The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific 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 1, 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 4, 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, Lessons 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 instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

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

• In Module 1, Topic 2, Lesson 3, students develop fluency with multiplication and division of fractions. In Activity 3.1, Fractional Fact Families, students investigate area models that show fraction-by-fraction, multiplication-division fact families. Students use cut-out diagrams from Lesson 1 to write multiplication-division fact families involving fractions. (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 1, students develop procedural skill and fluency as they determine whether a number makes an equation or inequality true. In Activity 1.2, Solutions from a Set, 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 instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level.

• In Module 1, Topic 2, 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 4, Topic 1, students independently demonstrate fluency with addition, subtraction, multiplication, and division of multi-digit decimals. In Skills Practice Section A, problems 1-8, students are provided with a number line that has determined points and answer a series of questions on which point is greater or less than a given value. (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 instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. The instructional 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 instructional 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 2014 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 instructional 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 instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

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 instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Standards for Mathematical Practice are referred to as Habits of Mind in this program. The Habits of Mind are identified in all lessons in both the teacher and student workbooks using an icon. There are four icons, only one represents a single MP, “attend to precision,” while the other three represent pairs of MPs, though generally one MP is the focus of the lesson. No icon is used for MP1, and it is stated in the Teacher’s Implementation Guide (TIG): “This practice is evident every day in every lesson. No icon used.” Each activity shows the practice or pair of practices being developed. Questions to facilitate the development of Habits of Mind are listed for both students and teachers throughout the program. The Habits are identified in the Overview in the Student and Teacher Editions, but not in the Family Guide that comes with the Topics. The icon appears within each lesson with questions listed in the Teacher Guide to facilitate the learning where they occur. 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. (Side note: persevere is spelled incorrectly throughout the program.) 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 “Questions to Ask” in the Teacher Guide and the corresponding Facilitation Notes in each lesson, MPs are used to enrich the content and are not taught as a separate lesson.

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.

MP4 - Model with mathematics.

• In Module 4, Topic 2, Lesson 3, students model with mathematics when creating a presentation of their solution. In Activity 3.8 The Diver, students “train” as a freediver to beat William Trubridge’s record from 2016 of freediving almost 407 feet into the ocean. Students decide what type of questions they would ask Trubridge, how long they would need to hold their breath in order to beat Trubridge, and then make a presentation for a similar problem.

MP6 - Attend to precision.

• In Module 2, Topic 2, Lesson 1, students attend to precision when completing several columns within a table involving ratios, fractions, and decimals. “The sixth grade class is planning a field trip to Philadelphia. To decide which historical site they will visit, the 100 sixth-graders completed a survey. The results of the survey are provided in the table. Complete the Ratio, Fraction, Decimal, and Grid columns with these representations of the survey results: a ratio using colon notation, a fraction in lowest terms, a decimal, a shaded grid, an equivalent percent.”

MP7 - Look for and make use of structure.

• In Module 1, Topic 1, Lesson 1, students make use of structure when writing equivalent equations. “Rewrite the original equation 5 x 27 = 135 with an equivalent equation to represent the model you drew. a. How can you rewrite the original product by substituting the sum of the two lengths making up the split side?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

Each activity asserts that a 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.

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 between two other rational numbers on a numberline.

MP2 - Reason abstractly and quantitatively.

• In Module 4, Topic 1, Lesson 1, students explain the meaning of zero, positive numbers, and negative numbers in a variety of contexts. The teacher is directed to ask: “What representation can I use to solve this problem?; How can this problem be represented with symbols and numbers?”

MP4 - Model with mathematics.

• In Module 2, Topic 3, Lesson 3, students identify statements that are true from a given set of information about the Duquesne Incline as commuter transportation. The teacher is directed to ask: “What expression or equation could represent this situation?; What representations best show my thinking?; How does this answer make sense in the context of the original problem?”
• In Module 3, Topic 3, Lesson 3, students create graphs, tables of values, and write algebraic equations to describe the relationship between two quantities. They also answer relevant questions about two real-world scenarios.

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 3, students choose standard and non-standard tools to measure length. “Use standard and non-standard tools to measure the lengths of the diagonals using six different units of measure and record them in the table. Be sure to include inches and centimeters as two of your units.” The teacher is directed to ask: “What tools would help me solve this problem?”

MP6 - Attend to precision.

