The materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. Students develop understanding throughout “Engage” and “Develop” activities, which typically activate prior knowledge and use manipulatives to introduce and build understanding of a concept. Students also have the opportunity to independently demonstrate their understanding in the “Demonstrate” questions at the end of each lesson where they attempt to synthesize their learning. 

For example: 

• In Module 2, Topic 1, Lesson 6, students develop an understanding of ratios by using various mathematical models to solve real-world problems. In Talk the Talk - “In Goes the Kitchen Sink,” students are given a ratio and use multiple representations (scale up/scale down, table, graph, double number line) to show equivalent ratios. (6.RP.A) 

• In Module 2, Topic 1, Lessons 5, students develop an understanding of the proportional relationship needed to solve a problem by using a table and a coordinate graph. In Activity 5.1 Analyzing Rectangle Ratios, students cut out and sort rectangles. They group and stack the rectangles according to the ratio of the side lengths. Then, they attach their rectangles to a coordinate grid. Students learn that graphs of equivalent ratios form a straight line that passes through the origin. (6.RP.3a, 6.RP.3b) 

• In Module 3, Topic 1, Lesson 3, students explore the use of properties of arithmetic in expressions and understand that these properties apply to expressions with variables. In Activity 3.2 Algebra Tiles and the Distributive Property, students use algebra tiles to multiply expressions using the Distributive Property, Order of Operations, and combining like terms. (6.EE.3) 

• In Module 4, Topic 1, Lesson 1, students develop an understanding of positive and negative numbers through the use of number lines to solve real-world problems. In Activity 1.1 Investigating Time on a Number Line - Human Number Line, students create a human number line and use it to show locations of time in the past, present, and future. Students analyze number lines and discuss the meaning of zero in the context of their number line. Negative numbers are described as the numbers to the left of zero on the number line. (6.NS.C) 

• In Module 4, Topic 1, Lesson 1, students engage in the application of mathematical skills when explaining how two integers are compared. In Talk the Talk, students communicate understanding through a situation: “Your sixth grade cousin goes to school in a different state. His math class has not yet started comparing integers. Write him an email explaining how to compare any two numbers. Be sure to include one or two examples and enough details that he will be able to explain it to his class.” (6.NS.7) 

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. 

• In Module 2, Topic 3, Lesson 2, students show conceptual understanding of ratio reasoning and relationships when solving real-world unit rate problems. In Activity 2.2 Writing Unit Rates, students write unit rates that compare the unit rates in two ways: the number of objects per dollar and the number of dollars per one object. Then students write two different unit rates for situations that do not involve money. They decide which unit rates are useful, both in general and to answer specific questions. (6.RP.3b) 

• In Module 2, Topic 1, Lessons 1, students demonstrate an understanding of ratios by creating and using ratio tables to solve real world problems. In Activity 4.3 Parts and Wholes in Ratio Tables, students use ratio tables to answer questions about mixing paint. Five pints of bluish green paint is made by using two pints of yellow paint and three pints of blue paint. Students use a ratio table to analyze student thinking about mixing paint and to determine various amounts of paint needed. (6.RP.3) 

• In Module 2, Topic 1, Lesson 5, students demonstrate an understanding of ratios by using various mathematical models to solve real world problems. In Activity 5.2 Graphing Equivalent Ratios, students analyze a time-to-distance rate scenario: Stephanie drives her car at a constant rate of 50 miles per hour. Students use a table, double number line, and the coordinate plane to determine the number of miles Stephanie drives over a period of time, and then they compare these different representations. (6.RP.3a, 6.RP.3b)

The materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill and fluency throughout the grade level. They also provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. This is primarily found in two aspects of the materials: first, in the “Develop” portion of the lesson where students work through activities that help them deepen understanding and practice procedural skill and fluency; second, in the MATHia Software, which targets each student’s area of need until they demonstrate proficiency. 

The materials develop procedural skill and fluency throughout the grade-level. 

• In Module 1, Topic 1, Lesson 5, students develop fluency with multiplication and division of fractions. In Getting Started, All in the Fact Family, students investigate area models that show fraction-by-fraction, multiplication-division fact families. (6.NS.1) 

• In Module 1, Topic 3, Lesson 4, students develop fluency when computing volume and surface area. Students learn and practice the standard algorithm for division, including division of decimals, in the context of volume and surface area. (6.NS.2, 6.NS.3) 

• In Module 3, Topic 1, Lesson 2, students develop procedural skill and fluency when evaluating algebraic expressions. In Activity 2.2, Matching Algebraic and Verbal Expressions, students play Expression Explosion to practice matching verbal and algebraic expressions. (6.EE.2) 

• In Module 3, Topic 2, Lesson 4, students develop procedural skill and fluency as they determine whether a number makes an equation or inequality true. In Activity 1, Identifying Solutions, students create equations from a list of given expressions. Then students decide which values from a set make each equation true. They investigate true and false equations and equations with no solutions and infinite solutions. (6.EE.5) 

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. 

