Program Information
- Copyright Type
- Proprietary
Math
Grade 8Publisher: Carnegie Learning
Copyright: 2015
The quality review is the result of extensive evidence gathering and analysis by Texas educators of how well instructional materials satisfy the criteria for quality in the subject-specific rubric. Follow the links below to view the scores and read the evidence used to determine quality.
Section 1. Texas Essential Knowledge and Skills (TEKS) and English Language Proficiency Standards (ELPS) Alignment
Grade |
TEKS Student % |
TEKS Teacher % |
ELPS Student % |
ELPS Teacher % |
Grade 6 |
100% |
100% |
100% |
100% |
Grade 7 |
100% |
100% |
100% |
100% |
Grade 8 |
100% |
100% |
100% |
100% |
Section 2. Concept Development and Rigor
Section 3. Integration of Process Skills
Section 4. Progress Monitoring
Section 5. Supports for All Learners
Section 6. Implementation
Section 7. Additional Information
Grade | TEKS Student % | TEKS Teacher % | ELPS Student % | ELPS Teacher % |
---|---|---|---|---|
Grade 8 | 100% | 100% | 100% | 100% |
The materials spend the majority of concept development of the primary focal areas in sixth grade, as outlined in the TEKS. The materials strategically and systematically develop students’ content knowledge as appropriate for the concept and grade-level as outlined in the TEKS. The materials provide practice opportunities for students to master the content.
Evidence includes but is not limited to:
All five of the modules cover one or more of the following: number and operations; proportionality; expressions, equations, and relationships; and measurement and data.
The design of the materials follows a cycle in each of the modules. Each module includes “Connections to Prior Learning,” which is a review of key concepts found in the Teacher's Implementation Guide, and a “Content and Alignment” section of the instructional materials. Each module includes an overview section outlining for the teacher all the learning that will take place within that module. The cycle then continues with a topic overview to detail the specific learning that develops within the module. Finally, lessons and activities are included to develop focused learning in each topic.
In addition, these key focal areas are spiraled among all modules. The materials showcase the focal concepts throughout by providing standard descriptions and essential ideas at the beginning of each module. The lessons include suggestions and materials to practice and reinforce the primary focal area skills in a variety of settings that include performance tasks, MATHia, a technology component, pre-tests, and post-tests. Questions and tasks within and across units build in academic rigor to meet the full intent of the primary focal areas. Problem sets across lessons scaffold from lower-level tasks (e.g., define, identify, or describe) to higher-level question types (e.g., evaluate, analyze, or create).
Lessons contain several activities, as well as the “Talk the Talk” for each lesson before getting to the actual assignment. The activities allow students to have several opportunities to master the content. MATHia is available for each lesson. This provides the students with additional practice to master the content. Students have a skills practice book to allow for more practice on each skill.
Module 2, Lessons 1.3 outlines the consistent cycle used in the materials and the systematic development of a broad topic to a specific lesson. In Topic 2.2, students create equations, tables, and graphs to analyze linear relationships. Multiple activities (1.1, 1.2, and 1.3) within one single lesson (2.2.1) show a progression of rigor to meet the full intent of primary focal areas. Every topic, lesson, and activity in the materials follows the same type of progression of rigor. In Topic 4.1, students learn about linear functions from multiple representations, and in Topic 4.2, students learn how to tell if those functions increase, decrease, or stay constant.
The materials build upon previously taught concepts across previous weeks and topics within each module to increase rigor and ensure students master the full intent of the concept. Module 3 guides teaching and learning math concepts through a variety of question types and tasks, such as “What is the most efficient way to solve the equation?” or “Are reverse operations used to solve equations?” Topic 1 introduces students to solving linear equations, and Topic 2 culminates systems of linear equations.
The materials include a variety of types of concrete models and manipulatives, pictorial representations, and abstract representations, as appropriate for the content and grade level. The materials guide teachers to help students move through the CRA continuum; however, there is limited evidence that the materials support teachers in understanding students’ progression along the CRA continuum.
Evidence includes but is not limited to:
The materials use models, manipulatives, and representations for concept exploration and attainment for the primary focal areas number and operations; proportionality; expressions, equations, and relationships; and measurement and data. Students work with models, manipulatives, and representations used for content exploration and attainment or have previous experience with the instructional materials through teacher guidance and instructions on activity sheets in the students’ Skills Practice Workbook. The materials include a variety of types of concrete models and manipulatives in only some of the lessons. It is more evident that pictorial and abstract representations are present throughout the materials. Every topic in the materials has an overview section to support the teacher before teaching each lesson.
In Module 3, students review the Distributive Property by calculating the product of two numbers using the area of a rectangle diagram. Then, to increase rigor, students simplify algebraic expressions using both the area model and symbolic representations. In Module 3, Topic 1, teachers are given background knowledge about students already having the essential skills and knowledge for solving one-variable linear equations from current and previous years based on what the student should understand; however, no guidance is provided for teachers to identify where a student's actual understanding is along the CRA continuum. Students are expected to reason abstractly and look for the structure of equations that lead to no solution, one solution, or infinite solutions. The lesson continues with abstract type problems and practice and does not have further teacher support for students to progress if they are not yet working at an abstract level.
In Module 4, the materials highlight the need for connecting concrete to abstract representations while solving the Pythagorean Theorem and its converse problems. Students learn to make sense of a problem situation by modeling it with a diagram as well as algebraic arguments and proofs. Module 4, Topic 2 begins with pictorial and abstract representations. Students are asked to use a picture of a baseball field and translate the information from the word problem to solve for the hypotenuse using the Pythagorean Theorem. Students use pictorial area models to prove aspects of the Pythagorean Theorem, and later in the topic, use the formula (abstract) to find the missing lengths of right triangles using the Pythagorean Theorem, which is a focal area for eighth grade. Students are explicitly taught how to use the representations they are given through teacher guidance and instructions on activity sheets in the students’ Skills Practice Workbook.
Module 5, Activity 1.2 provides guidance to the teacher for students struggling with the CRA continuum: teachers should require students to sketch a diagram of the cylinder, label its dimensions, and write the appropriate volume formula prior to substituting values and solving each problem. In Module 5, Topic 2, students use cardstock, modeling clay, birdseed, tape, scissors, and centimeter cubes to explore and learn about the volume of curved figures such as cylinders and cones. The facilitation notes in this lesson explain to the teacher how to guide the students in using the birdseed and dump it into the cylinder net using the cone net. From there, students note relationships and determine the formula for the volume of each of the shapes. Pictorial and abstract representations are evident throughout the lesson on volume. Students are shown multiple labeled cylinder, cone, and sphere pictures and are also asked to sketch their own shapes and then label and calculate the area of the base as well as the volume.
The materials support coherence and connections between and within content at the grade-level and across grade levels. The materials include support for students to build their vertical content knowledge by accessing prior knowledge and understanding concept progression. The materials include tasks and problems that intentionally connect two or more concepts as appropriate for the grade-level. The materials provide opportunities for students to explore relationships and patterns within and across concepts. The materials support teachers in understanding the horizontal and vertical alignment guiding the development of concepts.
Evidence includes but is not limited to:
At the beginning of the Teacher's Implementation Guide, a chart entitled “Middle School Math Solution Content at a Glance” explains how concepts are built during the current year and across multiple years. The materials provide a detailed “Introduction” at the beginning of each module and each topic. This section gives teachers a summary of what students should already know in the section called “What is the entry point for students?” This section supports teachers in knowing what the students already know and what they need to learn to build on the next concept. The materials include teacher supports and guiding documents that help teachers understand how concepts build over time. Teachers have access to extensive professional learning with a mobile and web-based app, LONG + LIVE + MATH, an online community, and a “Central Hub” with access to all products and resources included, in addition to the books and software. Additionally, the materials include guiding documents for teachers, which include content and alignment documents for each module, and a Teacher's Implementation Guide.
In Module 1, Lesson 1.6, teachers encourage students to reason about transformations using knowledge of the effects of rigid motions on the coordinates of the figures and the properties of the image learned in previous lessons. In Module 1, students transform geometric objects and establish facts about triangles and special relationships between special angle pairs. Module 2, “Developing Function Foundations,” combines students’ knowledge of transformations and similarity with similar triangles to explain why the slope between any two points is on the same line.
In Module 2, students use linear models to represent linear relationships using tables, equations, and graphs. This reappears in Module 3 as a tool to analyze and solve pairs of simultaneous linear equations. Developing function foundations provides the basis for the majority of high school algebra and statistics studies. In Lesson 1, before students start on the “Linear Relationships in Tables” lesson, students work on a warm-up activity to find the slope of two lines graphed on a coordinate plane. The materials remind students about their work in previous lessons in finding slope from similar right triangles and show how they can use a table to find the slope instead of creating a graph. Activity 1.2, “Bivariate Data,” includes a scenario with students selling sweatshirts and sweatpants with a school’s logo. Students determine if there is a relationship between sweatpant size and sweatshirt size. In this activity, students name a variable that could affect an article’s size. This lesson continues and shows the reason for bivariate data collection. In Topic 2, students use strategies from previous topics, such as equations, graphs, and tables, to connect the understanding of proportional relationships to linear relationships, slope, and slope formula. Students will use strategies from this module in future grades when students develop a deeper understanding of functions in Algebra I. In Topic 2.2, students use the angle-angle similarity theorem learned in Module 1 to study the slope’s properties. In Lesson 3.1, when introducing functions, students learn how to display a set of ordered pairs in different representations such as mappings, graphs, and tables. Students look at examples and non-examples of functions and then use the definition to identify their domain and ranges. To connect the mathematical idea of functions to linear relationships, the materials suggest that the teacher asks the students specific questions: “Does a function map each input to one and only one output?” Then, “Are all relations considered to be functions?” and “Are all functions linear?” Finally, materials support interconnections with the question, “Are linear relations also functions?”
In Module 2, Lesson 3.5, one practice problem describes four students’ bank accounts, where y-values represent the dollar amounts and x-values represent the time in months. The problem states, “The four students save money at a constant rate. Which of the four students will be the first to save $500?” Students look through four different representations in this problem: a table, an equation, a graph, and a verbal description. They are to apply mathematics to one of these representations. In Activity 1.4, the facilitation notes provide teachers with questions to increase depth, breadth, and complexity. No mention is made as to how this prepares students for next year’s work.
In Module 3, students model and solve equations first introduced in Topic 3, “Line and Angle Relationships,” when students calculate the missing measures of supplementary and complementary angles using equations and variables. Module 3 builds on students’ prior knowledge of solving and graphing linear equations. The overview informs the teacher of what students learned in sixth and seventh grade to support this new concept. In Topic 2, the overview states that students will use the tools developed in the previous topic, “Solving Linear Equations,” to solve linear systems using substitution, which is the current topic. The overview states how solving linear systems using substitution will prepare them for other methods of solving linear equations in high school mathematics. The Module 3 overview explains that students will learn how to solve equations using substitution, which will lead them to solve nonlinear equations and systems of inequalities in later math courses.
