The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, the instructional materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding.

Examples that show the development of conceptual understanding throughout the series include:

• A-SSE.1b: In Math 2, Module 4, Topic 1, Activity 5.2, students interpret how the first term of each root obtained from the quadratic formula is represented graphically. Students then consider why the second term of each root obtained from the quadratic formula is the same except for the sign and how this is represented graphically. At the conclusion of the activity, students generalize their findings by labeling the vertex, axis of symmetry, roots, and distance each root lies from the axis of symmetry for quadratics of the form $$y = ax^2 + bx + c$$ for a > 0 and a < 0 with two real roots and double real roots.
• F-IF.1: In Math 1, Module 1, Topic 1, Lesson 3, students develop their understanding of functions. In Activity 3.1, students analyze relations represented as an ordered pair, written description, graph, table, mapping, and equation to determine whether the relations are functions. In Activity 3.2, students identify the domain and range of functions by writing the domain and range in words or using inequality notation.
• G-CO.7: In Math 1, Module 5, Topic 3, Activity 2.1, students use a worked example to explain why a segment can be mapped onto itself in at most two reflections. In Activity 2.2, students use that criteria to demonstrate two triangles are congruent using the SSS congruence theorem.

Examples that show the materials providing an opportunity for students to independently demonstrate conceptual understanding throughout the series include:

• A-APR.2: In Math 3, Module 2, Topic 1, Activity 2.3, students divide polynomials and observe relationships between factors, divisors, dividends, and remainders to develop their understanding of the Remainder Theorem.
• G-SRT.6: In Math 2, Module 2, Topic 2, Activities 1.1 and 1.2, students use properties of similar right triangles to compare side length ratios for 45-45-90 and 30-60-90 triangles. These activities set the foundation for students to develop the definitions of trigonometric ratios later in the topic.
• S-ID.3: In Math 1, Module 4, Topic 1, Lesson 2 Assignment, students create a data set of 15 numbers where the mean and median are both 59 and the standard deviation is between 10 and 11. Students add an outlier to the data set and explain how the center and spread are affected.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Opportunities for students to independently demonstrate procedural skills across the series are provided in activities and assignments in the student materials, the online skills practice workbook, and MATHia software. 

Examples that show the development of procedural skills across the series include:

• A-SSE.1b: In Math 2, Module 3, Topic 2, Activities 2.2 and 2.3, students interpret a and b in exponential functions of the form $$f(x) = a(b^x)$$ within the context of population growth and decay. There are several additional practice problems in the Math 2, Module 3, Topic 2 skills practice workbook.
• A-SSE.2: In Math 3, Module 2, Topic 3, students rewrite rational expressions when graphing rational functions to determine asymptotes or discontinuities and when performing operations with rational expressions. Procedural skill practice is included in activities and assignments throughout the topic as well as in the Math 3, Module 2, Topic 3 skills practice workbook and MATHia.
• F-IF.7b: In Math 2, Module 3, Topic 1, Lessons 3 and 4, students graph piecewise-defined functions, including absolute value and step functions. In Math 3, Module 3, Topic 1, Activities 2.1 and 2.2, students graph square root functions. In Math 3, Module 3, Topic 1, Activity 2.3, students graph square root and cube root functions. The skills practice workbooks provide additional opportunities for students to independently demonstrate their procedural skills in graphing these functions.
• G-GPE.4: In Math 1, Module 2, Topic 4, Lesson 1, students use the distance formula and the slope criteria for parallel and perpendicular lines to classify triangles and quadrilaterals on a coordinate plane. In Math 2, Module 4, Topic 3, Lesson 2, students determine whether a given point lies on a circle on the coordinate plane. There are additional problems in the Math 1, Module 2, Topic 4, and Math 2, Module 4, Topic 3 skills practice workbooks.
• G-SRT.5: In Math 2, Module 2, Topic 1, Lesson 5, students solve problems using triangle similarity. Both MATHia and the skills practice workbook provide additional opportunities for students to independently demonstrate their procedural skills in using triangle similarity to solve problems.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. The instructional materials include multiple opportunities to engage students in routine and non-routine applications of mathematics throughout the series. Applications are included in single activities in each course and span several lessons and topics throughout a module. Performance task assessments corresponding to each topic often engage students in an application of mathematics in a real-world context.

