Cosenza Math – Grade 8 Advanced

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

The materials include a scope and sequence outlining the Texas Essential Knowledge and Skills (TEKS), the English Language Proficiency Standards (ELPS), and concepts taught in the course. This scope and sequence can be found under the Course Level Documents titled Grade 8 Advanced Mathematics Scope and Sequence. The scope and sequence includes two correlation tables. The first is a unit-to-TEKS/ELPS correlation table that allows the instructor to identify the standards in a specific unit. Additionally, TEKS/ELPS-to-unit correlation tables include the full language of the standards and allow the instructor to locate units for a specific standard.

The Grade 8 Advanced Mathematics Scope and Sequence document includes a bulleted list of concepts titled “Concepts & Knowledge to Be Learned in Grade 8 Advanced Mathematics.” For example, some bullet points include “All real numbers are either rational or irrational” and “At least one of the variables in an intercept always has a value of 0.”

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 materials include suggested pacing to support effective implementation for various instructional calendars. This pacing guide is available under the Course Level Documents. The guide includes 150-, 165-, and 180-day calendars. Each calendar includes columns for the instructional day, unit, lesson number, topic, TEKS, and ELPS.

At the end of each instructional calendar, the materials provide guidance for time adjustments that do not disrupt the instructional sequence or content coherence. For example, at the end of the 165-day calendar, a bulleted list outlines how the calendar may be adjusted, such as “The instructional day(s) spent on Diagnostic Assessments designated at the beginning of each of Units 1, 2, 3, 4, 5, 6, 7, 8, 9 and/or 10 can be reduced or skipped to recapture up to ten instructional days.” The length of the instructional calendar provides options for adjustment, with the longest calendar offering the most.

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 materials explain the rationale for the unit order and how the concepts connect throughout the course. The explanation is under the Course Level Document within the scope and sequence following the ELPS table. The page is titled Rationale for Unit Progression and contains the rationale for unit order and how the concepts connect throughout the course.

For Grade 8 Advanced Mathematics, the Rationale for Unit Progression provides detailed connections to both the Texas Response to the Curriculum Focal Points (TxRCFP) and to the TEKS process standards. An example from the rationale states, “Unit 3 focuses on symbolic manipulation of equations and inequalities to solve for a single variable. Unit 4 builds from the single-variable concepts and skills students have learned in previous grade levels and honed in the first three units of the course to begin to develop the idea of slope as a constant rate of change in linear relationships and the meaning of both x- and y-intercepts of linear functions.”

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

The materials include protocols with corresponding guidance for unit and lesson internalization. This guidance is in the Course Level Documents and includes a Unit Internalization Protocol and a Lesson Internalization Protocol for instructors. Each protocol is a checklist to help instructors internalize the content. The Unit Internalization Protocol guides instructors through three main actions: “Read through the digital unit pages and make notes for your ‘three weeks from now’ self,” “Connect unit learning to culminating tasks,” and “Anticipate where students may need support.” Under each action are deeper-level tasks, such as “Begin with the end in mind by unpacking the unit assessments.” Finally, the materials provide questions for the instructor to consider during unit internalization and prompts to support connections to student outcomes, such as “Think about what you have read before these resources as well as the additional information they provide. What will you do when students already know what you will be teaching?”

The Lesson Internalization Protocol presents the same instructor guidance but at the lesson level. The Lesson Internalization Protocol guides instructors through three main actions: “Read through the digital lesson pages and make notes for your ‘three weeks from now’ self,” “Connect lessons to culminating tasks,” and “Anticipate where students may need support.” Under each action are deeper-level tasks, such as “Begin with the end in mind by unpacking the lesson assessments.” Finally, the materials prompt instructors with questions to support continued lesson internalization and connections to student outcomes, such as “As you read through the lesson overview and detailed lesson plans, how does learning build and connect throughout the lesson?”

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

The materials include resources and guidance to help instructional leaders support instructors in implementing the materials. The Implementation Guide includes these resources and guidance under Course Level Documents. The guide includes various resources to help instructional leaders support instructors. For example, the resource Teacher Conversations to Support Implementation includes a short paragraph describing the “plan-do-study-act” and a table detailing how to accomplish it. The table columns include Design, Facilitate, Analyze, and Improve. Each column further breaks down what each step entails: Lesson Planning, Teaching and Learning, Student Learning Evidence, and Evaluate Outcomes. Each column also includes timing for each step and offers questions instructional leaders can ask their instructors.

The Implementation Guide also provides an Observation Protocol and Checklist page for instructional leaders. This page gives leaders an instructor observation checklist and includes observational support “Prior to the Lesson,” “During the Lesson,” and “After the Lesson.” Each category includes checklist criteria. In the Prior to the Lesson section, the checklist prompts leaders to check that instructors complete certain tasks, such as their daily lesson plan, and provides conversation starters to support leader conversations with instructors. In the During the Lesson portion, the checklist prompts the leader to look for various things, such as student placement in random groups and student engagement in mathematical discourse. In the After the Lesson section, the checklist prompts leaders to continue conversations with their instructors, such as sharing “glows,” providing feedback, and discussing “grows.”

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 knowledge and academic vocabulary needed to teach the unit’s concepts effectively. At the beginning of each unit in the course is a page titled, for example, “Unit 1: Algebraic Expressions, Equations, and Inequalities” that lists the unit, the lessons within the unit, the TEKS and ELPS covered, a Unit Introduction, a Unit Overview, and Prior Learning Supports. The Unit Overview page provides the background knowledge needed to teach the unit’s concepts effectively.

For example, the Prior Learning Supports section includes a column that provides previous years’ TEKS, the full language of the standards, and how they connect to the TEKS in the current grade. At the bottom of the page, under Resources, an 8AM Unit 1 Document provides the academic vocabulary needed to teach the unit’s concepts effectively.

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 materials include English supports for families in each unit, with suggestions for supporting their student’s progress. For example, at the bottom of the Unit Overview page for unit 3, “Slopes and Intercepts,” is a resource titled 8AM Unit 3 English Letter. The document includes suggestions for how families can support their students’ progress in that unit, such as involving their students in real-world applications and asking them to explain something they have learned.

The materials include Spanish supports for families for each unit, with suggestions for supporting their student’s progress. For example, at the bottom of the Unit Overview page for unit 3, “Slopes and Intercepts,” is a resource titled 8AM Unit 3 Spanish Letter. The document includes suggestions for how families can support their students’ progress in that unit, such as involving their students in real-world applications and asking them to explain something they have learned. This letter is in Spanish.

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, detailed lesson plans that specify daily objectives, questions, tasks, materials, and instructional assessments required to meet the content and language standards for the lesson in the Unit Overview page. For example, in unit 5, lesson 6, “Trend Lines,” the subsection Learning Outcomes provides the lesson objectives as “I can…” statements, such as “I can use a trend line to make predictions from a data set.” Within the same subsection, a table provides the TEKS and ELPS correlations with full indicator language for each.

The materials within the same Unit Overview page also contain questions aligned to the lesson objective in a subsection titled Lesson Overview. This is written as a Focusing Question and, in this example, is stated as “How can I use a trend line to approximate a linear relationship and make predictions?” Tasks within the lesson are found in the Lesson Plan subsection: Exploration, Explanation, and the Performance Task. Each task provides instructor guidance on implementation.

Finally, the Lesson Plan includes an instructional assessment in the form of one or more performance tasks. The Lesson Overview page includes the instructor version of the performance task(s), which include four levels based on student mastery of the lesson.

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 listing the educator and student materials needed to deliver the lesson effectively, along with suggested timing for each lesson component. On each Lesson Overview page, near the center, a subsection titled Pacing Guide suggests timing for each lesson component. Near the bottom of the Lesson Overview page are two subsections with the necessary Teacher Materials and Student Materials.

For example, in unit 7, lesson 3, “Multiplying Polynomials,” the Pacing Guide suggests the Exploration portion takes 20–25 minutes, the Explanation and Practice portion takes 20–25 minutes, and the Performance Task takes 30–35 minutes. Further down the page, the Teacher Materials section lists graphing technology and algebra tiles as necessary materials for educators. The Student Materials section lists pencil, paper, graphing technology, and algebra tiles as necessary materials.

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

The materials include guidance on effectively using lesson materials for extended practice through performance tasks. Within each lesson’s Overview page, under the Lesson Components table, a column provides one or more performance tasks. When an educator clicks the hyperlink to the performance task, the educator’s version opens and details the task’s various components. Each performance task has four versions, depending on the student’s proficiency level, and the lesson’s Overview page provides educator guidance on effectively using the performance task.

For example, in unit 2, lesson 1, “Identifying Functions,” the Lesson Overview page includes a subsection titled Lesson Plan. At the end of this plan is a bullet point for the performance task with sub-bullets. These sub-bullets guide educators on how and when to use the performance task. Examples of the sub-bullets include, “As a formative assessment, use the performance task to determine what students know about the topic,” “Performance tasks have four versions: on-level, simplified, enriched, and scaffolded,” and “If there are multiple performance tasks, select one or encourage self-differentiation by allowing students to select a task based on their interests and comfort levels.”

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 variety of instructional assessments at the unit and lesson level (including diagnostic, formative, and summative) that vary in task and question types. In Grade 8 Advanced Mathematics, each unit provides a diagnostic assessment at the beginning, followed by lesson-level formative assessments and concluding with a unit summative assessment. For example, unit 2 begins with a diagnostic assessment split into two parts. Problems 1-4 in section 2A contain previous-grade-level TEKS, and problems 1-8 in section 2B contain current-grade-level TEKS. Both parts include various task and question types, such as open-ended, fill-in-the-blank, and multiple-choice.

Each lesson contains various formative assessments. For example, in unit 2, lesson 4, “Domain and Range of Quadratic Functions,” materials provide two assessment options for educators. The Explanation portion of the lesson includes the first formative assessment and instructs educators to give students practice problems from specific workbook pages. Students are given both open-ended and multiple-choice questions. A second option for formative assessments at the lesson level is the performance tasks at the end of each topic. Unit 2, lesson 4, offers two performance task options: “Candles” or “Satellite Dish.” Both performance tasks use open-ended questions, and each task has four performance levels. The levels include simplified, scaffolded, on-level, and enriching tasks, which educators assign based on students’ mastery level.

The materials provide a summative assessment at the end of each unit, called, for example, unit 2 Test. Educators can print the assessment or assign it digitally. In the unit 2 Test, the main question types are multiple-choice and open-ended.

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

The materials include definitions and intended purposes for the instructional assessment types included. For example, the Grade 8 Advanced Mathematics Assessment Guide, available on the Course Level Documents page, includes a table listing the three assessment types: lesson-level formative, unit-level diagnostic, and unit-level summative. The table includes the definitions, purposes, administration guidance, and examples/formats for each instructional assessment.