In Module 4, Topic 1, the materials include a section that describes how students will use knowledge from expanding number systems in future learning. It continues with information about how the lessons in this topic will develop students' understanding of imaginary numbers in high school. In Module 1, students learn about similarity and scale drawing. Then, Module 4 requires students to recall congruent triangles’ formal definition and connect that prior knowledge to the current material, Pythagorean Theorem. The materials include tasks that require students to recognize and apply mathematics in contexts outside of mathematics. For example, in Module 3, “Getting Started: According to the Map,” students analyze a street map of Washington, D.C., and describe whether or not it is possible to be at an intersection of two intersecting streets, two parallel streets, or two streets that are the same street, but with different names.
In Module 5, Topic 2.2, students apply their prior knowledge of calculating the right pyramid’s volume to determine a cone’s volume. Connections are made that show that as the number of sides on the base of the pyramid increases, the shape of the pyramid approaches the shape of a cone. In Topic 3, students are tasked with verifying a student’s attempt to calculate how many basketballs they could fit into their locker, given only the locker’s dimensions and the radius of the basketball, a real-world application of what they’ve been learning in this topic.
The materials include quality tasks that address content at the appropriate level of rigor and complexity. Some tasks are designed to engage students in the appropriate level of rigor (conceptual understanding, procedural fluency, or application) as identified in the TEKS and as appropriate for the development of the content and skill. The materials clearly outline for the teacher the mathematical concepts and goals behind each task. The materials integrate contextualized problems throughout, providing students the opportunity to apply math knowledge and skills to new, but not varied, situations. The materials provide some teacher guidance on anticipating student responses and strategies. Lastly, the materials provide some teacher guidance on preparing for and facilitating strong student discourse grounded in the quality tasks and concepts.
Evidence includes but is not limited to:
The Teacher's Implementation Guide provides teachers guidance in each topic’s overview, including how students demonstrate understanding and how each topic’s tasks promote student expertise. The materials provide “Learning Together,” which is a subsection of the topic overview. It shows the progression of each lesson and the standards addressed, both target and spiral. The materials explain how each task builds student efficacy towards the goal of demonstrating mastery. For example, in the facilitation notes guide for each topic, the materials provide summaries of each activity and what students should demonstrate after completion. This section provides the teacher with a list of indicators to look for that constitute a student’s understanding of the topic’s standards. For example, the materials suggest that the teacher look for students’ identification of parts in a numerical expression using terms such as sum, factor, product, and coefficient to understand the distributive property.
The materials include meaningful tasks for students set in real-world contexts that allow them to demonstrate mastery of math concepts. For example, in introducing unit rates, the materials offer several real-world activities such as calculating miles per gallon on a road trip, comparing costs per number of toppings on different size pizzas, averaging weight loss per week for a dog that has been put on a diet by his veterinarian, finding the miles per hour based on a car’s odometer after a number of hours, and calculating the balance on a gift card for a local coffee shop after purchasing a number of coffees and teas. The materials do not guide the teacher on how to appropriately revise content to be relevant to their specific students, backgrounds, and interests. The materials provide teachers with possible student strategies to practice questions and tasks. For example, the materials include a problem-type called “Thumbs Up/Thumbs Down” to analyze problem solving in examples provided. When an incorrect response is presented, students look for errors in calculation and correctly solve it.
The materials’ online component, MATHia, provides students with multi-level hints to help them solve problems. No evidence was found that the materials provide teacher guidance throughout the topic or lesson overview on anticipated student strategies. The facilitation notes section in every lesson provides several open-ended questions to ask students and support discourse. The materials provide teacher responses to possible students’ answers, including how to direct students’ misunderstandings or misconceptions. The misconception section is included in most lessons, as are differentiation strategies to help students who may struggle. For students having difficulty finding the percent of a number, the materials suggest that the teacher use diagrams and models and explain that a percent is part of a whole and can be any size. The materials do not provide rubrics or keys with which teachers can evaluate and provide feedback to students while engaging in discourse. The materials partially support the teacher with setting up and reinforcing strong practices for student discourse. Although every topic provides an overview for the teacher that includes guided questions to support students through each topic’s learning, little to no detail is given to setting up and reinforcing these practices.
Module 1 begins with a conceptual understanding of transformation and continues to more abstract similarity topics. The materials increase in rigor in Topic 3 with line and angle relationships. In Lesson 1.1, the materials indicate that “Students may think that if figures are the same shape but not the same size that they are not truly the same shape.” Therefore, teachers draw a shape and use a projector or an interactive whiteboard to show that the same shape can be enlarged or reduced and still have the original shape. In Topic 2, students plot points on a coordinate grid and progress to determine which set of functions has a greater change rate. Students continue without their concrete models to the abstract activities of making decisions.
Module 2, Activity 2.3 requires students to match the designs and numberless graphs of six ceramic drinking cups that students created for an art project. Students use the cups’ shapes to determine how the height would be affected by the volume to match graphs to cup designs at the end of the lesson. No evidence was found that the materials guide the teacher on appropriately revising content relevant to their specific students, backgrounds, and interests. In Topic 4, the lesson structure builds in complexity from “Getting Started” to “Talk the Talk.” Students begin by analyzing data for the percent of book sales. Then, students continue to analyze data and use a line of best fit and two points to determine the line’s slope and its equation. Next, students practice drawing, writing, and using the equation of the line of best fit. Students write and interpret the equation of a trend line and use it to make predictions for a time in the future and the past. Last, students determine which of the three proposed lines provides the best fit for data, increasing and then decreasing trend. Students defend their choice of the line of best fit for the entire data set and then select which line they would use for interpolation and extrapolation purposes.
In Module 3, Topic 1.3, the facilitation notes explain that students will write equations, graph lines, and interpret points of intersection to solve this topic’s problems. The materials provide teachers with common misconceptions in student responses and strategies to combat those misconceptions for some of the lessons. In Module 3, Topic 2, the lesson starts with a scenario of animal trading at a county fair. Students analyze the trades for fairness. The theme of the county fair is carried throughout each of the lessons for this topic. There is no evidence that the materials guide the teacher on appropriately revising content relevant to their specific students, backgrounds, and interests.
In Module 4, Topic 2, students learn how to find the missing side length of a right triangle using the Pythagorean Theorem. In the first lesson, students discuss squaring the sides of given right triangles to look for patterns. The second lesson introduces them to the Pythagorean Theorem and asks them to prove the theorem, the third lesson teaches them to use the Pythagorean Theorem to calculate the length of the hypotenuse of a right triangle, and the final lesson teaches students to use the converse of the Pythagorean Theorem. In Topic 2.3, the following question is included for teachers to use to promote discourse: “Is it easier to compare fractions or decimals?”
The materials integrate fluency over the course of the year and include some teacher guidance and support for conducting fluency practice as appropriate for the concept development and grade; however, the materials do not include an explicit year-long plan for building fluency as appropriate for the concept development and grade. The materials integrate fluency at appropriate times and with purpose as students progress in conceptual understanding; however, the integration is not explicit and is not found throughout the materials. The materials include some scaffolds and supports for teachers to differentiate fluency development for all learners.
Evidence includes but is not limited to:
The Teacher's Implementation Guide provides detailed descriptions in the “Module and Topic Overview” that explain the plan for that module to build concept development and fluency and how it will address the targeted student expectation. This is consistently done throughout the materials for every module and topic; however, guidance is limited throughout the lessons and activities. The materials provide lessons and pacing structure that includes directions on helping students move through multiple fluency practice activities within a single lesson. Each assignment contains a review section that provides a spaced practice of concepts from the previous lesson and topic and the fluency skills important for the course. The materials include some opportunities within the facilitation notes of each activity for students to discuss their conceptual understanding behind the fluency practice by using questions listed for the teachers to guide the students through this process. The materials provide a “Stretch” section for every lesson for students to extend fluency when they have already met the fluency expectations. The materials provide a section in the facilitation notes titled “As students work, look for.” This section lists for teachers certain key things that help the teacher determine if students need differentiated supports.
The online component, MATHia, provides individual practice for students to build their fluency throughout the school year and track their progress within each module and targeted skill. The table of contents in the materials shows each module’s alignment, topic, lesson, and activity with the student expectation and the MATHia workspace. However, there is no guidance on how and when to use resources in the scope and sequence or individual lessons. The materials include the Skills Practice Workbook with activities that focus on building student fluency.
In Module 1, Activity 1.1, students use measuring tools and a given dilation with a scale factor greater than 1 to conclude that when corresponding angles are congruent and the ratios of corresponding sides are equal, the figures have the same shape but are different sizes. Students who struggle with fluency in dilation rules can compare and contrast concepts like the common use of the term dilation (e.g., having your eyes dilated for an eye exam) versus the mathematical definition
In the Module 2 Overview, the text highlights the importance of developing fluency with analyzing linear relationships, writing equations of lines, and graphing lines. In Lesson 2.6, the materials guide the teacher to look for students’ written rates and labels of numbers (e.g., calories/minute) when working with linear equations. Based on the teacher’s observations, students are identified for needing differentiated support during the fluency activities. In Module 2, the Stretch assignment prompts students to determine if the equation provided is linear or proportional and to justify their answer.
Module 3, Topic 1 provides a review section that requires students to solve equations using expression and equation fluency skills. In Lesson 1.1, students use multiple strategies to solve equations such as factoring, multiplying by the lowest common denominator, and properties of equality. In Lesson 1.3, students create an equation with at least one fractional coefficient and at least one negative coefficient with solutions x=0 and x=4/3. Because this activity already gives students the solutions and the constraints the equation must include, the students extend their fluency of creating linear equations. Activity 1.3 requires students to write equations, graph lines representing the equations, and interpret the point of intersection of two lines. Students who are struggling are prompted to use two ordered pairs to graph instead of graphing the line from the equation’s standard form. Conversely, to extend, students solve the problem using another strategy. In Lesson 2.3, the teacher asks students questions about using different variables for a given problem situation. The teacher provides discourse opportunities by asking the students if they could have used substitution to solve the same problem. This system of equations problem has more than two variables and can be solved using different strategies. In Topic 2, students work through linear equation problems and build fluency by writing and solving additional systems, using equations in the system to determine the most efficient solution strategy.
In Module 4, the Teacher's Implementation Guide explains how fluency in the real number system helps students in the concept development of the Pythagorean Theorem, its converse, and applying the theorem to calculate distances on a coordinate plane. In Lesson 1.2, students solve five warm-up questions where they convert a fraction to a decimal then place it on a number line. This activity supports the development of the conceptual understanding of rational and irrational numbers, which is the focus of the lesson. In Lesson 2.2, the materials guide the teacher through a 2-day lesson that includes an engaging activity and three fluency activities on the converse of the Pythagorean Theorem. The materials tell the teachers when to guide students, when they should be applying their knowledge, and when they should demonstrate what they have learned. These activities all support students in developing fluency. The directions explain when students should read aloud, what questions the teacher should ask, and when they should work in groups or pairs.
In Module 5, students continue to develop their fluency with powers in “Volume of Curved Figures” by applying the volume formulas to a variety of mathematical and real-world problems. In Lesson 2.3, students that struggle are prompted to find ⅔ of the volume of a sphere within a cylinder. The teacher gives students a drawing of a sketch showing ⅔ of a cylinder and then using words and the mathematical symbols of the formula for a sphere in comparison to a cylinder.
The materials integrate fluency at appropriate times and with purpose as students progress in conceptual understanding; however, the integration is not explicit and is not found throughout the materials. The materials include some scaffolds and supports for teachers to differentiate fluency development for all learners.