Examples where students engage in the application of mathematics throughout the series include:

• N-Q.2: In MATHia, Math 1, Searching for Patterns, Identifying Quantities Workspace, students watch an animation of a skier and decide on appropriate quantities to model the scenario.
• A-CED.1: In Math 1, Module 2, Topic 2, Activity 3.1, students write a linear equation to represent total sales as a function of the number of boxes of popcorn sold for a fundraiser. In Topic 2, Lesson 3 assignment, students write a linear equation to represent the cost of a trip as a function of the number of gallons of gas for the trip. In Topic 2, Lesson 4 assignment, students write an inequality to represent the cost to produce t-shirts in a month making a profit of at least $2,000 but no more than $10,000.
• F-IF.4: In Math 2, Module 3, Topic 1, Activity 3.2, students interpret key features of a graph of a piecewise function that models the percent of charge remaining on a cell phone battery over time. Students write a possible scenario that models the graph and explain what the slope, x-intercepts(s), and y-intercept represent in terms of the problem context. Students also write a scenario to model their own cell phone use during a typical day, graph the scenario, and determine the equation of the piecewise function.
• G-SRT.8: In the Math 2, Module 2, Topic 2, Performance Task, students use trigonometric ratios to compare the height of two drones, determine the distance between the two people controlling the drones, and calculate the angle of elevation from one person to their flying drone.
• S-ID.4: In Math 3, Module 5, Topic 1, Getting Started and Activity 3.1, students recognize that a data set representing the fuel efficiency for a sample of hybrid cars is normally distributed. Students use the Empirical Rule and a table to estimate areas under the normal curve.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the program materials. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding. Each topic includes: activities and assignments that develop students’ conceptual understanding and procedural skills in the student materials, skills practice worksheet that allows students additional practice to develop procedural skills, and performance tasks that assess students’ conceptual understanding and/or procedural skills often times in the context of a real-world scenario. 

Examples of where multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In Math 3, Module 2, Topic 1, Activity 1.2, students use chunking as a method to factor quadratics that have common factors in some of the terms, but not all the terms. Students use their procedural knowledge of chunking to identify and factor perfect square trinomials. In Module 2, Topic 1, Activity 2.2, students use their procedural knowledge of chunking to rewrite the difference of two cubes and build their conceptual understanding when determining the formula for factoring the difference of two cubes. 
• In Math 2, Module 4, Topic 2, Performance Task, students examine a table relating ticket prices and number of tickets sold for a baseball game, and students determine the linear regression equation that models the data. Students add a column to the table for the total amount of money earned from ticket sales and determine the quadratic equation that models the data. Students identify key features of the two graphs, describe these key features in the context of the problem, and explain why the data sets are modeled by different functions.
• In Math 1, Module 1, Topic 2, Performance Task, the materials provide two scenarios (written description and diagram) for how to track the growth of a tree. Students recognize that one scenario represents an arithmetic sequence, whereas the other scenario represents a geometric sequence. Students represent the two sequences using a table, graph, and equation. Students’ conceptual understanding of the differences between arithmetic and geometric sequences and their procedural knowledge of how to write recursive and explicit formulas are assessed in this performance task.
• In Math 3, Module 1, Topic 2, Activity 5.2, the materials provide a context involving cylindrical planters for city sidewalks and storefronts that come in a variety of sizes with specific height and radius requirements. Students generate a cubic function to model the volume $$V(x)=(\pi x^2)(2x)$$ from the base area function $$A(x) = \pi x^2$$ and the height function $$h(x) = 2x$$.

Examples where the instructional materials attend to conceptual understanding and procedural skills independently include:

• In Math 2, Module 3, Topic 2, Activity 1.4, students develop their procedural skills in rewriting expressions involving radicals and rational exponents using the properties of exponents.
• In Math 2, Module 4, Topic 2, Activity 3.3, Talk the Talk, students develop their conceptual understanding of systems of equations when considering the number of possible solutions for a system of equations consisting of a linear equation and a quadratic equation and two quadratic equations.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. MP1 and MP6 are used to enrich the mathematical content and demonstrate the full intent of these mathematical practices across the series.

