High School - Gateway 3

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.