The materials include some embedded opportunities to develop and strengthen mathematical vocabulary. The materials include some guidance for teachers to scaffold and support students’ development and use of academic mathematical vocabulary in context.
Evidence includes but is not limited to:
The Teacher's Implementation Guide and the Consumable Student Edition include key terms listed and defined in the “Module and Topic Overviews.” When appropriate, in the margins of the Consumable Student Edition, characters called “The Crew” use speech bubbles to highlight and define academic vocabulary. At the beginning of each lesson, the materials list the learning goals and key terms on the same page; however, the development of mathematical vocabulary is not addressed. Key terms are bolded in each activity’s text and accompanied by the formal definition, which is used within the lesson context. At the end of each topic, a summary is included for the students and teachers to review key terms learned throughout the series of lessons within that topic. The lessons embed the introduction and use of vocabulary within the context of mathematical tasks requiring students to communicate mathematical ideas. For example, the Student Edition embeds academic vocabulary within each lesson.
The materials provide repeated opportunities for students to listen, speak, read, and write using mathematical vocabulary within and across lessons. At the end of each lesson, students complete three sections: “Write,” “Remember,” and “Practice.” Students make connections to the previous activities in the lesson and also reiterate the formal language of mathematics. The materials do not explicitly build from students’ informal language to the formal language of mathematics by making explicit connections throughout the materials. The materials provide a section for differentiation in most lesson activities; however, vocabulary development is not one of the strategies provided. The Teacher's Implementation Guide prompts teachers to create a word wall of key vocabulary terms used through the materials.
In Module 1, Topic 1, one of the topic goals is “Students develop a formal understanding of translations, rotations, and reflections in the plane.” The key term congruent is defined, and two activities later, students revisit the definition of congruent figures. In Topic 3, students use the word congruent to identify angles of the same measure. In Lesson 1.2, students explain each term in their own words. In Lesson 1.6, the key terms are congruent line segments and congruent angles. One of the learning goals states that students will write congruence statements that can only be done by understanding the listed vocabulary. In Lesson 2, students create a vocabulary list with key terms at the beginning of the lesson. When the term is discussed throughout the lesson, students write down the definition in their notebooks. At the end of the lesson, students use each term in a sentence.
In Module 2, Topic 4, the “Getting Started” section encourages students to use the vocabulary they learned in the previous lesson to answer Question 1. The teacher provides a sentence stem to support students who have a hard time putting together a full sentence, especially students who are English Learners (EL). The following is a suggested stem for students: “The relationship between the explanatory and response variable is....”
Module 3, Lesson 1.3 prompts students to work in a group to complete questions and then share responses as a class. This activity is repeatedly used throughout the materials to promote the reading of math questions, listening to a group conversation, and speaking about their thinking process and response. In this same lesson, students demonstrate their learning by describing how they know when an equation has one solution, no solution, or infinite solutions. In Lesson 2, students read a formal equation definition system and write the correct term in the corresponding blank. Students are encouraged to use proper academic vocabulary when speaking with a partner or group. In Module 3, Topic 2, students are asked to explain why they may use substitution to solve a system of linear equations if they already solve the system by graphing. Students are reminded of the key concepts they learned in the lesson to guide them in their writing assignments. Finally, students read and solve five practice problems that only include formal language.
Module 4, Lesson 2 includes this differentiation strategy for struggling students: “The teacher connects the common use of the term closed and its mathematical meaning. If a door is closed, you cannot go outside. If you add integers, a and b, you should not have to go outside of its set to get the answer.”
In Module 5, Topic 1, students explain, in writing, how multiplying and dividing numbers in scientific notation is different from adding and subtracting numbers in scientific notation. The Teacher's Implementation Guide suggests possible correct explanations on the answer key section of that lesson, including vocabulary students should be using. In Lesson 1.1, students take notes to keep track of the vocabulary in the lesson on powers and integer exponents.
The materials provide opportunities for students to apply mathematical knowledge and skills to solve problems in new and varied contexts, including problems arising in everyday life, society, and the workplace. The materials include opportunities for students to successfully integrate knowledge and skills to problem-solve and use mathematics efficiently in real-world problems. Materials provide students opportunities to analyze data through real-world contexts.
Evidence includes but is not limited to:
The program requires students to integrate knowledge and skills together to make sense of a context and develop an efficient and successful solution strategy. For example, each topic includes an open-ended performance task that allows students to be creative in demonstrating what they learned. The task presents a scenario that includes student directions for acceptable work and a detailed rubric, teacher notes, and a sample answer. The materials include opportunities for students to solve real-world problems from a variety of contexts. For example, each lesson in the materials contains real-world problems such as buying back-to-school supplies, taking a bike ride home from school, and making brownies for the class.
Module 2 requires students to use ratios to describe a real-world problem situation. For example, students use enrollment data and the number of women at a certain university to write ratios to understand the data. In Lesson 2, students use a grade-appropriate graph to determine how far from home Greg was after driving for 2 hours and how fast Greg was driving during that time. In Activity 1.1, students complete a table comparing the number of students enrolled at a university and the number of male students enrolled and then analyze the data to determine proportionality. In Activity 1.4, students use the context of mixing paint colors to make the same orange shade in increasing quantities. In Lesson 2, students are presented with a basic context and two numberless graphs to represent two friends sharing a large bucket of popcorn. The characteristics of numberless graphs can be used to represent a real-world situation. In Module 2, Lesson 4.4, the materials provide students an activity where they are collecting data on a cognitive Stroop Test experiment where they study a person’s perception of words and colors using four different colors. After collecting this data, students analyze it by calculating the mean times and creating a scatter plot of the data collection. They then draw their trend line to fit the best scatter plot and make predictions using the line’s equation that best fits. They also interpret the slope and y-intercept of their lines.
In Module 3, Topic 1, “Assignment Practice,” students define a variable and write expressions from the information they are given about how many eggs a farmer gathers from several different chicken coops. In Activities 2.1 and 2.2, students solve equations around the context of songs on CDs and downloads of MP3s. In Lesson 2.4, students compare the cost of holding a roller skating event at three different locations. Students compare fees and costs per skates by completing tables, solving equations, and creating graphs for each of the three locations. In Topic 2, students are given two admission pricing options for a taco festival. Students determine which option Derick should choose if he wants to order a given number of tacos, has a given amount to spend, and when the VIP option becomes possible over the general admission option. In Topic 2, students use the Pythagorean Theorem to solve problems given a picture of a real-world context and a description of the problem represented by the picture, such as figuring out if a piece of rope will be long enough to anchor a volleyball net. In Activity 3.3, students write and solve a system of equations given a real-world scenario. Students determine how many adults and children attend the county fair, the cost a business must charge for tickets to make the desired profit, and how many open-ended and multiple-choice questions are on an exam.
In Module 5, Lesson 1.3, students calculate the number of times they have blinked in their lifetimes. Students use data about the average number of blinks a student accounts for in a day plus the number of hours a student sleeps on average per day and analyzes that data to determine how many times they have blinked in their lifetimes. This data is used to introduce students to write very large numbers in scientific notation. In Lesson 2.4, students investigate volume by determining the volume needed to fill a silo, the volume of a cone for yogurt, and the difference in volumes between cylindrical and rectangular prism-shaped popcorn containers. In Lesson 3, students calculate the volume of a sphere for multiple real-world objects like Earth, an NBA basketball, a Major League baseball, and the world’s largest ball of postage stamps. The data is developmentally and thematically appropriate for grade-level students. In Activity 4.3, students analyze the distance between the planets and the sun and then write the distance in scientific notation. Students then compare the numbers through a series of questions.
The materials do not show how they are supported by research on how students develop mathematical understandings. The materials do not include cited research throughout the curriculum that supports the design of teacher and student resources. The materials provide limited research-based guidance for instruction that enriches educators’ understanding of mathematical concepts and the recommended approach’s validity. Cited research that is current, academic, and relevant to skill development in mathematics, and applicable to Texas-specific contexts and demographics is not evident in the materials. A bibliography is not present.
Evidence includes but is not limited to:
At the beginning of the Teacher's Implementation Guide, there is a section titled “Our Research” where research is mentioned. Still, no actual research or synopsis of research is cited or included in the materials. Although the materials speak of their research, known in the materials as “The Carnegie Learning Way” founded by Carnegie Mellon University, teachers’ guidance does not include the physical research behind Carnegie Mellon’s research; the actual research is not cited.
On their home page, the materials briefly describe that educators from Pittsburg Public Schools teamed up with Carnegie Mellon University to create these materials. However, the materials have no evidence that their research is current, academic, and relevant to skill development in mathematics. The program does not describe the students’ context and demographics in the research used to design the program. It is not applicable to the Texas-specific context and demographics.
The instructional materials do not have a bibliography present but do include an acknowledgment page.
The materials provide guidance to prompt students to reflect on their approach to problem solving. The materials also provide guidance for teachers to support student reflection of approach to problem-solving. The materials do not prompt or guide students in developing and practicing the use of a problem-solving model that is transferable across problem types and grounded in the TEKS.
Evidence includes but is not limited to:
At the beginning of the Consumable Student Edition, students are introduced to a section called “Habits of Mind.” This section explains mathematical practices divided into five sections, four of which have an associated icon that can be found through the activities. Each section includes an overall approach to problem solving and lists 3–4 questions a student should ask themselves when they see the related icon. However, this is not a problem-solving model because it is not grounded in the TEKS mathematical process standards. It does not include analyzing given information, formulating a plan or strategy, determining a solution, justifying the answer, and evaluating the problem-solving process and the reasonableness of the solution in an organized format.
Although the Teacher's Implementation Guide speaks of problem solving several times as an important mathematical practice, the materials do not develop or practice a consistent and specific problem-solving model. The materials include activities grounded in the TEKS mathematical process standards where students analyze information, formulate a strategy, determine and justify a solution, and evaluate the problem-solving process and the solution, but a specific problem-solving model is not included. There are several guiding questions to help students problem-solve through each activity in the materials, but no specific model for them to use while problem-solving.
In Module 1, Activity 2.5, the Habits of Mind icon allows for the student to recognize this as a mathematical practice that is being developed. Students identify structure and regularity in their reasoning. Lesson structure design is grounded in the mathematical process where students “Engage,” “Develop,” and “Demonstrate” their reasoning about solving for unknown angle measures. In Module 1, Topic 3, “Questions to Ask,” students are asked to reflect on questions such as “Can you show me the strategy you used to solve this problem? Do you know another way to solve it?” and “Does your answer make sense? How do you know?”
In Module 2, Lesson 2.4, the materials provide an activity where students determine the y-intercept first from a graph, then from a table, and finally from the equation y=mx+b. The facilitation notes prompt teachers to ask students how the process for finding the y-intercept from a table is similar to finding it from a graph and check that their answers make sense. Students are asked if using the slope-intercept form or point-slope form is always acceptable to write the equation of a line. These questions allow students to reflect on their problem-solving approach and are evident in most lessons in the materials. In Activity 4.2, students graph a different equation to verify if the equation they solved for creates the same transformation.
In Module 3, Lesson 2.4, the materials provide an activity where students explore solving equations using graphing, substitution, and the equations’ inspection. They analyze all three methods and answer questions about when to use each of them. They are then given six different linear systems to solve by using their chosen method and then justify their selection and explain why they choose it. These questions allow students to reflect on their problem-solving approach and are evident in most lessons in the materials.