• In Module 1, Topic 1, students independently demonstrate fluency and procedural skill with multiplication and division of fractions through technology. In the MATHia Software, students calculate products and quotients of fractions, including mixed numbers and improper fractions. (6.NS.1) 

• In Module 1, Topic 3, students independently demonstrate fluency and procedural skill with addition, subtraction, multiplication, and division of decimals through technology. In the MATHia Software, students review Adding and Subtracting Decimals, Decimal Sums and Differences, Exploring Decimal Facts, Multiplying and Dividing Decimals, and Decimal Products and Quotients. (6.NS.3) In Module 1, Topic 3, students independently demonstrate fluency with addition, subtraction, multiplication, and division of multi digit decimals. In Skills Practice Section A, problems 1-6, students answer a series of questions on which value is greater or less than in a set of given values which includes fractions and decimals, problems 7-12, students order a list of given values from least to greatest which include fractions and decimals. (6.NS.3) 

• In Module 4, Topic 1, Lesson 2, students independently demonstrate fluency in solving mathematical problems involving absolute value. In Activity 2.2, Interpreting Absolute Value Statements, students complete tables of situations, absolute value statements, and numeric values described in given and student-generated situations. The tables include statements of equality and inequality. (6.NS.7) 

• In Module 5, Topic 2, students independently demonstrate the procedural skill of calculating Mean Absolute Deviation through technology. The MATHia Software provides multiple opportunities for the students to calculate and compare the mean absolute deviations with the spread of similar data sets. (6.SP.5c,d)

The materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. 

The materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. This is primarily found in two aspects of the materials: first, in the “Demonstrate” portion of the lesson where students apply what they have learned in a variety of activities, often in the “Talk the Talk” section of the lesson; second, in the Topic Performance Tasks where students apply and extend learning in more non-routine situations. 

The materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. 

• In Module 2, Topic 1, Lesson 6, students engage in the application of mathematical skills when analyzing ratios to solve real-world problems. In Activity 6.2, Choosing a Strategy to Solve Ratio Problems, students are given ratio situations and choose a strategy such as graph, scaling, or addition to solve the problem and analyze how their strategy worked. (6.RP.3) 

• In Module 5, Topic 1, Lesson 2, students engage in the application of mathematical skills when analyzing and writing equations to solve real-world problems. In Activity 2.1, Creating and Analyzing Dot Plots, students are given information about the medals won at the 2018 Winter Olympics and analyze the data, create a dot plot and a stem and leaf plot, as well as describe the distribution. Students take skills they’ve practiced and apply them to real data in order to analyze and report out on the questions that were generated such as, “What is the typical number of gold medals won by a country?” (6.SP.B) 

The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. 

• In Module 2, Topic 2, Lesson 3, students independently demonstrate the use of mathematics when solving real-world problems involving percent. In Talk the Talk, students demonstrate two different ways to determine the answer to questions given such as, “Leah’s goal is to score a 90 percent on the upcoming science test. If there are 40 questions on the test, how many does Leah need to answer correctly?” They then plan a presentation of the two solutions, making sure to talk about how they are the same and how they’re different. (6.RP.3) 

• In Module 3, Topic 3, students independently demonstrate the use of mathematics when analyzing and writing equations to solve real-world problems. In Performance task, Graphing Quantitative Relationships - Throw it in Reverse, students reverse the dependent and independent variables on a graph and analyze the impact that has. They discuss proportionality, rate of change, and generate questions that the new graph could answer. They also create tables and equations for both graphs. Finally, they compare and contrast the two versions of the data. (6.EE.9)

The materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

Within each topic, students develop conceptual understanding by building upon prior knowledge and completing activities that demonstrate the underlying mathematics. Throughout the series of lessons in the topic, students have ample opportunity to practice new skills in relevant problems, both with teacher guidance and independently. Students also have opportunities to apply their knowledge in a variety of ways that let them show their understanding (graphic organizers, error analysis, real-world application, etc.). In general, the three aspects of rigor are fluidly interwoven. 

