The materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials for Carnegie Learning Middle School Math Solution Course 3 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. 

• In Module 1, Topic 1, Lesson 4, students gain an understanding of translations, reflections, and rotations by manipulating two dimensional figures on a coordinate plane. In Activity 4.1 Mirror, Mirror, students see an image of an ambulance appearing backward like it would in a mirror. Students discuss why the image looks as it does and write their name in a similar manner. Students then use patty paper to reflect pre-images across the x-axis and y-axis and explore how the reflection affects the coordinates. (8.G.A) 

• In Module 2, Topic 1, Lessons 1, students explore slope and similar triangles to understand the slope-intercept form for the equation of a line. In Activity 1.4 Comparing Depth of Color, using concrete visual representations, students develop connections between tables, graphs, and expressions for a given situation. (8.EE.6) 

• In Module 3, Topic 1, Lesson 3, students demonstrate an understanding of a function when they create an input/output table and an x/y table. In Activity 3.1 Functions as Mappings from One Set to Another, students see examples of mapping ordered pairs and look at the relationship of x and y coordinates. Students write the ordered pairs shown by mapping, create their own mapping and then represent the numbers in an input/output table and an x/y table to visualize the relationships between them. (8.F.A) 

• In Module 4, Topic 2, Lesson 1, students develop an understanding of the Pythagorean Theorem through visual models. In Getting Started and Activity 1.1 Introducing the Pythagorean Theorem, students square the length of each side of a right triangle and describe patterns they see. They summarize the patterns to discover that, in a right triangle, the hypotenuse must be opposite the right angle. (8.EE.2, 8.G.6, 8.G.7) 

• In Module 4, Topic 2, Lesson 1, students develop an understanding of the Pythagorean Theorem when using manipulatives to create a proof. In Activity 1.2, Proving the Pythagorean Theorem, students use manipulatives (grid paper, cut-outs) to create a geometric proof of the Pythagorean Theorem three different ways. (8.EE.2, 8.G.6, 8.G.7) 

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

• In Module 1, Topic 2, Lesson 3, students demonstrate an understanding of similarity by justifying their answer with a rationale. In Talk the Talk - Summing Up Similar Figures, students determine if statements about similarity are always, sometimes, or never true and justify their answer based on prior learning. (8.G.A) For example, “The same order for a sequence of transformations can be used to map between two similar figures, regardless of which figure is used as the pre-image.” 

• In Module 2, Topic 1, Lesson 2, students demonstrate an understanding of proportional relationships when providing examples of connections between concepts. In Talk the Talk - A Web of Connections, students summarize what they have learned by connecting the steepness of a line to the concepts of slope, rate of change, unit rate, and the constant of proportionality. Students provide illustrations and examples showing the connections among these concepts. (8.EE.5 &.6) 

• In Module 3, Topic 1, Lesson 3, students create various representations of a function to demonstrate their understanding. In Talk the Talk - Function Organizer, students represent a function in different ways (problem, situations, graph, and ordered pairs) in a graphic organizer. (8.F.A) 

• In Module 2, Topic 1, Lesson 2, students demonstrate an understanding of proportional relationships when solving real-world problems. In Practice Questions, students are given situations and use their prior knowledge to choose a strategy that best represents the information, solve it using that method, and explain their solution. (8.EE.B)

The materials for Carnegie Learning Middle School Math Solution Course 3 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 1, Lesson 4, students develop procedural skill when determining coordinates of the vertices of two-dimensional figures. In Activity 4.2, Reflecting Any Points on the Coordinate Plane, students reflect a point across the x-axis and y-axis and record the coordinates of the images. Next, they are given the coordinates to graph a triangle which they reflect and record the coordinates of the vertices of the images. Finally, they are given the coordinates of the vertices of a triangle, and without graphing, they determine the coordinates of images resulting from different reflections. (8.G.2 & 3) 

• In Module 2, Topic 1, Lesson 1, students develop procedural skill when working with proportional relationships. In Activity 1.2, Comparing Ratios and Graphs, students connect ratios with graphs to show proportional relationships. They draw additional lines of equations with ratios greater than or less than the ratios graphed, comparing ratio magnitude with steepness of the lines. (8.EE.5) 

• In Module 3, Topic 2, Lesson 1, Activity 4,students develop procedural skill when using the Pythagorean Theorem to calculate side lengths. Students are given four right triangles with a missing side length. Students substitute values into the Pythagorean Theorem and solve the equation for the missing value. (8.G.6)