The materials state that MP1 is “evident every day in every lesson” and is not explicitly identified in either the teacher or student materials. The materials provide students opportunities to explain the meaning of problems, look for entry points when problem solving, plan solution pathways, and make conjectures about the form and meaning of a solution.

Examples of where and how the materials use MP1 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 3, Topic 2, Activity 3.2, students choose an appropriate function to model data of carbon dioxide concentration in the Earth’s atmosphere over time. Students consider what information would help them make a decision as to whether a linear or exponential function is best to model this context and data.
• In Math 2, Module 3, Topic 1, Activity 3.1, students develop a piecewise function from a scenario of pizza sales during the day. Students determine which piece should be used to determine the y-intercept of the function. 
• In Math 3, Module 1, Topic 2, Activity 5.1, students identify what x-intercepts represent and whether their values make sense within the context of a volume formula relating height, length, and width of a planter box.

Materials use a target icon to explicitly identify MP6 in teacher and student materials. Examples of where and how the materials use MP6 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 4, Activity 1.2, students classify triangles on the coordinate plane as acute, right, or obtuse, and scalene, isosceles, or equilateral by calculating distances and slopes.
• In Math 2, Module 1, Topic 1, Activity 1.1, students construct a circle, a diameter of a circle, and perpendicular bisector, and identify radii, arcs, central angles, chords, and secants using definitions of each geometric term.
• In Math 3, Module 1, Topic 1, Activity 5.3, students use the quadratic formula to determine how long it takes for a t-shirt to land on the ground after being launched and consider whether an exact solution or approximate solution is more appropriate for the context.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. MP2 and MP3 are used to enrich the mathematical content, and the materials demonstrate the full intent of these mathematical practices across the series. The materials use a puzzle piece icon to explicitly identify MP2 and MP3 in teacher and student materials. 

Examples of where and how the materials use MP2 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 1, Activity 5.3, students identify the quantities represented in an equation and table and determine which representation is converting Farenheit to Celsius and which representation is converting Celsius to Farenheit. Students compare the slope and y-intercept for each function within the context of the problem.
• In Math 2, Module 3, Topic 1, Activity 5.1, students use a table to show the conversion between the U.S. dollar and the Turkish lira. Students convert Turkish lira to the U.S. dollar and consider how the quantities for this conversion relate to the original conversion as an introduction to inverses.
• In Math 3, Module 3, Topic 3, Activity 5.2, students examine how many social media followers a quarterback and running back for a professional football team have. Students decontextualize the situation in order to think about the functions that represent the situation, and students contextualize the situation in order to interpret both equations to answer follow-up questions.

Examples of where and how the materials use MP3 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 1, Topic 1, Lesson 2, Getting Started, students analyze 19 graphs, sort them into at least two different groups, and provide a rationale for how their groups were created. In Activity 2.1, students consider four different groupings students created and determine the “rule” used to create their groupings, justify why a grouping is correct based on a provided rationale, or justify why a grouping is not correct based on a provided rationale. 
• In Math 2, Module 1, Topic 2, Activity 2.3, students analyze two students’ proof plans and determine which proof plan is correct.
• In Math 3, Module 5, Topic 2, Activity 2.1, students explore different types of biased samples as they consider who is correct in identifying a sampling procedure as a convenience sample or a subjective sample. Students also consider a student response regarding biased samples and explain why the student’s statement is correct based on their knowledge of sampling definitions.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. MP4 and MP5 are used to enrich the mathematical content, and the materials demonstrate the full intent of these mathematical practices across the series. Materials use a wrench icon to explicitly identify MP4 and MP5 in teacher and student materials. 

Examples of where and how the materials use MP4 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 3, Lesson 5 Assignment, students use information about how much a baker makes when she sells decorated cookies and cupcakes, her minimum profit goal, and the maximum hours she’d like to work to create a system of linear inequalities that model the constraints. Students modify their existing constraints to account for running out of supplies.
• In Math 2, Module 3, Topic 2, Activity 4.1, students use written descriptions of two methods of saving money. Students write a function to model each situation and add the functions. Students graph all three functions and make a connection to what students have previously learned about transformed functions.
• In Math 3, Module 4, Topic 1, Activity 2.1, students model the height of a rider on a Ferris wheel with a periodic function. Students create a graph and a table to represent the height of a rider above the ground as a function of the number of rotations of the Ferris wheel. 