For example: 

• In Module 4, Topic 2 Overview: “In The Four Quadrants, students explore the four quadrant coordinate plane. They use reflections of the first quadrant on patty paper and their knowledge of the rational number line to build their own four quadrant coordinate plane. They look for patterns in the signs of the ordered pairs in each quadrant and for ordered pairs that lie along the vertical and horizontal axes. After developing a strong foundation for plotting points and determining distances on the coordinate plane, students analyze and solve problems involving geometric shapes on the coordinate plane. They identify geometric shapes defined by given coordinates and determine perimeters and areas of geometric shapes in mathematical and real-world situations. Finally, students use the knowledge gained throughout the course to solve a wide range of problems on the coordinate plane, using scenarios, graphs, equations, and tables. Throughout this topic, students continue to develop their fluency with whole numbers, fractions, and decimals.” 

There are areas where an aspect of rigor is treated more independently, such as developing procedural skill and fluency in the MATHia software and Skills Practice or in the Performance Task where students work primarily with application.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Standards for Mathematical Practice are referred to as Habits of Mind in this program. The Habits of Mind are first introduced for teachers and students in the Front Matter of the MATHbook and Teacher’s Implementation Guide. For each practice or pair of practices, students are provided a list of questions they should ask themselves as they work toward developing the habits of mind of a productive mathematical thinker throughout the course. 

Materials state that MP1 aligns to all lessons in the Front Matter of the MATHbook and Teacher’s Implementation Guide. Generally, lessons are developed with activities that require students to make sense of mathematics and to demonstrate their reasoning through problem solving, writing, discussing, and presenting. Overall, the materials clearly identify the MPs and incorporate them into the lessons. All the MPs are represented and attended to multiple times throughout the year. With the inclusion of the Facilitation Notes for each lesson in the Teacher’s Implementation Guide, MPs are used to enrich the content and are not taught as a separate lesson. 

MP1 - Make sense of problems and persevere in solving them. 

• In Module 4, Topic 1, Lesson 3, students sort and classify rational numbers. They investigate how many rational numbers can fit between two other rational numbers on a numberline. 

MP2 - Reason abstractly and quantitatively. 

• In Module 2, Topic 3, Lesson 3, students reason abstractly when completing a table and constructing a graph to represent a unit rate. Using the graph, two unit rates are identified, one in which the x-value is 1 and one in which the y-value is 1, and the student explains the meanings of the unit rate in terms of the situation.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students are consistently asked to verify their work, find mistakes, and look for patterns or similarities. The materials use a thumbs up and thumbs down icon on their “Who’s Correct” activities, where students question the strategy or determine if the solution is correct or incorrect and explain why. These situations have students critique work or answers that are presented to them. 

Examples of students constructing viable arguments and/or analyzing the arguments of others include: 

• In Module 2, Topic 2, Lesson 2, “Noah and Dylan were assigned the numbers 0.06 repeating and 0.1 percent, but they disagreed on which was larger. Noah says that 0.06 repeating is less than 0.1, so 0.06 repeating is less than 0.1 percent. Dylan says that since 0.1 percent is the same as 0.001 and 0.001 is less than 0.06 repeating, 0.1 percent is less than 0.06 repeating. Who is correct? Explain your reasoning.” 

• In Module 3, Topic 2, Lesson 4, “Identify your equations that are always true, never true, and those equations where you don’t yet know whether they are true or false. Explain your reasoning.” and “Write an equation with variables that has no possible solution. Explain why the equation has no solution.” 

• In Module 5, Topic 1, Lesson 2, “Jessica asked, 'How many medals did the United States win? How many of those were gold?' Maurice thought a better set of questions would be, 'What is the typical number of medals won? What is the typical number of gold medals won by a country?' Who’s correct? Explain your reasoning.” 

• In Module 5, Topic 2, Lesson 1, student, “Analyze Abana’s statement. ‘The median number of points I scored is 10.’ Explain what Abana did incorrectly to determine that the median was 10. Then determine the correct median.”

The materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Throughout the teacher materials, there is extensive guidance with question prompts, especially for constructing viable arguments. 

• In Module 1, Topic 3, Lesson 3, the teacher is prompted to ask, “How is the area of a face of a cube measured? Analyze the two responses and explain why Leticia is incorrect in her reasoning.” 

• In Module 2, Topic 1, Lesson 1, students critique the reasoning of others when analyzing two predictions of a solution to a problem. “Robena and Eryn each predicted the final score of a basketball game between the Crusaders and the Blue Jays. Analyze each prediction. Describe the reasoning that Robena and Eryn used to make each statement.” Teachers are provided questions to ask, such as: "How can both predictions be correct when they are different? Do you think Robena’s or Eryn’s reasoning makes more sense for this situation? Explain your thinking.”” 

