TEKS Mathematical Models with Applications

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

The Table of Contents for Instructional Units outlines the concepts taught throughout the course and provides the corresponding Texas Essential Knowledge and Skills (TEKS) associated with those concepts. For example, lesson 1.1 “Data Types and Visualizations” shows TEKS.MMA.9A.

The materials include a Standards Correlations document that outlines the TEKS taught throughout the course. The TEKS compose the basis for the document’s organization, and the document shows the location and title of the part of the lesson that addresses a given standard. For instance, TEKS.MMA.10A corresponds to the Unit 2 Making Connections Task: Mathematical Models in Research.

The Suggested Pacing Guide with Scope and Sequence includes a table that breaks down each lesson with TEKS and English Language Proficiency Standards (ELPS). For example, the table for unit 3, topic c, lesson 3.5 shows TEKS.MMA.5B and ELPS 1.D.

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 Suggested Pacing Guide with Scope and Sequence includes a scope and sequence that outlines a specified number of instructional days for each unit in the materials to support effective implementation. The scope and sequence presents options for both 180- and 202-day instructional calendars.

The Suggested Pacing Guide with Scope and Sequence includes a suggested progression of one- and two-day lessons and assigns timing for each lesson component for both 50- and 90-minute periods. This section also suggests reducing practices for compressed instructional calendars.

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

The Suggested Pacing Guide with Scope and Sequence includes a rationale for the unit order, explaining that all nine units in the materials are broken into four strands. The rationale further explains that units 1–2 form a data research strand, units 3–4 form an algebraic modeling and probability strand, units 5–7 form a personal finance strand, and units 8–9 form a geometry and trigonometry strand.

The Suggested Pacing Guide with Scope and Sequence includes a rationale for each unit’s placement. For example, unit 3 follows the data research strand by building on the data collected. Unit 3 precedes units 4–7 because it introduces modeling techniques that those units will expand on.

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

The Program Implementation Guide provides a unit internalization protocol that includes seven steps, each with corresponding guidance, for thorough understanding and preparation to teach a unit. The steps build from step 1, “Read the unit pre-assessment,” to step 7, “Set the pacing.”

The Program Implementation Guide provides a lesson internalization protocol that includes eight steps, each with corresponding guidance, for thorough understanding and preparation to teach a lesson. The steps build from step 1, “Read the objective and TEKS in the lesson header and check the ELPS mapping on the ‘Enhanced Topic Overview,’” to step 8, “Plan the close.”

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

The materials include a Classroom Walk-Through Tool that provides campus and district administrators with a list of observable evidence cross-walked with mathematical process standards in the Advanced Quantitative Reasoning TEKS. Observable evidence includes “Students move between at least two representations,” “Guided practice models and representation before independent use,” and “Warm-up debrief connects prior learning to today’s task.”

The Program Implementation Guide provides PLC Facilitation Questions that guide administrators in leading educators in implementing the materials as designed during professional learning community meetings. For example, the guide prompts administrators to ask, “Which prerequisite skills will we pre-teach and when?”

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 provide an Introduction and Key Concepts for each lesson that outline the background content knowledge, prerequisite skills, and academic vocabulary needed to teach the lesson’s concepts effectively. For example, in lesson 1.1, the materials list and label two prerequisite skills as TEKS.6.12. The materials identify “categorical data” as a key concept. Several bullet points under this concept identify and define the vocabulary words “categorical data,” “ordinal data,” and “non-ordinal data.”

The materials provide Essential Knowledge and Skills for each topic that outline the unit TEKS and academic vocabulary needed to teach the concepts effectively. For example, unit 1, topic A identifies TEKS.MMA.9A and TEKS.MMA.9B for student mastery and a list of “Words to Know,” including definitions for “bar chart” and “bivariate data.”

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.

Each unit contains a “Conceptual Activities” section that lists interactive open education and/or Desmos resources. While students and families can use these resources at home, the materials do not suggest using these activities as family supports and do not provide the list in Spanish.

The materials provide educator supports for each topic, including Texas Essential Knowledge and Skills for the Topic, Essential Questions, Words to Know, and Recommended Resources. However, the materials provide only educator documents and do not include communications for families to support student progress.

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 a Topic Implementation Guide that breaks down each lesson in the topic. For each lesson, the guide includes essential questions aligned to the objective, TEKS and ELPS correlations, and a focus or lesson objective. For example, the Topic Implementation Guide outlines that unit 2, topic D, lesson 2.11 correlates to TEKS.MMA.9F and ELPS 3F and 4F. The guide states that lesson 2.11 teaches students to identify key features of linear and exponential functions in real-world scenarios and includes essential questions such as, “How can you use units to help interpret the slope and y-intercept?”

The materials include a Topic Implementation Guide with a table that aligns assessment items with common errors and educator prompts to address the misconceptions. For example, unit 4, topic C identifies the common error of incorrectly calculating the goodness-of-fit statistic and shows its alignment to progress assessment items 8, 9, and 10 and unit assessment item 15.

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 Topic Implementation Guide that breaks down each lesson in the topic. For each lesson, the guide includes suggested materials for educators and students as well as suggested timing for each lesson component. For example, unit 4, topic C lists the materials needed for lesson 4.8 as “Warm-Up 4.8, Scaffolded/Guided Practice 4.8 handouts, six-sided die or coin, and calculators.”

The materials include a Topic Implementation Guide with a table that outlines the suggested timing for each lesson component, separated into 50-minute periods or 90-minute blocks. For example, unit 5, topic C outlines that the warm-up and debrief should take five minutes, whereas the “Introduction and Key Concepts” should take eight minutes for a 50-minute period or 12 minutes for a 90-minute block.