• In Module 3, Topic 3, Lesson 1, students solve a real-world problem based on a pool party involving variables. “a. What two quantities are changing in this situation? b. Which quantity depends on the other? c. Define variables for each quantity and label them appropriately as the independent and dependent variables.”

MP7 - Look for and make use of structure.

• In Module 1, Topic 2, 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 2, Topic 1, Lesson 2, 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 bar models and formal properties to solve equations. The teacher is directed to ask: ”What characteristics of this expression or equation are made clear through this representation?; How can I use what I know to explain why this works?”

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. Students answer questions such as: “Use your example to explain what is meant by the phrase, 'multiply or divide an expression by a given value.' Use your example to explain what is meant by the phrase, 'product of two factors.' The teacher is directed to ask: How can I use what I know to explain why this works?; Can I develop a more efficient method?; How could this problem help me to solve another problem?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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 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 1, “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 1, Lesson 3, “Bella says, 'There are 5 buildings represented in the histogram since there are 5 bars.' Do you agree or disagree with Bella’s statement? If you do not agree with Bella, estimate how many buildings are represented in the histogram.”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the instructional 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 of these be thumbs up if their predictions are different? How is Robena’s reasoning different than Eryn’s reasoning?”
• In Module 3, Topic 2, Lesson 1, the teacher is prompted to ask, “Why is Rylee’s work incorrect? What feedback would you give Rylee about her strategy? What property did Clover use? Did anyone use the Reflexive Property of Equality as part of their strategy? Did Clover and Fiona use the same strategy? Is there more than one solution completing this equation?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that materials attend to the specialized language of mathematics.

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 2, Lesson 3, students accurately represent absolute value equations to calculate temperature changes, for example, calculating the temperature change between two points: |15.6| + |-10.6| = 26.2 degree drop.

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 difference between an expression and an equation?” A tip for helping ELLs differentiate expression and equation can be used as part of the lesson.
• 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 instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

The course has five modules with each module broken into topics. Each topic has a set of three to six lessons/activities. Each lesson consists of several sections, which may include Warm Up, Getting Started, Activities, Talk the Talk, and an Assignment. The Warm Up and Getting Started sections activate students’ prior knowledge and engage students in non-routine problem solving. The Activities develop students' understanding of concepts by exploring problems through both individual and whole group instruction. The students demonstrate their understanding of concepts by applying their knowledge to real-world problems in the Talk the Talk section. The Assignment includes five mini-sections that reinforce understanding of the new mathematical concept. Each lesson has a coordinating practice set called Skills Practice with exercises for students to solve using their new learning. MATHia (online) provides additional personalized exercises for students to show their understanding of the activity/lesson.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.

Lessons follow a consistent format that intentionally sequences assignments:

• “Warm Up” - exercises that activate students’ prior knowledge.
• “Getting Started/Engage” - students solve/think/share and notice others' work/thinking, usually for a non-routine problem.
• “Develop/Activities” - new learning takes place; students explore problems that engage them with examples and explanations of the targeted skill in a whole-class setting. Each Activity includes verbiage describing how the new knowledge relates to previous understanding.
• “Demonstrate/Talk the Talk” - students reflect on and connect what was learned.
• “Assignment” - five sections that review the lesson: Write - reviewing rules or vocabulary, Remember - summary of one or two key points, Practice - problems related to the activities, Stretch - an extension, and Review - looping in previous skills.

Students practice with “Learn Individually” lessons using the MATHia software or, if technology is not accessible, students use the Skills Practice workbooks.

Overall, each topic is sequenced to begin with prior knowledge and build upon that knowledge to develop conceptual understanding and procedural skill.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that there is a variety in what students are asked to produce. Students are asked to produce a variety of products in both digital and written form.

Some of these products include:

• Multiple representations including expressions and equations, models, arrays, number lines, etc.
• In Module 4, Topic 2, Lesson 2.2, students are given parts of quadrilaterals on a coordinate plane and use their knowledge of polygons and horizontal or vertical distances to complete the missing parts, then determine the area of the polygons.
• Justification of their thinking and others', critiquing others’ work, explaining why answers given are correct.
• Writing, reviewing, practicing, and stretching activities at the end of each lesson, for example, writing their own situations to model a given expression. (Module 1, Topic 2, Lesson 3.5)

Finally, each module includes a real-world connection where students produce solutions in a variety of ways to demonstrate their mathematical knowledge, such as plotting band member locations on a coordinate plane.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that manipulatives are faithful representations of the mathematical objects they represent and are appropriately connected to written methods.