In Module 5, Topic 1, questions guide students to develop and practice using a problem-solving approach in various problem types, such as tree diagrams, tables, and written models, when solving for powers and exponents.
The materials provide students with opportunities to select and use real objects, manipulatives, representations, and algorithms as appropriate for the stage of concept development, grade, and task. The materials provide opportunities for students to select and use technology (e.g., calculator, graphing program, virtual tools) as appropriate for the concept development and grade. The materials provide some teacher guidance on tools that are appropriate and efficient for the task.
Evidence includes but is not limited to:
The materials provide activities where students have to select from various tools, including a graphing calculator, an interactive whiteboard, and the online software MATHia for solving a problem. MATHia is an online, 1-on-1 adaptive math coaching software students can use throughout the course. MATHia includes various electronic tools, including virtual manipulatives, an expression editor, and a calculator that students can use. As students progress through MATHia, they choose the tools most appropriate for them to use on their current tasks. Students explore tools, use animations, categorize answers, and select problem-solving tools within MATHia.
MATHia allows students to use virtual representations for each of the topics in the materials that include grade-appropriate concepts such as area, circumference, proportions, graphing, integers, the order of operations, probability, angle measures, volume, and surface area. Within each module, the MATHia software provides students virtual manipulatives and embedded tutorial hints for lessons.
In Module 1, Lesson 1.1, students explore congruent figures by using either a word document or an interactive whiteboard to prove if two shapes are congruent or similar based on their side lengths and angles. In Topic 1, the materials provide six lessons where students learn how to use patty paper and the coordinate plane to introduce students to congruent figures with translations, reflections, and rotations. The patty paper helps students observe the characteristics of congruent figures after a transformation occurs on the coordinate plane. These six lessons within the materials also provide multiple activities for students to practice the use of these tools by the time students learn about dilations, similarity, and sums of angles, which are topics that follow in this same module.
In Module 2, students use MATHia to determine whether a function is linear, write an equation given a scenario, and write an equation given two points or the slope and y-intercept. Then, students model scenarios using equations, tables, and graphs within the software program.
In Module 3, Topic 1, students learn the algorithm for solving equations with variables on both sides of the equal sign. In Topic 2, “Talk the Talk,” students determine which method for solving systems of equations is best and to provide justification for the selected method. The answer key for this activity states as long as students can justify their answers, a number of methods would be appropriate for selection.
In Module 4, Lesson 1, students search for patterns and sort objects into different groups. From 30 number cards, students analyze and sort numbers into groups. They can group the numbers any way they feel is appropriate but in more than one group. Students then have to provide a rationale for why they grouped the numbers the way they did. The materials offer students the opportunity to select from using unit cubes, Venn diagrams, and number lines to help them develop an understanding of the Real Number system. Unit cubes are used for students to explore perfect squares as well as cube roots for numbers that are not perfect cubes. Number lines are used to place whole numbers, integers, and rational numbers as a visual tool. The Venn diagram is offered as a tool to categorize real numbers and their subcategories as well as irrational numbers. In Module 4, Topic 2, the Teacher's Implementation Guide explains, in detail, the tools that will be introduced to students in the topic of Pythagorean Theorem. The materials give the teacher the background knowledge needed to teach students how to use a protractor, visual proofs on grid paper, graphing calculator, and the MATHia software to guide students in understanding the Pythagorean Theorem and its converse.
In Module 5, Topic 1, the lesson overview explains to the teacher that using a graphing calculator would be an appropriate and efficient tool to guide students in developing an understanding of powers and Scientific Notation. Once students can generalize rules for positive and negative exponents, the materials guide the teacher in using a graphic organizer to note these “rules'' or properties of powers that can be used for any Scientific Notation problem with a negative or positive exponent.
The materials prompt students to select a technique (mental math, estimation, number sense, generalization, or abstraction) as appropriate for the grade-level and the given task. The materials support teachers in understanding the appropriate strategies that could be applied and how to guide students to more efficient strategies. The materials provide opportunities for students to solve problems using multiple appropriate strategies.
Evidence includes but is not limited to:
The “Habits of Mind” are tips used throughout the materials to guide teachers and students towards the appropriate strategies for problem-solving. The Habits of Mind are introduced and explained at the beginning of the Teacher's Implementation Guide for teachers to refer to all year long.
In Module 1, Topic 1, students are introduced to using patty paper to investigate congruent figures. This strategy is used several times to learn about reflections, rotations, translations, and dilations on a coordinate plane.
In Module 2, the teacher facilitation notes suggest having half the class determine the equation using point-slope form and the other half of the class to determine the equation using the slope-intercept form. Then, the class discusses which process is more efficient. Students make connections to the slope-intercept form of a linear equation and use their understanding of similar triangles to illustrate why the line’s slope is the same between any two points. Students can also find slope using the slope formula or a graph. In Lesson 3.5, the materials provide teachers facilitation notes on how to guide students through different strategies when identifying and working with functions. Teachers are guided through function representations such as sequences, mappings, tables, graphs, equations, and ordered pairs and strategies they can teach students to help them identify functions. In Module 2, Activity 4.3, students graph two sets of linear equations, each consisting of lines with the same slope. Students are guided through linear transformations throughout the activity and then select a strategy for verifying the relationship among the lines. In the teacher facilitation notes, differentiation strategies are suggested for students who are struggling. Students can always start with the basic line, y=1x+0, or graph the line to visualize the transformation effect after each column or table is completed.
In Module 3, the materials provide teachers with a detailed module overview with information about the strategies students will explore in developing function foundations. Teachers are given information about using previously learned concepts like ratios, tables of values, and graphing proportional relationships to analyze linear relationships. The materials also include information about how similar triangles will be used to explore the slope of lines. In Lesson 1, the facilitation notes guide the teacher to support students in rewriting equations without fractions or decimals before solving them, using Properties of Equality, and factoring and multiplying by the LCD as strategies for solving equations with variables on both sides. Students learn multiple strategies for solving equations with variables on both sides of the equal sign, based on the equation’s coefficients. In Topic 2, students write a system of equations from the given information and then solve it using their chosen method. Students learn different ways to solve a system of linear equations based on how the equations are written. There is a focus on selecting the correct technique for each problem type. In Lesson 2.4, the materials ask students to solve linear equations by graphing, by inspection, and by substitution and then determine which system works best for the activity’s problems.
In Module 4, students use the perimeter, area, and number sense to find many of the eight Pythagorean triples for a given perimeter. In Lesson 2.1, students are introduced to the Pythagorean Theorem by first exploring the area sums of three squares, then later are introduced to the formula a2 + b2 = c2. The activity prompts students as side notes on the activity pages to remind them of the generalization that the area of two smaller squares should be equal to the larger square area for it to create a right triangle. It also reminds students to use the correct formula to prove a right triangle can be created. In Lesson 2.2, the materials provide questioning prompts for the instructor to help students generalize their techniques to solve for the converse of the Pythagorean Theorem. The materials guide the teacher in having the students select substitution to check that the equations are correct and prove equivalency.
The materials support students to see themselves as mathematical thinkers who can learn from solving problems, make sense of mathematics, and productively struggle. The materials support students in understanding that there can be multiple ways to solve problems and complete tasks. The materials support and guide teachers in facilitating the sharing of students’ approaches to problem-solving.
Evidence includes but is not limited to:
The materials include tasks designed to support students in productive struggle as they make sense of the problem and solve it. “Habits of Mind” encourages experimentation, creativity, and false starts to help students tackle difficult problems and persevere when they struggle. For instance, the materials provide students with worked examples throughout that have a thumbs up and thumbs down icon next to it. At times, students determine if the thumb should be up or down based on the example. If the solution is correct, students identify connections between the steps. If the example is incorrect, students indicate a thumbs-down icon, identify the error made, and then correct it.
The materials challenge beliefs and biases that conflict with all students seeing themselves as mathematical thinkers. For example, a section called “Myth” at the beginning of each topic debunks common misconceptions about math. The following are examples of math myths busted that demonstrate all students are mathematical thinkers: “I don’t have the math gene,” “Asking questions means you don’t understand,” “There is one right way to do math problems,” “I’m not smart,” and “Faster = smarter.” Each module also includes a family guide. One topic of the family guide focuses on debunking common myths that students might believe about being a poor mathematician and replacing the myth with positive and encouraging statements about being a successful mathematician.
The materials provide suggestions for sequencing the discussion of student strategies for solving the problem. For example, each activity includes facilitation notes in the Teacher's Implementation Guide, a detailed set of guidelines that walk teachers through implementing the “Getting Started,” “Activities,” and “Talk the Talk” portions of the lesson. Each lesson includes an activity overview, grouping strategies, guiding questions, possible student misconceptions, differentiation strategies, student look fors, and an activity summary. The materials provide instructional supports for facilitating the sharing of student’s approaches. The materials provide teachers with “listen fors,” so teachers can be prepared for how to respond to students’ thinking while solving problems. Each activity contains an extensive list of sequential questions for teachers to ask students to work through learning tasks individually, in small groups, or in large groups.
The materials provide instructional supports for facilitating the sharing of student’s approaches. For instance, alternative grouping strategies, such as whole class participation and the jigsaw method, are sometimes recommended for specific activities under differentiation strategies in the Teacher's Implementation Guide. Additionally, grouping suggestions appear to help chunk each activity into manageable pieces and establish the lesson’s cadence.
In Module 1, Lesson 5, “Getting Started,” students are presented with jigsaw puzzle pieces, and two are missing. Students match each missing piece to the puzzle’s open spot and describe the sequence of translations, reflections, and rotations that move the puzzle piece to the matching open spot. The goal of the activity is for students to describe and solve real-world situations using rigid motion transformations.
In Module 2, Lesson 1.1, the materials provide an activity where students use tables, equations, and graphs to represent proportional relationships. Students are given plenty of opportunities through four activities within the topic to practice each of the pathways to solve proportional relationship problems. In Activity 2.2, students are shown three tables with arrows to calculate the slope. Students must determine if the slope was calculated correctly for each case and explain any errors that may have occurred when the arrows were drawn. In Lesson 3.2, students analyze the graph’s characteristics to include its slope, whether it’s linear or nonlinear, and whether it's a function. In the facilitation notes, teachers are guided in the “As students work, look for...” section to look for the graph’s labeling by placing the numbers at the endpoint and not the line segment. The materials also ask teachers to look for the different ways students determine the slope of the line. In Lesson 5, “The Whole Fruit Basket,” materials provide questions to guide discussion on how students summarize the strategies they use to compare functions. In Lesson 6, students transition from graphing equations using tables to creating an equation to graph a linear relationship, a more efficient method.
In Module 3, the materials provide a section called, “Myth: Just give me the rule. If I know the rule, then I understand math.” This section gives students information about this myth. It explains that rules without meaning will not be remembered nor learned. Activity 1.1 begins with an example that shows two solution strategies for factoring equations. In Activity 3.1, students play a Tic Tac Bingo game to create equations with given solution types and then summarize the strategy they used to create them. In Lesson 4, students select from various tools and strategies to solve systems of equations: inspection, tables, graphs, and substitution.
