6th Grade - Gateway 3

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The Facilitation Notes at the end of each topic provide differentiation strategies, common student misconceptions, and suggestions to extensions. The course also provides a Front Matter section intended to provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. 

The Front Matter section includes:

• Guiding Principles of Carnegie Learning.

• Content Organization including Modules, Topics, and Pacing.

• Course standards overview chart.

• A Table of Contents on the Module level provides connection to prior learning, connection to future learning, a chart of the CCSSM sorted by topic, and a list of materials needed within the module.

• A Table of Contents on the Topic Level provides MATHia recommended lessons for each topic and a pacing guide of MATHbook and MATHia lessons.

• Guidance for implementing MATHbook which is structured consistently as ENGAGE, DEVELOP, and DEMONSTRATE.

• Guidance for implementing MATHia.

• Guidance on assessing students by checking readiness, monitoring learning, and measuring performance.

• Planning resources that include pacing guidance, topic planners, lesson planners, and lesson-level facilitation notes.

• Guidance for supporting students in their language development and social emotional learning.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objective.

• In Module 1, Topic 1, Lesson 1, Getting Started, Teacher’s Implementation Guide, the materials provide sets of questions to support student discourse within the lesson. Questions are identified by type, such as Gathering, Probing, Seeing Structure or Reflecting and justifying. A sample Seeing Structure question states, “Why does everyone get the same total area even though they divided the walkway differently?”.

• In Module 2, Topic 1, Lesson 5, Lesson Planning, Teacher’s Implementation Guide, teachers are given different options for students who are on target and for students who are not there yet. Students who are not on target yet will use MATHia to practice using graphs to determine equivalent ratios. Students who are on target will complete a different activity and present how they used graphs to solve problems.

• In Module 4, Topic 2, Lesson 2, Getting Started, Teacher’s Implementation Guide, differentiation strategies are provided to support student learning in an activity. In the original activity, students represent points and are placed on a human coordinate plane. Students are then asked to plot and label the point where the student is standing and record the coordinates of the point in a table. The materials provide an alternative to the human coordinate plane for teachers who have smaller rooms. Instead, teachers are instructed to display a sizable coordinate plane on the board and use colored dots stickers to plot the ordered pairs for each location. A second suggestion for differentiating instruction is to use string to connect the points in the human coordinate plane so students could more clearly see the shapes being formed.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

The materials provide an overview at the beginning of each module and for each topic within the module. The Module Overview provides an explanation for the naming of the module, research for why the module is included as part of the scope and sequence, connections to prior learning, and connections to future learning. The Topic Overview provides an explanation of how key topics are developed, an examination of the entry point for students to connect to prior learning, and identification of the importance of the topic for future learning. 

Examples of how the materials support teachers to develop their own knowledge of more complex, course-level concepts include:

• In Module 2, Topic 1, Lesson 5, the materials provide an adult-level explanation in determining equivalent ratios using a coordinate plane. The materials state, “Just as you can represent equivalent ratios using tables and double number lines, you can represent them on the coordinate plane. The ratio \frac{y}{x} is plotted as the ordered pair (x,y). When you connect the points representing the equivalent ratios, you form a straight line that passes through the origin.”

• In Module 3, Topic 1, Lesson 4, the materials provide an adult-level explanation in determining if two expressions are equivalent. The materials state,“You can use a graph or properties to verify two expressions are equivalent. You can use a table of values to show two expressions are not equivalent; however, a finite set of values in a table is not enough to verify equivalency.”

Examples of how the materials contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject, include:

• In Module 3, Topic 2, Topic Overview, Connection to Future Learning, the materials state, “Equations provide students with their first formal encounter with variable equations. They will continue writing and solving equations and writing inequalities as constraints in Graphing Quantitative Relationships. In grade 7, students will expand their ability to solve equations to one- and two-step linear equations with positive and negative values.” The materials then provided a mathematical representation showing how to use a double number line to solve 2j+10=46.

