The materials for Carnegie Learning Middle School Math Solution Course 2 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. 

• In Module 2, Topic 1, Lesson 1, students show their understanding of adding and subtracting rational numbers using a visual representation. In Talk the Talk, Mixing Up the Sums, students create addition problems from a given sum using number lines to explain and demonstrate their understanding. (7.NS.A) 

• In Module 2, Topic 1, Lesson 2, students show their understanding of adding and subtracting rational numbers using a visual representation. In Activity 1 Walking the Number Line, students walk along a number line to develop a conceptual understanding using physical and visual representations. (7.NS.1b) 

• In Module 3, Topic 2 Lesson 1, In Activity 2.2 Mathematics Gymnastics - Rewriting Expressions Using the Distributive Property, students develop their understanding of the distributive property by drawing area models to represent expressions. (7.EE.A) 

• In Module 5, Topic 2, Lesson 1, students are introduced to and explore the concept of a cross-section when they create different shapes by slicing a three-dimensional figure. In Activity 1. Slicing a Cube - students build a cube in clay and slice their clay cubes to create six cross-sectional shapes (a triangle, a square, a rectangle that is not a square, a parallelogram that is not a rectangle, a hexagon, and a pentagon). They discuss with other students where and how they sliced the cubes to make the cross-sections. (7.G.3) 

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

• In Module 2, Topic 2, Lesson 1, students interpret models and write explanations to demonstrate an understanding of multiplication of rational numbers. In Activity 1.1 Equal Groups - Modeling the Multiplication of Integers, knowing that multiplication can be represented as repeated addition, students are shown an example of 3 x 4 with both a number line and circles with positives in them. Students explain how these represent the multiplication problem. For each integer problem, students explain their understanding. (7.NS.2, 7.NS.3) 

• In Module 3 Topic 1, Lesson 1, students demonstrate conceptual understanding of expressions when writing an explanation of their solution. In Talk the Talk - Business Extras, students write and evaluate expressions for given real-world problems. (7.EE.1, 7.EE.2) 

• In Module 5, Topic 2, Lesson 1, students demonstrate an understanding of a cross-section by categorizing each sliced cross-section. In Activity 3 Cross-Sections of Right Rectangular Prism, students create a graphic organizer of cross-sections they sliced in a rectangular prism. They cut out diagram cards and description cards related to the different cross-sections and tape them into the appropriate rows of the organizer. (7.G.3)

The materials for Carnegie Learning Middle School Math Solution Course 2 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 throughout the grade level. They also provide opportunities to independently demonstrate procedural skill 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; 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 2, Lesson 1, students develop procedural skill when solving problems involving unit rate. In Getting Started, students complete given tables and include the unit rate of lemon-lime for each cup of punch for each recipe. Students then draw a graph for each recipe on the coordinate plane. Then students label each graph with the person's recipe and the unit rate. (7.RP.1) 

• In Module 3, Topic 2, Lesson 1, students develop procedural skill when writing expressions using the distributive property. In Mathematics Gymnastics, students rewrite algebraic expressions with rational coefficients using the distributive property. They also expand linear expressions. Students factor linear expressions in a variety of ways, including by factoring out the greatest common factor and the coefficient of the variable. (7.EE.1) 

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

• In Module 2, Topic 2, students demonstrate procedural skill when they make conjectures about the rules for multiplying and dividing integers. In the MATHia Software, students fill in the blanks to show the understanding of multiplying and dividing integers. (7.NS.A) 

• In Module 3, Topic 1, students demonstrate procedural skill when solving two-step equations through technology. In the MATHia Software, students determine unknown values and enter values into tables to recognize patterns. Students express the patterns in two-step expressions and use the solver tool to solve two-step equations. (7.EE.4a) 

• In Module 4, Topic 1, students demonstrate procedural skill when determining the probability of an event. In the MATHia Software, students work to build probability models and determine probabilities of simple and disjoint events and use proportions to make predictions based on samples and theoretical probabilities. (7.SP.6 & 7)

The materials for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 2, Lesson 1, students engage in the application of mathematical skills when calculating unit rates to solve real world problems. In Talk the Talk - Getting Unit Rate-ier, students are asked to determine whether each statement is true or false, and provide an example to justify their answer. (7.RP.1) 

• In Module 1, Topic 2, Lesson 2, students engage in the application of mathematical skills when using proportional relationships to solve real-world problems. In Activity 1.3, Proportional or Not?, students explore tables and graphs to discover that graphs of proportional relationships are straight lines and tables have a constant ratio. (7.RP.2b) 

•  (7.EE.4) 