For example, the lesson-level formative assessment row defines it as “Assessment during the lesson (consistently during the lesson).” The purpose states, “Provide instructors with feedback to make instructional decisions for the remainder of the lesson and/or next lesson.” The unit-level diagnostic assessment row defines it as “Assessment at the beginning of the unit of instruction (once per unit)” and states the purpose as “Uncover misconceptions prior to instruction.” The unit-level summative also states the definition and purpose.

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

The materials include educator guidance to ensure consistent and accurate administration of instructional assessments. For example, under the Grade 8 Advanced Mathematics Course Level Documents, the materials include a Performance Task Rubric and Scoring Guide. The document opens with a rubric for all course lesson-level performance tasks. The rubric contains three main scoring sections: procedural, conceptual, and communication, with each section contributing zero, one, or two points each, for a total of six points possible for each performance task. Breaking down the rubric further, the zero-, one-, and two-point columns each have three options for the procedural, conceptual, and communication sections.

For example, in the procedural section, each column breaks down into three options. An example of a statement in the zero-point column is “Little to no evidence of mathematical reasoning is present or the reasoning is scattered and incomplete.” An example of a statement in the one-point column is “Uses a strategy that leads to a solution of the problem. Some errors in the procedure may result in an incorrect solution, but the errors are minor and the procedures are appropriate.” An example of a statement in the two-point column is “Uses a very efficient and organized strategy leading to a solution.” The conceptual and communication sections are also broken down into three options each for the zero-, one-, and two-point columns.

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

The materials include diagnostic assessments aligned to the TEKS and objectives of the course, unit, or lesson. For example, in Grade 8 Advanced Mathematics, each unit begins with a diagnostic assessment, such as the Unit 2 Diagnostic Assessment. The Grade 8 Advanced Mathematics Test Answer Key & Diagnostic Guide provides educators with the answer key for the Diagnostic Assessment and includes the item number, TEKS, correct answer, and diagnostic information. The Diagnostic Assessment for unit 2, problem number 4 under section 2B, for example, shows an exponential function with an open-circle restriction. The problem states, “Part of an exponential function is graphed on the grid. Which inequality best represents the domain of the part of the function shown?” followed by multiple-choice options as inequalities. This question aligns with one of the unit 2, lesson 5 objectives, which states, “I can determine the domain and range of an exponential function.”

The materials include formative assessments aligned to the TEKS and objectives of the course, unit, or lesson. For example, the unit 2, lesson 1 performance task is a formative assessment aligned with the TEKS and the lesson objective. In the Instructor Edition, the first page lists the TEKS the task assesses. The objective for unit 2, lesson 1 states, “I can identify a function from a set of ordered pairs, table, mapping, or graph.” The performance task provides students with an image of a continuous graph, a mapping, a discrete graph, a set of ordered pairs, a table, and a secondary continuous graph. The problem reads, “Review the different representations presented. Select those illustrating y as a function of x. For those not illustrating a function of y in terms of x, describe a change that could be made to create a function. Justify your reasoning.”

The materials include summative assessments aligned to the TEKS and objectives of the course, unit, or lesson. At the end of each unit, educators can assign a summative assessment called the unit test. The test answer key includes a table with columns for the item number, reporting category, readiness or supporting, content student expectation (TEKS), process student expectation (also TEKS), and the correct answer. For example, when viewing the unit 2 Test Answer Key, educators are able to see each of the TEKS the test is aligned to. When viewing the printable test, for example, problem 2 reads, “Which of the relationships shown below is not a function?”, followed by multiple-choice options of different relations. This question aligns with an objective from unit 2, lesson 1, which states, “I can identify a function from a set of ordered pairs, tables, mapping, or graph.”

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

Instructional assessments include TEKS-aligned items evaluating varying levels of complexity. In Grade 8 Advanced Mathematics, instructional assessments include three types: diagnostic, formative, and summative. Each contains varying levels of complexity. For example, in unit 2, the diagnostic assessment is split into two sections. The first section covers TEKS-aligned items from previous grade levels, while the second covers current-grade-level TEKS. Within each section, questions vary in complexity, ranging from finding dependent quantities from a table to finding the domain and range of a graph. Each lesson includes two formative assessments: the performance task and practice problems from the book. The performance task can be given based on students’ level of mastery and can vary in complexity, with options such as an enriched, scaffolded, on-level, or simplified task. The practice problems in the book also offer a range of complexity levels. Finally, the unit summative assessment, called the unit test, provides varying levels of complexity, and the questions are based on similar problems students encounter in the book, practice problems, the performance task, and the grade-level section of the diagnostic test.

Instructional assessments also include a variety of TEKS-aligned item types. For example, in unit 2, the diagnostic assessment contains open-ended and multiple-choice item types. The performance task contains only open-ended questions, but the practice questions in the book include both open-ended and multiple-choice item types. The unit test includes multiple-choice and open-ended items.

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

The materials include scoring information to ensure accurate scoring of instructional assessments. For example, in Grade 8 Advanced Mathematics under the Course Level Documents, a Performance Task Rubric and Scoring Guide provides a rubric for all course lesson-level performance tasks. The rubric contains three main scoring sections: procedural, conceptual, and communication, with each section contributing zero, one, or two points for a total of six points possible for each performance task. Breaking down the rubric further, each column has three options for the procedural, conceptual, and communication sections. For example, in the procedural section, the zero-point column lists three options. An example of a statement in this column is “Little to no evidence of mathematical reasoning is present or the reasoning is scattered and incomplete.” An example of a statement in the one-point column is, “Uses a strategy that leads to a solution of the problem. Some errors in the procedure may result in an incorrect solution, but the errors are minor and the procedures are appropriate.” An example of a statement in the two-point column is “Uses a very efficient and organized strategy leading to a solution.” The conceptual and communication sections also break down into three options each for the zero-, one-, and two-point columns.

The materials include guidance for interpreting student performance on instructional assessments. In Grade 8 Advanced Mathematics, under the Course Level Documents,” is a document titled Grade 8 Advanced Mathematics Assessment Guide, and within this document is a section titled Guidance for Interpreting Student Assessment Performance. This section explains how to interpret student performance across different assessments and outlines steps educators can take. For example, one paragraph in this section states, “For assessment scoring and its relationship to student progress, consider the aggregate of all questions related to the same TEKS. Did the student master this TEKS by answering the vast majority of questions aligned to the TEKS correctly? What pattern(s) do you see in their incorrect answers? (e.g., a student answers all computation questions that provide a decimal are answered correctly but items that provide a fraction are answered incorrectly).”

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

The materials provide educators with guidance on using the included tasks and activities to respond to student performance trends on assessments. In Grade 8 Advanced Mathematics, each unit contains one or more performance tasks as a formative assessment option. For example, unit 1, lesson 2, “Solving One-Variable Inequalities” contains one performance task titled “Band.” This task contains four levels of complexity: simplified, scaffolded, on-level, and enriching. The page Performance Task: Unit 1, Lesson 2 guides educators on assigning tasks at each level and to “Differentiate tasks based on instructional assessments in the previous portions of the lesson,” such as the Exploration and Explanation portions.

The Unit 1 Diagnostic Assessment answer key also provides guidance. When opened, the Grade 8 Advanced Mathematics Test Answer Key & Diagnostic Guide is set up in a table format, with two sections as the document’s main focus. The first section contains “assessment of prerequisite content knowledge,” while the second contains “assessment of preconceptions & misconceptions.” Within each section, the table is further broken down into item number, content TEKS, correct answer (with work shown), and diagnostic information. The diagnostic information column guides educators in responding to student performance trends. An example Explanation in the diagnostic column is, “Item 1 addresses understanding of a Big Idea about equations. Students should be able to explain how they know a value makes an equation true, as they have encountered this in previous grade levels for simpler equations than they will encounter in Lesson 1 of Unit 1. If students are unable to do so, it is recommended that this item, along with an example equation and possible values, be used as a warmup question for Lesson 1 and students be purposefully paired to discuss their responses.”

Evaluation for 2.2c
Materials include tools for teachers to track student progress and growth, and tools for students to track their own progress and growth.

The materials include tools that help educators track student progress and growth. For example, in Grade 8 Advanced Mathematics, the Course Level Documents include a guide titled Grade 8 Advanced Mathematics Teacher Tracking Document. The document is divided into five main columns, with the last further divided into six sub-columns. The first column is for educators to enter student names; the second lists student strengths; the third notes what students are working on now; and the fourth lists the student’s next goal(s). The last column contains each grade-level TEKS, broken down into five assignment sub-columns and an overall progress column.

The materials include tools that help students track their own progress and growth. For example, in Grade 8 Advanced Mathematics under Course Level Documents, the Grade 8 Advanced Mathematics Assessment Guide includes the Printable Student Progress Tracking Tool, followed by a paragraph that explains how to use it. The tracking tool is a table with multiple columns. The first column lists the TEKS in the course with full standards language. Students use the next five columns to list any assignments that cover that specific TEKS within the course. Students use the final column to enter their overall progress for that TEKS. After the TEKS table, students list their Strengths, their “What I am Working On Now,” and “What My Next Goal Will Be.”

Another example of an available tool that lets students track their progress is on each unit’s Overview page. For example, on the Grade 8 Advanced Mathematics unit 2 Overview page, the 8AM Unit 1 English Letter describes the content taught in unit 2 and how families can support their students’ progress. Finally, a table at the bottom of the letter allows families and students to track student progress within the unit. The table has three columns: unit activity, assignments, and grades earned. Students and families can track students’ progress within each lesson.

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 materials include educator guidance for differentiated instruction, activities, and paired (scaffolded) lessons for students who have not yet reached proficiency on grade-level content and skills. For example, in “Grade 8 Advanced Mathematics, under the unit 1, lesson 1 Overview page, a section titled Paired (Scaffolded) Lessons & Guidance for Small Group Instruction contains a short worksheet and answer key for students who have not yet reached proficiency. The section states, “For students who have not yet reached proficiency on grade-level content and skills, use the paired (scaffolded) lesson below during the Explanation lesson phase and follow the teacher guidance below the table that contains the paired lesson activity. Convene a small group of students based on information from the Unit Diagnostic Assessment and Exploration activity for you to work with while the remainder of the class is working on the Explanation activity.” The worksheet and answer key include explicit educator guidance on using the activity sheet to best support students.

The materials include educator guidance for differentiated activities that support students who have not yet reached proficiency in grade-level content and skills. For example, in Grade 8 Advanced Mathematics, each lesson provides one or more performance tasks as formative assessments. Each task offers four levels of complexity: a simplified task for students who have not yet reached proficiency, a scaffolded task for students who are somewhat proficient, an on-level task for students who are proficient, and an enriching task for students who are highly proficient. The performance task’s Overview page gives educators guidance on how to distribute each task.