Manipulatives are embedded in activities and the MATHia Independent Digital Lessons. Number lines, patty paper, equivalency cards, etc. are used throughout the year in connection to the mathematics being presented and are faithful representations. For example, students cut, fold, and tape a cube net and answer questions about the net, then make conjectures on different nets based on creating this one. (Module 1, Topic 3, Lesson 3, Getting Started: Breaking Down a Cube)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

In the Teacher Edition, facilitator notes for each activity include questions for the teacher to guide students' mathematical development and to elicit students' understanding. The material indicates that questions provided are intended to provoke thinking and provide facilitation through the mathematical practices as well as getting the students to think through their work. The Note provided on page FM-21 of the Teacher’s Implementation Guide Volume 1 reads, “When you are facilitating each lesson, listen carefully and value diversity of thought, redirect students’ questions with guiding questions, provide additional support with those struggling with a task, and hold students accountable for an end product. When students share their work, make your expectations clear, require that students defend and talk about their solutions, and monitor student progress by checking for understanding.”

Each lesson guide in the Teacher Edition provides quality questions to help guide students' mathematical development.

For example:

• “Did you calculate the area of both figures, or did you take a shortcut? If you took a shortcut, what was it?”
• “If you didn’t have a rectangle with the same dimensions as a parallelogram, how could you determine the area of the parallelogram?”
• “Which formula did you learn first, the area of a parallelogram or the area of a rectangle?”
• “Explain how knowing the area formula for a rectangle helped you derive the area formula for a parallelogram.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the materials contain a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning such as how to use and read data in the MATHia software.

In the Lesson Resources, the teacher guide provides information including a lesson overview, lesson structure and pacing facilitation notes, questions to ask, connections to standards, a materials list, essential ideas, facilitation notes, what to look for when students are working, and a summary of the lesson.

As part of the blended learning approach, there is Learning Individually with MATHia software. There is ample support for students and teachers to engage with this software such as the Getting Started guide, a table of contents, an RTI table of contents, and MATHia system requirements.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation for containing a teacher edition (in print or clearly distinguished/accessible as a teacher edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematical concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Within MyPL, teachers can view instructional videos that provide adult-level explanations and examples for teachers to enhance their own knowledge of the content. The instructional videos address textbook lessons, MATHia, mathematical content, and classroom strategies. For example, in the video, March MADness - Mean Absolute Deviation (Course 1, Module 5, Topic 2, Lesson 3), teachers view suggestions for implementing the lesson. Course 1 contains 51 lesson videos. MyPL also includes 33 videos addressing mathematical content that are not lesson-specific, and the advanced mathematics concepts addressed by the videos include, but are not limited to: ellipses, hyperbolas, and discontinuities and asymptotes of rational functions. The Teacher’s Implementation Guide for each course provides detailed information regarding how mathematical content fits into the series overall, and the materials include module overviews that describe the mathematics of the module and how the content is connected to prior and future learning.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials contain a teacher edition (in print or clearly distinguished/accessible as a teacher edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for Kindergarten through Grade 12.

• The Module Overview includes information for the teacher with explanations that build the teacher’s understanding of how the lesson content fits into the curriculum. It tells what mathematics is in the module and how the module connects to prior and future learning.
• Each Topic Overview provides information on the mathematical content in the lessons as well as where it fits in the scope of mathematics from Kindergarten through Grade 12. Knowledge required from prior chapters and/or grades is explicitly called out in this section.
• The Topic Overview also connects each lesson to standards from a previous grade. These previous grade standards are embedded into each lesson through the Warm Up and Getting Started sections as well as the Activity sections.
• The Topic Overview describes the entry point or prior experience with the mathematical concept for students, why what is being learned is important, and how the activities in the topic promote student expertise in the MPs.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

• There is a pretest for every topic in each module that addresses standards that will be taught. The post-test for the topic is the same test.
• The Topic Overview provides a list of Prerequisite Skills needed for the topic, which creates an indirect opportunity for teachers to gather information about students’ prior knowledge although there is no direct guidance provided to the teacher about how to use the information.
• The MATHia software is used as an assessment and progress monitoring tool, providing personalized data about where a student stands on various skills.
• In every assignment in the textbook, there is a Review section. Students practice two questions from the previous lesson, two questions from the previous topic, and two questions that address the fluency standards outlined in the Standards. This provides teachers information about students' learning gaps as they work through the instructional materials.