• In Module 3, Topic 1, Lesson 4, Activity 2, students engage in the application of mathematical skills when creating equations to solve real-world problems. The materials provide the following scenario, “Felicia’s Pet Grooming charges $15 for each dog washed and groomed on the weekend. The cost of the dog shampoo and grooming materials for a weekend’s worth of grooming is $23.76. Felicia wants to know her weekend profits.”(7.EE.4a)

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

• In Module 1, Topic 3, Lesson 1, students independently demonstrate the use of mathematics when using proportional relationships to solve real-world problems. Percent models, proportions, and the constant of proportionality are revisited to solve percent problems with simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error. (7.RP.3) 

• In Module 2, Topic 1, students independently demonstrate the use of mathematics when addition and subtraction of rational numbers is used to solve real-world problems. In Performance Task, students represent rational numbers as a sum and a difference of two rational numbers, then develop their own real-world problems that model their representations. (7.NS.3) 

• In Module 4, Topic 3, Lesson 3, students independently demonstrate the use of mathematics when describing how to use a dot plot or box-and-whisker plot to determine variation of data and the mean. In Talk the Talk, students write one to two paragraphs to summarize the key points in the lesson by explaining how it is possible to determine the mean and the variation of data for two populations from a dot plot or box-and-whisker plot. They include answers to questions such as: “How can you compare the mean and the spread of data for two populations from a dot plot? If the measures of center for two populations are equivalent, how can the mean absolute variation show the differences in variation for two populations?” (7.SP.B)

The materials for Carnegie Learning Middle School Math Solution Course 2 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 5, Topic 1, Lesson 1 Overview, “Students distinguish measuring tools from construction tools as they differentiate the concepts of sketch and construct. They learn how to correctly draw, sketch, and name each of the essential building blocks of geometry. Students use a compass to construct circles and arcs. They then duplicate line segments and angles using only construction tools.” 

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 2 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. 

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 2, Topic 2, Lesson 2, students make sense of problems when classifying numbers. “What types of numbers are the quotients in Question 1? Use the definitions of the different number classifications to explain why this makes sense.” 

MP2 - Reason abstractly and quantitatively. 

• In Module 3, Topic 1, Lesson 1, students reason quantitatively when looking at various expressions. “The expressions and $−(\frac{1}{2})xy$ are examples of expressions that are not linear expressions . Provide a reason why each expression does not represent a linear expression.”

• In Module 1, Topic 1, Lesson 1, students reason abstractly when they use a formula to compute circumference. In Activity 1.3, students create a formula for the circumference of any circle and use it to compute unknown values.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 2, Lesson 2, problems include: “Dontrell claims that the number of bird feeders Bob builds is proportional to the number of bird feeders Jake builds. Do you agree with Dontrell’s claim? Explain your reasoning.” and “Vanessa thinks that there are only two: one with a width of two inches and a length of six inches, and another with a width of three inches and a length of four inches. Is she correct? Explain your reasoning.” 

• In Module 2, Topic 1, Lesson 5, Activity 1, students are given a table showing when some elements reach freezing point. Then, students are given the following questions, “Patricia and Elliott are trying to figure out how much temperature would have to increase from the freezing point of hydrogen to reach the freezing point of phosphorus. Patricia says the temperature would have to increase $545.7\degree F$, and Elliott says the temperature would have to increase $322.3\degree F$. Who is correct? Explain your reasoning.”

• In Module 4, Topic 1, Lesson 3, students construct viable arguments when evaluating the probability of the landing position for a cup in a game two friends are designing called Toss the Cup. “1. Predict the probability for each position in which the cup can land. 2. List the sample space for the game. 3. Can you use the sample space to determine the probability that the cup lands on its top, bottom, or side? Explain why or why not. 4) Do you think all the outcomes are equally likely? Explain your reasoning.” 

• In Module 5, Topic 1, Lesson 3, “Sarah claims that even though two segment lengths would form many different triangles, she could use any three segment lengths as the three sides of a triangle. Sam does not agree. He thinks some combinations will not work. Who is correct? Remember, you need one counterexample to disprove a statement.” 

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 2. Lesson 3, Activity 2, teachers are prompted to ask, “Why did Parker multiply each side by 3200? How did Nora get the fraction $\frac{1}{40}$? What is similar and different between the two solution strategies? How can you solve the problem using Nora’s method? How could you use scaling to solve this proportion?”

• In Module 1, Topic 3, Lesson 4, Activity 4.1, teachers are prompted to ask, “What is the difference in the steps to calculate the percent increase and percent decrease? Which value goes in the denominator? Provide an example of a 50% decrease.” 