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 materials include pre-teaching of unfamiliar vocabulary and in-text references. In Grade 8 Advanced Mathematics, each unit and lesson includes a vocabulary list for educators at the bottom of the page, along with guidance on supporting students’ acquisition of new vocabulary terms. For example, unit 5, lesson 1 lists the vocabulary terms to preteach as constant of proportionality, constant of variation, and direct variation. The description above these terms states, “One way to support students’ acquisition of new vocabulary terms is to preteach the vocabulary terms using a strategy such as a Frayer model.”

The materials also include educator guidance in the Course Level Documents titled Grade 8 Advanced Mathematics Teacher’s Guide. This guide provides educators with a six-step process for building academic vocabulary: “1. Introduce the new term(s) by providing a student-friendly example, description, and/or explanation. 2. Students restate the example, description, and/or explanation in their own words. 3. Students create a non-linguistic representation of the term(s) (pictures, graphic, symbol, etc). 4. Engage students periodically in activities that deepen their understanding of the term(s). 5. Periodically involve students in discourse during which they discuss the term(s) with one another. 6. Involve students periodically in games that allow them to play with the term(s).”

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.

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The materials include educator guidance for differentiated extension activities for students who have demonstrated proficiency in grade-level content and skills. In Grade 8 Advanced Mathematics, each lesson within each unit includes one or more performance tasks assigned to students based on mastery of the lesson’s content. For example, unit 9, lesson 6, provides the performance task “Sales,” with four levels for each task: a simplified task for students who are not yet proficient, a scaffolded task for somewhat proficient students, an on-level task for proficient students, and an enriching task for highly proficient students. The Teacher Notes for Task prompt educators to “Differentiate tasks based on instructional assessment in the previous portions of the lesson.”

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 materials include explicit prompts to help the educator model and explain the concepts to be learned. For example, in Grade 8 Advanced Mathematics, in the unit 5, lesson 4 Explanation, there is a subsection titled Instructional Hints that provides the educator with the direct prompt, “Correlation is a quantitative statistical term that is used when one calculates the relationship between two variables and the strength of that relationship (correlation coefficient). Association is a qualitative term that describes how two variables are associated without calculating the exact relationship or strength of that relationship.” A second example is in the Exploration for unit 7, lesson 5 in the Instructional Hints, which provide the direct prompt, “Use the table feature of graphing technology to check answers. Enter the original expression into Y1 of the function editor and then answer into Y2. Compare the tables for equivalence.”

The materials include guidance to help the educator model and explain the concepts to be learned. For example, in Grade 8 Advanced Mathematics, in the unit 5, lesson 4 Explanation, there is a subsection titled Instructional Hints. This section provides the educator with guidance such as “In this lesson, focus on the descriptive term association and the types of association students can visually see in a graph and then use words to describe.” A second example is in the Exploration for unit 7, lesson 5, where the Instructional Hints state, “When using Algebra tiles, make explicit connections between the tiles and the algebraic representations. Also, make explicit connections among the operation, properties of algebra, and actions taken with the tiles.”

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

The materials include educator guidance and recommendations for effectively delivering and facilitating lessons using a variety of instructional approaches. For example, in the Grade 8 Advanced Mathematics course, all lessons follow the same structure. Each lesson features an Exploration section, an Explanation section, and one or more performance tasks. During the Exploration portion, guidance directs educators to group students in a visibly random way and have these groups watch a short video. Students then collaborate within their groups on an activity while the educator applies provided Questioning Strategies such as “When you say ___________, what do you mean?” and “How do you know your answer is correct?” This section also offers educators Instructional Hints and specific supports for emerging bilingual students.

After students complete the Exploration section, they move on to the Explanation section. Here, educator guidance instructs students to read specific sections of a book, then independently solve the “You try it!” problem. Educators support reasoning and productive struggle by using Questioning Strategies provided in this section. Educators also bring students back together as a whole group to discuss their answers before having students complete practice problems independently. This section provides educators with additional instructional hints, questioning strategies, and ideas for formative assessments.

Lastly, educators review the results from formative assessments in previous sections and assign a performance task to students. Each performance task contains four levels, and the guidance helps educators determine which level to assign based on students’ content mastery. For example, educators assign the simplified task to students who are not yet proficient, the scaffolded task to students who are somewhat proficient, the on-level task to students who are proficient, and the enriching task to students who are highly proficient.

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

The materials support multiple practice types and include guidance for educators and recommended structures to support effective implementation. In Grade 8 Advanced Mathematics, the unit 2, lesson 2, Exploration provides educators with recommended structures and practice types. For example, “Group students in a visibly random way (e.g., playing cards, drawing straws, spinner, etc.) into groups no larger than three students per group,” followed by collaborative activities. In this same lesson, the Explanation section directs educators to “Bring students back together as a whole group to debrief the content and examples” following a guided lesson.

Another example is found in the unit 5 Spiral Review 2 Overview page. The subsection Teacher Directions prompts educators to give individual students the Spiral Review and have them work on the task individually. Next, it states, “Students stand up and circulate around the room, partnering with as many different people as needed to give their answer and get their partner’s answer in return. Simple exchange of answers is not acceptable. Explanation should accompany the answer.”

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 materials include educator guidance on providing linguistic accommodations for various levels of language proficiency as defined by the ELPS. Support for Emergent Bilinguals provides prompts, accommodation recommendations, scaffolding, and Questioning Strategies for educators to implement. For example, in the unit 4, lesson 2 Exploration, under Support for Emergent Bilinguals, the materials guide the educator to “Ask students to read the debriefing questions in the exploration activity aloud. Ensure emergent bilingual students pronounce the high-frequency word ‘linear’ with accuracy (stressing the first syllable). Help students correct any mispronunciation and provide them with an opportunity to try again.” The materials provide scaffolding prompts for Pre-Production, Beginning, Intermediate, High-Intermediate, and Advanced proficiency levels. The prompt for the Beginning level states, “Accept drawn responses and/or spoken responses in the student’s native language to a same language partner.”

Additionally, unit 5, lesson 2, Support for Emergent Bilinguals provides a table with Spanish-English cognates to “foster student cross-linguistic connections in oral and written mathematical discourse and increase comprehension.” For example, “equation/ecuacion, fraction/fraccion, and solution/solucion.” Lastly, the materials include Questioning Strategies to support productive struggle and reasoning through questions. For example, in unit 5, lesson 2, “Clarifying Question: When you say __________, what do you mean?” or “Advancing Question(s) [ask and walk away] Do you know the y-intercept? If so, what is it? How could you use the slope formula?”

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

The materials include implementation guidance to help educators use the materials effectively in state-approved bilingual/ESL programs. The Grade 8 Advanced Mathematics Teacher’s Guide, Guiding Principles to Support Emergent Bilinguals, states, “Planning multimodal lessons that grow students’ listening, speaking, reading, and writing skills as they acquire more complex English language is recommended. Each lesson is aligned to at least one ELPS, and many suggestions for supporting emergent bilinguals can be found in lesson plans.” The materials provide a “six-step process for building academic vocabulary” that can be “used with the vocabulary terms listed in each Lesson Overview.” Each Lesson Overview contains Frayer models that support the six-step process via hyperlinked vocabulary terms. Additionally, Guidance for Teachers to Provide Linguistic Accommodations and Support Emergent Bilinguals provides a table that “outlines examples of teacher actions that support emergent bilingual students at any level of English language proficiency to make cross-linguistic connections through oral and written discourse.” For example, in the cross-curricular language acquisition domain of Reading and at the Pre-Production level of language proficiency, the educator’s actions are “Demonstrate gestures & provide pictures/ concrete objects related to academic vocabulary.”

The unit 8, lesson 3 Exploration and Support for Emergent Bilinguals sections include implementation guidance. For example, “Support students as they use routine language for classroom communication by engaging in a classroom discussion about the content in the video and their experiences with the activity sheet.” Prompts guide educators to ask, “How do you know when to talk with your partner and when to focus on the whole-class discussion?” and “What are the key differences between multiplying like bases and dividing like bases?”

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 materials include embedded guidance for educators to support emergent bilingual students in developing academic vocabulary and increasing comprehension through oral and written discourse. For example, in the unit 6, lesson 1 Exploration, under Support for Emergent Bilinguals, the materials state, “Addressing ideas and opinions is important in mathematical discourse. Placing emergent bilinguals in small groups with their peers provides a safe environment for them to sharpen their skills in listening to and speaking the English language while they are learning about mathematics.” Next, the materials direct educators to “Pose the following questions one at a time, directing students to show you with gestures or pictures or tell you in their own words: How do you see the table changing from row to row? How do you see the graph changing as the x-values increase?”

The materials include embedded guidance for educators to support emergent bilingual students in developing academic vocabulary, building background knowledge, and making cross-linguistic connections through oral and written discourse. In the unit 2, lesson 2 Exploration, under Instructional Hints, the materials provide guidance and prompts for educators such as “Remind students to look for the input values (independent variable) and output values (dependent variable)” and “Use technology to generate graphs from tables or symbolic representations as appropriate. Often, students will see the multiple output values in a graph before they will in a list, mapping, or table.” In the unit 2, lesson 2 Exploration, Support for Emergent Bilinguals offers additional guidance and prompts, such as “Addressing ideas and opinions is important in mathematical discourse. Placing emergent bilinguals in small groups with their peers provides a safe environment for them to sharpen their skills using the English language while they are learning about mathematics. In formal classroom interactions with their peers, provide a word bank of newly acquired content area vocabulary, including function and relation, for emergent bilingual students to refer to as they demonstrate their acquisition of academic language using sentence stems such as: This _____ (is/is not) a _____ because....” and “To evaluate this function, you need to....” The materials direct educators to “Use Spanish-English cognates to foster student cross-linguistic connections in oral and written mathematical discourse and increase comprehension.” For example, in unit 5, lesson 2: solution/solucion and fraction/fraccion.

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 throughout a lesson and/or unit (including instructional assessments) that require students to demonstrate depth of understanding aligned with the TEKS. Exploration activities frequently scaffold from mathematical contexts to real-world applications. For example, in TEKS 8AM.3A, Unit 3, Lesson 1, “Developing Slope with Similar Right Triangles,” the Blackline Master (Student Pages) Exploration document develops the concept of slope from patterns in similar triangles oriented in the same direction. In the Explanation phase of the lesson, the ePub provides students the opportunity to respond to a “You Try It!” prompt and practice questions that continue to require students to develop further and demonstrate a depth of understanding aligned with the TEKS. Finally, the performance task provides differentiated opportunities for students to demonstrate their depth of understanding of the TEKS.