While there are opportunities to collect information about students’ prior knowledge, the materials do not provide strategies about how to utilize the information in the classroom.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide strategies for teachers to identify and address common student errors and misconceptions.

• In the Topic Guide, lessons regularly have a section titled “Misconceptions” with suggestions for teachers to identify and address common student errors and misconceptions.
• Example: “Students may get confused over the varied uses of the term scale—such as scaling up, scaling down, scale on a number line, even bathroom scale. Take the time to explain how all the uses of the term are related.” (M2-37J)
• Teachers are encouraged to engage students in mathematical conversations to address student errors and misconceptions with phrases such as, “Remind the students…, Discuss with students…, Point out that….”
• MATHia software provides a solution pathway to common student misconceptions. “Like a human tutor, MATHia re-phrases questions, re-directs the student, and hones in on the parts of the problem that are proving difficult for the student. Hints are customized to address the individual student, understanding that there are often multiple ways to do the math correctly.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The materials provide several opportunities for ongoing review and practice:

• Students practice with “Learn Individually” lessons using the MATHia software or, if technology is not accessible, the Skills Practice workbooks. In the Skills Practice book, odd number answers are provided, so students know if they’re solving problems correctly; and in the MATHia software, feedback is continually given for both correct and incorrect answers.
• The MATHia software includes “Hints” which students can select while reviewing and practicing skills. There are three types of “Hints”:
• Just-in-Time Hints automatically appear when a student makes a common error.
• On-Demand Hints are hints that a student can ask for at any time while working on a problem.
• Step-by-Step demonstrates how to use the tools in a lesson by guiding step-by-step through a sample math problem.
• Each lesson ends with Talk the Talk, a few questions that capture the learning of all of the activities the students have engaged in with the lesson.
• Each lesson also has a short review section that provides a spiral review of previous concepts.
• Standardized Practice Test that the teacher can use at any time to review and practice concepts and skills learned throughout the course.
• Prior to each lesson there is a Warm-Up that reviews previous topics.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet the expectation for assessments clearly denoting which standards are being emphasized. The series offers several types of assessments, print and digital:

• MATHia provides information for each student based on standards.
• Performance tasks clearly note which standards are being assessed. 
• The student-facing versions of the Pretest, Post test, and the End of Topic Test do not denote which standards are being emphasized.
• The digital overview contains assessments and an assessment overview document. The document contains each assessment as well as which standard is assessed for each individual problem.
• The Carnegie Edulastic Assessments Suite displays standards for each problem within each assessment provided. These standards are not student-facing.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. Materials include some guidance for teachers to interpret student performance. Answer keys are provided for all assessments. Performance Tasks include a detailed scoring rubric for teachers to use when interpreting student performance; however, no other assessment provides guidance for teachers about scoring student performance. MATHia reports provide teachers with detailed information about student performance in relation to progress on standards and suggestions on the skills that require additional support. Teachers can monitor students working in MATHia and view in-the-moment guidance that indicates to teachers which students need additional support. The materials also offer teachers an APSLE (Adaptive Personalized Learning Score) report which is a predictor for year-end summative assessments. Videos within MyPL explain this report in more detail while outlining the research and models behind the report.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The materials include a detailed Scope and Sequence of the course, including pacing. The lesson summary and the essential ideas provide further information on sequencing of the lessons. There is a chart in the Teacher’s Implementation Guide that includes a table with a column entitled, “Connections to Prior Learning,” which enhances the opportunity to scaffold instruction by identifying prerequisite skills that students should have.

All lessons include instructional notes and classroom strategies that provide teachers with key math concepts, sample questions, differentiation strategies, discussion questions, possible misconceptions, what to look for from students, and summary points providing structure for the teacher in making content accessible to all learners.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 partially meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

A primary strategy for meeting the needs of all learners in this program is MATHia software. MATHia provides differentiation, providing varying levels of support based on how students solve the problems. The software adapts to the specific responses and “tutors” each student to meet their needs.