• In Module 2, Topic 1, Lesson 3, Activity 2, teachers are prompted to ask, “How is using two-color counters related to using a number line to represent integer addition?” 

• In Module 3, Topic 1, Lesson 3, Activity 1, teachers are prompted to ask, “Why do you think Brent started this way? What is another way you could have solved the equation? How can you check your solution?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 4, Topic 1, Lesson 4, Talk the Talk, students develop a simulation to model different situations and describe one trial. They then conduct the simulations and answer related questions. Supplies such as note cards for the experiments are provided for students to decide how they will simulate the situation to answer the questions. 

MP5 - Use appropriate tools strategically. 

• In Module 5, Topic 1, Lesson 4, students use appropriate tools when constructing triangles with given angles. In Activity 4.1 A Triangle Given Three Angles, students construct various triangles. Materials list for this lesson include: Compasses, Patty paper, Protractors, Rulers, and Straightedges. 

• In Module 1, Topic 1, Lesson 1: Talk the Talk, students are asked to use what they have learned during previous activities to now draw two circles, one with a radius length of 3 centimeters and one with a diameter length of 3 centimeters and compare their characteristics. No guidance is given on what tools are to be used when doing this.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 5, Topic 1, Lesson 1.3, The terms angle, sides of an angle, vertex, and ray are defined for students. “An angle is formed by two rays that share a common endpoint. The angle symbol is $\angle$. The sides of an angle are the two rays. The vertex of an angle is the common endpoint the two rays share.” Students copy an angle using a compass and a straightedge. They then identify the angles and vertices of angles in their constructions. 

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

• In Module 3, Topic 1, Lesson 1, the teacher guide provides detailed definitions to help with explanations. “A variable represents an unknown quantity. You can model the relationship between variable expressions with the same variable on a number line.” The student book condenses this definition to a more student-friendly version. “In algebra, a variable is a letter or symbol that is used to represent an unknown quantity.” 

• In Module 4, Topic 1, Lesson 3, students explain the difference between experimental and theoretical probability in their own words.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 3, Topic 2, Lesson 1, students analyze an expression in order to write equivalent expressions using the distributive property. This lesson states, “Often, writing an expression in a different form reveals the structure of the expression. Meaghan saw that each expression could be rewritten as a product of two factors.” 

• In Module 5, Topic 2, Lesson 4, students develop a strategy for calculating the areas of regular polygons, generalize it by decomposing regular polygons, and determine that they can calculate the area of one of the n congruent triangles in the n-gon and multiply the area by n to calculate the area of the regular polygon. They transfer their work to pentagons and any regular polygonal base of a prism or pyramid. 

MP8 - Look for and express regularity in repeated reasoning. 

• In Module 2, Topic 2, Lesson 1, students look for and express regularity in repeated reasoning, when they notice repeated calculations to understand algorithms and make generalizations. In Activity 2, students describe and extend two different patterns of equations, students then make generalizations that can be applied to different pairs of integers and eventually describe a rule that can be used to multiply any two integers.  

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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, Talk the Talk, 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 Reflecting and justifying question states, “How did you use the circumference formula to support your response?”. 

• In Module 2, Topic 2, Lesson 3, Lesson Planning, Teacher’s Implementation Guide, students evaluate numeric expressions with signed rational numbers and identify the properties to justify their results. Students who are not on target yet will use MATHia to practice using number properties to evaluate expressions with signed numbers. Students who are on target will complete a different activity and present how they used number properties to work with signed numbers.

• In Module 4, Topic 2, Lesson 2, Getting Started, Teacher’s Implementation Guide, students are asked to calculate probabilities of a litter of 3 puppies being all female. The materials include a Language Link to support English Language Learners by explaining the definition of a litter of puppies.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 1, Lesson 1, the materials provide an adult-level explanation in describing how to use ratios to analyze the properties of geometric figures. “The constant pi () represents the ratio of the circumference of a circle to its diameter, or $\pi=\frac{c}{d}$ and is a constant value equal to approximately 3.14 or $\frac{22}{7}$. The circumference of a circle is the distance around the circle. The formulas to determine a circle’s circumference are $C=\pi d$ or $C=2\pi r$, where d represents the diameter and r represents the radius. 

• In Module 3, Topic 2, Lesson 3, Activity 3, Additional Facilitation Notes, the materials provide teachers with a common misconception for students when multiplying and dividing by negative numbers. The materials state, “Students may overgeneralize this rule and think they must reverse the inequality symbol whenever there is multiplication or division involving a negative number. Clarify that this rule applies only when the inverse operation of multiplication or division is applied on both sides of the equation with a negative value.”