• In Module 5, Topic 1, Topic Overview, Connection to Future Learning, the materials state, “Students will use their knowledge of variability, the statistical process, and data displaying in the remaining grade 6 topic, Numeric summaries of Data. In that topic, they will add box-and-whisker plots to their knowledge of data displays. Students will also connect the shape of the distribution of a data set to the relative locations of the mean and median of a data set.” The materials then provide a mathematical representation with images representing skewed right data, symmetric data, and skewed left data. The materials then state, “In grade 7, students will use the data displayed learning in The Statistical Process to compare data distributions. They will use statistical problem-solving to investigate and draw inferences about populations. In grade 8, students will move into comparing data in two variable, bivariate data.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Examples of materials providing correlation information for the mathematics standards addressed throughout the grade level include: 

• Each Module within the courses contains a Module Teacher’s Implementation Guide Overview. The Module Teacher’s Implementation Guide Overview provides the standards for each topic as well as the standards for each MATHia workspace that is paired with each topic.

• Found under each topic’s Teacher Materials section, the Front Matter in the Teacher’s Implementation Guide provides a Standards Overview chart. The chart identifies lesson standards and spaced practice standards. Each module with the Teacher’s Implementation Guide also has a standard overview represented as a dot matrix that identifies the standards addressed in each module, topic, and lesson. Additionally, each topic has a Topic Overview which lists the standards for each lesson. 

• Each topic also has a Topic Overview under the Teacher Materials section that identifies the standards in each lesson. A session log is also available that identifies the sessions MATHia will be utilized.

Module Overviews located at the beginning of each Module identifies specific grade-level mathematics. The Topic Overview located at the beginning of each Topic identifies the role of the mathematics present within the Module. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:

• In Module 1, Topic 3, Topic Overview, the materials provide an overview of how key concepts of Decimals are developed. The topic begins with building on students’ prior knowledge of plotting decimals on a number line and comparing and ordering decimal values. Students develop toward adding, subtracting, and multiplying with decimals. Students divide decimals by whole numbers and finally divide decimals by decimals using long division.

• In Module 3, Teacher’s Implementation Guide, Module Overview, the materials provide a connection to prior and future learning. Students apply their previous understanding of Order of Operations and arithmetic properties to the set of non-negative rational numbers. Students will build upon their reasoning to determine unknown values when solving one-step equations later in the course.

• In Module 4, Topic 2, Topic Overview, the materials provide an overview of how key concepts of The Four-Quadrants are developed. The topic begins with students building a four-quadrant coordinate plane. Students develop toward solving problems involving geometric figures on the coordinate plane and solving a wide range of problems on the coordinate plane that involve scenarios, graphs, equations, and tables.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Front Matter in each course provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted, all examples are found in the Front Matter of the Teacher’s Implementation Guide.

Examples of the materials explaining the instructional approaches of the program include: 

• The Front Matter of the Teacher’s Implementation Guide includes the programs, “Guiding Principles.” The four guiding principles state, “All students are Capable Learners.” “Learning by Doing™”, “Learning Through Assessments,” and “Education is a Human Endeavor.”

• The program’s instructional approach is, “...based on a scientific understanding of how people learn, as well as an understanding of how to apply the science to the classroom.” There are three phases to the instructional approach: ENGAGE, DEVELOP, and DEMONSTRATE. The materials provide an explanation for each instructional approach. ENGAGE is intended to, “Activate student thinking by tapping into prior knowledge and real-world experiences.” DEVELOP is intended to, “Build a deep understanding of mathematics through a variety of activities, and DEMONSTRATE is intended to, “Reflect on and evaluate what was learned.”

• “Introduction to Blended Learning,” explains how MATHbook and MATHia are designed to be used simultaneously to support student learning. Students will “Learn Together” using the MATHbook approximately 60% of the time and “Learn Individually” using MATHia 40% of the time. 