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

The Practice Retrieval and Extension Guide for each unit includes a section that outlines which elements of the materials serve as homework, extension for students close to proficient, enrichment for students who have demonstrated proficiency, and example activities for monitoring progress. For example, under “Monitoring Progress,” the materials state that the warm-up is the “daily check on retention” and the topic progress assessment is the “checkpoint that drives grouping decisions.”

The Practice Retrieval and Extension Guide for each unit includes a section that outlines every practice resource in the unit, along with guidance for when to assign it. For example, the chart shows that Practice B can act as homework, a second attempt after reteach, or extended practice. Guidance also states to assign this when Practice A shows partial mastery.

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.

The materials include a diagnostic pre-assessment in each unit with varied item types, such as multiple-choice and free-response. For example, the Unit 2 Pre-Assessment has nine multiple-choice questions and one open-ended response with three parts.

The materials provide a summative assessment at the end of each unit with varied tasks and item types. For example, the unit 1 assessment contains multiple-choice, graphing, and open-ended response item types. The tasks in the unit 1 assessment vary from “Which group has the least range?” and “What type of plot is this?” to “Which graph would be most appropriate?” and “How might the council use data from a study to encourage community members to vote?”

The materials include formative assessments at the lesson level that vary in task and question types, such as warm-ups, Problem-Based Tasks, and practice sets. For example, Warm-up 1.2 provides a table of 12 rivers and their lengths, then asks students to find the mean and median lengths. The 1.2 Problem-Based Task: Which Bike gives tables of two different bike models and their annual maintenance costs and asks students to determine which bike Zahara should buy, stating, “Compare the data using measures of center and spread, and include a graph.”

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

The Unit Structure defines the course assessments and explains their intended purpose. For example, the document states that the unit pre-assessment “can be used to gauge students’ prior knowledge” and the unit assessment “can be used to document the extent to which students grasped the concept and skills.”

The Unit Structure document lists each student activity, including formative assessments, and explains their purpose. For example, the document describes the performance tasks as “opportunities [for students] to apply concepts and skills,” while educators “informally assess and plan.”

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

The Unit Structure provides a description of unit assessments that reads, “Each unit ends with 12 multiple-choice questions and three extended-response questions that incorporate critical thinking and writing components. This can be used to document the extent to which students grasped the concepts and skills of each unit. Targeted TEKS are listed in the answer key.” The materials lack clear guidance on how to administer assessments, including suggested time allotments.

The Unit Structure identifies Problem-Based Tasks as activities to use at the end of instruction. The document also states that the optional coaching questions for the Problem-Based Tasks “should be used at the discretion of the educator for students requiring additional support.” However, the materials do not clearly guide educators on administering instructional assessments.

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

The pre-assessment key for each unit lists the TEKS addressed by every question. For example, the Unit 1 Pre-Assessment—Answer Key states the item aligns with the TEKS.MMA.9A. These diagnostic assessments also align with the unit objectives. For example, item 1 on the Unit 1 Pre-Assessment aligns with the unit 1 objective to “summarize data.”

The materials provide an answer key for all formative instructional assessments at the lesson level, which identifies the correlating TEKS at the beginning of the answer key. These assessments include warm-ups, Problem-Based Tasks, and practice sets that align with the objectives of the lesson and unit. For example, the “1.1 Warm-Up Debrief” begins by stating the aligned TEKS standards.MMA.9A and TEKS.MMA.9B.

The materials include summative assessments at the topic and unit levels, with multiple-choice and open-ended questions aligned to the objectives of the topic or unit. The materials also identify each summative assessment item’s alignment with the course TEKS by including the TEKS number next to each item on the answer keys. For example, the unit 1 assessment answer key shows the item 1 answer is c and aligns with standard TEKS.MMA.9A.

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

The materials include instructional assessments with TEKS-aligned items that assess different levels of complexity, such as procedural tasks, applying multiple skills to problems and tasks, and open-ended questions. For example, the unit 2, topic B assessment contains questions that require calculations, contextual explanations, the determination of the validity of an answer, and the drawing of conclusions.

The materials include interactive unit assessments, along with a written assessment option that contains TEKS-aligned multiple-choice, fill-in-the-blank, inline choice, and graphing questions. For example, the unit 7 interactive assessment contains seven multiple-choice, five fill-in-the-blank, and three questions that include a combination of fill-in-the-blank statements, inline choice, and graphing.

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

The materials include scoring information for all instructional assessments as answer keys. These answer keys provide the correct answer to multiple-choice problems and sample responses for open-ended questions. For example, the unit 2 assessment asks students if any sensor readings are significant, and the provided exemplar response states, “Although the readings for sensor A are more significant, they still fall below the 95% confidence level threshold.”

The materials include scoring information but do not provide guidance for interpreting student performance on the assessment. For example, the unit 1 assessment answer key lists the correct answers but does not identify potential misconceptions in incorrect answers or offer support for educators to make informed instructional decisions.

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

The Unit Structure states the unit pre-assessments “can be used to gauge students’ prior knowledge and to inform instructional planning.” The materials do not provide further guidance on how educators can use student performance to inform instructional planning.

The materials include summative assessments at the topic and unit level with answer keys that identify each assessment item’s alignment to the course TEKS by including the TEKS number directly next to each item on the answer keys. For example, the unit 1 assessment answer key shows the item 1 answer is c and aligns with standard TEKS.MMA.9A. The materials do not include guidance on the use of tasks and activities to respond to student performance on assessments.

