen students’ understanding of common factors and common multiples.

In unit 9, lesson 9, “Calentamiento: ordenar sándwiches,” the materials include prompts and guidance to support educators in providing explanatory feedback based on student responses and anticipated misconceptions about interpreting inequalities in real-world contexts. The lesson anticipates that students may incorrectly conclude that 13.86 sandwiches should be rounded up to 14 and guides educators to address this misconception by asking students to calculate the cost of 14 sandwiches and compare it to the $83 budget, demonstrating why 14 is not a valid solution. Educators are prompted to poll the class, have students explain why each statement is true or false, and record student reasoning to clarify the meaning of the inequality and distinguish between the mathematical solution and the real-world context. The guidance also emphasizes that while 13.86 satisfies the inequality mathematically, a sandwich shop would not sell fractional sandwiches, so the greatest whole-number solution is 13 sandwiches. These embedded prompts support educators in providing explanatory, concept-focused feedback that addresses misconceptions and strengthens students’ understanding of inequalities in context.