• The Front Matter of the Teacher’s Implementation Guide provides a rationale for the sequence of the modules, topics, and lessons within the course and series.

• “Comprehensive Assessment,” includes checking student readiness using the MATHia Ready Check Assessments and the MATHbook Getting Ready resources, monitoring learning by question to support discourse, and measuring performance using pre-tests, post-tests, end of topic tests, standardized tests, and performance tasks. 

Examples of materials including and referencing research based strategies include:

• In the Front Matter of each course in the Teacher’s Implementation Guide, the materials state, “The embedded strategies, tools, and guidance provided in these instructional resources are informed by books like Adding It Up, How People Learn, and Principles to Action.”

• In the Front Matter of each course in the Teacher’s Implementation Guide, the materials state, “MATHia has its basis in the ACT-R (Adaptive Control of Thought-Rational) theory of human knowledge and cognitive performance, developed by John Anderson - one of the founders of Carnegie Learning (Anderson et. al., 2004; Anderson, 2007).”

• Each Module Overview includes a section on “The Research Shows…” citing research related to a strategy, tool, or content matter within the module. In Grade 6, Teacher’s Implementation Guide, Module 1 Overview, the materials cite research from Navigating through Measurements, page 4 by stating,”Understanding of and proficiency with measurement should flourish in the middle grades, especially in conjunction with other parts of the mathematics curriculum.” 

• The materials of each course provides a link to a website referencing more extensive research on the research-based strategies incorporated in the program.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Examples of where materials include a comprehensive list of supplies needed to support the instructional activities include:

• The online materials for each course provides a “Course Materials List” located in the General section of the Teacher Materials. The list contains the supplies needed for each Module.

• In each course, the Front Matter in the Teacher’s Implementation Guide provides Module pages in the Table of Contents. The Module pages specify materials needed for each module in the right corner of the page. 

• The list of materials is also provided in the Topic Overview included at the beginning of each topic.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meets expectations for having assessment information included in the materials to indicate which standards are assessed. The materials state, “Assessment is an arc and not a one-time event. It is a regular part of the instructional cycle. Ongoing formative assessment underlies the entire learning experience driving real-time adjustments, next steps, insights, and measurements. Check Readiness > Monitor Learning > Measure Performance.” The materials identify the following as assessments: 

• Check Readiness

  • Module Readiness is in the MATHia Readycheck Assessment and measure, “student readiness of concepts and skills that are prerequisite for any upcoming content. The scoring guide informs student instructional needs.” The MATHbook Getting ready reviews prior experiences with mathematical content that will be built upon in the module. 

• Monitor Learning

  • MATHia contains LiveLab where teachers can monitor student work for “real-time recommendations on how to support student progress.” The MATHbook contains Lesson Overview listing learning goals, review questions, and making connections to prior learning. The MATHbook contains Questions to Support Discourse for each activity to assess, “students’ sense-making and reasoning, to gauge what they know, and generate evidence of student learning.” The MATHbook also contains Talk the Talk tasks to allow students to reflect on their learning from the lesson and provide teachers with information on whether students can demonstrate the learning outcomes. 

• Measure Performance

  • MATHis provide Skill Reports monitor skill proficiency of students in mastery workspaces, Standard Reports provide an overview of students’ proficiency on specific standards, and Predictive Analytics allow teachers to monitor student progress to predict students’ year-end outcomes.

• MATHbook contains Summative Assessments in the form of Pretest, Post-test, End of Topic Test, Standardized Test, and Performance Tasks. 

Examples of how the materials consistently identify the standards for assessment include: 

• In Module 1, Topic 1, Assessment Overview, the materials identify, “Numbers and Operations - Fractions, The Number System, and Expressions and Equations” as the standard domains for the assessments.  The materials then provide a specific standard for each question in the pre-test, post-test, End of Topic Test, Standardized Test, and the Performance Task.