In Module 4, students categorize 30 different numbers using any criteria they choose. Students learn several strategies for working with the real number system through discussion, some structured and others open-ended, while working with partners and in small groups. In Activity 1.2, students complete the geometric proof assigned to them, record their findings on a graphic organizer, and then share their results with the class. In Lesson 1.3, students learn about square roots and cube roots and where these roots land in the Venn diagram that classifies the real number system. Students solve a few problems in partners; the facilitation notes guide teachers to have students shape their responses as a class and then guides teachers in asking students how many different solutions the inequality has. The materials constantly show this type of question throughout the modules to share their solving process and listen to their classmates and their approaches.
In Module 5, Lesson 2.3, the facilitation notes in the Teacher's Implementation Guide explain an activity to introduce calculating the volume of a sphere and a cylinder. The guide starts with a whole group activity where students use clay to make a cylinder and a sphere with the same height and investigate each shape’s volume formula. The guide then provides suggestions for the teacher to move students into partners to use the volume formula and solve problems. As an extension, the materials challenge students with questions about finding the water volume inside a sphere that is only 87 percent full.
The materials allow students to communicate mathematical ideas and solve problems using multiple representations, as appropriate for the task. The materials guide teachers in prompting students to communicate mathematical ideas and reasoning in multiple representations, including writing and the use of mathematical vocabulary, as appropriate for the task.
Evidence includes but is not limited to:
The materials include multiple opportunities within every module, topic, and lesson to communicate mathematical ideas. Students are able to solve problems and communicate their thinking process with small groups, partners, and the whole class. At the end of each lesson, the materials provide a “Talk the Talk” and a “Write” section where students are usually asked to communicate their ideas in written format. Throughout the materials, teachers engage students with higher-level questions during the teaching process. The questions happen in small groups and in large groups. Students communicate their learning within those groups in a variety of ways.
Throughout the materials, in every module, topic, and activity, the materials provide a series of questions for teachers to ask students. Sometimes students answer those questions in writing in their Consumable Student Edition, and other times they answer the questions orally in their small groups. Both oral and written responses are included multiple times during each topic. The materials provide students a Write section at the end of every lesson throughout every module. Every topic within each module starts by giving the teacher an overview of the topic’s focus to include essential vocabulary. Within every topic, the lessons begin with vocabulary development. To review each topic, the materials include a section where the vocabulary is summarized for students.
In Module 1, Lesson 1, the Teacher's Implementation Guide prompts the teacher to have students recall geometry vocabulary and notation as they describe provided shapes. The teacher guide provides an EL tip for teachers to provide a dictionary for vocabulary and language support before the lesson. In Activity 1.1, materials provide teachers with differentiation strategy guidance to have students explain their work without using specific numbers, which supports students in recognizing the structure of the problems and understanding rate. In Activity 1.2, students are asked to compare angle measures of logos and make a conjecture. Students then test their conjecture by measuring various angles in the original and new logos and describing their conclusion.
In Module 2, students learn about proportional relationships through teacher questions prompting them to show their answers in multiple ways: tables, equations, and graphs. Students answer questions 1 through 4, comparing three different college students’ driving speeds, with a partner or group. When finished, they share their answers, and the teacher is provided with the following questions to ask as student work is monitored: “Could this graph be used to create a table of values? How?” and “Could this equation be used to create a table of values? How?” In Lesson 3.5, the materials provide a lesson on comparing functions using different representations. Students solve problems using equations, tables of values, graphs, and verbal descriptions to develop their understanding and solve problems related to functions. Students write a scenario that represents a table of value given. Students then write an equation that would have a less steep slope than the one in the table given. Finally, students are given a graph line to compare the slope on the graph to the one on the table and the one they created. In Activity 4.3, students work in groups to analyze cards with linear relationships, then discuss problems and solutions with all group members. In Activity 5.3, students match tables, graphs, contexts, and equations and explain how they determined the equations with the representations.
In Module 3, Lesson 1, students work on production cost and profit activity. Teachers ask how to conclude linear equations in three representations that have the point of intersection as the common solution. In Activity 3.1, students work with a small group to answer six questions about linear equations. The first question requires an equation, the third question requires a written explanation, the fourth question requires a table and a graph, and the sixth question requires students to estimate.
In Module 4, the materials give students several activities to introduce and develop the Pythagorean Theorem concept. The materials guide the teacher to support students through the Pythagorean Theorem’s use and reasoning using area models, proofs, the formula, and applying the theorem on a coordinate plane. The facilitation notes support teachers with guided questions to ask students as they reason through the different representations.
In Module 5, students create graphic organizers to demonstrate their knowledge of five different exponent properties. One of the questions to consider in creating each graphic organizer is “How would you describe this property to a friend?” In Activity 1.1, students learn three key terms. Definitions for those terms are given in the Student Consumable Edition. The Teacher's Implementation Guide includes differentiation strategies to assist students in the understanding of those key terms. In Lesson 2, “Talk the Talk,” students create a graphic organizer to organize their thoughts regarding the six properties learned in the lesson, work through stations to evaluate each group's graphic organizer, and add input or make corrections to other groups’ work.
The materials provide opportunities for students to engage in mathematical discourse in a variety of settings (e.g., whole group, small group, peer-to-peer). The materials integrate discussion throughout to support students’ development of content knowledge and skills appropriate for the concept and grade-level. The materials guide teachers in structuring and facilitating discussions as appropriate for the concept and grade-level.
Evidence includes but is not limited to:
Throughout the materials, in every activity, there is at least one opportunity for students to engage in mathematical discussions with a partner, a small group, or the whole class. Most activities include multiple opportunities for these discussions. Every activity includes at least one opportunity for discussion in various groups, including the beginning, middle, and end of concept and skill development. The Teacher's Implementations Guide provides teachers with a series of questions to ask students to prompt discussions in every activity throughout the materials. Sometimes those discussions are guided by the teacher, and other times the discussions are guided by the students.
Each lesson is designed to have students work with a partner or in groups to answer specific lesson questions and then share out responses to the class. Each lesson is designed to have an active discussion engagement. Materials are designed to move from student discussion to classroom discussion with teacher guidance. Each lesson guides differentiated grouping strategies, such as whole group or jigsaw, in the “Differentiated Strategies” section of the teacher guide. The materials offer teachers guidance on how to structure discussion that is appropriate for the grade level and choose a grouping structure for discussion that will support students in developing content knowledge and skills.
In Module 1, the materials include three lessons and ten separate activities within those lessons for the concept development of similarity, emphasizing dilations of figures on and off the coordinate plane. As students work out each of these ten activities with the teacher’s guidance, the concept moves from introducing dilations with essential vocabulary and related rules to the figures’ effect after a scale factor is applied. The next lesson moves into developing dilations and similarity on a coordinate plane. The last lesson and activity move students to verify and prove similarity using angles and side lengths. Every one of the ten activities includes opportunities for students to discuss in the whole group, small groups, and partners at each phase of the concept development (beginning, middle, and end). In Lesson 2.2, the materials provide students a lesson on dilating figures on the coordinate plane. The facilitation notes guide the teacher to have a student read the introduction to the lesson aloud and analyze the given worked examples in the whole group. The materials then prompt the teacher to move students into partners or small groups to answer questions 1–2 in the student workbook. After they have finished, students come back to the whole group and share responses as a class to check for understanding. Finally, the students return to work with their partners or groups to work on questions 3–5 and again share responses as a class when they finish. In Lesson 2.3, the materials provide activities to develop the concept of similarity by exploring equal angles and proportional side lengths to different figures. The facilitation notes guide the teacher in starting this activity in the whole group and ask a student to read the introduction aloud and discuss as a class. The materials then suggest having students work with a partner or in a group to complete questions 1–6 then share responses as a class. This type of example is evident throughout the materials in every lesson.
In Module 3, Lesson 2, students analyze a map of Washington, D.C., and begin exploring systems of equations with a partner or group. In the next activity, students read the introduction aloud and then complete question 1 as a class. At the conclusion of the lesson, students work with a partner or in a group to complete questions 1–3 and then share with the whole class about how to write systems of equations from graphs.
In Module 4, students use Pythagorean Theorem to calculate and compare the two-dimensional diagonal path’s lengths and the three-dimensional diagonal path of the fly with a partner or a group and then share responses as a class.
The materials reviewed for grade 8 meet the criteria for the indicator. The materials provide opportunities for students to construct and present arguments that justify mathematical ideas using multiple representations. The materials assist teachers in facilitating students to construct arguments using grade-level appropriate mathematical ideas.
Evidence includes but is not limited to:
The materials follow the same structure for all topics and lessons. This structure includes introducing a concept in a whole-class setting, then moving students into partners or small groups to answer questions. Within these group settings, teachers use the materials’ facilitation notes, which provide prompts for the teacher to assist students when constructing arguments. Each module of the Teacher's Implementation Guide includes “Questions to Ask” that structure the facilitation of constructing arguments. Students consistently have to reflect and ask themselves, “How can I justify my answer to others,” indicating that whenever students explain their reasoning, they need to be able to justify their answer.
The materials provide a “Thumbs Up/Thumbs Down” section throughout the lessons that give students a problem that has been worked out by a fictional student. Sometimes the problem shown is worked out correctly, and sometimes it is incorrect. The student analyzes the method used to solve the problem shown and either gives it a thumbs up or a thumbs down. Students write if they agree with the method and the solution or not and explain their reasoning behind their stand. Based on the student's answer, teachers help students redirect their misconceptions or confirm their understanding of a concept.
In Module 2, Lesson 1.2, the materials provide three activities to help students understand slope as the steepness of a line. Students use multiple representations, such as drawing triangles on a coordinate grid, using patty paper to translate lines to a new graph and compare steepness, and symbolically learning the notation of writing equations in slope-intercept form y=mx+b. The materials ask students to answer questions like if a given equation could be written as a proportion and why or why not, and if a given line represents a proportional or non-proportional relationship. Students refer to one of the representations they have learned to justify their answers. In Activity 3.2, students use a graph and right triangles to justify that the line’s slope is the same between any two points on the line. In Lesson 5, students use multiple representations when comparing functions and materials, including tasks for students to justify their reasoning with a partner, group, and class. Activity 5.3 has students order functions from least to greatest rate of change. Students justify their ordering for a verbal description, table, equation, and graph.
In Module 4, students are given a practice section at the end of a lesson on the real number system where a fictional student has classified a set of numbers based on the real number system. Three sets of numbers are classified. The student analyzes the list of numbers and determines if they agree or not with the fictional student’s grouping. Students explain their reasoning and justify their answers. Before constructing a statement, a student needs to be able to identify, define, and classify each section of the real number system and use their knowledge to prove that their statement is correct. The side note on the activity gives the teacher the correct answer. This allows the teacher to assist students when it is observed that they do not fully understand the concept.
In Module 5, Topic 1, Activity 2.1, students justify properties selected for use when simplifying problems involving exponents.
The materials include a variety of diagnostic tools that are developmentally appropriate (e.g., observational, anecdotal, formal). The materials do not guide teachers to ensure consistent and accurate administration of diagnostic tools. The materials include some tools for students to track their own progress and growth; however, some of those tools were not accessible for review. The materials include diagnostic tools to measure all content and process skills for the grade level, as outlined in the TEKS and Mathematical Process Standards.