The materials provide practice opportunities throughout a lesson to demonstrate depth of understanding aligned with the TEKS. For example, in unit 7, lesson 3, “Multiplying Polynomials (TEKS 8AM.10B),” the Explanation component provides questions that progress in rigor. For example, the first set of questions asks students to multiply given polynomials, then a subsequent question asks, “Kami was simplifying the expression (2x – 3)(4x + 5) on a math exam. She got the answer 8x - 15, but that answer was not one of the choices. What mistake did Kami make in simplifying, and what should her answer have been?”

Instructional assessments require students to demonstrate depth of understanding aligned to the TEKS over the course of a unit. For example, in alignment with TEKS 8AM.10.E and 8AM.10.F, Unit 7 Polynomial Expressions (Laws of Exponents & Factoring), Diagnostic Assessment, section 7B, question 4 directs students to “Write a simplified expression that you think is equivalent to 1.25(8n – 2p) – 3.5(4n + 6p).” In unit 7, lesson 5, the 8AM.10E Performance Task—Quilts (formative assessment) states, “Tena made a 3-foot by 4-foot rectangular baby quilt for her best friend. She had and used all 16 square feet of a solid fabric to make a border around the quilt. When making the border, Tena made its width the same all around the quilt. To be sure the quilt looked the same no matter which side of it was facing up, she made sure she had the same amount of border fabric on the top and bottom sides of the quilt. If Tena used all 16 square feet of the solid fabric to make the border, how wide was the border? Justify your reasoning.” The unit 7 Test requires students to demonstrate depth of understanding aligned to the TEKS. For example, a multiple-choice question asks, “A rectangle has a width of (2x – 5) and a length of (x – 3). What is the expression that represents the area of the rectangle?”

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

The materials provide questions and tasks that progressively increase in rigor and complexity, leading to grade-level proficiency in the mathematics TEKS. The diagnostic assessment, the performance task for each lesson, and the Unit Test include questions and tasks that demonstrate progression. For example, a question in the unit 8 Diagnostic Assessment states, “The area of a square rug is 210 square inches. What is the approximate length of one side of the rug?” (TEKS 8AM.2B) A final question demonstrates an increase in rigor: “A right triangle has a leg with a length of 4 meters and a hypotenuse that is 13 meters long. What is the length of the other leg of the right triangle in meters?” (TEKS 8AM.7C). In Unit 8, “Quadratic Equations,” in lesson 1, “Solving Quadratic Equations by Factoring,” in the 8AM.8A Performance Task—Golf Course, the task description states, “Students evaluate a real-world situation about the projectile motion of a golf ball that can be modeled using a quadratic function. Students use a method for solving the equation to find how long the ball was in the air and how long it takes to reach maximum height.” Lastly, the TEKS-aligned, summative Unit Assessment allows students to demonstrate mastery of learning objectives. For example, the unit 2, “Functions Introduction” Unit Test presents questions that progress in rigor and complexity. The first set of multiple-choice questions assesses understanding of the concept of “a function as a special dependency relationship between two variables in which any given value of the independent variable will only generate one corresponding value of the dependent variable.” Increasing in complexity, a final question states, “The height of a football kicked into the air is given by the function below, where h is the height of the ball in feet and t is time in seconds. What is the range of the function?”

The Explanation component of each lesson demonstrates questions and tasks that progressively increase in rigor and complexity, leading to grade-level proficiency in the mathematics TEKS. For example, in unit 7, lesson 3, “Multiplying Polynomials,” the Explanation component links to the ePub: Multiplying Polynomials. The first set of questions asks students to multiply given polynomials. As questions continue, they progress in rigor. A final question states, “On a recent road trip, Patrick traveled at an average rate of 3x + 5 miles per hour and reached his destination in 4x - 1 hours. Write an expression in standard form that represents the distance Patrick traveled (d = rt).”

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

The materials demonstrate coherence across units by explicitly connecting patterns, Big Ideas, and relationships between mathematical concepts. The Grade 8 Advanced Mathematics Scope and Sequence, Rationale for Unit Progression, explains that the materials are directly correlated with the Texas Response to the Curriculum Focal Points (TxRCFP) “to create TEKS-aligned instructional materials in a logical progression,” and the process standards are incorporated throughout the entire course. For example, to demonstrate a strategic sequence, the rationale states, “Unit 1 equips students with the tools they will need to work with real numbers that are a TxRCFP grade-level connection to rational numbers from previous grade levels, but for the first time including irrational numbers that are needed for application of the Pythagorean Theorem. In Unit 2, students calculate and interpret data with a single variable using visual and symbolic representations. Unit 3 focuses on symbolic manipulation of equations and inequalities to solve for a single variable. Unit 4 builds from the single-variable concepts and skills students have learned in previous grade levels and honed in the first three units of the course to begin to develop the idea of slope as a constant rate of change in linear relationships and the meaning of both x- and y-intercepts of linear functions.”

The materials demonstrate coherence across units by explicitly connecting patterns, Big Ideas, and relationships between mathematical concepts in the unit 8 Overview page. Within the Unit Overview section, the materials state, “Quadratic functions are new to students in Grade 8 Advanced Math—Algebra 1. Students have seen some of the building blocks that lead up to this unit. Students began solving equations (one-variable, degree one) in Grade 6 Advanced Math, extended their knowledge in Grade 7 Advanced Math, and further extended their knowledge in Unit 1 of Grade 8 Advanced Math—Algebra 1. In this unit, students apply what they know about solving equations with one variable that are of degree one to solving equations with one variable that are of degree two.”

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

The materials demonstrate coherence across units by connecting content learned in previous courses/grade levels to content to be learned in the current course/grade level on each lesson Overview page. For example, on the Grade 8 Advanced Mathematics unit 6, lesson 2 Overview page, a Prior Learning Supports section connects the current unit to 6th- and 7th-grade expectations. An example provided from the 6th-grade expectations states, “In Grade 6th Advanced Mathematics, students graphed the solution set of one-variable, one- and two-step inequalities on a one-dimensional number line,” and in 7th grade, “Students graphed the solution set of one-variable, two-step inequalities on a one-dimensional number line.”

The materials demonstrate coherence across units by connecting the language learned in previous courses/grade levels to the content to be learned in the current course/grade level within each lesson Overview page. For example, the Grade 8 Advanced Mathematics unit 6, lesson 2 Overview page features a Vocabulary section that introduces new terms and encourages students to engage with them using a Frayer model template. In unit 6, lesson 2, students encounter and connect the terms intersection point and system of inequalities.

The materials demonstrate coherence across units by connecting what students will learn in future courses/grade levels to the content in the current course/grade level within each lesson Overview page. For example, on the Grade 8 Advanced Mathematics unit 6, lesson 2 Overview page, a Prior Learning Supports section connects the Grade 8 topic to Algebra 1 topics. An example provided from Grade 8/Algebra 1 states, “In Unit 4 of Grade 8 Advanced Mathematics, students graph the solution set of one, two-variable linear inequality on a two-dimensional coordinate plane. In doing so, they connected the dashed boundary line on a coordinate plane with the open circle point on a number line and the shaded region above or below the line on a coordinate plane with the ray to the left or right of the endpoint on a number line.”

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

The materials demonstrate coherence at the lesson level by connecting students’ prior knowledge of concepts and procedures from the current grade level to new mathematical knowledge and skills within each lesson’s Overview page. For example, in Grade 8 Advanced Mathematics, the unit 7, lesson 1 Overview page features a Prior Learning Supports section that highlights connections within Grade 8 Advanced Mathematics. This section states, “In unit 8, students will simplify radical expressions with numbers, and then extend the laws of exponents to include rational exponents where the numerator represents the power and the denominator represents the root.” Unit 7, lesson 1 covers simplifying expressions with integral powers.

The materials demonstrate coherence at the lesson level by connecting students’ prior knowledge of concepts and procedures from the prior grade levels to new mathematical knowledge and skills within each lesson’s Overview page. For example, in Grade 8 Advanced Mathematics, the unit 7, lesson 1 Overview page features a Prior Learning Supports section that highlights connections from Grade 6 Advanced Mathematics to Grade 8 Advanced Mathematics. Unit 7, lesson 1 covers simplifying expressions with integral powers. The Grade 6 Advanced Math connection says, “In Grade 6 Advanced Math, students were introduced to exponents while they generated equivalent numerical expressions using the order of operations.” In Grade 8 Advanced Mathematics, students are expected to “apply what they know about exponents to look for number patterns and generate the laws of exponents for integral exponents.”

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

The materials provide spaced retrieval opportunities with previously learned skills and concepts across lessons and units. For example, in Grade 8 Advanced Mathematics, most units contain one or more Spiral Reviews. These provide educators with information on which units, lessons, and TEKS are spiraled through the review. The unit 7 Spiral Review states, “This spaced retrieval and interleaved practice opportunity across lessons and units addresses the following TEKS from previous instruction,” followed by a list of units, lessons, and TEKS. Examples provided from this unit Spiral Review include unit 2, lesson 2: 8AM.12C, unit 7, lesson 1: 8AM.11B, unit 7, lesson 5: 8AM.10E, and unit 7, lesson 6: 8AM.10C. The Spiral Review is a small-group activity with students in groups of three or four. Teams compete to have the most points by the end of the “trashketball” game. The review contains open-ended questions specific to each group of students.

Another example can be seen in the unit 6 Spiral Review 2, which states, “This spaced retrieval and interleaved practice opportunity across lessons and units addresses the following TEKS from previous instruction,” followed by a list of units, lessons, and TEKS. Examples provided from this Spiral Review include unit 1, lesson 1: 8AM.5A, unit 2, lesson 3: 8AM.2A, unit 4, lesson 4: 8AM.2C, unit 6, lessons 3 and 4: 8AM.2I, and unit 6, lessons 5 and 6: 8AM.5C.

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

The materials provide interleaved practice opportunities with previously learned skills and concepts across lessons and units. For example, in Grade 8 Advanced Mathematics, most units contain one or more Spiral Reviews. These provide educators with information on which units, lessons, and TEKS are spiraled through the review. The unit 4 Spiral Review states, “This spaced retrieval and interleaved practice opportunity across lessons and units addresses the following TEKS from previous instruction,” followed by a list of units, lessons, and TEKS. Examples from this unit Spiral Review include unit 3, lesson 5: 8AM.3D; unit 3, lesson 5: 8AM.3C, 8AM.3D; unit 4, lesson 2: 8AM.3F; and unit 4, lesson 3: 8AM.3F, 8AM.3H.