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

• In Module 2, Topic 3, Lesson 3, Students determine when equations have one solution, no solutions, or infinitely many solutions. Finally, students write their own equations: one that has one solution, one that has no solutions, and one that has infinite solutions. (8.EE.7a)

• In Module 3, Topic 3, Lesson 2, students demonstrate procedural skill when analyzing and writing systems of linear equations. In The Road Less Traveled, Systems of Linear Equations, students write and analyze systems of linear equations. They informally calculate the solutions to systems of linear equations and then graph the systems of equations. Students conclude when parallel lines comprise the system the lines will never intersect, so there is no solution to the system. (8.EE.8a) 

• In Module 5, Topic 1, Lesson 1, students demonstrate procedural skills when solving expressions with exponents. In Activity 3, students solve expressions with exponents using the product of power rule and the power to a power rule. In Activity 4, students solve expressions with exponents using quotients of powers. (8.EE.1)

• In Module 5, Topic 2, students demonstrate procedural skill when calculating the volume of three-dimensional figures. In the MATHia Software, students have multiple opportunities to determine the volume of cylinders, cones, and spheres and use the volume of a cylinder or sphere to determine its radius. (8.G.9)

The materials for Carnegie Learning Middle School Math Solution Course 3 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 3, Topic 1, Lesson 4, students engage in the application of mathematical skills when using linear functions to solve real world problems. In Activity 3.4, students construct linear functions and analyze the graphical behavior of linear and nonlinear functions in situations such as Little Red Riding Hood’s journey, climbing cliffs, plant growth, bank accounts, etc. (8.F.5) 

• In Module 3, Topic 1, Lesson 5, students engage in the application of comparing functions to solve real-world problems. In Activities 5.1 & 5.2, students compare functions presented in different ways, such as tables, graphs, equations, and context, to explore the rate of change in real-world situations. (8.F.2) 

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

• In Module 1, Topic 1, students independently demonstrate the use of mathematics when working with transformations to solve real world problems. In the Performance task, students are given a piece of a pattern. Students use transformations to create an original complete pattern. This work is done within the context of creating a quilt. (8.G.2 &.3) 

• In Module 2, Topic 2, students independently demonstrate the use of mathematics when analyzing and writing equations to solve real-world problems. In the Performance Task, given pricing data for three health clubs, students generate equations and determine the best value for various lengths of memberships. (8.EE.7) 

• In Module 5, Topic 1, Lesson 2, students independently demonstrate the use of mathematics when using exponents to solve real world problems. In Talk the Talk - Organize the Properties, students organize their learning regarding six rules for exponents. For each rule, they write a definition, list facts and characteristics, develop and solve example problems, and then write a generalized rule for each property. (8.EE.1)

The materials for Carnegie Learning Middle School Math Solution Course 3 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 1, Topic 3 Overview, “In Line and Angle Relationships, students use their knowledge of transformations, congruence, and similarity to establish the Triangle Sum Theorem, the Exterior Angle Theorem, relationships between angles formed when parallel lines are cut by a transversal, and the Angle-Angle Similarity Theorem for similarity of triangles. Students use hands-on tools to make and justify conjectures about the sum of the interior angles of a triangle, the relationship between triangle side and angle measures, and the value of exterior angles of triangles. They then apply their results to new problems. Next, students use patty paper and translations to form parallel lines cut by a transversal. They determine and informally prove the relationships between the special angle pairs formed when parallel lines are cut by a transversal and use these relationships to solve mathematical problems, including writing and solving equations. Finally, students use parallel line relationships and tools to establish the Angle-Angle Similarity Theorem and use the theorem to determine if triangles in complex diagrams are similar.” 

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 3 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 4, Topic 1, Lesson 2, students have to make sense of a set of numbers and the size of that set of numbers in order to answer the following three questions. “The first set of numbers that you learned when you were very young was the set of counting numbers, or natural numbers. Natural numbers consists of the numbers that you use to count objects: {1, 2, 3, …}. Consider the set of natural numbers: 1. Why do you think people call this set of numbers the set of counting numbers? You have also used the set of whole numbers. Whole numbers are the natural numbers and the number 0, the additive identity. 2. Why is zero the additive identity? 3. Explain why having zero makes the set of whole numbers more useful than the set of natural numbers. Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses 4. What is the additive inverse of a number? 5. Represent the set of integers. Use set notation and remember to use three dots to show that the numbers go on without end in both directions.”

MP2 - Reason abstractly and quantitatively. 