Examples of where and how MP5 is used to enrich the mathematical content and demonstrate the intentional development of the full intent of MP5 across the series include:

• In Math 1, Module 1, Topic 3, Activity 2.2, students use technology to construct a scatter plot, determine the linear regression equation, and compute the correlation coefficient for a data set.
• In Math 2, Module 1, Topic 1, Lesson 4, students use constructions to determine the appropriate place for building an information kiosk at a zoo. Students choose from dynamic software, a compass, or other appropriate tools. Students are not told which tool to use and are expected to choose based on availability and/or appropriateness.
• In Math 3, Module 5, Topic 1, Activity 3.1, students use technology to find z-scores and percentiles for situations that can be modeled by normal distributions.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. MP7 and MP8 are used to enrich the mathematical content, and the materials demonstrate the full intent of these MPs across the series. The materials use a box icon to explicitly identify MP7 and MP8 in teacher and student materials.

Examples of where and how the materials use MP7 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 1, Topic 2, Lesson 2, Getting Started, students consider several sequences on cut out cards, determine the unknown terms of each sequence, and describe the pattern of each sequence. Students use the patterns they observed to group the sequences and provide a rationale as to why they created each group. Students re-group the sequences as arithmetic, geometric, or neither.
• In Math 2, Module 3, Topic 3, Activity 2.1, students use a table to calculate first and second differences for two linear equations and two quadratic equations and graph each equation. Students notice patterns to discern a relationship between the first differences for a linear function and whether the graph is increasing or decreasing as well as a relationship between the second differences for a quadratic function and whether the parabola opens up or down. 
• In Math 3, Module 4, Topic 1, Activity 4.2, students explore patterns in the unit circle coordinates and use their knowledge of unit circle coordinates in the first quadrant and symmetry to label the coordinates in quadrants II, III, and IV.

Examples of where and how the materials use MP8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 1, Activity 1.4, students express regularity in repeated reasoning to verify that constant differences in arithmetic sequences written explicitly are equivalent to the slope of the arithmetic sequence written in function notation of f(x) = ax + b.
• In Math 2, Module 3, Topic 2, Lesson 2, Talk the Talk, students express regularity in repeated reasoning by differentiating exponential growth and decay when identifying equations that are appropriate exponential models to represent a growing population.
• In Math 3, Module 2, Topic 3, Activity 2.1, students express regularity in repeated reasoning to sketch transformed rational functions based on the general form of transformed functions.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the underlying design of the materials distinguishes between problems and exercises. 

Each course in the series is divided into five modules and each module is divided into topics. Students engage in several lessons and activities related to that topic. The structure of each lesson is consistent throughout all courses: Students engage in Warm Ups, Getting Started tasks to activate prior knowledge, a variety of activities that allow students to develop their understanding of mathematical concepts, Talk the Talk tasks where students can reflect on and communicate their understanding of mathematical concepts they learned in the activities, and an assignment. Each assignment consists of five sections: Write, Remember, Practice, Stretch, and Review. 

MATHia provides additional personalized exercises to support student understanding of concepts. Also, Skills Practice workbooks provide exercises for students to reinforce their learning and build mastery.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the design of assignments is not haphazard and exercises are given in intentional sequences.

Exercises within student assignments are intentionally sequenced to build understanding and knowledge. Assignments consist of five sections:

• Write: Students reflect on terms and concepts from the lesson,
• Remember: Key terms and concepts from the lesson are identified for the student,
• Practice: Students use the key concepts learned in the lesson to solve problems,
• Stretch: Students are pushed to think beyond the skills addressed in the lesson activities to solve problems, and
• Review: Students reinforce conceptual understanding and fluency of skills from previous lessons and topics.

Each section of the assignment reinforces student understanding of concepts from the corresponding lesson, provides practice to support student understanding of concepts from previous lessons, or assists students in making connections among concepts from several lessons.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that there is a variety in how students are asked to present the mathematics. 