Most lessons provide “Differentiation strategies,” “Questions to ask,” and a “Misconception” section. Most of the suggestions and the questions included in the “Questions to ask” section are intended for all students rather than geared toward helping students who struggle or challenging students ready to go deeper. For example, in Module 4, Topic 2, Lesson 3.1, Questions to ask: “What quantities are being compared? Which quantity depends on the other? How did you decide which quadrants you will need in order to graph the data? How did the signs of the numbers in the table help you decide which quadrants are needed in order to graph the data? How did you determine the weight differentials in Question 2?”

However, in the “Differentiation strategies” section, suggestions are limited but more specific. For example, in Module 1, Topic 2, Lesson 3.1: “For students who struggle, reduce the number of diagrams for which they must create fact families.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

• Each Topic Overview includes a section called, “What is the entry point for students?” For example, in Module 1, Topic 1, the Factors and Area overview states: “Students enter Grade 6 with a conceptual understanding of area and fluency in computing the perimeter and area of rectangles. They have used tiling to relate area to multiplication and addition, and they have used informal statements of the number properties. Students have also used area models to represent multiplication. Factors and Area draws on this prior knowledge to formalize the Distributive Property, to reinforce connections between area models and multiplication, and to determine the area of new shapes.”
• Some application tasks, particularly the Performance Task, allow for multiple solution strategies or representations. For example, in Module 5, Topic 1, “The Statistical Process,” students create a histogram for each display given and then describe how they selected the size of the bin. Students make a conclusion about the amount of sleep that children, teenagers, and adults get.
• Some assessment questions allow for multiple entry points. For example, in Module 3, Topic 3, Post-test question 5, students write a story that matches a graph shown, allowing answers to vary.
• Lesson activities provide limited opportunities for students to create their own solution paths since strategies are often provided.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

ELL Tips are specifically cited throughout the materials. For example, “ELL Tip: In the first sentence, there are several words that may be challenging for English Language Learners (analyze, situation, corresponding). Have students read the paragraph independently and circle words that they do not know. Then have students find people in the class that have different circles. The students should help each other define words that they have circled. If numerous students are circling the same words, define these as a class.” (M3-197)

Additional differentiation strategies included in the materials are often general, such as providing additional examples, using manipulatives, or using a graphic organizer.

Some suggestions are specific to the lesson, but don’t necessarily further knowledge, such as in Module 2, Topic 2, Lesson 1.3: “Differentiation strategy: To support students who struggle, provide a hint sheet to support their learning while they play the game with peers who may think faster than they do."

There are differentiation suggestions that do not include a rationale as to how they would provide support. For example, in Module 2, Topic 1, Lesson 1.1: “For students who struggle, provide another example to compare/contrast. For example, scores for two games are 99-100 and 1-2. Which game was probably 'closer'?” It is unclear how this suggestion would support a student's development of additive and multiplicative reasoning.

However, there are numerous examples that do support accommodations for special populations such as:

• “For students who struggle with base and height, it may be helpful to have them cut out figures. That way, they can turn the paper to see different sides can be the base. They can then use folds to determine the height.” (M1-15E)
• “For students who struggle, transferring measurements from Question 2 into the diagram may be confusing. You may want to have pre-labeled diagrams available in those cases.” (M1-29D)
• “For students who struggle, using the terms greatest and least may sound counter-intuitive compared to their answers. Stress that they should concentrate on key terms. In GCF, factor is the key term, and common factors cannot be larger than both values. In LCM, multiple is the key term and multiples may be larger than both values.” (M1-39G)
• “Questions 1 through 4 and Questions 5 through 6 require the same knowledge; however, Questions 5 through 6 are less scaffolded and different information is provided. Assign Questions 1 through 4 to students who struggle and assign Questions 5 through 6 to students who need a challenge.” (M4-73H)
• “For students who struggle, the use of inequality symbols often adds more confusion. You may want to adjust the directions so that students circle the larger number or label the values on a number line.” (M1-71G)
• “To support students who struggle, it may be difficult to draw tape diagrams with rectangles of the same size. For this reason, you may want to provide a rectangle stencil or pre-drawn tape diagrams.” (M2-37E)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing a balanced portrayal of various demographic and personal characteristics.

• No examples of bias were found.
• Pictures, names, and situations present a variety of ethnicities and interests.
• The text is black and white with green as the only color. The people are gray with black hair, but still appear to represent many ethnicities.
• Problems include a wide span of international settings, as well as situations in urban, suburban, and rural settings.
• There is a wide variety of names in the problems, from James, Ben, and Haley to Keirstin, Miguel, and Miko, representing a variety of cultures.