Another example is found in the unit 5 Spiral Review 1, which states, “This spaced retrieval and interleaved practice opportunity across lessons and units addresses the following TEKS from previous instruction,” followed by a list of units, lessons, and TEKS. Examples provided from this unit Spiral Review include unit 1, lesson 1: 8AM.5A, unit 2, lesson 3: 8AM.2A, unit 4, lesson 4: 8AM.2A, 8AM.2C, unit 4, lesson 5: 8AM.2H, and unit 5, lesson 1: 8AM.2A, 8AM.2D.

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

Questions and tasks require students to interpret models and representations for mathematical concepts in the unit 4, lesson 5 performance task of the Grade 8 Advanced Mathematics course. For example, the enriching performance task, “Glasses and Speakers,” directs students to read a real-world application problem about virtual reality glasses profits and Bluetooth speaker profits. The question states, “The company wants to show a profit greater than $600 in the first week.” Students use this information to create a graphical representation of the problem, then interpret it to answer various questions. One question asks, “Will the company meet its goal for profits if they sell 2 pairs of virtual reality glasses and 11 speakers in the first week?” Another question states, “If the company sells 10 speakers in the first week, what is the fewest number of virtual reality glasses they must sell to meet the profit goal?”

Questions and tasks require students to analyze and evaluate models and representations for mathematical concepts and situations in Exploration: unit 7, lesson 5 of the Grade 8 Advanced Mathematics course. For example, the Exploration section includes an activity sheet in which students use algebra tiles to factor a trinomial when a = 1. The activity sheet has three columns: symbolic (where the quadratic expression is written), concrete (where students sketch the algebra tiles they use to represent the situation), and verbal description (where students write, in words, what their algebra tile representation means). One example is already provided, with all areas of the table completed for reference. The next four problems are blank, where students will complete the tables on their own. Further down in the worksheet, students choose two factors of a given quadratic expression. Later, students factor independently.

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

Questions and tasks require students to create models to represent mathematical situations in Exploration: unit 6, lesson 1 of the Grade 8 Advanced Mathematics course. The Exploration section provides educators with an activity sheet for students to graph systems for real-world problems. The first question states, “The Tucker family will rent a car for a day trip. Mrs. Tucker has picked out two rental car companies to choose from.” Students are given information on two car companies, A and B. Each company provides the per-day cost and the per-mile cost. Students create a table for each car company that lists the miles, the mathematical process, and the daily cost. The Miles portion of the table is already filled out for students. From there, students graph each equation on a coordinate plane. Students use the graphed system to answer questions.

Questions and tasks require students to create models to represent mathematical situations in Exploration: unit 8, lesson 4 of the Grade 8 Advanced Mathematics course. The Exploration section includes an activity sheet on completing the square. The first question directs students to use algebra tiles to model a given quadratic equation and sketch the tiles used. The following questions have students change their algebra tiles into a square, sketch the result, and use their new algebra tiles to record the resulting polynomial. Students use this algebra tile process to factor additional polynomials, sketching each step with their algebra tiles to help them find their Combined Expression.

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

Questions and tasks provide opportunities for students to apply conceptual understanding to new problem situations and contexts in the on-level performance task: unit 10, lesson 2 in the Grade 8 Advanced Mathematics course. Each lesson includes one or more performance tasks, and educators assign students different levels based on their understanding and ability. The Performance Task “Yeast Culture Cells” covers graphing exponential functions and states, “A scientist is studying a yeast culture. The scientist models the yeast population mathematically to examine the number of yeast cells in the culture per hour using the function f(x) = 85(2)^x/2.” The problem asks students questions about the situation, such as, “What is the meaning of the asymptote of the function in terms of the situation?” and “Describe the domain and range related to the yeast culture.”

Questions and tasks provide opportunities for students to apply conceptual understanding to new problem situations and contexts in Explanation: unit 6, lesson 2, in the Grade 8 Advanced Mathematics course. Educators assign students practice problems from the student’s course workbook. The assignment covers graphing systems of linear inequalities, and one problem states, “A system of linear inequalities is created by all points where the y-values are less than zero and the x-values are greater than zero. In which quadrant of the coordinate plane will the solution to the system of linear inequalities lie?” A graph is available for students to use. Another question states, “Susan has up to $60 to spend at the used book store annual sale and plans to buy at least 1 book. All hardback books cost $12 and all paperback books cost $4. This relationship is shown in the graph below,” and a graph follows. Then the question asks, “If Susan decides to buy 3 books in hardback form, what numbers of paperback books are possible for her to buy?”

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

The materials provide tasks designed to build the automaticity and fluency students need to complete grade-level tasks. In Grade 8 Advanced Mathematics, students participate in Spiral Reviews for each unit. Some units have more than one Spiral Review. These reviews indicate which units and TEKS are present in the specific Spiral Review, then assign a task. For example, the unit 1 Spiral Review states, “This spaced retrieval and interleaved practice opportunity across lessons and units addresses the following TEKS from previous instruction: ‘Unit 1, Lesson 1: 8AM.5A’ and ‘Unit 1, Lesson 2: 8AM.5B’.” The Spiral Review activity asks students to practice prior skills in solving one-variable equations and inequalities. Another example comes from the unit 5 Spiral Review 1, which states, “This spaced retrieval and interleaved practice opportunity across lessons and units addresses the following TEKS from previous instruction: ‘Unit 1, Lesson 1: 8AM.5A’, ‘Unit 2, Lesson 3: 8AM.2A’, ‘Unit 4, Lesson 4: 8AM.2A, 8AM.2C’, ‘Unit 4, Lesson 5: 8AM.2H’, and ‘Unit 5, Lesson 1: 8AM.2A, 8AM.2D’.” This review activity asks students to practice prior skills in writing linear equations and inequalities.

The materials provide tasks designed to build the automaticity and fluency students need to complete grade-level tasks on the Grade 8 Advanced Mathematics unit 4, lesson 4 Overview page. This page provides educators with a Prior Learning Supports section and states, “Students began working with variables in equations in 4th grade and 5th grade where they represented multi-step problems involving the four arithmetic operations with whole numbers using equations with a letter standing for the unknown quantity (4.5A, 5.4B). In Grade 6 Advanced Math, students extended their understanding of using variables to represent an unknown to represent equations with two variables. Here, students represented additive relationships and multiplicative relationships using multiple representations (6AM.9C). The student is expected to represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. In Grade 7 Advanced Math, students combined additive and multiplicative relationships into the generic y = mx + b, or slope-intercept, form of a linear relationship (7AM.9A). The student is expected to represent linear proportional and non-proportional relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b. In this Grade 8 Advanced Mathematics lesson, students write linear equations from verbal descriptions, tables, and graphs. In Unit 5, students will write linear equations in multiple symbolic forms, including slope-intercept, point-slope, and standard forms.” The first task to build fluency and automaticity is in the Exploration section of this Overview page. Students are given an activity sheet on writing linear equations. They read an application problem, then create a table, write a function, and graph the situation. Students move on to the Explanation section, which covers their coursework and again emphasizes practicing writing linear equations. Finally, students complete a performance task on shoes.

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.

The materials give students opportunities to practice applying efficient, flexible, and accurate mathematical procedures within the lesson and/or throughout a unit. For example, in Grade 8 Advanced Mathematics, the unit 8, lesson 1 Overview page contains an activity sheet for students on solving quadratic equations by factoring. One question asks students to “Look at the given example and then try one on your own. Solve by factoring: 2x^2 – 5x – 3 = 0.” The steps to solve by factoring follow, including both a written explanation and a visual for students to reference. Then the question has students “Solve by factoring: x^2 – 3x – 10 = 0.” A second question on the activity sheet asks students to “Solve each of the following by factoring” and provides four different types of quadratic equations to factor. The four equations include a regular quadratic equation, a difference-of-squares equation, a binomial where you can take out only a greatest common factor, and an equation with a greatest common factor that can be factored further, leaving students to decide which method best fits each question. Continuing on in unit 8, the lesson 4 Overview page contains an activity sheet for students on solving quadratic equations by taking the square root. The sheet provides students with a worked-out example of solving a quadratic equation by taking the square root, and two questions follow: “Check these answers by substituting the values into the original equation to see if it makes a true statement” and “Solve 3x^2 + 7 = 55 using the method shown in the example.”

The materials give students opportunities to practice applying efficient, flexible, and accurate mathematical procedures within the lesson and/or throughout a unit. For example, in Grade 8 Advanced Mathematics, the unit 8, lesson 5 Overview page includes an activity sheet for students to solve quadratic equations using the quadratic formula. The directions provide the quadratic formula and ask students to recall it, ensure the equation is in standard form, and then substitute the values for a, b, and c into the formula. In the first two questions, students practice plugging the a, b, and c values into the quadratic formula without simplifying or solving. The second question is not in standard form, and a hint states, “Rewrite so that the squared term is positive.” The next two questions ask students to substitute the values into the quadratic formula, then simplify just the expression under the square root. Neither equation is in standard form, but this time no hint is given. The last question on this page requires students to fully use the quadratic formula to solve an equation already in standard form. An example from a different unit appears in the unit 9 Spiral Review 3. The activity sheet provides practice in solving quadratic equations. One question asks students to solve three different equations by factoring, and another asks them to solve three equations by taking square roots, completing the square, or using the quadratic formula.

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.

The materials provide opportunities for students to evaluate procedures, processes, and solutions for efficiency, flexibility, and accuracy within the lesson and throughout a unit in the Exploration: unit 6, lesson 2 section. Educators assign students an activity sheet on graphing systems of linear inequalities. For example, one question states, “For one of the graphs of two linear inequalities, select a point in each of the four regions. Substitute the x- and y-coordinates into both inequalities. Which of the four points makes both inequalities true? Find a partner who chose the other graph and check one another’s work.” Another question states, “How many other points would make both inequalities true if you substituted the coordinates into both inequalities?”

The materials provide opportunities for students to evaluate procedures, processes, and solutions for efficiency, flexibility, and accuracy within the lesson and throughout a unit in the Explanation: unit 6, lesson 5 section. In the Instructional Hints, Support for Emergent Bilinguals, and Questioning Strategies sections, educators are assisted in guiding student thinking by providing Questioning Strategies and hints such as, “Use a problem-solving template as needed to support students as they move through the problem-solving process,” “Encourage students to determine when each algebraic method (substitution or elimination) is more appropriate for which types of equations,” “Engage students in a discussion about the possible number of solutions for a system of linear equations that they noticed and applied,” “How are the equations the same? How are they different?,” “How do you know which method is best for a particular system of linear equations?,” and “How do you know your solution is reasonable?”