• In Module 3, Topic 1, Lesson 4, Activity 4.2, students reason abstractly and quantitatively when they create equations from word problems, solve the equations, and then give their answers within the context of the original problem.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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

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

• In Module 2, Topic 2, Lesson 2, Activity 1, “Analyze Cely’s calculation of the slope using the table of values. Cely: I used the ratios of $\frac{y}{x}$ to calculate the slope of this line. $\frac{y}{x}=\frac{2.3}{1}=2.3$. Explain why Cely did not calculate the slope of this line correctly.”

• In Module 3, Topic 1, Lesson 4, “You and your friends are rock climbing a vertical cliff that is 108 feet tall along a beach. You have been climbing for a while and are currently 36 feet above the beach when you stop on a ledge to have a snack. You then begin climbing again. You can climb about 12 feet in height each hour. Does this situation represent a function? Explain your reasoning.” 

• In Module 4, Topic 2, Lesson 2, “Orville and Jerri want to put a custom-made, round table in their dining room. The tabletop is made of glass with a diameter of 85 inches. The front door is 36 inches wide and 80 inches tall. Orville thinks the table top will fit through the door, but Jerri does not. Who is correct and why?” 

• In Module 5, Topic 1, Lesson 3, “Kanye and Brock each tried to write the number 16,000,000,000 in scientific notation. Analyze each student’s reasoning. Who is correct?” 

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

• In Module 1, Topic 3, Lesson 1, teachers are prompted to ask, “Is there another way to arrange your three angles? Do you get the same result? Is the sum of the interior three angles the same for everyone’s triangle?" 

• In Module 2, Topic 3, Lesson 5,Getting Started, teachers are prompted to ask, “Why did you choose to write your equation in slope-intercept form? What is another equation that represents each line? In general, when two lines intersect at one point, do their slopes have to be opposites? Explain your thinking.”

• In Module 3, Topic 1, Lesson 5, Getting Started, teachers are prompted to ask, “How can you determine the greater rate of change without making any calculations? Will this method work if the scales on the axes are not the same? Why not? How can you calculate the actual slopes from the graphs?”

• In Module 5, Topic 1, Lesson 2, teachers are prompted to ask, “For expression B, what is the rule when rewriting negative exponents as positive exponents? For Expression A, what clarifications could you make to the rule so that Adam and Shane apply it correctly? For Expression B, why does the 2 remain in the numerator?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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

MP4 - Model with mathematics. 

• In Module 3, Topic 2, Lesson 3 Assignment, students model situations with equations. The materials state, “Analyze the scatter plot relating a male chihuahua’s weight with its age. a)Draw the line of best fit for the set of data values. Then determine the equation for the line of best fit. b)Interpret the meaning of the slope and y-intercept for your equation. c)Use your line of best fit to predict the weight of a 9-week-old male chihuahua.”

• In Module 5, Topic 5, Lesson 1, students complete a tree diagram detailing the puppy’s lineage back seven generations. They express the number of sires and dams for each generation in expanded notation and power notation and answer related questions. 

MP5 - Use appropriate tools strategically. 

• In Module 2, Topic 3, Lesson 4, Activity 2, students solve systems of equations using a graph, inspection, or substitution. Students, “Solve each linear system. State which elements of each system led to your chosen solution strategy.”

• In Module 5, Topic 1, Lesson 4, Activity 3, students choose a method to calculate each sum or difference. In Activity 3, problem 6 reads, “Calculate each sum or difference using any method. Write your answer in scientific notation. a. $3.7\times10^5+2.1\times10^6$.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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 1 Topic 1, Lesson 6, Talk the Talk, students justify their answer by using the term transformation. “Suppose a point (x, y) undergoes a rigid motion transformation. The possible new coordinates of the point are shown. Assume c is a positive rational number. $(y,-x) (x,y - c) (x,-y) (x+c,y) (x-c,y) (-y,x) (-x,-y) (-x,y) (x,y+c)$ 1. Record each set of new coordinates in the appropriate section of the table, and then write a verbal description of the transformation. Be as specific as possible. 2. Describe a single transformation that could be created from a sequence of at least two transformations. Use the coordinates to justify your answer.” 

• In Module 4, Topic 1, Lesson 2, the term bar notation is defined for students. “A repeating decimal is a decimal with digits that repeat in sets of one or more. You can use two different notations to represent repeating decimals. One notation shows one set of digits that repeats with a bar over the repeating digits. You call this bar notation.” Students are also shown how to write repeating decimals as a fraction or with ellipses. 