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 answer keys for diagnostic and summative assessments, including correct answers and correlated TEKS. For example, the Unit 6 Pre-Assessment—Answer Key shows that question 1 has an answer of b and aligns to TEKS.MMA.3B. However, the materials do not provide a tool that allows educators to track student progress and growth.

The materials include interactive pre-assessments, practice sets, topic assessments, and unit assessments that give students immediate scores and show missed points. For example, after completing the Unit 6 Interactive Pre-Assessment, the student sees a box with the attempt number, time spent on the pre-assessment, percent score, and a review button. The review lets the student view the answers they submitted, with a red “incorrect” or green “correct” below each question. The results of interactive activities do not include TEKS or a tool for students to track their own progress and growth.

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.

The Unit Structure includes a description of Scaffolded Practice (Printable Practice). The document states, “This set of 10 printable practice problems provides introductory-level skill practice for the lesson. This practice set can be used during instruction time.” The materials do not provide further guidance on differentiating lessons for students who have not yet reached proficiency.

Units 3, 8, and 9 include 20 Ramp Up lessons that reteach prerequisite material from previous courses. For example, the lesson 9.4 “Evaluating Trigonometric Functions” activity reteaches geometry and eighth-grade TEKS involving right triangles and finding trigonometric ratios. It provides no guidance for educators to differentiate instruction or activities in these lessons for students who have not yet reached proficiency.

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).

The Enhanced Unit Overview for each unit includes a table in sections 1–3 that identifies language traps to preempt and cognates to make explicit for each topic. For example, the unit 2, topic A Enhanced Overview shows “the word random usually means haphazard while in this context a simple random sample is a deliberate, defined process.” Section 5 also includes guidance for unfamiliar academic vocabulary. For example, the unit 2, section 5 Enhanced Overview suggests that educators guide students to create a Frayer model for the topic’s highest-confusion term, using non-examples to surface the language trap.

The Instructional Strategies document provides strategies like the Concept-Picture-Word Wall to help educators develop academic vocabulary. The materials provide a general description of the strategy and generic ways to use it in a lesson. For example, in the Instructional Strategies document, the “How Do You implement Concept-Picture-Word Walls in the Classroom?” section states, “Postings on the Concept-Picture-Word Wall can be arranged strategically to connect concepts, units of study, or groups of words where appropriate.” The materials do not include pre-teaching or embedded supports for unfamiliar references in text.

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

The Unit Structure identifies a performance task as “an extension activity to provide students with opportunities to practice, reinforce, and apply mathematical skills and concepts to a real-world task.” The materials do not provide further guidance on differentiating this activity or other instruction to enrich students who have demonstrated proficiency.

Each unit contains a Conceptual Activities document that lists interactive open education and/or Desmos resources. The document provides a synopsis of what students will do for each activity. For example, the unit 1 Conceptual Activities includes Standard Deviation in a Normal Distribution and its synopsis states, “In this activity, students will become acquainted with percentages under the normal curve by dragging lines around a graph.” The materials do not provide guidance on using the activities as enrichment or extension for students who have demonstrated proficiency.

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.

The 2.1 Guided Practice provides prompts the educator can read to help explain the sample spaces represented in question 1 of the guided notes. To support modeling in question 1, which asks students to “list the sample space and then describe each of the following events in terms of the student’s activities,” the educator prompt states, “Describe the event {Eva, Nate}. Eva is in the debate club and the German club.”

The materials include explicit prompts in the Problem-Based Task: Coaching Questions for each lesson. The coaching questions are direct prompts the educator may read to help explain the task to students. For example, lesson 2.1 Problem-Based Task asks students to organize 10 sets of outcomes into equal groups, using each set exactly once, and the coaching questions include, “What are the outcomes in the complement of D?”

The materials include guidance to support the educator in modeling and explaining the concept(s) students will learn in the Introduction and Key Concepts of each lesson. For example, the lesson 4.4 Introduction and Key Concepts includes guidance to support the educator in modeling and explaining the concepts of experiments and events. The guidance “Note that the words ‘experiment’ and ‘event’ in probability have very specific meanings that are not always the same as the everyday meanings” helps educators explain the differences in these terms and similar terms in everyday use.

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

The Topic Implementation Guide for each topic includes several sections with guidance and recommendations for effective lesson delivery using a variety of instructional approaches. For example, in the guide, unit 4, topic C, section 3 identifies a strategy for each of the mathematical process standards covered in the topic; for example, “Have students physically run a small trial (coin flips, die rolls) to generate real empirical data before relying on given data sets.” Section 5 provides several monitoring and coaching suggestions, such as “Listen for students treating a single unlikely outcome (e.g., four heads in a row) as ‘impossible’ rather than ‘unlikely,’ and redirect them to the language of probability rather than certainty.” Section 8 outlines some recommendations for alternate strategies; for instance, “Use color-coding to distinguish theoretical values from empirical/observed values throughout a shared table or graph.”

The materials include multiple types of practice in each lesson that support guided, independent, and collaborative practice. Each lesson includes a warm-up, written practice sets, and interactive practice sets that support independent practice. Scaffolded practices, guided notes, and enhanced instruction support guided practice. Problem-Based Tasks support collaborative practice. The materials include student-facing materials, answer keys, and instructional guides for effectively facilitating lesson delivery through guided, independent, or collaborative practice within the lesson components.

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

The materials include multiple types of practice in each lesson that support guided, independent, and collaborative practice. Each lesson includes a warm-up, written practice sets, and interactive practice sets that support independent practice. Scaffolded practices, guided notes, and enhanced instruction support guided practice. Problem-Based Tasks support collaborative practice.