• In Module 3, Topic 1, Lesson 1, the Talk the Talk assesses the standard 6.EE.1 by having students use Order of Operations to analyze evaluated expressions and determine if the solution is correct.

Standards for Mathematical Practice are referred to as habits of mind within the materials. The habits of mind are only identified within the activities in the MATHbook.  Within the activities the Questions to Support Discourse are used to assess the activities. Examples include:

• In Module 3, Topic 1, Lesson 4, Activity 1, the summary in the Teacher’s Implementation Guide states “You can use a graph or properties to verify two expressions are equivalent. A table of values can only show two expressions are not equivalent.” The Habits of Mind listed are, “Model with mathematics. Use appropriate tools strategically.” Within the activity, the Questions to Support Discourse states, “Do you think both expressions will create the same value regardless of the value of x? Explain your thinking. Why do you think you should connect the points? What do you think the graph would look like with expressions that are not equivalent? Why do the properties show that the expressions are equal for any value? What methods can you use to determine the equivalency of two expressions?”

• In Module 4, Topic 2, Lesson 1, Activity 1, the summary in the Teacher’s Implementation Guide states, “You can use the signs of an ordered pair to identify its quadrant location. A point lies on an axis when one of the coordinates is zero.” The Habits of Mind listed are, “Look for and make use of structure. Look for and express regularity in repeated reasoning.” Within the activity, the Questions to Support Discourse states, “In what direction do you travel when the x-coordinate is positive? In which direction do you travel when the y-coordinate is negative? Why do all the points that lie on the y-axis have zero as their x-coordinate? If a point has a negative x-coordinate, in which quadrants might it lie? What do all points that lie in Quadrant II have in common?"

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Answer keys are provided to determine students’ learning and reports provide teachers’ guidance on interpreting student performance. Suggestions for follow-up are provided through LiveLab, which alerts the educator to students who may need additional supports in specific skills, and the Skills Practice which provides suggestions on how students can re-engage with specific skills.

Examples include:

• In the MATHia Group Skills Report, teachers can view each student’s skill mastery progress organized by module, unit, and workspace. The materials state, “For each skill, a student can be in one of the following categories: Proficient: The student has a greater than or equal to 95% probability of understanding and correctly executing that skill. Near Proficient: The students has a 70%-94% probability of understanding and correctly executing that skill. Remediation Suggested: The students has a <70% probability of understanding and correctly executing that skill. In Progress: The student is currently completing problems that address this skill. Not Started: The student has not encountered workspaces that address this skill.” In the digital material, Help center, Math, LiveLab, At Risk Student Alert in LiveLab, the MATHia Report states, “The At-Risk Student Alert tells a teacher when a student is at risk of not mastering a workspace, as he/she is struggling with the understanding of a specific math concept. The warning will appear as a life preserver icon next to the student's current status on the main Class Dashboard. Click to the Student Details screen to review which workspace he/she is struggling with and specific math skills covered in the workspace to better understand how to provide targeted remediation for this student. You can review the skills in the Mastery Progress section of the Student Dashboard to help you provide that targeted remediation.”

• Summative Assessments are provided in the form of Pre- and Post Tests, End of Topic Tests, and Standardized Tests. The materials provide answer keys with the correct answers for each of the summative assessments. Performance Tasks provide a sample student solution and a scoring rubric to interpret student performance. The materials provide Skills Practice located in the Additional Facilitation Notes at the end of each lesson. The materials state, “After working through MATHbook lessons and MATHia workspaces, some students may need to re-engage with specific skills. You can use the Skills Practice problem sets to support small group remediation.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. Assessments include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. 

The Summative Assessment Suite provides opportunities for students to demonstrate understanding of the standards. The End of Topic Test assesses the full range of standards addressed in the topic using short-answer and open ended questions. Standardized Tests include multiple-choice and multiple-select questions. The Performance Task given for each topic provides open-ended questions allowing students to demonstrate learning of standards and mathematical practices. 