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 sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term in a sequence is an individual number, figure, or letter in the sequence.” 

• In Module 2, Topic 3, Lesson 1, Talk the Talk - How Do You Choose? “Throughout this topic, you have solved systems of linear equations through inspection of the equations, graphing, and substitution. How do you decide when each method is most efficient? Create a presentation or a poster to illustrate your decision-making process when you solve a system of linear equations. Consider these questions to guide the content of your presentation.  What methods do you know for solving systems of linear equations?  What visual cues about the equations in the system guide your decision?  How do the slope and y-intercept of the equations affect your decision?  Does the form of the equations in the system affect your choice? Use the systems of linear equations you solved throughout this lesson to support your reasoning and as examples of when you would choose each solution method. List at least three key points that you want to include in your presentation.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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 4, Topic 1, Lesson 1, students look for and make use of structure when sorting numbers into different groups. “Searching for patterns and sorting objects into different groups can provide valuable insights. Cut out the 24 number cards located at the end of the lesson. Then, analyze and sort the numbers into different groups. You may group the numbers in any way you feel is appropriate. However, you must sort the numbers into more than one group. In the space provided, record the information for each of your groups; Name each group of numbers; List the numbers in each group; Provide a rationale for why you created each group.” 

MP8 - Look for and express regularity in repeated reasoning. 

• In Module 1, Topic 1, Lesson 3, students look for regularity in repeated reasoning when performing translations of figures. In Activity 3.1, students copy figures and the coordinates of their vertices onto patty paper and perform translations of the figures. They record the coordinates of the original and translated figures and explore how the translation affected the coordinates of the pre-image. Students make a general conjecture about the effect of a horizontal or vertical translation on an ordered pair.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

The Front Matter section includes:

• Guiding Principles of Carnegie Learning.

• Content Organization including Modules, Topics, and Pacing.

• Course Standards Overview Chart.

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

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

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

• Guidance for implementing MATHia.

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

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

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

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

• In Module 1, Topic 1, Lesson 1, Getting Started, Teacher’s Implementation Guide, the materials provide sets of questions to support student discourse within the lesson. Questions are identified by type such as Gathering, Probing, Seeing Structure or Reflecting and justifying. A sample Probing question to support discourse states, “What strategies did you use to categorize your shape? How are your categories similar to or different than your classmates’ categories?”.

• In Module 3, Topic 2, Lesson 2, Activity 2, Teacher’s Implementation Guide, the materials include annotations with suggestions on how to chunk the activity. The “Chunking the Activity” suggests the following: “ Read and discuss the situation, Group students to complete 1 - 5, Check-in and share, Group students to complete 6 - 8, and Share and Summarize”. 

• In Module 3, Topic 1, Lesson 4, Lesson Planning, Teacher’s Implementation Guide, students explore and identify characteristics of functions. Students who are not on target yet, will “practice identifying key features of graphs”. Students who are on target will complete a different activity and share how they identified key features of functions in graphs.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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 3, the materials provide an adult-level explanation in describing how to model translation on the coordinate plane. The materials state, “A horizontal translation impacts the x-coordinate of a point. A point with coordinates $(x,y)$, when translated horizontally by $c$ units, has new coordinates $(x+c,y)$. A vertical translation impacts the y-coordinate of a point. A point with the coordinates $(x,y)$, when translated vertically by $c$ units, has new coordinates $(x, y+c)$.”

• In Module 3, Topic 2, Lesson 1, the materials provide an adult-level explanation describing essential ideas of the lesson. The materials state, “Scatter plots display associations and are a tool to identify patterns in bivariate data. A scatter plot may have a linear or nonlinear association, or it may not have an association. Some scatter plots have a positive or negative association and outliers. An outlier is a point that varies significantly from the overall pattern of the data. Without an outlier, a scatter plot may have an association.”

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 4, Teacher’s Implementation Guide, Module Overview, Connection to Future Learning, the materials state,” As students solve a wider range of equations, they will encounter the need for imaginary numbers. Just as in Expanding Number Systems, students will reason about the properties of imaginary numbers. Students use the Pythagorean Theorem when solving volume problems in the next module. They will also use the Pythagorean Theorem throughout the high school curriculum: modeling and solving problems…”. A mathematical representation is provided showing a chart of the complex number system.