The Unit Structure states that printable practice and independent practice “can be used in any combination of educator-led instruction, cooperative learning, or independent application of knowledge.” However, the materials do not contain any guidance for educators on recommended structures to support implementation of any activities or lessons.

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.

The Enhanced Unit Overview for each unit includes a section that includes a strategy or sentence stem for each level of linguistic proficiency. For example, the unit 4 Enhanced Unit Overview lists options for each of the five levels for guided practice, including “Point to the next step” for pre-production, “First __, next, __” for beginning, and “I chose __ over __ because __, and I verified it by __” for advanced proficiency.

The materials include one instructional document with references to support emergent bilingual students. The “Instructional Strategies” document provides generic guidance on research-backed instructional strategies and includes a statement about using these strategies with emergent bilingual students. For example, the guidance provided for text-to-speech technology is: “Additionally, for ESL students, hearing the English mathematical language, especially referring to mathematical representations and notation, can help put English words to the ideas they see. Some text-to-speech technology can translate written and mathematical text into other languages, so students can hear the text in their natural language and see the English highlighted on the screen as they hear it.”

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

The Program Implementation Guide includes a “Using These Materials in State-Approved Bilingual and ESL Programs” section. This section includes a table that aligns different program models with what each model involves and how to use the materials within it. For example, the table shows that for the ESL pull-out model, educators should share the topic’s key words and the enhanced topic overview language trap with the ESL educator before the topic begins to front-load vocabulary rather than provide it through remediation.

The materials include an instructional document with references to support emergent bilingual students. The “Instructional Strategies” document provides generic guidance on research-backed instructional strategies and includes a statement about using these strategies with emergent bilingual students. For example, the guidance for the “Concept-Picture-Word Wall” states, “Concept-Picture-Word Walls can also be written in multiple languages.” Especially for students who are on-grade-level in their native language, a multi-lingual Concept-Picture-Word Wall can help students connect the content they already know in another language to the English vocabulary necessary for success on English-language math activities and tests.”

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.

The Enhanced Unit Overview for each unit includes an Embedded Discourse Guidance—Oral and Written with a table that breaks down educator moves, with ELPS citations, for building academic vocabulary, increasing comprehension, making cross-linguistic connections, and increasing background knowledge in both oral and written discourse. For example, the unit 6 Enhanced Unit Overview suggests the use of “text-to-speech” for building cross-linguistic connections in oral discourse by playing the passage in the student’s home language and then in English with the English text highlighted.

The materials provide embedded guidance for educators to support increasing comprehension through oral discourse in the Problem-Based Task coaching questions. Scaffolded questions help guide students to comprehend and complete the task. For example, the 4.4 Problem-Based Task: Recognizing Equal Sets: Coaching Questions includes questions for educators to guide students in discussing “What is the sample space?” and “Now list the 10 sets you need to place into groups of equal sets, showing the outcomes in each.” The discussion continues until students develop the understanding to organize the sample spaces into equal sets.

The Instructional Strategies document provides strategies like the Concept-Picture-Word Wall to help educators develop academic vocabulary. The materials give a general description of the strategy and generic ways to use it in a lesson. For example, under the heading “How Do You implement Concept-Picture-Word Walls in the Classroom?” the Instructional Strategies document states, “Postings on the Concept-Picture-Word Wall can be arranged strategically to connect concepts, units of study, or groups of words where appropriate.” The document also includes ELPS citations.

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.

In the 8.12 Problem-Based Task: Fighting Flames from a Distance assessment, students use trigonometric ratios to calculate distances and determine whether firefighters need to stand closer to the building based on their water-hose restraints.

The materials allow students to analyze marketing claims based on graphs and statistics. The materials provide opportunities to demonstrate depth of understanding throughout unit 2, topic B, as shown in 2.6 Practice A, where students analyze data and draw conclusions about the outcomes presented.

Over the course of unit 4, topic A, the materials include opportunities for students to interpret information from various graphs and draw conclusions from the data. For example, the 4.3 Problem-Based Task: Spin Away gives students information about a spinner and asks them to create a list of outcomes from that spinner and answer multiple open-ended questions about the outcomes and events.

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

In the 2.9 Scaffolded Practice, students plot data from a table into a scatterplot, describe the shape of the points, draw a line of best fit to estimate the data, and finally write the equation of the line that best models the data. Then, in the 4.8 Guided Practice, students identify the sample space for a bag of marbles, calculate theoretical and empirical probabilities for outcomes, and finally analyze and compare the probabilities.

In the 2.9 Problem-Based Task: Sleepy Time—Coaching Questions, students plot data into a scatterplot, describe the shape of the points, draw a line of best fit to estimate the data, and finally write the equation of the line that best models the data. Students then compare their line to the line a graphing calculator linear regression tool gives to determine which is a better fit for the data. Then, in the 4.7 Guided Practice, students plot a scatterplot of speed and fuel economy, graph two given equations, determine which function better estimates the data, and use the best-fit function to calculate fuel economy at a particular speed.

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

The Suggested Pacing Guide with Scope and Sequence provides a rationale for unit order that explicitly states how patterns, big ideas, and relationships between mathematical concepts connect across the units. For example, units 8 and 9 close the course with geometry and trigonometry, moving from similarity and right-triangle trigonometry into the unit circle and periodic modeling.

The Suggested Pacing Guide with Scope and Sequence provides a rationale for each unit’s placement and explicitly states how patterns, big ideas, and relationships between mathematical concepts connect across units. For example, unit 4 follows two modeling-heavy units and returns to the statistics/research strand. It extends the data-collection vocabulary from unit 1, and comparing theoretical to empirical probability directly parallels the residual/model-fit thinking from unit 2.