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

The materials include embedded supports to guide educators in leading students toward increasingly efficient approaches across various lessons in Unit 8 of the Grade 8 Advanced Mathematics course. For example, each lesson Overview page features a section titled Overview of Process Standards that explains how the lesson incorporates certain process standards. On the unit 8, lesson 1 Overview page, this section states, “In this lesson, students select tools, such as pencil and paper or technology, and techniques, including mental math, number sense, and fact fluency, to solve quadratic equations by factoring the quadratic polynomial when the equation is in standard form. Students may: Select technology as a tool to check their answers. Select pencil and paper, along with mental math, to factor a quadratic expression and then solve the equation. Select a technique like fact fluency to look for patterns in numbers, particularly factor pairs.” Another section in each lesson, Prior Learning Supports, describes how the lesson builds on similar content from previous lessons and grade levels. The example from unit 8, lesson 1 states, “Solving quadratic equations is a new skill to students in Grade 8 Advanced Mathematics. However, some antecedent concepts and skills support students as they do. In Unit 7 of Grade 8 Advanced Mathematics—Algebra 1, students factored quadratic (degree two) expressions using a variety of techniques, including greatest common factors (inverse of the distributive property), perfect square trinomials, difference of squares binomials, and good old-fashioned ‘guess and check’ or ‘trial and error’ (8AM.10D). The student is expected to rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property. (equivalent to A.10D) 8AM.10E The student is expected to factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two (equivalent to A.10E) (8AM.10F). The student is expected to decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial (equivalent to A.10F). In Unit 1 of Grade 8 Advanced Mathematics—Algebra 1, students solved linear equations (degree one) with one variable, building on their middle school experiences of solving one-variable equations of degree one (8AM.5A). The student is expected to solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides (equivalent to A.5A). A third section on each Lesson Overview page, Teaching Hints, gives educators hints and information to assist students, for example: “Connect back to students’ prior experiences with factoring. If necessary, rewrite the quadratic equation in standard form and then look for patterns such as: Perfect squares and Common factors. This lesson focuses on solving quadratic equations that can be factored into linear terms, and then applying the Zero Product Property to determine what value of x (or whichever variable is in the equation) makes each factor equal to 0. Subsequent lessons explore additional methods to solve quadratic equations. Use technology to check answers. Graph the original quadratic expression as though it were an equation and see where it crosses the line y = 0 (x-axis).”

The materials include embedded supports to guide educators in leading students toward increasingly efficient approaches in various Unit 8 lessons of the Grade 8 Advanced Mathematics course. For example, each Lesson Overview page features a section titled Overview of Process Standards that explains how the lesson incorporates certain process standards. On the unit 8, lesson 5 Overview page, this section states, “In this lesson, students apply what they know about solving a quadratic equation to solve problems that present themselves in everyday life, society, and the workplace. Students may: Solve a quadratic equation using the quadratic formula in a geometric context such as area or volume. Solve a quadratic equation, using the quadratic formula, involving a scientific context such as projectile motion.” Another section in each lesson, Prior Learning Supports, describes how the lesson builds on similar content from previous lessons and grade levels. The example from unit 8, lesson 5 states, “Solving quadratic equations is a new skill to students in Grade 8 Advanced Math—Algebra 1. However, some antecedent concepts and skills support students as they do so. Also, earlier in this unit, students solved quadratic equations using factoring and inverse operations (including completing the square when necessary). In Unit 1 of Grade 8 Advanced Mathematics—Algebra 1, students solved linear equations (degree one) with one variable, building on their middle school experiences of solving one-variable equations of degree one. 8AM.5A The student is expected to solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides (equivalent to A.5A). In Unit 7 of Grade 8 Advanced Mathematics—Algebra 1, students factored quadratic (degree two) expressions using a variety of techniques, including greatest common factors (inverse of the distributive property), perfect square trinomials, difference of squares binomials, and good old-fashioned ‘guess and check’ or ‘trial and error.’ In this lesson, in order to take the square root of both sides of the equation as an inverse operation, the quadratic equation must be written in vertex form, a(x - h)2 + k, so that there is a binomial square that can be unsquared in order to get x by itself without an exponent (8AM.10E). The student is expected to factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two. (equivalent to A.10E) Previously in Unit 8, students solved quadratic equations by factoring and inverse operations. In this lesson, students formalize the quadratic formula, which involves completing the square for a generic quadratic polynomial in standard form.” A third section on each lesson Overview page, Teaching Hints, gives educators hints and information to assist students, for example: “The quadratic formula works for equations in standard (polynomial) form that are equal to 0. You may need to review symbolic manipulation for students to rewrite equations in that form as needed. The quadratic formula will work for any quadratic equation. If the discriminant is positive, the equation has real solutions. If the discriminant is a perfect square, the equation has rational solutions. If the discriminant is negative, avoid telling students the quadratic equation has no solutions; that’s an incorrect statement. The equation has no real solutions and when students learn about complex numbers in Algebra 2, they will be able to solve this type of quadratic equation.”

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

The materials explicitly state how the lesson addresses the conceptual and procedural emphasis of the TEKS. The lesson components, including a Lesson Overview, Learning Outcomes, an Overview of the Process Standards, and Teaching Hints, describe how the lesson addresses the TEKS’s conceptual and procedural emphasis. For example, in unit 4, lesson 3, “Transformations of Linear Functions,” the Lesson Overview—Focusing Question states, “How can I relate all linear functions to the parent linear function?” The Learning Outcomes state that students will “graph linear functions, identify key features of linear functions, and explain the effects of various transformations on a linear function.” The materials address the procedural emphasis of the TEKS in the Overview of the Process Standards. The overview states, “In this lesson, students display, explain, and justify mathematical ideas and arguments about how different parameters affect the graph of the linear parent function, y = x. Students may employ these processes to master content in the following ways: display their thinking about how a parameter impacts the graph of the linear parent function using words, symbols, and visual representations such as graphs; explain how a parameter impacts the graph of the linear parent function; and justify arithmetically or graphically why that parameter impacts the graph of the linear parent function in that way.”

In unit 3, lesson 1, “Developing Slope with Similar Right Triangles,” the Learning Outcomes state the conceptual emphasis of the TEKS in a student-friendly statement: “I can understand slope as a rate comparing the change in y-values to the change in x-values for a linear relationship between the two variables, x and y.” The Overview of Process Standards section states the procedural emphasis of the TEKS. For example, students will construct several similar right triangles to determine slope, and use a table to organize and record data.

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.

Questions and tasks include the use of concrete models and manipulatives as required by the TEKS. The materials include a Blackline Master for Concrete Models and an answer key that provides a manipulative exemplar. For example, in unit 4, lesson 2, “Graphs and Features of Linear Functions,” the Blackline Master for independent practice directs students to cut out cards, sort them, and match them to “each description, coordinate point, slope, and interpretation with its appropriate situation.” Questions and tasks include pictorial representations (figures/drawings) and abstract representations, as required by the TEKS. The 8AM.3F Performance Task—Serenity Village states, “To solve for the function’s zero and other specific values, students can graph the function rule on a graphing calculator to use the table of values and calculator functions, or they can apply algebraic processes to the function rule.”

Questions and tasks include the use of concrete models, pictorial representations (figures/drawings), and abstract representations, as required by the TEKS. For example, in unit 7, lesson 3, “Multiplying Polynomials,” under Exploration, the materials direct students to “use Algebra Tiles to model and carry out operations on polynomials, recording the concrete, verbal, and symbolic representations of each step.” The Lesson Overview and Teaching Hints show how the questions and tasks align with using concrete models and abstract representations. The hints include, “When using algebra tiles, make explicit connections between the tiles and the algebraic representation. Make explicit connections among the operation, properties of algebra, and actions taken with the tiles (e.g., remove 3 tiles means subtract 3 of what the tile represents). Emphasize connections to the area model for multiplication. Students saw this for whole numbers in elementary math, rational numbers earlier in middle school math, and with polynomials in Grade 8 Advanced Math—Algebra 1.”

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

The 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 7, lesson 5, “Factoring Polynomials,” the Overview of Process Standard(s) defines how students will “analyze mathematical relationships involving polynomial expressions and their factors.” The materials encourage students to “Analyze number patterns and factor patterns to determine if the terms of a trinomial represent perfect squares, connecting the representation to communicate the factors of the trinomial. Analyze factor patterns to determine if a trinomial has two real-number factors. Analyze number patterns to determine if a binomial is a difference of two squares, and if it is, communicate the factors of the binomial.” Additionally, the Teaching Materials and Student Materials direct educators and students to use graphing technology, algebra tiles, paper, and pencil. In the performance task lesson component, 8AM.10E Performance Task—Ladders, students must “apply operations with polynomials to solve a problem involving a ladder leaning against a house wall.” To solve, students can “draw and label a picture to visualize the problem.” They can “use a graphing device to solve for the zeros of the trinomial rather than factoring, or they can use a guess and check strategy.” As the students complete each step to solve the problems, the materials direct students to explain and justify their reasoning.

The materials provide further opportunities for the students to connect, create, define, and explain concrete and representational models to abstract (symbolic/numeric/algorithmic) concepts, as required by the TEKS. For example, in unit 1, lesson 1, “Solving One-Variable Equations,” in the Overview of Process Standards, the instructions direct students to “Select an appropriate tool such as a manipulative like algebra tiles, graphing technology, or paper and pencil. Select a technique such as mental math when adding, subtracting, multiplying, or dividing integers. Select a technique, such as properties of algebra or real numbers, to isolate the variable and solve the equation.” The Exploration titled Solving Equations allows students to define and explain. For example, a question asks, “How can you determine whether a one-variable equation has no solution, one solution, or many solutions?”

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

The materials provide opportunities for students to develop academic mathematical language using visuals, manipulatives, and other language-development strategies on the Grade 8 Advanced Mathematics Lesson Overview page. For example, the unit 5, lesson 3 Overview page contains Teaching Hints that state, “Graphing technology may allow you to create additional sets of parallel and perpendicular lines in order to explore the relationships among their slopes.” Further into the Overview page is a section titled Vocabulary that states, “One way to support students’ acquisition of new vocabulary terms is to preteach the vocabulary terms using a strategy such as a Frayer model. For these key terms, we provide partially completed Frayer models for students to add to as they move through the lesson and develop a deeper understanding of each term. Click on the term to see the Frayer model.” Students receive a Frayer model template along with a list of important vocabulary terms. The lesson includes an Exploration section where students complete an activity sheet on parallel and perpendicular lines. This sheet contains linear graphs in different colors, and students determine the slope of the lines and use this information to answer questions. In the Explanation section, students read about parallel and perpendicular lines and see visuals of both.