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, Teacher’s Implementation Guide, Module Overview, Connection to Future Learning, the materials state, “This module strengthens students’ reasoning and fluency in solving equations. In grade 8, students analyze and solve systems of linear equations, which involve equations with variables on both sides and rational coefficients.” A mathematical representation is provided to show how to solve a system of equations by graphing when comparing the prices of two items.

• In Module 1, Topic 2, Topic Overview, Connection to Future Learning, the materials state, “The characteristics of proportional relationships, their graphs, and their equations provide the underpinnings of algebra and functions. In grade 8, students will connect proportional relationships, lines, and linear equations. Students will connect the constant of proportionality with the slope of a line that passes through the origin.” The materials then provide a mathematical representation describing characteristics of slope.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 Proportional Relationships are developed. The topic begins with students using their knowledge of proportionality to solve real-world problems involving tips, gratuities, commissions, simple interest, taxes, markups, and markdowns.

• In Module 2, Teacher’s Implementation Guide, Module Overview, the materials provide a connection to prior and future learning. Students build on their fluency with adding, subtracting, multiplying, and dividing whole numbers, decimals, and fractions in order to perform operations with rational numbers. Students will build upon their rational numbers knowledge as they evaluate expressions, and solve equations and inequalities involving rational, irrational, and complex numbers in later modules and courses.

• In Module 4, Topic 2, Topic Overview, the materials provide an overview of how key concepts of Compound Probability are developed. The topic begins with students building upon their knowledge from Introduction to Probability (the previous Topic) to use arrays, lists, and tree diagrams to organize sample spaces from compound events. Students learn about compound events that use “and” and “or” and then design and conduct simulations for compound probability problems.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 7, Teacher’s Implementation Guide, Module 4 Overview, the materials cite research from the GAISE Report, p. 8, states, “Probability is an important part of any mathematical education. It is a part of mathematics that enriches the subject as a whole by its intersection with other uses of mathematics. Probability is an essential tool in applied mathematics and mathematical modeling. It is also an essential tool in statistics.” 

• 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 2 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 2 meet 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 profice 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 3, Topic 1, Assessment Overview, the materials identify, “Expression 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 2, Topic 1, Lesson 2, the Talk the Talk assesses 7.NS.1b by having students “recognize patterns about the sum of numbers based on the signs of the numbers you are adding.”

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 1, Topic 3, Lesson 3, Activity 2, the summary in the Teacher’s Implementation Guide states, “The relationship between an order amount and the total amount, including a percent fee, is proportional. You can represent it as $y=kx$.” The Habits of Mind listed are, “Look for and make use of structure. Look for an express regularity in repeated reasoning.” Within the activity, the Questions to Support Discourse states, “How did you calculate the amount with the shipping fee? How does your graph show that this is a proportional relationship? Which x-value and y-value did you use to write the constant of proportionality as $\frac{y}{x}$?” 

• In Module 5, Topic 2, Lesson 4, Activity 1, the summary in the Teacher’s Implementation Guide states, “You can use formulas to calculate the volume and surface area of geometric solid with bases that are not rectangles.” The Habits of Mind listed is, “Attend to precision”. Within the activity, the Questions to Support discourse state, “What is the difference between a right rectangular prism and other rectangular prisms? How is a pyramid related to a prism that has the same base and height? How is calculating the surface area of a pyramid similar to calculating the surface area of a prism? Do you need to calculate the area of the base, or is it given to you? How does that change what you need to do? Why is the unit of measure that describes volume different from the measure used to describe the surface area?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 2 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 7.G.4. Students are given the following scenario, “Pete’s Plumbing was just hired to replace the water pipes in the Johansson’s house. Pete has two types of pipes. He can use a pipe with a radius of 8 cm or a pipe with a radius of 4cm. The 4 cm pipes are less expensive than the 8 cm pipes for Pete to buy, so Pete wonders if there are a number of 4 cm pipes he could use that would give the same amount of water to the Johansson’s house as one 8 cm pipe.” Then students respond to the following statements, “Explain how to compare an 8 cm pipe and a 7 cm pipe in this situation. Determine the difference in the amount of water each pipe gets to the Johansson’s house. Pete says two 4 cm pipes will equal an 8 cm pipe because 2 times 4 equals 8. Pete’s assistant, Jeff, says it will take four 4 cm pipes to equal one 8 cm pipe. Who is correct? Explain your reasoning.”