MATHia provides formative assessment data on standards aligned to each topic using the following item types: Grapher tool, Solver tool, Interactive diagrams, Interactive worksheets, Sorting Tools, and short answer questions. 

Examples include: 

• In Module 1, Topic 1, the Performance Task develops the full intent of standard 6.NS.1. Students are given the following scenario, “Summer is making fabric garlands to hang as decorations in a photo booth. She makes the garlands with pink and gray fabric strips that are each \frac{1}{6} foot wide and \frac{5}{6} foot long and lace strips that are \frac{1}{6} foot wide and \frac{3}{4} foot long. Each garland has 20 pink strips, 20 gray strips, and 10 lace strips. Summer has this material available: 

  • pink fabric that is 1 foot wide and 8 \frac{1}{2} feet long 

  • gray fabric that is \frac{1}{2} foot wide and 12 \frac{1}{2} feet long

  • lace that is \frac{1}{6} foot wide and 22 \frac{1}{2} feet long.” 

Students are then asked the following question, “If summer uses all the material she has to make the photo booth garlands, which material will she use up first? How many garlands can she make?”

• In Module 4, Topic 2, End of Topic Test Form A develops the full intent of standard 6.G.3.  Problem 10 gives students the following scenario and questions, “Gina plotted the points (-3, 4), (4, 4), (-3, -2), and (4, -2) on the coordinate plane. a)Determine the height of the quadrilateral. b)Determine the length of the quadrilateral. c)Gina said the points formed a square. Is she correct? Explain your reasoning.” Then, Problem 11 gives the following scenario and questions, “Plot the points (5, -5), (-3, 2), (-5, -5), and (3, 2) on the coordinate plane. Connect the points to create a quadrilateral. a)Identify the type of quadrilateral that you graphed. b) Determine the height of the quadrilateral. c)Determine the area of the quadrilateral.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics. The materials identify strategies to support language development, productive skills, and interactions throughout the series. The materials include “Additional Facilitation Notes” at the end of each lesson to assist teachers as they support students. The “Additional Facilitation Notes” include differentiation strategies, common student misconceptions, and suggestions to extend certain activities. 

Examples of the materials regularly providing strategies, supports, and resources for students in special populations to support their regular and active participation in grade-level mathematics include: 

• The materials identify strategies to support language development of all students. An academic glossary, including written definitions and visual examples, is available in MATHbook and MATHia. MATHia uses Google Translate and Text-to-Speech to support students with assignments. The Teacher Implementation Guide incorporates “Language Link” to support language development for students. Examples of “Language Link” In the Teacher’s Implementation Guide include:

  • In Module 2, Topic 1, Lesson 2, the materials state, “Because of the number of words in these situations, students may struggle to understand the situations. For the first problem, have students engage in a Think-Pair-Share activity, but with three students per group, pairding each ELL student with a native English speaker. Assign each student one of the three parts. Having them talk through their part will help them understand the question and practice their spoken language skills.”

  •  In Module 3, Topic 2, Lesson 5, the materials state, “Students may be familiar with the everyday use of terms literal or literally. There is no connection between the mathematical meaning of literal equation and the meaning of those terms.”

• The materials include “Additional Facilitation Notes” at the end of each lesson to support struggling students and advanced learners. For each differentiation strategy, the materials identify when to utilize the strategy in the lesson, the intended audience, and details of implementing the strategy. Examples of differentiation strategies in the Teacher’s Implementation Guide include: 

  • In Module 3, Topic 2, Lesson 2, Activity 3, the materials suggest supporting struggling students as they work on Question 1 by having teachers, “Solve Question 1, part (a) with students, using both a double number line and an equation.” 

  • In Module 5, Topic 1, Lesson 2, Activity 3, the materials suggest supporting all students as they work on Question 11 by having teachers, “Suggest that students turn the stem-and-leaf plot so that the stems are at the bottom of the graph. This strategy may help them identify the terms related to distributions.”