Evidence includes but is not limited to:
Based on the Texas Education Agency’s definition of diagnostic tools, “systematic assessments and instruments to gather information to monitor progress and identify learning gains,” the materials reviewed include diagnostic tools. The homepage of the material’s website has a brief description of two assessments used to check student progress. One of them is “MATHia,” which is a computer-based component of the materials. MATHia includes a formative assessment for each skill. There is an indication that MATHia, may provide the opportunity for students to track their own progress and growth throughout. The second is a partnership with “Edulastic,” described as a separate addition to the materials that include summative assessments addressing K-12 standards. The MATHia component was available for review in a limited demo. Edulastic was not available for review.
Formative assessments are provided throughout each lesson and workspace, providing the teacher with ongoing student performance feedback. A variety of topic-level summative assessments are provided to measure student performance, a designated set of standards. Before, during, and after each topic, the materials offer opportunities to assess students’ learning appropriately. This is done throughout the materials in the Teacher's Implementation Guide through suggested questions for the teacher to check for understanding during the activities within each lesson. The materials also provide pre- and post-assessments by topic.
The materials are designed to allow students to demonstrate understanding in a variety of ways. For example, in the “Demonstrate” portion of each lesson, students solve questions in multiple ways. In Module 1, Topic 3, students are given a diagram and determine the eight unknown angle measures inside the figure and then list the labeled side lengths in order from least to greatest. In Module 4, Topic 1, students place sets of numbers in the given Real Numbers Venn Diagram. They then use the graphic organizer to justify whether a statement is true or false about real numbers.
For each module, students learn the lesson goals and set their own goals. Students reflect on the lesson’s main idea, ask questions to clarify their learning, and revisit questions posed at the lesson’s opening. There is no explicit student tracker for data; it is suggested that students track progress independently. Diagnostic tools are not included to measure all content and process skills, as outlined in the grade-level TEKS. Still, various topic-level summative assessments are provided to measure student performance on a designated set of standards.
The materials include diagnostic tools; however, some of the tools were not accessible to review. The materials support teachers with some guidance and direction to respond to individual students’ needs in all mathematics areas, based on measures of student progress appropriate to the developmental level. Diagnostic tools yield meaningful information for teachers to use when planning instruction and differentiation. Materials provide some resources and teacher guidance on how to leverage different activities to respond to student data. The materials do not provide guidance for administrators to support teachers in analyzing and responding to data.
Evidence includes but is not limited to:
Materials include “LiveLab” data monitoring tools to analyze student usage of MATHia. The tool provides teachers with a monitoring section where students are listed in order from those struggling the most to the students that have mastered the concept. When a teacher clicks on a specific struggling student, LiveLab provides an at-risk predictive warning at the top of the page with remediation suggestions on skills or concepts needing to be re-taught or reviewed to be successful in that specific topic. LiveLab provides teachers with a tour video to explain how to understand the diagnostic tools’ results. Although limited access was provided to the Texas Resource Review to evaluate the videos, it is clear that more videos similar to the tour video exist for teachers to access. The MATHia software component includes a reporting platform where data can be analyzed for each student, class, and the whole school. This platform’s reports are the Adaptive Personalized Learning Score (APLSE) report, session report, standard report, and detailed student report.
In the “Edulastic Assessment Suite,” reports are color-coded to help identify areas for improvement and strengths. The materials provide various suggestions and activities for teachers to use to address the results of student assessments. For example, in MATHia, access, facilitation, and follow-up suggestions after a formative assessment are available via the “CL Online Resource Center.” The Edulastic Assessment Suite allows teachers to view assessment summaries, sub-group performance, question analyses, response frequency, performance by standards, and students’ performance. Furthermore, teachers can view individual student assessment profiles and mastery profiles.
Based on the limited access to MATHia, Livelab, and the Edulastic Assessment Suite, it is not evident that the materials meet the indicator.
The materials include routine and systematic progress monitoring opportunities that accurately measure and track student progress. The frequency of progress monitoring is appropriate for age and content skills.
Evidence includes but is not limited to:
Formative assessments are provided throughout each lesson and workspace, providing the teacher with ongoing feedback on student performance. A variety of topic-level summative assessments are provided to measure student performance over a designated set of standards. For example, before, during, and after each topic, the materials offer multiple opportunities to assess students’ learning appropriately. This is done throughout the Teacher's Implementation Guide through suggested questions for the teacher to check for understanding during the activities within each lesson. The materials provide pre- and post-assessments by topic. The materials also provide progress monitoring through MATHia and the Edulastic Assessment Suite; however, these were not accessible to review.
Each lesson includes a “Demonstrate” section known as “Talk the Talk.” This section is essentially an ongoing formative assessment that helps the teacher make decisions about helpful connections that need to be made in future lessons. Each topic includes specific aligned assessments: pre-test, post-test, end-of-topic test, standardized test practice, and performance task. In the Teacher's Implementation Guide, each lesson includes “As students work, look for” sections that note specific language, strategies, and errors to look and listen for as the teacher circulates and monitors students working in pairs or groups. Each lesson embeds “Questions to Ask” and “Misconceptions to Look For” that help teachers monitor progress.
In Module 1, there are various options for progress monitoring, including patty paper and other manipulatives, the use of the MATHia software, and the use of paper/pencil assessments. In Lesson 1, students use patty paper to verify the congruence of two shapes and then explain how to use one figure’s transformations to obtain the second figure, demonstrating the knowledge of everything taught during this lesson with appropriate pacing and use of patty paper. In Module 2, Lesson 2, students are presented with two contexts represented by piecewise graphs and calculate the change rate for each piece of the graph. The “As students work, look for” section notes to look for inaccurate labeling of the graph by placing letters at the endpoints of the line segment rather than at the line segment itself and various determining slope methods.
The “LiveLab” data monitoring tools in the materials are used to analyze the students’ usage of MATHia. The materials provide teachers a tour video to explain how to understand the results of the diagnostic tools. Access to reviewers is limited to just a tour video. Materials also include the Edulastic Assessment Suite; however, the Texas Resource Review did not have complete access to evaluate assessment guide checkpoints, timelines, or teacher tips for tracking progress.
Materials include some guidance, scaffolds, supports, and extensions that maximize student learning potential. Materials provide recommended targeted instruction; however, the materials lack sufficient activities for students who struggle to master content. Materials provide recommended targeted instruction and activities for students who have mastered content. Materials provide some additional enrichment activities for all levels of learners.
Evidence includes but is not limited to:
The materials provide some recommended targeted instruction; however, there is a lack of evidence that the materials include activities for students who struggle to master content. Throughout the materials, the lessons provide differentiation strategies, tasks, and questions for struggling students. The Teacher's Implementation Guide provides a “Facilitation Notes” section embedded at the beginning of each lesson, including “Differentiation Strategies,” such as additional scaffolding or alternative methods to help struggling learners. Materials help teachers identify and provide students with opportunities to develop precursor skills and concepts necessary to the content and aligned to the “Where Have We Been?” section of each lesson. There is limited evidence of additional lessons or activities for targeted instruction that include differentiated instructional approaches. The materials include limited instructional strategies that address various accessibility needs such as vision or hearing impairment; for example, the materials’ digital component (MATHia) includes audio narratives.
Module 1, Lesson 1.4 provides a differentiated strategy for students that encounter confusion in a given question because the vertices to the images are not labeled. The materials state, “Use visual tactics by looking at general side lengths and angle measures to determine corresponding vertices.” In Activity 2.1, students who need more support use a graphic organizer.
In Module 3, Activity 3.1, students who struggle with solving systems of equations use substitution and interact with the worked-out example by labeling the equations “first” and “second” equation. Then students draw a box around 8x in the equation y=8x to make it explicit what they are substituting for y. Students redo the steps in the margin by following the directions to solidify the process in their minds. Finally, students write the answer as an ordered pair.
In Module 5, Lesson 2.1, students struggling with differentiating volume formulas for cylinders and prisms use stencils that have circles and ovals to sketch drawings of the 3-dimensional figures. Students point out congruent bases in both the prism and the cylinder. The teacher reminds the students of the dimensions similar in both shapes, such as the height, and also the dimensions that are different, like length, width, radius, and diameter. Teachers are guided to help make connections from the diagrams to the volume formulas.
The materials provide recommended targeted instruction and activities throughout for students who have mastered content. Each lesson includes differentiation strategies that include additional challenges for students ready to advance beyond the scope of the activities within the lesson. The materials include a “Stretch” section in each lesson for advanced learners who have already mastered the lesson’s concepts. Furthermore, the materials include recommendations for upward scaffolds or extensions to deepen grade-appropriate learning by providing “Questions to Ask'' in the material’s facilitation notes.
In Module 1, Activity 2.1, students who are ready to extend can create a type of translation that goes beyond the scope of the student expectations as identified in the TEKS. In Module 1, Activity 2.2, students extend the activity by performing a reflection then translation, or translation then reflection, and compare the images.
In Module 3, Activity 1.3, students write equations, graph lines representing the equations, and interpret the point of intersection of the two lines. To extend this activity, students solve a given equation using another strategy. One option is to divide 2 from all terms in 2p+2d=34, resulting in 1p+1d=17. When comparing the equation to 1p+3d=23, it is more visible that 2 additional drinks cost $6, so one drink costs $3. By substitution, one pizza costs $14.
The materials include some additional enrichment activities for all levels of learners. The materials provide students opportunities to explore and apply new learning in various ways by including activities that allow students to analyze and internalize target instruction. Each topic includes a performance task, or an open-ended question where students can be creative in showing how they can demonstrate their learning.
In Module 3, students analyze a street map of Washington, D.C., and describe whether or not it is possible to be at an intersection of two intersecting streets, two parallel streets, or two streets that are actually the same street but with different names. This activity is designed to engage students in thinking about graphs of linear equations with one solution, no solutions, and infinitely many solutions.
In Module 5, Lesson 2.4, the materials provide an enrichment activity where students calculate the mean, interquartile range, and the mean absolute deviation of two different graphical representations. After analyzing the data and calculating those values, students are asked to prepare a presentation of their data analysis to give to the class.
The materials include a variety of instructional approaches to engage students in the mastery of the content. The materials support developmentally appropriate instructional strategies. The materials support some flexible student grouping. Although the materials provide routines and activities across all modules, the routines and activities are primarily designed for large and small group instruction. The materials lack teacher guidance for students who need one-on-one attention for a particular skill or concept acquisition. The materials support multiple types of practices and provide some guidance and structures to achieve effective implementation.
Evidence includes but is not limited to:
The materials include hands-on, concrete practice with manipulatives, visual representations, and symbolic abstractions throughout all modules. Students use models to solve problems and then solve problems without the use of models. “Think-Pair-Share,” hands-on activities, and discovery-based learning are examples of the different instructional approaches used to engage students in the mastery of the content across all instructional materials’ modules.
The materials provide an online component called MATHia, where students work on self-paced instruction incorporated for individual exploration. MATHia provides unit overviews, step-by-step instructions, hints, and a glossary. The materials offer learning experiences included for individual exploration.
The Teacher's Implementation Guide provides suggestions for teaching strategies but includes limited support on when to use a specific strategy. Throughout the materials, there are examples of several different teaching strategies that include but are not limited to teaching with multiple representations, accessing prior knowledge, creating and using models, and students participating in authentic mathematical discourse. Common misconceptions are provided for the teacher to address throughout the lesson, but there is a lack of activities to meet all individual students’ needs. Differentiation strategies in each lesson are geared primarily for students who struggle; however, these strategies generally suggest a grouping rearrangement and offer limited indicators to help teachers understand when a student needs these interventions.