• In Module 5, Topic 1, Topic Overview, Connection to Future Learning, the materials state, “Students will continue to expand the complexity of powers that they can evaluate. In high school, students evaluate rational number exponents.” The materials then provide a mathematical representation showing the rational exponent representation and the radical form representation of the function $f(c)=2^x$ using $f(0)$, $f(\frac{1}{2})$, $f(1)$, and $f(\frac{3}{2})$. The materials also state, “Exponent and Scientific Notation prepares students for a more rigorous and abstract exposure in high school. Scientific notation will arise in students’ science courses in middle school and high school, particularly in the study of chemistry.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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 Line and Angle Relationships are developed. The topic begins with students using their knowledge of transformations, congruence, and similarity to investigate and establish the Triangle Sum and Exterior Angle Theorems. Students develop their understanding of relationships of special angle pairs formed when parallel lines are cut by a transversal and use these relationships to establish the Angle-Angle Similarity Theorem.

• In Module 2, Topic 2, Topic Overview, the materials provide an overview of how key concepts of Linear Relationships are developed. The topic begins with students using their prior knowledge of graphing linear equations and determining slope from a graph to calculate slope of linear relationships in tables and from contexts. Students derive and write equations in slope-intercept form and point-slope form.

• In Module 5, Teacher’s Implementation Guide, Module Overview, the materials provide a connection to prior and future learning. Students build on their fluency with evaluating numeric and algebraic expressions with whole-number exponents. Students will build upon the complexity of powers they can evaluate when evaluating rational number exponents in high school.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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 8, Teacher’s Implementation Guide, Module 2 Overview, the materials cite research from Developing Essential Understanding of Expressions, Equations, and Functions: Grades 6-8, p. 71 states, “It is advantageous for students to develop fluency in the use of multiple strategies for solving equations and to develop the ability to select the most appropriate strategy for a given problem.” 

• 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 3 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 3 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 5, 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 1, Topic 2, Lesson 2, the Talk the Talk assesses 8.G.3 and 8.G.4 by having students explain how the location of the center of dilation affects the coordinates of the dilated figure, and describe how to verify the similarity of figures on the coordinate plane using coordinates.

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 2, Topic 3, Lesson 1, Activity 2, the summary in the Teacher’s Implementation Guide states, “ A change in the cost or income structure affects the break-even point.” The Habits of Mind listed is, “Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others.” Within the activity, the Questions to Support discourse state, “Do you think the organization will have to sell more or fewer shirts given these changes? Will they need to sell more shirts or fewer shirts to make a profit? If you rewrite your profit expression by combining like terms, what does each term represent? Why does it make sense that the break-even point is the same as the point of intersection? How does the break-even point compare to the break-even point from the previous activity? Why does that make sense?”

• In Module 4, Topic 1, Lesson 3, Activity 3, the summary in the Teacher’s Implementation Guide states, “You can estimate an irrational cube root with a decimal value by locating it between two perfect cubes’ cube roots.” The Habits of Mind listed is, “Look for and make use of structure. Look for and express regularity in repeated reasoning.” Within the activity, the Questions to Support discourse state, “What type of number is the result of the cube root of a perfect cube? Why is that the case? Can you take the cube root of a negative number? Explain your thinking. What equation models this situation? Why? How is the strategy to estimate cube roots related to the strategy to estimate square rots? Why is the estimate closer to 3 than to 4?  How did you verify that your response is the best estimation to the nearest tenth? What equation models this situation? What is the only number that is a whole number but is not a natural number?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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 3 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 8.G.1. Students are given the following scenario and question, “Delaney is making a quilt for her bed. The design she wants to use is a sunburst. However, Delaney only has a pattern for one part of the design as shown. Show 2 different ways to use rigid motions to create the entire sunburst quilt. Can you use all 3 rigid motions (translations, reflections, rotations) to create the entire sunburst quilt?”

• In Module 4, Topic 2, End of Topic Test Form A develop the full intent of the standards 8.G.7 and 8.G.8. Problem 2 states, “Karen is planting a tree. She wants to use two guy wires to stabilize the tree. If Karen places the guy wires 8 feet up the trunk and 7 feet from the base, how much total wire will she need for both guy wires? Write an equation to determine the unknown side. Then, solve the equation. Express your answer as a decimal approximation.” For Problems 14 and 15, students must, “determine the distance between each pair of points.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics. The materials identify strategies to support language development, productive skills, and interactions throughout the series. The materials include “Additional Facilitation Notes” at the end of each lesson to assist teachers as they support students. The “Additional Facilitation Notes” include differentiation strategies, common student misconceptions, and suggestions to extend certain activities. 