Throughout lesson activities and assignments and in MATHia workspaces, students create and present mathematics in a variety of ways, such as writing and graphing equations and functions, creating and interpreting tables, analyzing and interpreting data sets, constructing geometric models, and justifying their reasoning and critiquing the reasoning of others. Examples include:

• In Math 1, Module 1, MATHia, Introduction to Function Families Workspace, students interpret a graph and equation modeling exponential growth of a bank account earning compound interest to complete a table of outputs, identify the maximum and/or minimum of the function, explain why certain points don’t make sense given the context provided, and identify the asymptote.
• In Math 1, Module 2, Topic 3, Activity 1.3, students write a system of linear equations to represent the amount of money Marcus and Phillip save each week, use substitution to solve the system of linear equations algebraically, and graph the system to verify their solution.
• In Math 2, Module 3, Topic 1, Lesson 4, Talk the Talk, students write a piecewise function to model a given scenario, select one of two graphs that best represent the situation, and explain their reasoning.
• In Math 3, Module 3, Topic 1, Activity 4.4, students compare two correct solution methods for multiplying radicals and explain the difference in their methods and critique the reasoning of an incorrect solution method for dividing radicals. Students also consider four solution methods for adding and subtracting radical terms and justify the correct methods and critique the incorrect methods.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Manipulatives, such as graphing technology, grid paper, patty paper, rulers, compasses, and straightedges, are embedded in activities throughout the course. Examples include:

• In Math 1, Module 3, Topic 2, Activity 3.2, students use technology to choose the correct function (linear or exponential) and write a regression equation to model a data set.
• In Math 2, Module 1, students use patty paper, compass, and straightedge to make geometric constructions.
• In Math 3, Module 1, Topic 2, Activity 1.1, students draw a rectangle, disc, and isosceles triangle on an index card, cut out the shape and tape it along the center to a pencil, and then spin the pencil to rotate the shape to determine what three-dimensional solid is formed by spinning the two-dimensional shape.

MATHia workspaces incorporate animations, explore tools, classification tools, and problem solving tools (graphing tools and virtual manipulatives).

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each lesson contains Getting Started, Activities, and Talk the Talk sections. Facilitation Notes are provided for each of these sections and contain a list of suggested questions, labeled Questions to Ask, to move student learning forward.

Also, the materials support teachers in planning and providing effective learning experiences by providing module, topic, and lesson overviews that contain guidance for ways in which students can demonstrate understanding and probing questions to be used during instruction.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for containing a teacher’s edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. The introductory pages of the first volume of the Teacher’s Implementation Guide for each course provide detailed information regarding the instructional design of the series, lesson structure, assignment structure, problem types students will work with, thought bubbles to promote student self-reflection, mathematical habits of mind, the academic glossary, the modeling process, and assessments. Detailed information regarding content alignment within a given course complement a general overview of standards addressed within that course.

The Facilitation Notes embedded within every lesson provide ample and useful annotations and suggestions regarding an overview of the mathematical concepts addressed, standards addressed, essential ideas, pacing, what to look for from students, Questions to Ask, grouping strategies, common student misconceptions, differentiation strategies, and a summary statement of the mathematical ideas addressed.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. The introductory pages of the first volume of the Teacher’s Implementation Guide for each course provide guidance for using the MATHia software and collecting data reports generated by MATHia. The table of contents explicitly identifies certain MATHia workspaces that correlate to specific modules in the given course.

The instructional materials for Carnegie Learning Math Solutions Integrated series  meet expectations for containing a teacher’s edition that contains full, adult-level explanation and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.

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

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for containing a teacher’s edition that explains the role of the specific mathematics standards in the context of the overall series. For each course, the Module Overview provides information on how the mathematical ideas from that module connect to prior learning and future learning. The Topic Overview provides the entry point or prior experience with the mathematical ideas for students, as well as how the activities in that topic promote student expertise in the mathematical practice standards. 

Additionally, each Teacher’s Implementation Guide includes a Course Module Overview table that provides an overview of the mathematical ideas of each module, how the lessons within that module connect to students’ prior learning, and how students use the mathematical ideas from that module in later modules within the course or series. For example, in the Integrated Math 1 Module Overview for Module 3: Investigating Growth and Decay states:

• Connections to Prior Learning: “The lessons in this module build on students’ experience with geometric sequences and common ratios. They have learned the effects of dilations and horizontal and vertical translations functions. Students have also used horizontal lines to solve linear functions graphically.”
• Module Overview: “Students learn that certain geometric sequences can be defined as exponential functions. They transform exponential functions and distinguish between growth and decay scenarios. Students solve exponential equations algebraically using common bases or graphically using a horizontal line.” 
• Connections to Future Learning: “Exponential functions are the first nonlinear functions that students have studied in depth. Throughout this module students compare and contrast linear and exponential functions, an important skill as students encounter more complicated function types in high school mathematics.”