• In Module 3, Topic 2, Lesson 1, the teacher is prompted to ask, “ What feedback would you give Rylee about her strategy? Why would Clover want to write $8+4$ as $7+5$? Why would Fiona want to write $8+4$ as $7+4+1$?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Each activity asserts that a practice or pair of practices are being developed, so there is some interpretation on the teacher’s part about which is the focus. In addition, what is labeled may not be the best example; i.e., using appropriate tools strategically (MP5) is sometimes weak where it’s labeled, but student choice is evident in Talk the Talk and Performance Tasks, which are not identified as MP5. Over the course of the year, the materials do attend to the full meaning of each mathematical practice. 

MP4 - Model with mathematics. 

• In Module 3, Topic 2, Lesson 4, Activity 2, students model with mathematics when creating an inequality graph. Students are given scenarios where they must, “Define a variable and write a mathematical statement to represent each situation. Then sketch a graph of each inequality.”

MP5 - Use appropriate tools strategically. 

• In Module 2, Topic 1, Lesson 6, given a scenario, students choose to solve problems by using the graph, by scaling, or by using an addition strategy. 

• In Module 3, Topic 3. Lesson 2, students are given a graph to represent the total cost of an item with taxes. Students are instructed to, “Write an equation to represent the relationship between the cost of an item, x, and the total cost, y.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.  

Each Topic has a “Topic Summary” with vocabulary given with both definitions and examples (problems, pictures, etc.) for each lesson. There is consistency with meaning, examples, and accuracy of the terms. 

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. 

• In Module 3, Topic 3, Lesson 3, Talk the Talk, students complete a graphic organizer describing the advantages of using verbal, tabular, graphical, and equation representation. 

• In Module 4, Topic 1, Lesson 2, students accurately represent absolute value equations to calculate evaporation changes, for example, calculating the evaporation change between two points: $|2.1| + |-0.9| = 3$ evaporation change. 

The materials use precise and accurate terminology and definitions when describing mathematics and include support for students to use them. 

• In Module 1, Topic 1, Lesson 1, key terms are identified as numeric expression, equation, and Distributive Property. Students describe the expression $4(2+15)$ in different ways. Teachers are prompted to ask, “What is the purpose of the arrows in the example? Draw a diagram to represent this expression. Could you write the expression as $(2 + 15)4$? Explain your thinking.” 

• In Module 2, Topic 3, Lesson 3, the Topic Overview describes how students deepen their understanding of converting units of measurement using ratio reasoning and strategies for determining equivalent ratios. The term convert is defined, and students use approximate conversion rates to estimate measurement conversions before engaging with formal methods of converting. Converting among units of measurement in the same system is recast in terms of conversion ratios, which can also be called conversion rates.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

MP7 - Look for and make use of structure. 

• In Module 1, Topic 1, Lesson 1, students make use of structure when writing equivalent equations. Reflect on the different ways you can rewrite the product of 5 and 27. Select one of your area models to complete the example. How did you take apart the side length of 27? What are the factors of each smaller region?  What is the area of each smaller region? What is the total area?”

• In Module 1, Topic 3, Lesson 1, students are given two strategies to order rational numbers, then look for and make use of structure to identify and evaluate the most efficient strategies for a solution. 

• In Module 1, Topic 2, Lesson 1, students decompose a parallelogram to create a rectangle and conclude the two shapes have the same area. The same formula can be used to determine the area of either figure. In the process of reconstructing the rectangle from the parallelogram, students make use of the structure of rectangles to discover the relationship between the parallelogram and rectangle. 

• In Module 3, Topic 2, Lesson 3, students use double number lines and formal properties to solve equations. The teacher is directed to ask: ”How did you use the double number line to write your equations? How do you know that all of the equations have the same solution? Use the double number line to create another equation with the same solution. Analyzing the structure of the given equation, how can you tell whether the unknown within the algebraic expression is smaller or larger than the value of the constant it equals?” 

MP8 - Look for and express regularity in repeated reasoning. 

• In Module 3, Topic 1, Lesson 3, students use the Distributive Property to factor algebraic expressions, rewriting expressions as a product of two factors, including expressions where the coefficients of the original terms do not have common factors. The teacher is directed to ask: Use your response to explain what is meant by the phrase, 'product of two factors’.

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.”