In Module 1, Lesson 1.2, students use patty paper to design a logo by copying a figure and “use rigid motion such as transformations such as sliding, flipping, and spinning.” A differentiation strategy suggested in this lesson is to have students create a graphic organizer for terms throughout the lesson.
In Module 2, Topic 1.1, the materials include activities that guide students to access prior knowledge; later in the same lesson, visual representations of graphs of functions are used to teach proportional relationships. In Activity 2.3, students independently write stories describing the context of two numberless graphs by interpreting the information; no teacher guidance is provided on how to support students who need one-on-one intervention to understand the skill or concept needed to complete the task. In Module 2, Topic 3, teachers are encouraged to work problems together as a class and allow students to work in small groups during the teaching process. The materials suggest the students work with a partner or in groups to complete questions and share responses with the whole group. While the materials suggest students work in pairs or groups to solve a given problem, there is a lack of evidence that the materials for eighth grade support students who need one-on-one interventions for skill or concept acquisition. In Module 2, Lesson 4.1, students work as a whole class to create one large human chain to study the speed of nerve impulses and make predictions about the relationship between the number of people and the speed. This activity moves from the whole class to smaller groups to record the data in a scatter plot and look for trends.
In Module 5, Activity 1.1, the materials provide questions for teachers to ask in each chunk of the activity and guide teachers to observe as students work and provide scaffolds based on student responses. In Activity 2.2, students create a graphic organizer for exponents’ properties, write definitions, list facts and characteristics, develop and solve example problems, and write a general rule for each property.
The materials include some accommodations for linguistics; however, the accommodations are broad for all English Learners, and accommodations commensurate with various English language proficiency levels are limited. The materials do not offer scaffolds for English Learners. The materials provide limited opportunities to encourage the strategic use of students’ first language to develop linguistic, affective, cognitive, and academic skills in English.
Evidence includes but is not limited to:
The eighth-grade course materials include no reviewable evidence of various linguistic accommodations for students who are learning English, particularly regarding their English language proficiency level. The materials provide mostly general EL tips that are not specific to developmental levels of English language proficiency.
The Teacher's Implementation Guide serves as the resource for teachers to provide extra modifications within the lessons and activities. The materials include EL tips to make these modifications intentional and natural to the specific activity and lesson. For example, the materials provide multiple opportunities for interaction between students while working on problems in small groups or partners. Some of the EL tips guide the teacher in grouping strategies that are appropriate for specific activities. Although there are several suggestions throughout the materials for EL students to be paired or work in a small group that includes native English speakers, these groups’ focus is not to develop language development. The instruction is not sequenced to support students at varying levels and does not allow for playful and interactive repetition.
In Module 2, Topic 1, ELs may struggle with the word translation but may already know how to translate a word from their native tongue into English. The Teacher's Implementation Guide suggests explaining that a translation occurs by sliding an object up, down, right, or left in math. In Lesson 1, the Teacher's Implementation Guide suggests making the vocabulary and language support readily available to ELs. A dictionary placed under the student’s desk could be beneficial and allow the student to quietly look up the meaning of a word if they are unsure. In Lesson 5, the Teacher's Implementation Guide suggests pairing a beginner and advanced EL student. The teacher then assigns which partner will go first (usually, the student who is more advanced). Then, the student that goes second is encouraged to mimic the sentence structure and frames their partner used in their responses.
In Module 3, Activity 2.3, the Teacher's Implementation Guide suggests having students answer questions 1 and 2 of the activity verbally with a partner to improve their speaking and communication skills. Students will read the problem within their pairs, and then each takes turns answering questions. It is recommended for struggling ELs to mimic their partner’s sentence structures.
In Module 4, Lesson 2, the Teacher's Implementation Guide suggests teachers work with EL student groups and create examples of their reasoning using a sentence stem. In Module 5, Lesson 2, the Teacher's Implementation Guide suggests teachers help EL students improve speaking and communication skills by reading problems with their partner and taking turns answering questions and mimicking their partner’s sentence structures.
The materials include a cohesive, year-long plan to build students’ mathematical literacy skills and consider how to vertically align instruction year to year. The materials provide review and practice of mathematical skills throughout the span of the curriculum.
Evidence includes but is not limited to:
The instructional materials include a cohesive, year-long plan to build students’ concept development and consider how to align instruction that builds year to year vertically. The materials include a year-long plan of content delivery based on 160 instructional days. The Teacher's Implementation Guide provides a map that shows the sequence of topics and the number of blended instructional days (1 day is 50 minutes). The materials include a content plan that is cohesively designed to build upon students’ current level of understanding with clear connections within and between lessons and grade levels. For instance, each module includes three sections: “Connections to Prior Learning,” “Overview,” and “Connections to Future Learning.” Additionally, the Teacher's Implementation Guide provides a course content map that shows connections between lessons in the modules. The materials include guidance that supports the teacher in understanding the vertical alignment for all focal areas in Math Texas Essential Knowledge and Skills in preceding and subsequent grades. In the Teacher's Implementation Guide, each module overview begins with connections to prior learning and connections to future learning.
Within the Teacher's Implementation Guide, there is a resource map, “Middle School Math Solution Content at a Glance,” that shows the instruction sequence for the year and the number of instructional days allocated for each year. In the Teacher's Implementation Guide, the “Course Standards Overview” document provides a mapping of how standards are targeted and reviewed within each lesson. The material includes the following five overarching modules and their corresponding, focused topics, and lessons that require a complete year-long plan to deliver the content: “Module 1: Composing and Decomposing”—3 subtopics and 12 lessons and multiple activities within each lesson; “Module 2: Relating Quantities”—3 subtopics and 12 lessons and multiple activities within each lesson; “Module 3: Determining Unknown Quantities”—3 subtopics and 13 lessons and multiple activities within each lesson; “Module 4: Moving Beyond Positive Quantities”—2 subtopics and 6 lessons and multiple activities within each lesson; “Module 5: Describing Variability of Quantities”—2 subtopics and 7 lessons and multiple activities within each lesson. These five modules are sequenced to develop students’ understanding by providing a plan that connects modules and lessons. The Teacher's Implementation Guide includes a description of the connections made with a table showing each of the five modules and the Connections to Prior Learning, Overview of the module, and the Connections for Future Learning.
The materials include a detailed module and topic overview in the Teacher's Implementation Guide, which teachers can read before lesson delivery to learn the vertical alignment in previous and subsequent grades. These module and topic overviews do not mention specific TEKS, but they do all fall under one of the focal areas that align directly to the TEKS. At the beginning of each module, a section explains how each topic is connected to prior learning. At the beginning of each topic, there is a section entitled “What is the entry point for students?” that clearly states the connections to prior learning. At the beginning of each topic, there is also a section entitled “Why is the topic important?” that clearly explains when students will use the knowledge learned in the topic in subsequent lessons during the current school year and future math courses. At the end of each lesson, there is a section called “Talk the Talk,” which serves as a cumulative review of all of the lesson’s activities. Each lesson within each topic provides opportunities for practice through teacher questioning, the Consumable Student Edition, the Skills Practice Workbook, and the MATHia software. The practice materials within a topic build upon previously taught content from within that topic. The final section of each assignment is “Review” and includes problems from previous lessons and modules. The Review for the lessons in Module 1 includes questions from material learned in seventh grade.
In Module 1, the overview notes that students will learn to transform geometric objects. Their previous geometric knowledge, starting from kindergarten through elementary and middle school, will help build on transformation. Previous concepts taught in grade 7, such as proportionality and scale drawing, will help develop students’ understanding of the effects of a 2-D shape transformation on a coordinate grid. In Topic 1, the Teacher's Implementation Guide supports teachers in understanding how students use proportional relationships and scale factors to understand dilations in terms of coordinates and determine if figures are similar. In Module 1, Lesson 1, students practice dilations of triangles with a given scale factor. In Topic 3, students practice using the Angle-Angle Similarity Theorem to demonstrate if each pair of triangles are similar.
In Module 4, Topic 1, the materials provide four different focused lessons and 13 activities where students will explore and practice working with the Pythagorean Theorem, the Pythagorean Theorem’s converse, distances on a coordinate system, and the side lengths in two and three dimensions. In Lesson 1.1, students learn about the Pythagorean Theorem. In Lesson 1.2, students build on that knowledge and focus on the converse of the Pythagorean Theorem. In Lesson 1.3, students take their knowledge from the previous two lessons and apply it to a coordinate plane. Finally, in Lesson 1.4, students find side lengths to two and three dimensions, which could not be done without first exposing students to the previous three lessons. In Module 4, Lesson 2.3, students learn about applying the Pythagorean Theorem to a coordinate plane. At the end of the lesson, students review and practice problems previously taught in the materials, such as classifying rational and irrational numbers, solving equations, and determining if three side lengths would create a right triangle.
The materials are not accompanied by a TEKS-aligned scope and sequence outlining the essential knowledge and skills that are taught in the program, the order in which they are presented, and how knowledge and skills build and connect across grade levels. The materials include supports to help teachers implement the materials as intended. The materials do not have resources and guidance to help administrators support teachers in implementing the materials as intended. The materials include a school years’ worth of math instruction, including realistic pacing guidance and routines.
Evidence includes but is not limited to:
The materials include a webpage with a link to the alignment to standards by state. The TEKS alignment to the scope and sequence of the materials can be accessed through this page. The materials include a table that shows the Course 1 materials’ correlations to the 2012 Texas Essential Knowledge and Skills. The table is a 15-page list of all the TEKS included within the materials. The TEKS listed show that they are all content SEs; however, the process standards are not shown on this table. The table shows the textbook number, the module, topic and lesson, and the MATHia software location where the student expectation will be presented. The materials include a standards overview in the Teacher's Implementation Guide and provide a mapping document that identifies where the standard will be targeted and where it will be reviewed within each module, topic, and lesson. This standards overview does not list the standards using TEKS. A Texas teacher would have to reference the correlation table to align the materials’ standards overview to the TEKS. The materials do not include a scope and sequence that describes how the essential knowledge and skills build and connect across the grade levels. However, each module in the materials includes an overview that provides a connection across grade levels described in detail, but this is not listed on the scope and sequence document that aligns with the Texas standards.
The instructional materials for grade 8 include support to help teachers implement the materials as intended. The materials support teachers in understanding how to use the resource as intended. The Teacher's Implementation Guide provides teachers with consistent lesson structure and walks teachers through key features of each lesson: “Learning Goals,” “Connections,” “Getting Started,” “Activities,” “Talk the Talk,” and “Assignments.” The Teacher's Implementation Guide provides facilitation notes by activity, a detailed set of guidelines that walks the teacher through implementing the various components of the lesson. These guidelines include an activity overview, grouping strategies, guiding questions, possible student misconceptions, differentiation strategies, student look fors, and an activity summary. The Teacher's Implementation Guide describes the depth of understanding that students need to develop for each standard and a pathway for all learners to succeed. The facilitation notes provide detailed support for the planning process and are the primary resource for planning, guiding, and facilitating student learning. Materials can be accessed online or in print.