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

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

  •  In Module 2, Topic 3, Lesson 1, Activity 1, the materials state, “The phrase break-even point may be new to students. To help them make sense of it, connect it to phrases coined when companies kept records by hand, saying that they were “in the red” when they used red ink to denote negative earnings and “in the black” when they used black ink to denote when they were profitable.”

  • In Module 4, Topic 1, Lesson 2, the materials state, “Ensure that students understand the term rational has both an everyday use, meaning reasonable, and a mathematical meaning. Connect the mathematical definition of rational number with the root word, ratio.”

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

  • In Module 3, Topic 1, Lesson 4, Activity 1, the materials suggest supporting students who struggle as they discuss the definition of linear function by having, “students circle and number the two criteria necessary for a linear function in its definition.”

  • In Module 5, Topic 1, Lesson 4, Activity 4, the materials suggest supporting students who struggle as they work on the activity by having teachers, “Provide a table of metric linear unit conversions to support students as they calculate with centimeter, meter, and kilometer measurements.”

• 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 3 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 2, Topic 1, Lesson 2, Activity 1, students are asked “What is the unit rate? Explain what the unit rate means in terms of this situation.” The materials suggest extending the question for advanced learners by having students “Calculate Jack and Jill’s rate in MPH. Determine how many minutes it takes Jack and Jill to walk one mile.”

• In Module 3, Topic 1, Lesson 4, Activity 3, the materials represent the area of a square as $A=s^2$. Students are instructed to calculate the domain and range of the equation. Then, students are told the area of a square can be modeled as $y=x^2$. Students are instructed to calculate the domain and range and describe how it is different from the previous domain and range. The materials suggest extending the question for advanced learners by showing “students how to express the domain and range using inequalities with infinity.”

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. Students are instructed to, “Use what you know about interior and exterior angles to show why it is possible to tessellate with a regular hexagon but not with a regular pentagon.”

• In Module 5, Topic 2, Lesson 3, Assignment, students are given an optional “Stretch” question. Students are given the following scenario and question, “A typical orange has 10 segments and water composes about 87% of it. Suppose an orange has a diameter of 3 inches. 1) Determine the volume of water in each segment.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 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, Talk the Talk, the language link provided in the Teacher’s Implementation Guide states, “Remind students to refer to the Academic Glossary to review the definition of consider and related phrases. Suggest they ask themselves these questions: Do I see any patterns? What happens if the shape representation, or number change?”

  • In Module 4, Topic 2, Lesson 2, Activity 3, the language link provided in the Teacher’s Implementation Guide states, “Ensure students understand the use of stretcher bars in Question 5. Canvas is a cloth material. Just as you can stretch your clothing when you tug on it, you can stretch a canvas. Sherie uses the stretcher bars to pull the canvas tight to create a smooth surface to paint. This question asks how Sherie can use a ruler and measure a length to make sure the frame’s corners make right angles while stretching the canvas.”

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 3 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

• In Module 1, Topic 3, Lesson 1, Getting Started, students are instructed to, “Draw any triangle on a piece of patty paper. Tear off the triangle’s three angles. Arrange the angles so they are adjacent.” Students use the visual to, “Write a conjecture about the sum of the three angles in a triangle.” 

• In Module 2, Topic 2, Lesson 6, Activity 1, students, “Carefully cut out the graphs, tables, contexts, and equations located on pages 299 and 301.” First, students match the equation card with the appropriate table, graph, or scenario. Then, students analyze the graphs. 

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

• In Module 1, Topic 1, MATHia Software Workspaces, Rigid Motions on the Coordinate Plane, Translating Plane Figures, students are given a pre-image and image on the coordinate plane and need to select the appropriate transformation (options include reflection, rotation, dilation, horizontal translation, and vertical translation) and how to move the image (i.e. reflect over the x or y-axis, how many degrees to rotate, scale factor for a dilation, or number of units to move horizontally or vertically for a translation). Once students map the pre-image onto the image, students answer the following question, “Does the transformed pre-image match the image target?” and determine whether the pre-image and image are congruent and/or similar.

• In Module 4, Topic 2, MATHia Software Workspaces, The Pythagorean Theorem, Exploring the Pythagorean Theorem, students engage with an “Explore Tool” that provides a right triangle with squares along the edge of each triangle to investigate the Pythagorean Theorem. Students drag the corners to build triangles of certain lengths and see how the squares change, while examining how the areas of the two smaller squares compare to the area of the largest square.