• The MATHia User Guide Implementation Tools state MATHia uses technology to,”...differentiate to create a personalized learning path for each student.” MATHia supports all students through “Step-by-Step” guided worked examples, “On-Demand Hints” providing multi-level hints, and “Just-in-Time Hints” to help correct common errors.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity. At the end of each lesson, the Teacher’s Implementation Guide has “Additional Facilitation Notes” to assist teachers in supporting all students, especially struggling students and advanced learners. The “Additional Facilitation Notes” include differentiation strategies, common student misconceptions, and suggestions to extend specific activities. 

Examples of suggestions in the Teacher’s Implementation Guide to extend student learning to provide opportunities for advanced students to investigate grade-level content at a higher level of complexity include:

• In Module 4, Topic 1, Lesson 2, Activity 2, students are instructed to “Complete the table with an appropriate situation, absolute value statement, or number. For the last row, assign the correct units to the numeric example based on your situation.” The materials suggest extending the problem for advanced learners by having teachers ask students to generate original situations and responses for Activity 2. 

• In Module 5, Topic 1, Lesson 3, Activity 1, students are given a histogram showing the number of floors in the tallest buildings in the Twin Cities. Students are instructed to “ 3) Describe the range of floors included in each of the remaining bins shown on the horizontal axis.” The materials suggest extending the problem for advanced learners by suggesting “students use inequality notation to respond to his question. For example in part (a), 20\leq f<30, where f represents the number of floors.”

An article titled “Using the Assignment Stretch with Advanced Learners'' located in Help Center, Math, Teaching Strategies states, “Each Assignment includes a Stretch that provides an optional extension for advanced learners that stretch them beyond the explicit expectation of the standards. The Teacher’s Implementation Guide provides suggestions for chunking the assignment for each lesson, including the Stretch. These suggestions consider the content addressed in each session and recommend corresponding Practice, Stretch, and Mixed Practice questions. To ensure that advanced learners are not doing more work than their classmates, consider substituting the Stretch for Mixed Practice questions. When there are no Mixed Practice questions aligned with the Stretch, substitute the Stretch for the Journal or Practice questions.” 

Examples of the materials including “Stretch” questions to extend leaning of the grade-level topic/concept include:

• In Module 2, Topic 1, Lesson 2, Assignment, students are given an optional “Stretch” question. Students are provided three recipes to make chocolate chip cookies, and instructed to “Order the recipes from the least chocolate chips per cookie to the most chocolate chips per cookies. Explain your answer.”

• In Module 3, Topic 2, Lesson 2, Assignment, students are given an optional “Stretch” question. Students are instructed to, “Solve each equation. Check each solution. 1) 34=x-17 2) a-25=92 3) r-3.4=13.1 4) 24\frac{1}{2}=t-5\frac{1}{4}.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. The materials state, “Everyone is an English language learner. Whether it’s learning the language itself or the specialized, academic language of mathematics, students consistently use various strategies to make sense of the world.”Additionally, the materials state that the design and recommended implementation of MATHbook and MATHia provides students with the structure to address all four domains of language (listening, speaking, reading and writing).

In the digital materials, Help Center, Math, Teaching Strategies, an article titled “Supporting ELL Students” provides strategies used in the materials aligning to best practices. The following are strategies the article states are used in the materials: “Text-to-speech and Google Translate make the text accessible in MATHia. Throughout the text, students demonstrate that vocabulary can have multiple meanings. Comprehensive monitoring of student progress via MATHia. The adaptive nature of MATHia. Worked examples in the software and textbook provide a way to explain and model the thought process in which problems can be solved; this also applies to the step-by-step problem in MATHia. Clear learning goals are stated in each lesson, and the narrative statement at the beginning of each lesson provides an opportunity for students to anticipate how the new information will connect to previous learning.”