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

The materials connect content and language learned in previous courses/grade levels to the content students will learn in Mathematical Models and Applications. Each lesson includes an Introduction and Key Concepts document that identifies prior knowledge and links prior course vocabulary to the current course. For example, the 2.9 Introduction and Key Concepts references the skill of plotting points on a coordinate plane from grade 6 and connects the language of slope and y-intercept to finding the equation of a line of best fit. The materials do not connect future courses/grade levels to content students will learn in Mathematical Models and Applications.

The materials connect language learned in previous courses/grade levels to the content students will learn in Mathematical Models and Applications. Each topic within the unit includes an essential knowledge and skills document that lists the vocabulary from previous courses and the current course necessary for the topic. For example, in the unit 5, topic A “Essential Knowledge and Skills” section, the vocabulary list contains “equation” from grade 1, “coordinate plane” from grade 6, and “exponential decay” from Algebra I.

Each lesson includes an “Introduction and Key Concepts” section with a box labeled “Prerequisite Skills.” The prerequisite skills include a list of previous grade-level skills and vocabulary with previous grade-level TEKS. However, the materials do not connect future courses/grade levels to the content students will learn in Mathematical Models and Applications.

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

The materials connect concepts and procedures from the current course to new mathematical knowledge and skills. For example, in lesson 2.9, students find linear regressions and later, in lesson 3.4, find exponential regressions. The concepts of regression and the procedures for finding it are similar, so students connect learning from one lesson to the next.

Unit 4, topic A addresses probability by representing sample spaces and events. The 4.1 Introduction and Key Concepts connects drawing and interpreting Venn diagrams to representing and understanding experimental outcomes using lists and tree diagrams from grade 7.

The materials connect concepts vertically from prior grade levels and horizontally through the course. For example, in unit 1, lesson 1.1, students connect finding medians and interquartile range from grade 6. Then, in lesson 1.3, students create a boxplot from medians, quartiles, minimums, maximums, and outliers, and then interpret the data in context.

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

The materials include unit assessments that contain questions about concepts and skills learned throughout the unit. For example, the unit 4 assessment includes questions about identifying a sample space from lesson 4.2, analyzing a Venn diagram from lesson 4.1, and calculating the probability of a compound event from lesson 4.6. Students only practice spaced retrieval during the summative assessment. The materials do not provide regular opportunities for spaced retrieval in lessons or across units.

The materials do not provide spaced retrieval opportunities with previously learned skills and concepts across units. While different units cover parts of standards, each lesson includes isolated skill practice. For example, both units 1 and 4 address MMA.9A, but the lessons are independent of each other with no explicit spaced retrieval opportunities between the units.

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

The materials do not include interleaved skills or concept practice across lessons and only focus on current learning. For example, each lesson in unit 4 includes multiple practice problems. Students analyze Venn diagrams in lesson 4.1, identify events in lesson 4.3, and calculate combinations and permutations in lesson 4.5. Practice opportunities include only the concepts and skills from the current lesson, with no interleaved practice.

The materials do not provide interleaved practice opportunities with previously learned skills and concepts across units. While different units cover parts of standards, each lesson includes isolated skill practice. For example, units 1 and 4 both address MMA.9A, but the lessons are independent of each other, with no explicit interleaved practice opportunities between the units.

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

In the 3.3 Problem Set A, students interpret contextual situations, such as the half-life of elements, and corresponding function equations to identify the domain and sketch a graph. Students also analyze graphs of populations, bacterial growth, diseases, and other contextual situations to identify key features of exponential graphs in the context of an insect population.

In the 1.1 Scaffolded Practice, students evaluate which graph is most useful for a given purpose. Students analyze the audience’s needs, evaluate which representation provides the correct measures, and explain their choices.

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

In the 2.7 Problem-Based Task: Movie Buzz, students analyze a data set from the first four days of a movie’s release and create a regression equation to predict the number of attendees on the fifth day.

In Mathematical Models and Applications, students use regression models available through technology to create linear and exponential function models, interpret correlations, and make predictions. For example, in the 2.11 Guided Practice, students answer questions given a variety of tables and graphs and use technology to create best-fit regression equations and describe values in the context of the problem.

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

In the 5.2 Scaffolded Practice, students relate the meaning of the slope and y-intercept of a line to the corresponding parameters of a net-pay equation and then, in another context, apply the meaning to hiring employees.

In Mathematical Models and Applications, students use scale factors with two- and three-dimensional objects to demonstrate proportional and non-proportional changes in surface area and volume. The materials include questions and tasks that give students opportunities to apply conceptual understanding to new problem situations and contexts. In the 8.1 Problem-Based Task: Prettying Up the Pentagon, students apply their learning of dilations and parallel lines to determine the perimeter of a walkway in the courtyard of the Pentagon building.

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

Across lesson 4.3, students repeatedly construct dot plots, histograms, box plots, frequency polygons, and other displays from numerous data sets, making the routine procedures automatic. This repetition frees attention for the grade-level demand of interpreting representations and drawing and evaluating conclusions. Explicit procedural supports, such as step-by-step graphing-calculator routines for histograms, box plots, and scatter plots, and a repeatable computational rule for determining the number of histogram classes, reinforce fluency.

Each lesson includes two practice problem sets to develop fluency. In lesson 3.9 Practice Set A and Practice Set B, students factor binomial and trinomial expressions of the same type. Each practice set provides several opportunities to practice only this skill. This builds automaticity in selecting a strategy when looking at a problem and fluency in the procedure.