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing strategies for gathering information about students’ prior knowledge within and across courses. There is a pre-test for every topic in each module that addresses standards that will be taught. The post-test for the topic is similar to the pre-test, although not the same questions. The Topic Overview provides a list of Prerequisite Skills needed for the topic, which creates an indirect opportunity for teachers to gather information about students’ prior knowledge, although there is no direct guidance provided to the teacher about how to use the information. MATHia software is used as an assessment and progress monitoring tool. In every assignment in the textbook, there is a Review section. Students practice two questions from the previous lesson, two questions from the previous topic, and two questions that address the fluency standards outlined in the standards.  In the Module Overview, there is a connection to student’s prior learning. This explains to the teacher what students should know or be able to do based on previous learning.

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

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing support for teachers to identify and address common student errors and misconceptions. Materials highlight common student errors and/or misconceptions for teachers in the lesson Facilitation Notes in the Teacher’s Implementation Guide for each course. The materials also provide strategies to teachers that are mathematically sound for addressing common student errors and/or misconceptions in the lesson Facilitation Notes in the Teacher’s Implementation Guide. Examples include:

• In Math 1, Module 2, Topic 1, Activity 3.3, Facilitation Notes, the materials identify the following misconception when students vertically dilate a linear function: “Students may look at a graph that is vertically stretched and see it as horizontally compressed. Address the fact that without more information, the graph could be described both ways; however, the function provides additional information. The A-value deals with vertical movement. Therefore, the y-values are affected by the transformation, while the x-values stay constant. Have students take notes with stretches for a graph that is vertically stretched and a graph that is vertically compressed. Horizontal transformations will be addressed with exponential and quadratic functions.”
• In Math 3, Module 4, Topic 1, Activity 4.5, Facilitation Notes, the materials identify the following misconception when students rewrite a quadratic in vertex form to solve problems: “Students may think that because they used one method to start on the solution path that they cannot use a different method at any point. For example, when a quadratic equation is written in general form, the zeros may be easiest to identify using graphing technology. Encourage students to try different solution paths.”
• In Math 3, Module 1, Topic 3, Activity 1.3, Facilitation Notes, the materials identify the following misconception when students investigate characteristics of even and odd functions, “Students may confuse even functions with even-degree polynomials and odd functions with odd-degree polynomials.” Materials suggest asking the following questions to help students: “What is the difference between an even function and an even-degree polynomial? What is an example of an even function that is not an even-degree polynomial? What is the difference between an odd function and an odd-degree polynomial? What is an example of an odd function that is not an odd-degree polynomial?”

Additionally, the MATHia software identifies and addresses common student errors and misconceptions. MATHia re-directs a student and provides customizable hints to address individual student difficulties.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide several opportunities for ongoing review and practice for students in learning both concepts and skills:

• Warm Ups and Getting Started tasks provide review of skills from previous lessons.
• Talk the Talk includes a few questions related to the learning students engaged in during the lesson.
• The Assignment includes a Review of several exercises that provide ongoing practice of previous concepts and skills for students.
• The MATHia software or, if technology is not accessible, the Skills Practice workbooks provide ongoing practice and review opportunities for students. The MATHia software continually provides feedback for correct and incorrect answers, and the Skills Practice workbooks include answers to the odd number exercises for students to check their progress. 
• The MATHia software includes “Hints” that students can select when reviewing and practicing concepts and skills. Just-in-Time Hints automatically appear when a student makes a common error and On-Demand Hints are provided when a student asks for a hint while working on a problem. Step-by-Step Hints demonstrate how to use the tools in a lesson by providing step-by-step guidance through a sample problem.

The instructional materials for Carnegie Learning Math Solutions Integrated series meet expectations for assessments clearly denoting which standards are being emphasized on assessments. The series provides five types of assessment for every topic: Pre-test, Post-test, End of Topic Test, Standardized Practice Test, and Performance Task. The Performance Task clearly denotes which standards are being assessed. The other assessments address the set of standards included in each topic (as outlined in the Teacher’s Implementation Guide). Materials describe which standard aligns with which problem in the Assessment Overview. Assessments do not provide standards on student-facing materials. The Edulastic Assessment Suite provides standards for each assessment, and MATHia also provides students with standards-identified content. 