The materials give students opportunities to develop academic mathematical language using visuals, manipulatives, and other language-development strategies on the Grade 8 Advanced Mathematics Lesson Overview page. For example, the unit 9, lesson 1 Overview page contains Teaching Hints that state, “As you move through the different transformations, use graphing technology to look for patterns in the graphs and tables. Students may explain the transformation from the graph, but sometimes they need the values in the table to justify why the transformation happened. A vertical and horizontal dilation (compression or stretch) can look the same. Use the table of function values to distinguish between the two.” Further into the Overview page is a section titled Vocabulary that states, “One way to support students’ acquisition of new vocabulary terms is to preteach the vocabulary terms using a strategy such as a Frayer model. For these key terms, we provide partially completed Frayer models to which students should add as they move through the lesson and develop a deeper understanding of the vocabulary term. Click on the term to see the Frayer model.” Students receive a Frayer model template along with a list of important vocabulary terms. In the Exploration section of this lesson, students complete an activity sheet on parent quadratic function transformations. The directions state, “1. Examine the given function and record the value of a. 2. Sketch what you think the graph will look like. 3. Graph the function using a graphing calculator. 4. Sketch your graph and describe how it is different from the graph of f(x) = x^2.” A table follows, split into four columns based on the four steps above. The first example is completed for students. Students also receive blank graphs.

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

The materials include embedded educator guidance to scaffold and support students’ development and use of academic mathematical vocabulary in context. For example, in Grade 8 Advanced Mathematics, educators can access a Teacher’s Guide on the “Course Overview” page. Within this document is a section on scaffolds that states, “Clarifying and focusing questions are included in multiple portions of each lesson to support educators in scaffolding student learning. Supports for emergent bilinguals are provided throughout the course to scaffold listening and speaking during mathematical discourse, as well as reading and writing to acquire and express understanding, including academic and classroom vocabulary. Prior learning supports are included for each lesson to connect prior learning that may be essential for understanding the new learning taking place in the classroom.” A little more than halfway through the Instructor’s Guide is a section on building vocabulary using a six-step process: “1. Introduce the new term(s) by providing a student-friendly example, description, and/or explanation. 2. Students restate the example, description, and/or explanation in their own words. 3. Students create a non-linguistic representation of the term(s) (picture, graphic, symbol, etc.). 4.Engage students periodically in activities that deepen their understanding of the term(s). 5. Periodically involve students in discourse during which they discuss the term(s) with one another. 6. Involve students periodically in games that allow them to play with the term(s).

The materials include embedded educator guidance to scaffold and support students’ development and use of academic mathematical vocabulary in context in the Unit 8 Advanced Mathematics course under the Exploration section of unit 3, lesson 5. A section is labeled for emergent bilingual students, but these scaffolds can be used for all students. This section states, “Provide students with the opportunity to seek clarification of spoken language as needed. Provide adequate wait time. Ask students how much more time they need to think. Direct students to use sentence stems to seek clarification of spoken language. Read each of the questions/sentence stems aloud. Would you please repeat that? So you’re saying...Can you help me to… Ensure emergent bilingual students respond to your oral directions with accuracy by making use of the provided questions/sentence stems. For the second debriefing question at the end of the exploration activity, students write the content-area text of their response and justification using a variety of sentence lengths. Support emergent bilinguals’ writing skills development by providing sentence stems of varying lengths. The list from questions 2 and 3 are… When I compare the list from question ___ to the list from question ___, I notice that… The lists from questions 2 and 3 (are/aren’t) the same because....” Another section, Instructional Hints, states, “Slope, the steepness of a line, is a special case of the rate of change of a linear function. Emphasize the connections among slope, rate of change, and unit rate to anchor the notion of slope in students’ prior knowledge. The slope of a line is an important precalculus notion, as students in calculus will investigate functions in terms of the slope of a tangent line at a given point (instantaneous rate of change) for non-linear functions. The rate of change for linear functions is always constant, but the rate of change for other functions is not. Invest time in helping students develop a robust understanding of slope so that when they get to quadratic functions, they will have a non-example of a function with a constant rate of change.”

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 materials include embedded educator guidance to support appropriate mathematical language, including vocabulary, syntax, and discourse, and guidance to support mathematical conversations that give students opportunities to hear, refine, and use math language with peers and build their math language toolkit over time. The materials also guide educators in supporting student responses using exemplar responses to questions and tasks. The Exploration section titled “Support for Emerging Bilinguals” provides educator guidance and prompts to support vocabulary development and mathematical discourse. For example, in unit 6, lesson 5, “Solving Mathematical Problems with Systems,” under Exploration, the Support for Emerging Bilinguals materials guide educators to “Engage students in a discussion about the possible number of solutions for a system of linear equations that they noticed and applied. Provide sentence stems that focus on transition words to help emergent bilingual students bridge big ideas.” The suggested sentence stems include, “____ is the same as _____; Systems with ____ solution(s) ____, but systems with ___ solution(s)....; ____ differs from ____ in that ____.” Additionally, the materials provide Teacher Hints and Questioning Strategies to help foster vocabulary development and mathematical discourse.

The Exploration lesson component includes a Blackline Master answer key to support vocabulary development during students’ independent practice. It includes an exemplar answer to help educators ensure students use appropriate mathematical language in their written and/or verbal responses. For example, in unit 6, lesson 5, “Solving Mathematical Problems with Systems,” a question states, “Determine without solving the system whether it has no solution, one solution, or an infinite number of solutions. Explain how you know you are correct.” The suggested exemplar response states, “The system of equations has one solution. Because the slopes of the lines are different, the two lines will intersect at a single point on the coordinate plane.” Additionally, the Lesson Overview’s Vocabulary section provides the following educator guidance: “One way to support students’ acquisition of new vocabulary terms is to preteach the vocabulary terms using a strategy such as a Frayer model. For these key terms, we provide partially completed Frayer models to which students should add as they move through the lesson and develop a deeper understanding of the vocabulary term.” The Frayer model is a graphic organizer that instructs students to define the vocabulary word in their own words and list facts, characteristics, examples, and non-examples. The Frayer model helps students build their math language toolkit over time.

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

The TEKS process standards are integrated appropriately into the materials. In Grade 8 Advanced Mathematics, each lesson includes an Overview page with a Learning Outcomes section that lists the TEKS in the lesson and whether they are process standards, along with an overview of the lesson’s process standards. For example, on the unit 7, lesson 2, “Adding and Subtracting Polynomials” Overview page, the Learning Outcome TEKS states, “8AM.1G Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.” The Overview of Process Standards section states, “In this lesson, students display and explain the mathematical ideas of adding and subtracting polynomial expressions using both concrete/pictorial models and then justify those actions using the properties of algebra.” Students may “Display the idea of adding and subtracting polynomial expressions using visual models such as algebra tiles. Explain the idea of adding and subtracting polynomial expressions using visual models such as algebra tiles, and then explain the argument of how the tiles combine like they do to create an equivalent expression. Justify how the tiles combine like they do using properties of algebra (e.g., a ‘zero pair’ works because it applies the additive inverse).”

In the Exploration section of unit 7, lesson 2, “Adding and Subtracting Polynomials,” students add and subtract polynomials of degree one and degree two. The directions state, “Answer all questions to the best of your ability. Justify your responses. For items 1–4, use Algebra Tiles to model and carry out operations on polynomials, recording the concrete, verbal, and symbolic representations of each step.” A table comprises three columns: symbolic (where the expression is given), concrete (where students draw the algebra tiles used to represent the problem), and verbal description. Debriefing Questions state, “How does working with positive and negative tiles follow the patterns established by integer operations?” As students work collaboratively to determine the solution, they will apply the TEKS process standards 8AM.1G.

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

The materials include a description of how TEKS process standards are incorporated and connected throughout the course in the Grade 8 Advanced Mathematics Scope and Sequence document found on the Course Overview page. The description of the scope and sequence document states, “The scope and sequence document provides information concerning alignment to the Texas Essential Knowledge and Skills (TEKS) for grade 8 advanced mathematics and the English Language Proficiency Standards (ELPS), along with a rationale for the order of units of instruction, including the knowledge and concepts to be learned during the course.” The scope and sequence document includes a table at the beginning that lists each unit in the course. Under each unit is a list of the TEKS found within that unit, including the process standards. For example, under unit 1, “Algebraic Expressions, Equations, & Inequalities,” the TEKS list includes “8AM.1C, 8AM.1E, 8AM.5A, 8AM.5B, and 8AM.12F,” and the process TEKS within that list are 8AM.1C and 8AM.1E. Those same process standards are found in unit 3 (8AM.1C), unit 4 (8AM.1C and 8AM.1E), unit 5 (8AM.1E), unit 6 (8AM.1E), unit 7 (8AM.1C), unit 8 (8AM.1C), and unit 9 (8AM.1E).

The materials include a description of how TEKS process standards are incorporated and connected throughout the course in the Grade 8 Advanced Mathematics Scope and Sequence document found on the Course Overview page. The description of the scope and sequence document states, “The scope and sequence document provides information concerning alignment to the Texas Essential Knowledge and Skills (TEKS) for grade 8 advanced mathematics and the English Language Proficiency Standards (ELPS) along with a rationale for the order of units of instruction, including the knowledge and concepts to be learned during the course.” The document includes a table listing all 8th-grade advanced math TEKS in full and indicating whether each is a process standard. Next to each TEKS are the course units, and if the TEKS appears in that unit, an “x” is placed in the box under the unit next to the TEKS. This chart helps the user see which TEKS appear in which unit and which units overlap TEKS 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.

The materials include a description for each unit of how the TEKS process standards are incorporated and connected throughout the unit. The Unit Overview Resources section includes a document with a TEKS and ELPS Alignment table. The table notes the mathematical process standards, the knowledge and skills student expectations, and the ELPS student expectations. Additionally, the document lists Objectives that include “I can” statements and describe what students will know and be able to do across the unit. For example, in unit 4, “Linear Functions & Linear Inequalities,” the 8AM Unit 4 Document includes the process standards 8AM.1C, 8AM.1D, 8AM.1E, 8AM.1F, and 8AM.1G. An example of the alignment between the TEKS process standards and the Objectives includes “8AM.1C—The student is expected to select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.” The Objective states, “I can write a linear equation in two variables from a table, graph, or verbal description.”

The Course Level Documents, called Course Resources, include a scope and sequence and Implementation Guide. Each document includes a description for each unit of how the TEKS process standards are incorporated and connected throughout the unit. For example, the Grade 8 Advanced Mathematics Scope and Sequence provides a year-at-a-glance table that shows the distribution of TEKS, including the TEKS process standards, across two semesters. A separate chart demonstrates the distribution of TEKS process standards by unit. For example, unit 1 includes TEKS process standards 8AM.1C and 8AM.1E. Unit 2 includes 8AM.1D and 8AM.1F. Unit 3 includes 8AM.1C, 8AM.1D, and 8AM.1F. This table allows educators and administrators to determine where the TEKS process standards are addressed and whether they are adequately addressed. The Implementation Guide’s Embedded Process Standards section provides the following teaching guidance: “Each Lesson Overview page in our online system includes an overview paragraph and bullet points that describe how the process standards specifically connect to the content standards of the lesson… For more details regarding which process standards are highlighted in each lesson, please refer to the Lesson Overview pages in the online platform or the Pacing Guide document.”