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 the 2.9 Problem-Based Task: Sleepy Time—Coaching Questions, students plot data into a scatterplot, describe the shape of the points, draw a line of best fit to estimate the data, and write the equation of the line that best models the data. Students then create a regression equation with a graphing calculator, compare their best-fit estimate to the calculator’s regression, and evaluate which fits the data better. Students practice with flexible procedures and can determine which is more efficient and accurate for the given problem. In this way, students practice applying efficient, flexible, and accurate mathematical procedures within lessons.

In unit 8, students investigate scale factors, apply similarity to different angle and side criteria, and solve problems using similarity and congruence. Students gain accuracy with scale factors and similarity in the first lessons and then use efficient and flexible mathematical procedures in lesson 8.5 by proving triangle similarity using either side-angle-side (SAS) or side-side-side (SSS) similarity when appropriate. In this way, students practice applying efficient, flexible, and accurate mathematical procedures throughout the unit.

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.

While the materials allow students to practice accurate, efficient, and flexible mathematical procedures, they do not give students opportunities to evaluate procedures, processes, and solutions for efficiency, flexibility, and accuracy. In unit 8, students investigate scale factors, apply similarity to different angle and side criteria, and solve problems using similarity and congruence. Students use efficient and mathematical procedures to prove triangle similarity using side-angle-side (SAS) and side-side-side (SSS) similarity. Students use different methods to solve problems but do not evaluate procedures for efficiency, flexibility, or accuracy.

In the 2.9 Problem-Based Task: Sleepy Time—Coaching Questions, students plot data into a scatterplot, describe the shape of the points, draw a line of best fit to estimate the data, and write the equation of the line that best models the data. Students calculate the line of best fit with a graphing calculator to determine which is a better fit for the data. Students practice flexible and efficient procedures, but the materials do not ask students to evaluate the flexibility or efficiency of the different methods, despite exposing them to them.

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

The materials do not contain embedded supports for educators to guide students toward increasingly efficient approaches. For example, in the 2.9 Problem-Based Task: Sleepy Time—Coaching Questions, students plot data into a scatterplot, describe the shape of the points, draw a line of best fit to estimate the data, and finally write the equation of the line that best models the data. Students then compare the line to the regression found with a graphing calculator to see which fits the data better. Even so, the materials do not provide guidance or specific prompts for educators to guide students toward this more efficient method.

In lesson 8.6, students use ratio segments to solve for unknown side lengths. The 8.6 Scaffolded Practice includes problems such as “Explain the reflexive property of congruent segments” and “State whether the Triangle Proportionality Theorem or the Triangle Angle Bisector Theorem is being used.” However, the materials do not provide embedded prompts or guidance for the educator to explicitly connect ideas and tasks or to make the connection to move students toward an increasingly efficient approach.

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

The materials explicitly state how they address the TEKS procedural emphasis. For example, the 6.3 Guided Practice outlines the steps to solve a problem in individual boxes. The steps for one problem, finding the fourth term of a sequence, include “Determine whether the sequence is explicit or recursive” and “Use the given formula to find the first four terms.” These steps emphasize the procedures necessary to address TEKS.MMA.3A, which is listed at the top of the practice.

The Introduction and Key Concepts for each lesson outline the procedures taught in the lesson and how they align with the TEKS. For example, the 6.1 Introduction and Key Concepts explains the variables and formula for simple interest loans but does not explain why the formula works. The document also provides a box labeled “Common Errors/Misconceptions” with three bullet points. Educators could address the misconceptions by explaining the conceptual emphasis, but the document does not explicitly state the conceptual emphasis of the TEKS.

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.

The 2.9 Guided Notes asks students to determine if given functions are a “good fit” for a given set of data. Students plot points on a Cartesian graph to visualize the trend of the points before creating an abstract representation, an algebraic equation, to model the data. The Mathematical Models and Application TEKS do not require concrete models and manipulatives, but the materials provide data tables and use real-world connections, such as occupants in a living area, to support student understanding.

Lesson 5.2 includes questions and tasks that utilize pictorial and abstract representations. In 5.2 Practice A, students write an equation to model each net-pay situation and then create a graph, pairing an abstract equation with its pictorial graph. In the 5.2 Scaffolded Practice, students graph a linear equation and label its x- and y-intercepts on the coordinate plane. The Mathematical Models and Applications TEKS do not require the use of concrete models and manipulatives, but the materials provide visuals of data and real-world connections to concrete examples, like money and paychecks, to support student understanding.

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

Lesson 4.4 supports students in creating diagrams of sample spaces, connecting sample spaces with set notation, and explaining sample spaces given by diagrams and set notation. For example, in the 4.4 Guided Practice, students use a table to connect the meaning of set notation symbols and define outcomes listed in set notation. Students also create a Venn diagram and tree diagrams to represent sample spaces and connect outcomes listed in set notation to the sample space. Throughout the practice, students explain how the models connect to outcomes listed in set notation.

Lesson 5.2 supports students in connecting graphs to abstract symbolic notation, creating graphs that model the equations, and explaining how the graphs connect to equations. For example, in the 5.2 Problem-Based Task, students create a net-income equation for each of two jobs, create corresponding graphs, and explain how the models help Mike decide about the jobs. These tasks connect, create, define, and explain the relationship between representational graphs and their corresponding equations, slopes, and intercepts.

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

The materials give students opportunities to develop academic mathematical language using visuals. For example, the 3.3 Enhanced Instruction shows graphs of exponential functions with the x-intercept and y-intercept identified to help students visualize the key academic terms. Then, in the 3.3 Guided Practice, students use a graph to identify key features and then create a graph from a list of key features.