• In Module 5, Topic 1, End of Topic Test Form A develops the full intent of standards 7.G.2 and 7.G.5. Problem 14 has students solve the following scenario with algebra, “If the measure of ∠A is one-eighth times the measure of its supplement, what is the measure of ∠A and its supplement?”Problem 18 asks students to, “Use a compass and a straightedge to construct a line segment twice the length as the segment shown.” Problem 19 asks students to, “Use a compass and a straightedge to construct an angle with twice the measure as the given angle.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 2, Lesson 3, the materials suggest for teachers to, “Remind students to refer to the Academic Glossary (page FM-20) to review the definition of estimate and related phrases. Suggest they ask themselves these questions: 

    • Does my reasoning make sense? 

    • Is my solution close to my estimation?”

  • In Module 3, Topic 1, Lesson 4, Getting Started, the materials suggest for teachers to, “Create a word wall or other space to display the Properties of Equality. Encourage students to frequently reference that space to support their use of the properties in their explanations.”

• 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 1, Topic 2, Lesson 2, Activity 2, the materials suggest supporting struggling students as they work on Question 3 by having, “students label each point on the graph with its corresponding ratio to see how the ratios relate to the scenario and that they are not the same.”

  • In Module 3, Topic 2, Lesson 1, the materials suggest supporting all students as they work on Question 4 by having, “students annotate the distribution on the expression to help them remember to apply division to all terms in the numerator.”

• 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 2 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 3, Topic 1, Lesson 3, Activity 1, the materials suggest extending the worked example for advanced learners by challenging “students to solve the problem another way, basically to divide first and then subtract. Check that students divide both terms in the expression by 2. Compare solutions and methods.”

• In Module 5, Topic 2, Lesson 4, Activity 3, the materials suggest extending the activity for advanced learning by challenging “students to identify where the perimeter is evident in the area formula. See whether they can write the area formula in terms of the perimeter.”

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 1, Topic 3, Lesson 1, Assignment, students are given an optional “Stretch” question with a scenario about two cashiers applying discounts. The scenario states “The cashier on their left is taking 20% off the total bill and then subtracting $10.00. The cashier on their right is subtracting $10.00 first and then taking 20% off the total.” The students are asked “In order to get a better deal, should Emma and Jacob go to the cashier on the left or the right? Or does it not matter? Show all of your work and explain your reasoning.”

• In Module 2, Topic 2, Lesson 1, Assignment, students are given an optional “Stretch” question where they evaluate the expression $(-3)^3$, $(-4)^2$, $(-2)^5$, and $(-\frac{1}{4})^2$.  Students then explain what they noticed when evaluating the expressions.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Getting Started, the language link provided in the Teacher’s Implementation Guide states, “Remind students to refer to the Academic Glossary to review the definition of analyze and related phrases. Suggest they ask themselves these questions: Do I see any patterns? What happens if the shape, representation, or numbers change?”

  • In Module 5, Topic 1, Lesson 1, Getting Started, the language link provided in the Teacher’s Implementation Guide states, “Outside of mathematics, the terms sketch and draw are often interchangeable. Students may not recognize the mathematical difference. Draw students’ attention to the terms and have them explain the difference when using them in mathematics.”

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 2 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 2, Topic 1, Lesson 2, Activity 1, students walk a number line to calculate the values of 1+3, 0+(-4), -3+5, and -1+(-4). The Note in the Teacher’s Implementation guide provides the following instructions, “Before class, use masking tape to set up a human number line on the floor from at least -5 to 5. Spaces between tick marks should be about one step in width.”

• In Module 5, Topic 1, Lesson 3, Activity 1, students use patty paper to investigate the possibility of creating a triangle knowing the lengths of two sides. 

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

• In Module 3, Topic 1, MATHia Software Workspaces, Modeling Equations by Equal Expressions, Analyzing Models of Two-Step Linear Relationships, students are given a scenario that can be modeled by a two-step linear equation and complete a drag and drop activity matching expressions with their corresponding description from the scenario. For example, students are given, “Tre and Mark invested $26 in drink mix, sugar, and paper cups and opened a lemonade stand. They decide to sell each cup of lemonade for $2. The situation can be modeled by the equation y=2x-26.” Students match up “x,” “2,” “26,” “y,” “2x,” and “2x-26” to the following: “the number of cups,” “the amount Tre and Mark have after investing in supplies,” “the total amount made selling ups of lemonade,” “the cost of a cup of lemonade,” and “the amount Tre and Mark have made in profits.”

• In Module 4, Topic 2, MATHia Software Workspaces, Compound Probability, Introduction to Compound Events, students watch an animation that demonstrates how simple events can be combined into compound events using the example of the game, “Rock, Paper, Scissors.” Students answer questions about the definitions of simple and compound events after viewing the animation.