The materials include a digital version of the materials and modules that include a “MyPL” Professional Learning app with videos to give teachers background knowledge for a particular lesson and ideas on how to implement it. MyPL includes custom learning sessions, led by Master Math Practitioners, in an online video library accessible. They also include a PowerPoint presentation for each lesson that could be used by teachers. This is also available in Google Slide format. The materials provide links on their homepage to a “LiveLab Tour” for teachers to learn how to use this assessment tool within the materials. There is also a “Help Center” link on the homepage that supports teacher implementation and provides quick “how-to” guides. The materials are available in print and digital format. Accessing the materials on the materials’ webpage is easy to follow. Each module’s format, topic, and lesson materials follow the same format, making it consistent and organized.
The instructional materials for grade 8 do not include resources and guidance explicitly stated to help administrators support teachers in implementing the materials as intended. Nor do the materials provide tools explicitly stated to support the administrator in recognizing best instructional practices and arrangements in a math classroom. However, administrators can use many of the materials mentioned above to support teachers in implementing the materials. For example, the scope and sequence, including the list of math essential knowledge and skills, can be used by administrators to support teachers in their implementation of the materials.
The eighth-grade materials include five modules that span for the pacing of 160 days of the standard 180 days of instruction to allow for flexibility for assessments, an extension of lessons, and differentiation for students. The Teacher's Implementation Guide contains “Middle School Math Solution Content” at a glance with pacing outlined by unit and lessons to allow extra days to address assessments, differentiation, and extended opportunities. Each lesson includes pacing suggestions. Additionally, each module overview includes pacing information and a topic overview, which includes more detailed information to help teachers with pacing. The lesson structure and pacing are included for each lesson and are provided for the entire course to be completed in a school year.
The materials provide some guidance for strategic implementation without disrupting the content sequence that must be taught in a specific order following a developmental progression. The materials are not designed in a way that allows LEAs the ability to incorporate the curriculum into district, campus, and teacher programmatic design and scheduling considerations.
Evidence includes but is not limited to:
The instructional materials for grade 8 provide guidance for strategic implementation without disrupting the sequence of content that must be taught in a specific order following a developmental progression. The materials include strategic guidance on implementation that ensures the sequence of content is taught in the order consistent with the developmental progression of mathematics. For instance, the Teacher's Implementation Guide provides a table in the “Content and Alignment” section that demonstrates how the sequence of modules, topics, and lessons are developed to coherently build new understanding onto the foundations developed in prior grades or previous lessons of the course. Additionally, supporting standards are positioned to reinforce the readiness standard of the grade. In grade 8, students are taught solving linear equations before solving systems of linear equations. Each course materials’ Table of Contents delineates modules and topics to ensure students learn about precursor concepts first. All courses include all focal areas within the years’ instruction without disrupting the content’s sequence.
However, the materials do not include guidance that supports teaching focal areas aligned to a classroom/school context without disrupting the content’s sequence. While this is the case, at least one of the focal areas for eighth grade (proportional relationships, expressions and equations including the Pythagorean Theorem, and making inferences from data) is included in each of the five modules designed for eighth grade. The module and topic overview supports teachers on how that module/topic aligns with focal areas in mathematics; the materials do not guide teachers on how lessons could be taught if the given sequence is disrupted.
The material design offers some variety to allow for easy implementation of school designs. Printed materials and digital materials are available. The Teacher's Implementation Guide allows for extensions of the lessons, and grouping options are often mentioned. Suggestions for co-teaching, multi-grade classrooms, and varying lengths of times for mathematics instruction are not included in the materials. According to the materials, the materials are designed for daily instruction in a 50-minute class period. There is no guidance on implementing the materials in any other school setting.
In Module 2, the materials focus on developing the foundations of functions that is the prerequisite to working on Module 3, which focuses on modeling linear equations. The Module 5 Overview in the Teacher's Implementation Guide supports teachers in identifying how students build on their existing knowledge of exponents to develop new rules for operating with integer exponents.
The materials provide some support for the development of strong relationships between teachers and families. The materials specify some activities for use at home to support students’ learning and development. The materials reference literature for parents and at-home support; however, some of the materials referenced were not accessible to the reviewers.
Evidence includes but is not limited to:
The materials provide “Home Connections” in the MyCL portal, including a video detailing basics about the materials and the use of it in class and at home. The MyCL portal includes a “Community Tools” section with helpful study skills and tips and additional math practice exercises. It also includes a parent/caregiver webinar offering strategies for supporting student learning at home. This webinar was not available for a deep evaluation by the Texas Resource Review.
The materials include a “Math Power for Parents Handbook” to help parents understand the course materials by giving them a summary and examples of each math topic. This is offered in both English and Spanish. The example available to the reviewers online was for elementary. It is unclear if this is also offered for middle school students. The online component of the materials, called MATHia, is available to students anytime and anywhere. However, it is unclear if the parents have access to MATHia to view their student’s progress and work or if they would just help support them as they work on it at home.
The materials’ website notes “Family Math Nights,” where students can present their work; however, examples of what is included in the materials are not evident. The materials provide a “Family Guide” at the beginning of each topic intended to be sent home to keep parents informed of what their child will be learning in that topic. Family Guides, available in English and Spanish, for each topic, overview the mathematics that will be taught, what the student should have previously learned, and how it will be used for future learning. They include real-world examples, standardized test question examples, and some key vocabulary students will learn.
Each Topic within each module includes a Family Guide. The Family Guide includes an overview of the topic, “Where Have We Been?” and “Where Are We Going?” sections, examples of problems to be learned in the topic, key terms to be introduced during the topic, talking points, and an explanation of a false myth that many people believe about math. However, there is no guidance or recommendation for teachers to use the Family Guide or develop strong relationships between teachers and families.
In Module 3, Topic 2, the Family Guide states that students will learn about equations and solving systems, and it outlines what students have learned before and where their learning is going. There is a scale drawing that is used to solve problems, and an explanation is included. The Family Guide also includes a debunked myth, key terms, and talking points parents can discuss with their students about the topic.
In Module 5, Topic 1, the Family Guide suggests parents support student learning by helping them recognize that learning is a slow process and encourage a positive self-image as math learners.
The materials include appropriate white space and design that supports and does not distract from student learning. Pictures and graphics are supportive of student learning and engagement without being visually distracting.
Evidence includes but is not limited to:
The Teacher's Implementation Guide follows a clear and consistent design for information. This guide includes a module and a topic overview, facilitation notes, materials needed for the lesson, lesson and activity overviews, standards addressed, essential ideas, lesson structure, pacing, differentiation strategies, and a summary.
The materials include intentional white space in each margin so that teachers can make additional planning notes or reflect on the lesson’s implementation. This appears in every lesson throughout the material next to the facilitation notes section of the activity overviews and the section labeled, “Questions to Consider.” The Student Edition is consumable; the pages include an appropriate amount of white space to show work. The graphics are simple, clear, and concise. Visuals are placed with purpose throughout the materials, i.e., tables, graphs, models.
The material includes fonts that are clear and easy to read. Headings are bold and identified through a colorful box for emphasis. Lessons and activities include pictures, tables, and graphs that do not distract from the text on the page or interfere with learning. Numbers within the tables, grids, and graphs are large enough for students to read easily.
The Student Edition includes “The Crew” images that have thought bubbles with reminders about previous content, questions to help students think about different strategies, and fun facts. The teacher aide’s images guide students by making connections and reminding students to think about the details. Models are placed throughout the materials, and “Habits of Mind” icons trigger students to ask themselves reflective questions as they work.
The materials adhere to User Interface Design guidelines. For instance, the materials have a consistent layout and color scheme: green, black, and white; it is not distracting. The overall aesthetic of the materials is visually appealing and includes a minimalist design. Workspace and note sections are provided in the Teacher's Implementation Guide and Student Edition to provide space for reflection or for documentation purposes.
In MATHia, animation offers user control and freedom to rewatch demonstrations of various math concepts. MATHia software contains familiar options such as play, volume, enter full screen, and print icon images and terminology for easy student use. Resources also provide a glossary with definitions and graphic images, such as for x-axis and solid volume.
In Module 2, Lesson 2.1, using tables, graphs, and equations to represent linear relationships includes several tables and graphs that are large enough for students to read clearly. The titles for the graphs and the titles for the x-axis and y-axis are easily readable, and so are the numbers displayed on the graphs. The tables include enough space for the students to write inside and enough space around the graphics to show their mathematical thinking process.
In Module 3, Lesson 2.1, the students can see the warm-up section as the starting point because the font is bold and purple and requires them to work on the problem. The learning goals and key vocabulary terms are to the side in black font, which indicates that they do not have to write or answer anything in that section. The rest of the lesson follows the same pattern with questions and problems to answer in purple and directions in black font. The lesson includes tables and graphs that are large enough to read and white space to work the problems out and answer the questions. Side notes are included throughout the lesson to help students answer questions or remind them of something important. The topic number, title, and page number are included at the bottom of each page for easy access. At the end of each activity, the materials intentionally provide an empty blank page for students to show math work, make notes, or summarize their thinking throughout the activity. The Teacher's Implementation Guide includes facilitation notes with support, such as a detailed description of what is needed for students to calculate costs and profits from real-world situations and what questions to ask during the lesson. These sections are labeled with larger bold fonts that allow for easy identification.
In Module 4, Lesson 1.3, the materials include an activity on the real number system. On the activity pages, pictures of Venn diagrams are shown with directions for the students to classify 30 different numbers in the correct section of the Venn diagram. This picture is essential to the lesson and the standard being targeted and supports student learning. The materials also include pictures of students with a quote to give students important tips and reminders to help them answer the questions on that page in the activity.
The materials include online components that are grade-level appropriate and provide learning support. The online elements align with the curriculum’s scope and provide support and enhance student learning as appropriate, as opposed to distracting from it, and include teacher guidance.
Evidence includes but is not limited to:
The materials reviewed include a computer-based software called MATHia that can be accessed anytime and anywhere by students and teachers. Each lesson in the materials has a corresponding lesson in MATHia and is included in the pacing guide for each module, topic, and lesson. MATHia is a 1-to-1 adaptive math coaching program that provides a personalized learning path and ongoing formative assessment. MATHia contains a mini video, “Why this Matters,” that students watch to see a real-world connection to learning. MATHia technology supports include videos, immediate feedback in practice problems, interactive tools, and manipulatives. Additional activities and assessments for each topic and lesson are found in MATHia. It creates a personalized learning path for each student with feedback and hints that are embedded within the software. It also provides teachers with reports of class and individual student’s progress by standards.
In the Teacher's Implementation Guide, the materials outline how MATHia can be incorporated into each module and what topics it will cover. Additionally, there are “Learning individually with MATHia” or “Skills Practice'' sections in the “Module Overview” that summarize how MATHia will be used for each topic and standard and how many instructional days are needed using the software. The individual student practice helps support their learning in the classroom when interacting with the materials since the concepts in MATHia overlap and reinforce the targeted learning standards. The MATHia computer-based software in the materials promotes student participation by providing additional practice for each topic. Since MATHia is accessible at any time and any place, students can continue participating and interacting with the materials at home and even on the weekend to enhance their learning. The Teacher's Implementation Guide provides teachers information about MATHia, how it is structured, how it is aligned to each lesson, and the type of problems included within the software that students will work on. The guide also includes a description of the reports and how to access them. The materials provide teachers with a MATHia browser in the MyCL portal to view and experience the software as a student would. This allows teachers to read a report, review, and reteach a student who is not successful in a MATHia module.