The materials include a list of links to virtual manipulatives, games, and other activities in the “Conceptual Activities” sections. For example, in unit 2, the “Conceptual Activities” section lists one Illustrative Mathematics resource. These links point to external resources and are not embedded in the program’s lessons. The materials do not provide opportunities for students to develop academic mathematical language using manipulatives.

The Instructional Strategies document includes a few general language development strategies, such as “close reading” and “text-to-speech technology,” but it lacks specificity on when and how to integrate these strategies into each lesson. This exemplifies how the materials do not provide students with opportunities to develop academic mathematical language using other language development strategies.

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

Each lesson’s Introduction and Key Concepts lists key vocabulary with definitions. For example, the 3.3 Introduction and Key Concepts lists the words intercepts, intervals, and extrema in bold print and provides definitions. However, the materials do not provide explicit guidance for educators on how to scaffold and support students as they develop academic vocabulary in context.

The materials do not include embedded educator guidance to scaffold and support students’ development and use of academic mathematical vocabulary in context. The Instructional Strategies document outlines vocabulary development strategies, such as the Concept-Picture-Word Wall and Novel Ideas, but the document provides only a generalized outline of the strategy without detailed guidance on how to use the strategies in context within specific lessons.

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.

The Coaching Sample Responses provide scaffolded questions with exemplar responses that guide educators to support discourse with appropriate mathematical vocabulary and give students opportunities to hear and use math language with peers. For example, in the 6.2 Problem-Based Task: Financing Options Coaching Sample Responses, the statement “Both options show increasing balances for the first six months,” follows the question “Do you notice any patterns?” as part of an exemplar response. These questions and responses guide educators to give students opportunities to hear and use math language with peers. The 3.3 Problem-Based Task: Cold Light, Not Bright Coaching Sample Responses provides prompts and exemplar responses to move students from “It is getting smaller” to “The function is decreasing.” These prompts and exemplar responses guide educators in supporting the use of appropriate mathematical language during whole-class and small-group discussions, allowing students to hear and use mathematical vocabulary, like “increasing” and “decreasing,” throughout the course. However, the materials do not contain embedded educator guidance to support appropriate syntax.

The Enhanced Instruction presentations present questions to engage students in mathematical conversations, and exemplar responses appear after each question. For example, the 5.1 Enhanced Instruction includes the question “How much money will he earn if Albert works 40 hours this week?” On the same slide, the exemplar response
“Substitute 40 for x in the expression from the previous expression,” and the calculations appear directly below the question. While educators may use the question to engage students in mathematical conversations with provided exemplar responses, the materials do not contain embedded educator guidance to support syntax.

The Enhanced Unit Overview for each unit provides educator guidance to support mathematical conversations and gives students opportunities to hear, refine, and use mathematical language with peers to develop their mathematical toolkit over time. For example, in section 5, the Unit 4 Enhanced Overview includes eight different embedded strategies and educator moves such as “Students answer True/False or Always-Sometimes-Never guiding questions individually, then reconcile aloud in pairs before Guided Practice begins.” However, the materials do not contain embedded educator guidance to support appropriate syntax.

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

The materials appropriately incorporate the TEKS process standards in the Problem-Based Tasks. For example, in the 4.3 Problem-Based Task: Spin Away, students use a given spinner to determine the number of outcomes for given events. This requires students to apply mathematics to everyday situations, communicate mathematical ideas using multiple representations, create and analyze mathematical relationships, and analyze given information. Then, in the 5.2 Problem-Based Task: Fishing Net Pay, students apply mathematics to a real-world decision, analyze two income options, formulate a plan by creating a net-income equation for each job and graphing both, and justify which job Mike should choose while considering non-pay factors by reflecting the process standards of applying mathematics to everyday life, using multiple representations, and communicating a justified conclusion.

The material provides Making Connections tasks in most units that integrate the TEKS process standards. For example, in the Unit 3 Making Connections Task, students explore how a car’s speed affects its stopping distance under certain road conditions. Students apply their mathematical knowledge about graphing and exponential relationships to predict the stopping distance based on a car’s speed. Students analyze given stopping distances and speeds to determine a solution and select appropriate techniques and tools as part of their problem-solving model. Students communicate their mathematical ideas to explain their reasoning using representations they created.

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

The Introduction to the Program lists the TEKS process standards and states that included activities “incorporate concept and skill development and guided practice, then move on to the application of new skills and concepts in problem-solving situations.” However, the materials do not describe how they incorporate and connect TEKS process standards throughout the course.

Each unit in the materials identifies the process standards addressed in the unit. For example, in the unit 5, topic A, “Essential Knowledge and Skills” section, a box labeled “MPS” at the top right lists the process standards 1A, 1B, 1E, and 1G. Although the unit identifies and incorporates the standards, the materials do not describe how they incorporate and connect TEKS process standards 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 each topic in a unit, the Essential Knowledge and Skills identifies process standards addressed in the topic, emphasizing how each standard connects to the essential questions. For example, in the unit 2, topic C, the “Essential Knowledge and Skills” section, a box labeled “MPS” at the top right lists all seven process standards, helping students connect these standards to answering the three essential questions provided at the top of the document. However, the materials do not describe how they connect the TEKS process standards throughout the unit.

Each lesson in a unit contains Problem-Based Tasks that identify the process standards connected to the task, emphasizing how the unit’s activities incorporate the process standard. For example, the 2.1 Problem-Based Task: Song Requests lists process standards 1B, 1E, and 1F in a box at the top right labeled “MPS.” However, the materials do not describe how they connect the TEKS process standards throughout the unit.