The instructional materials for Carnegie Learning Math Solutions Integrated series meet expectations for providing sufficient guidance to teachers for interpreting student performance on assessments and providing suggestions for follow-up. Materials include some guidance for teachers to interpret student performance. Answer keys are provided for all assessments. Performance Tasks include a detailed scoring rubric for teachers to use when interpreting student performance; however, no other assessment provides guidance for teachers about scoring student performance. MATHia reports provide teachers with detailed information about student performance in relation to progress on standards and suggestions on the skills that require additional support. The materials also offer teachers an APSLE (Adaptive Personalized Learning Score) report which is a predictor for year-end summative assessments. Videos within MyPL explain this report in more detail while outlining the research and models behind the report.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners. The materials provide a Content at a Glance pacing guide for each course as well as a more detailed pacing guide for lessons and activities within each module in the Teacher’s Implementation Guide. Each module overview in the Teacher’s Implementation Guide identifies how the content in that module is connected to students’ prior learning and future learning. All lessons include instructional notes and classroom strategies that provide teachers with sample questions, differentiation strategies, common student misconceptions, what to look for from students, and summary points. Each of these components provide structure for the teacher to help scaffold lessons so that the content is accessible to all learners.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing teachers with strategies for meeting the needs of a range of learners. MATHia, an adaptive online component of the series, differentiates learning for every learner. The level of support is individualized to each student as the program adapts to student answers. 

The Teacher’s Implementation Guide provides differentiation strategies, Questions to Ask, and common misconceptions for most lessons. Most Questions to Ask are intended for all learners, while the differentiation strategies provided are specifically intended for all students, students who struggle, or students who are ready to extend their learning. For example, in Math 1, Module 2, Topic 1, Activity 5.2, the Teacher’s Implementation Guide lists the following differentiation strategies in the Facilitation Notes: 

• “To support students who struggle, have them make a table for the graph so that they have the same representations to compare.”
• “To extend the activity, ask students to compare the two recycling centers using equations.”

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. In the Instructional Design section of the Teacher’s Implementation Guides, the materials state, “Carnegie Learning recognizes the importance of connecting multiple representations of mathematical concepts. Lessons present content visually, algebraically, numerically, and verbally.”

Each Topic Overview in the Teacher’s Implementation Guide includes a section titled: “What is the entry point for students?” The Topic Overview for Math 2, Module 4, Topic 1 states, “Importantly, students have extensive experience locating solutions to equations using a graphical representation. Connecting solutions to graphs helps students to understand why quadratic equations have two solutions that are symmetric about the line of symmetry. In middle school, they used the Properties of Equality and square roots when applying the Pythagorean Theorem and when solving for a side length given the area of a figure. This serves as the launching point for solving more complicated quadratic equations.”

Performance tasks often allow for students to solve a problem using a variety of solution strategies or representations. For example, in Math 2, Module 1, Topic 2, Performance Task, students use two methods to determine the distance between Earth and a satellite in orbit. Lesson activities provide limited opportunities for students to develop their own solution path, as specific strategies are often provided for students.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing support, accommodations, and modifications for English Language Learners (ELL) and other special populations that will support their regular and active participation in learning mathematics. 

ELL Tips are regularly incorporated in lessons in the Teacher’s Implementation Guide. Most of the ELL Tips address explicitly teaching vocabulary that may pose challenges and impede problem solving for students. For example, in Math 1, Module 1, Topic 3, Activity 2.1, materials offer the tip, “Help students connect the meaning of correlation with its root word relate. Then, connect the terms positive correlation and negative correlation with the slope of the line, and no correlation with no relationship.” A limited number of ELL Tips address other strategies to help ELL students, such as honoring the use of students’ native languages, building relevant background, simplifying sentences, and modifying vocabulary.

Suggestions for supporting other special populations (i.e., struggling students and advanced students) are included in the Teacher’s Implementation Guide in the Facilitation Notes for each activity. Some of these suggestions for other special populations are general, and some are specific to the content of the lessons in which they are found.