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

The materials include an overview of the TEKS process standards incorporated into each lesson. The Lesson Overview provides a table that includes the TEKS process standards that align to the lesson. Each Lesson Overview also includes the Overview of Process Standard(s), which describes how the TEKS process standards are incorporated. For example, in unit 9, lesson 3, “Linear Factors and Zeros of Quadratic Functions,” the Overview of Process Standard(s) states, “In this lesson, students communicate mathematical ideas (in this case, the relationship between the linear factors and zeros of a quadratic function) and their implications using multiple representations, including graphs, tables, and symbols.” In unit 10, lesson 1, “Geometric Sequences,” the Overview of Process Standard(s) states, “In this lesson, students create and use representations such as diagrams, symbols, tables, and graphs to organize, record, and communicate ideas about geometric sequences.”

The lesson components Exploration and performance task incorporate the TEKS process standards. For example, in the unit 9, lesson 3, Exploration, “Blackline Master: Linear Factors and Quadratic Functions (AK),” the materials ask the student, “A quadratic function can have 0, 1, or 2 real zeros. Draw an example graph on the grids provided below to illustrate the three scenarios. Explain each of the graphs drawn” (TEKS 8AM.1D). The 8AM.7B Performance Task—Windows demonstrates alignment with the TEKS process standard 8AM.1A in the Mathematically Speaking section. The materials state, “In this task, students evaluate a real-world situation about an arched window. Given the function rule in vertex form, students will determine the location of the zeros of the function, or the x-intercepts of the related function graph, and use those base-points of the window to determine the width of the window. Students will also interpret the vertex point as the height of the window at its center.”

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

The Exploration component of each lesson gives students the opportunity to think mathematically, persevere through problem-solving, and make sense of mathematics. The materials direct educators to randomly group students into groups of no more than three. Students watch an instructional video and work collaboratively to complete the Exploration assignment while the educator monitors to “facilitate student group exploration and reflection.” The materials provide Instructional Hints to guide educators with suggested scaffolding and instructional strategies. For example, in unit 4, lesson 5, “Graphing Linear Inequalities,” the Instructional Hints state, “The focus in this activity is generating a graphical representation given a symbolic representation. Sometimes, it is useful for students to use intermediate representations; for example, it may be helpful to generate a table of values for the boundary line and then use that table to graph the boundary line.” The materials provide Questioning Strategies to “support student reasoning and productive struggle through questioning.” For example, a Clarifying Question asks, “When you say __________, what do you mean?” An Advancing Question [ask and walk away] asks, “What relationship(s) do you see in whether the boundary line is solid or dashed/broken?” The activity sheet Graphs of Linear Inequalities directs students to match an inequality with a graph, graph given inequalities, and graph an inequality on a coordinate plane to solve a real-life word problem. The answer key provides multiple possible responses for the educator to reference and support productive struggle.

The Performance Task component of each lesson offers multiple task levels that allow educators to differentiate based on student performance in earlier components of the lesson. This differentiation ensures that students have the opportunity to think mathematically, persevere through problem solving, and make sense of mathematics at a level that allows for productive struggle. The materials provide Teacher Notes for Task that “provide additional educator guidance for scaffolding and opportunities to extend student thinking. For example, unit 9, lesson 1, “Parent Quadratic Function Transformations” includes a 8AM.7C Performance Task—Archway Onlevel for which the notes state, “The Designing the Archway task provides students with a starting equation, a series of transformations, and then asks them to write the equation for the fully transformed quadratic function. The Moving the Garland task provides students with descriptions of two transformations and a graphical representation, then asks them to write the equation of the fully transformed function. If time is a concern, select the task that is most appropriate for your students.” The materials provide educators with question prompts to “support student reasoning and productive struggle through questioning.” For example, a Clarifying Question asks, “Can you rephrase what the problem is asking?” A Focusing Question asks, “What representations might you use?” And an Assessing Question asks, “What evidence from the problem/task supports your answer?”

The Teacher Guide’s Providing Students Effective Feedback section provides educator guidance to ensure students have the opportunity to think mathematically, persevere through solving problems, and to make sense of mathematics through effective educator feedback. The materials state, “The focus of feedback should be to support students’ productive struggle - not to remove it. Effective feedback leads students to recognize their next step(s) and how to take them to achieve the learning objective(s).” The materials explain that educators should create a classroom environment that fosters students’ belief in their self-efficacy. The guidance states, “If feedback needs to be extensive, it should be chunked into smaller pieces so that students can digest each one and work toward improvement...”

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. The Teacher Materials include a Unit Introduction and a Unit Overview. Both resources describe the vertical coherence between grade levels, within a course, and within a unit. This coherence shows how skills and concepts build on each other and gives students the opportunity to understand that there are multiple ways to represent and solve problems and tasks. For example, in unit 6, “Linear Equations,” the Teacher Materials’ Unit Introduction explains, “In previous units, students represented and solved linear equations and linear functions. Unit 6 focuses on two simultaneous linear equations, or a system of two linear equations.” The Unit Overview explains, “In Grade 6 Advanced Math, students solved one-variable, one- and two-step equations and inequalities and then represented the solutions on a one-dimensional number line. In Grade 7 Advanced Mathematics, students solved one-variable equations with variables on both sides of the equal sign and represented the solutions on a one-dimensional number line.” The Teacher Materials provide a Unit Document that includes all Objectives as “I can” statements. The Objectives also demonstrate how students will understand multiple representations to solve problems and complete tasks. For example, “I can write a system of linear equations from a table of values or a graph,” and “I can write a system of linear equations from a verbal description.”

The Spiral Review gives students the opportunity to explain that there are multiple ways to represent and solve problems and complete tasks. For example, in unit 6, “Systems of Equations and Inequalities,” the Teacher Directions state, “This spiral review utilizes an instructional strategy designed to foster student mathematical discourse.” Students work in pairs to determine the solution to a given problem within a time limit. After students determine the answer, they move to the corner of the room with the corresponding solution posted. If students are in each corner, they are directed to explain their solution. The educator guidance states, “Hold a discussion about how they know their answer is correct.”

The Exploration and Performance Task lesson components support students in justifying that there can be multiple ways to represent and solve problems and complete tasks. For example, in unit 10, lesson 2, “Graphs and Key Attributes of Exponential Functions,” under Exploration, a question states, “Does this situation involve exponential decay or exponential growth? Explain how you know.” In the 8AM.9D Performance Task—Caffeine Onlevel, students determine the solutions for several questions and tasks. The final question states, “When will the amount of caffeine reach 0 mg? Justify your reasoning.” The educator guidance, dubbed Look For, states, “Look for student justification of choices of solution strategy.”

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.

The materials are designed to help students make sense of mathematics through multiple opportunities to do, write about, and discuss math with peers and educators. For example, in Grade 8 Advanced Mathematics, in the Exploration section of unit 5, lesson 7, students work in pairs or groups. First, student pairs complete an activity sheet on trend lines and lines of best fit, as well as one on the correlation coefficient. As pairs of students work, educators can use the Questioning Strategies section within the Exploration to help students write about and discuss math. Some questions include, “When you say __________, what do you mean?,” “How can you keep track of the squares you have already tried?,” and “Is there a formula that could help find the distance between two points?”

The materials are designed to help students make sense of mathematics through multiple opportunities to do, write about, and discuss math with peers and educators. For example, in Grade 8 Advanced Mathematics, in the Exploration section of unit 9, lesson 4, students work in pairs or groups. Students watch a video on vertex form, first watching an example, then working through a problem on their own in their math journals. Students complete an activity sheet in their pairs on vertex form. Once students finish, educators are to “facilitate student group exploration and reflection.” Support for emergent bilingual students includes pairing students to speak and listen as they work through the Exploration activity, write responses to specific items, and justify their reasoning. Educators can provide students with sentence stems such as “How do you…?” and “Where do you place…?” While students work, educators can use questions from the Questioning Strategies section such as “When you say ____________, what do you mean?” and “How do you know the standard form and vertex form really are equivalent?”

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

The materials support educators in guiding students to share and reflect on their problem-solving approaches, including explanations, arguments, and justifications. For example, in Grade 8 Advanced Mathematics, the materials provide educators with embedded guidance on Questioning Strategies in the Exploration and Explanation sections of each lesson. The Questioning Strategies found in unit 6, lesson 5, “Solving Mathematical Problems with Systems” include: “How are the equations the same? How are they different?,” “How do you know which method is best for a particular system of linear equations?,” “How do you know your solution is reasonable?,” and “What do you mean by substitute?”

The materials support educators in guiding students to share and reflect on their problem-solving approaches, including explanations, arguments, and justifications. For example, in Grade 8 Advanced Mathematics, educators have embedded guidance on Questioning Strategies in the Exploration and Explanation sections of each lesson. The Questioning Strategies found in unit 4, lesson 2, “Graphs and Features of Linear Functions” include: “What formula would be helpful in this situation?,” “How do you determine which is the x-intercept and which is the y-intercept from the context of the problem?,” “Which formula is called for in this problem?,” and “How did you solve the problem?”

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

The materials include prompts and guidance to help educators provide explanatory feedback based on student responses and anticipated misconceptions. For example, in unit 7 Spiral Review 2: Polynomial Operations and Equivalent Forms of Polynomials, in the “trashketball” activity’s Teacher Directions, the materials include guidance to support educators in providing explanatory feedback based on student responses and anticipated misconceptions, such as “Divide the class into teams comprised of 3 or 4 students each. The winner of trashketball is the team with the most points at the end of the game. If desired, print the blackline master of the Spiral Review for Polynomial Operations and Equivalent Forms of Polynomials for each student, and display the Spiral Review for Polynomial Operations and Equivalent Forms of Polynomials slides. For each question, provide teams of students with time to determine a solution. Recommended times are: Question 1: 2 minutes, Question 2: 2 minutes, Question 3: 4 minutes, Question 4: 3 minutes, Question 5: 5 minutes. Once all teams have submitted their answers, teams who answered correctly then have the opportunity to shoot to score points. Take time before the next question to address any common misconceptions. Repeat for each of the 5 questions.”

The materials include prompts and guidance to help educators provide explanatory feedback based on student responses and anticipated misconceptions. For example, in the unit 7 Spiral Review, the Teacher Directions instruct educators to “Take time before the next question to address any common misconceptions.” This supports educators in identifying errors from student responses and providing feedback during instruction.