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

Each Problem-Based Task in each lesson includes the TEKS process standards in a box labeled “MPS,” reinforcing how each problem or task connects to specific standards. For example, the Problem-Based Task 5.1: Gross Calculations identifies process standards 1A, 1B, 1E, and 1G. The task pertains to Maurice’s paycheck and incorporates applying problems in the workforce. Using the remaining identified standards, students analyze information about Maurice’s pay to write an equation, evaluate it, compare the results to the table of Maurice’s hours and pay, and determine whether Maurice is underpaid.

In each topic in a unit, the “Essential Knowledge and Skills” section identifies process standards addressed within the two to four lessons in the topic. For example, the unit 9, topic C “Essential Knowledge and Skills” section lists process standards 1A, 1B, 1C, 1E, 1F, and 1G and provides four essential questions to emphasize how lessons 9.7 and 9.8 incorporate these standards.

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

The materials give students opportunities to think mathematically, persevere through problem-solving, and make sense of mathematics in each lesson through the Problem-Based Tasks. In the 5.1 Problem-Based Task: Gross Calculations, students think mathematically to determine whether Maurice is underpaid by identifying his overtime weeks, writing separate earnings equations for non-overtime and overtime weeks, computing his pay, and comparing their calculations to the values in a pay table. Coaching questions such as “What do you need to know?” and “Explain your reasoning” help students persevere through the task and make sense of the result in context. The coaching questions scaffold the reasoning without removing the problem-solving demand.

The materials include conceptual tasks that feature multipart problems with open-ended questions. These tasks require students to make sense of math by observing and making connections. For example, in the unit 3, topic C, Conceptual Task: Weighing Job Offers activity, students compare two job offers based on earning potential after 30 years. The materials include guiding questions like “How are the two offers different?” “How are they the same?” and “What are the unknown variables in this problem?” which support students as they think mathematically and persevere through solving problems.

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

The materials support students in understanding that there can be multiple ways to represent and solve problems and complete tasks. For example, in the 8.12 Enhanced Instruction presentation slides and in the 8.12 Guided Practice, students use the Pythagorean theorem and trigonometric ratios to solve problems. In the guided practice example 3, the materials state that students may use either the Pythagorean theorem or the tangent ratio to solve the problem. Throughout the enhanced instruction and guided practice, students understand how to use visual representations, such as diagrams, and algebraic representations, such as equations, to explain the problems and solutions.

The Facilitating Productive Struggle and Feedback Guide describes a reflection routine that supports students in explaining and justifying that multiple ways can represent and solve problems. In the second move, students compare their approach with another student’s to surface alternative explanations. In the third move, students argue or justify one approach over the other. And in the fourth move, students refine their own justifications after considering multiple methods.

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 the Conceptual Task: Smart Phone Surge, students create a linear regression model and make predictions about how many smartphone users there will be in the future. Throughout the conceptual task, students do math to create a regression and calculate values. Students also write about what the calculations mean in context and explain their thought process. The “Introduction to the Program” guides educators to “circulate, observe, speak to individuals and small groups, and assess” during the conceptual task so that students discuss math with their peers and with educators.

Lesson 5.1 provides students with multiple opportunities to create gross-pay equations in the 5.1 Scaffolded Practice and 5.1 Guided Practice. Students write about conclusions, observations, explanations, and justifications in the 5.1 Problem-Based Task: Gross Calculations and 5.1 Scaffolded Practice. Students discuss gross pay in the Recommended Closure Activity at the end of the 5.1 Problem-Based Task: Gross Calculation Coaching Sample Responses.

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 the 6.3 Problem-Based Task: Repayment Schedule, students evaluate the effects of making an additional payment to an amortization table. Students share explanations of how to calculate the principal balance, argue whether an extra payment in month 10 was applied to principal or interest, and justify the calculation of a new balance after the extra payment in month 10. The coaching questions and sample responses that accompany the Problem-Based Task help educators guide students through the problem-solving approach to determine and explain the effect of an extra payment.

The Facilitating Productive Struggle and Feedback Guide supports educators in guiding students to reflect on their explanations, arguments, and justifications. The guide outlines a four-move reflection routine for use after a Problem-Based Task or conceptual task. The second move focuses on student reflection on other student explanations, while the third move asks students to reflect on arguments for one method over another. Finally, the fourth move focuses on reflection on the student’s own justification.

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

The Facilitating Productive Struggle and Feedback Guide for each unit includes prompts and guidance for educators to give explanatory feedback on anticipated misconceptions and common errors. Each lesson in the unit has a unique table with the common misconception, a prompt for educators to ask, and guidance on how to respond. For example, the Unit 4 Facilitating Productive Struggle and Feedback Guide lists confusing “union” and “intersection of sets” as a common misconception in lesson 4.4. The guide then prompts, “Does this description require being in at least one set or being in both sets at the same time?” and guides the student to translate the word “or” to union and the word “and” to intersection before writing set notation.

The materials do not include prompts to help educators provide explanatory feedback based on student responses. The 3.11 Problem-Based Task: Parabolic Party Streamers Coaching Sample Responses includes sample student responses to educator prompts to guide students in finding the maximum height of a streamer tossed through the air. The materials provide only correct sample responses, not incorrect responses or misconceptions. For example, the sample response “Yes, the equation h(t) = –16t2 + 32t + 56 is written in the form f(x) = ax2 + bx + c, which is the standard form of a quadratic equation” follows the prompt “Is the formula written in standard form?” The materials provide independent prompts that do not support educators in providing feedback to student responses.