The MATHbook allows students to “...highlight, annotate, and even write words in their native language.” The materials also provide graphic organizers to show relationships between concepts and terms. The materials also suggest, “Grouping students provides structured opportunities for ELL to practice speaking in English. Pairing an ELL with more proficient English speakers allows opportunities to communicate their thinking in a low-stress way.” Although MATHbook provides strategies/supports for students to read, write, and/or speak in a language other than English, the strategies/supports are only available in English and Spanish. 

General strategies/supports the materials provide in MATHbook for students who read, write, and/or speak in a language other than English: 

• The Academic Glossary provides definitions, ask yourself prompts, and related phrases for terms that will help students think, reason, and communicate ideas. The materials state, “There is strong evidence backing the importance of teaching academic vocabulary to students acquiring English as a second language.” 

• The Glossary provides written definition and visual examples for mathematic specific vocabulary. The MATHbook glossary is only available in English and Spanish.

• The materials provide Language Links throughout lessons to support student language. Examples include:

  • In Module 1, Topic 1, Lesson 2, Activity 3, the language link provided in the Teacher’s Implementation Guide states, “Connect the terms multiple and multiply. You can create multiples of a number by multiplying it by 1, 2, 3, etc. Discuss how the meaning of the everyday term commute relates to the Commutative Property. Commute means to travel; according to the Commutative Property, terms can travel or move to a different order.”

  • In Module 3, Topic 2, Lesson 4, Activity 1, the language link provided in the Teacher’s Implementation Guide states, “Relate the everyday meaning of identity to the Identity Properties of Addition and Multiplication. Your identity is what makes you, you. In mathematics, an identity property is an operation and value that allows a number to equal itself. When you add zero to a number or multiply a number by one, the result is the original number.”

General strategies/supports the materials provide in MATHia for students who read, write, and/or speak in a language other than English: 

• The Glossary provides written definitions and visual examples for mathematic specific vocabulary. The MATHia glossary is only available in English and Spanish.

• MATHia Software Workspaces are available in English and Spanish. Students can use the text-to-speech feature in MATHia to hear the problems read aloud in several languages while customizing the speed and pitch at which the voice reads. Additionally, students are able to change the problems to all languages available within Google Translate.

• All MATHia videos are fully closed captioned and are available in English and Spanish.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Example of how Mathbooks manipulatives are accurate representations of mathematical objects and are connected to written methods:

• In Module 1, Topic 2, Lesson 1, Activity 2, students use the area of a parallelogram to investigate the area of a triangle. Students are instructed to, “Use a separate piece of patty paper to trace each triangle. a) Rotate the patty paper to create a parallelogram composed of two identical triangles. b) Draw the parallelogram you created on your patty paper and label its base and height.” 

• In Module 3, Topic 2, Lesson 3, Activity 3, students use a double number line to solve the equations 34x=25 and 27x=49.

Example of how MATHia’s manipulatives are accurate representations of mathematical objects and are connected to written methods:

• In Module 1, Topic 2, MATHia Software Workspaces, Deepening Understanding of Volume, Determining Volume Using Unit Fraction Cubes, students watch an animation that shows how to determine the volume of a right rectangular prism using unit fraction cubes and the least common multiple of the denominators for the length, width, and height of the prism. Students answer questions related to the mathematics described in the animation and extend their understanding when answering questions about the number of cubes that will fit along the length of the rectangular prism and the volume of the rectangular prism.

• In Module 4, Topic 2, MATHia Software Workspaces, Extending the Coordinate Plane, Exploring Symmetry on the Coordinate Plane, students use an “Explore Tool” to investigate where positive, negative, and 0 values for the x-coordinate and y-coordinate appear on the coordinate plane. Students then complete a drag and drop activity placing nine points on the x-axis, y-axis, both axes, or neither axis and generalize their findings by completing four summary statements after using the “Explore Tool.”