8th Grade - Gateway 3

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

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

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.

LLessons follow a consistent format that intentionally sequences assignments:

• “Warm Up” - exercises that activate students’ prior knowledge.
• “Getting Started/Engage” - students solve/think/share and notice other’s work/thinking, usually for a non-routine problem.
• “Develop/Activities” - new learning takes place; students explore 5-10 problems that engage them with examples and explanations of the targeted skill. These are typically problems to solve together as a class with instructor guidance. Each Activity includes verbiage describing how the new knowledge relates to previous understanding.
• “Demonstrate/Talk the Talk” - students reflect on and connect what was learned.
• “Assignment” - five sections that review the lesson: Write - reviewing rules or vocabulary, Remember - summary of one to two key points, Practice - problems related to the activities, Stretch - an extension, and Review - looping in previous skills.

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

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

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that there is a variety in what students are asked to produce. Students are asked to produce a variety of products in digital and written form.

Some of these products include:

• Multiple representations through writing equations and expressions, drawing models, creating arrays, drawing and placing numbers on number lines, etc.
• In Module 2, Topic 3, Lesson 5 Talk the Talk, students analyze a table of values, create a situation that represents the data in the table, create a new equation with a slope that is less steep, compare the slopes, then explain how they created their linear function and compared the slopes.
• Justification of their thinking and others', critiquing others’ work, explaining why answers given are correct.
• Writing, reviewing, practicing, and stretching activities in each assignment given at the end of each lesson, such as creating a Venn diagram showing the relationship between the six sets of numbers shown. (Module 5, Topic 1, Lesson 3 Talk the Talk)

Finally, each module includes a real-world connection where students produce solutions in a variety of ways to demonstrate their knowledge, such as listening to classmates, then clearing up misunderstanding about concepts or proving the Pythagorean Theorem.

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that manipulatives are faithful representations of the mathematical objects they represent and are appropriately connected to written methods.

Manipulatives are embedded in activities and the MATHia Independent Digital Lessons. Number lines, patty paper, equivalency cards, etc. are used throughout the year in connection to the mathematics being presented and are faithful representations. For example, students work with a city map of Washington, DC to identify and explain transversal lines and the angles formed. (Module 1, Topic 3, Lesson 2)

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that the materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

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

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

For example:

• “How does the length of side AB compare to the length of side A’B’?”
• “How does angle A compare to angle A’?”
• “Are all of the corresponding sides congruent?”
• “Are all of the corresponding angles congruent?”
• “How does the x-coordinate for point A compare to the x-coordinate for point A’? Is this true for all of the x-coordinates of the corresponding vertices?”

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that the materials contain a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning such as how to use and read data in the MATHia software.

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

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

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation for containing a teacher edition (in print or clearly distinguished/accessible as a teacher edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematical concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Within MyPL, teachers can view instructional videos that provide adult-level explanations and examples for teachers to enhance their own knowledge of the content. The instructional videos address textbook lessons, MATHia, mathematical content, and classroom strategies. For example, in the video, The Right Triangle Connection - the Pythagorean Theorem (Course 3, Module 4, Topic 2, Lesson 1), teachers view suggestions for implementing the lesson. Course 3 contains 54 lesson videos. MyPL also includes 33 videos addressing mathematical content that are not lesson-specific, and the advanced mathematics concepts addressed by the videos include, but are not limited to: ellipses, hyperbolas, and discontinuities and asymptotes of rational functions. The Teacher’s Implementation Guide for each course provides detailed information regarding how mathematical content fits into the series overall, and the materials include module overviews that describe the mathematics of the module and how the content is connected to prior and future learning.

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that materials contain a teacher edition (in print or clearly distinguished/accessible as a teacher edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for Kindergarten through Grade 12.

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

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

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

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

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that materials provide strategies for teachers to identify and address common student errors and misconceptions.

• In the Topic Guide, lessons regularly have a section titled “Misconceptions” with suggestions for teachers to identify and address common student errors and misconceptions.
• Example: “Students sometimes believe that any point (x, y) means that both the values of x and y are positive, and the coordinate pair always lies in Quadrant I. The introductory sentence and questions that follow may somehow validate that misconception. If that is the case, complete a second question as practice, placing (x, y) in a different quadrant and answering Question 2 again, referring to the other quadrants.” (M1-39E)
• Teachers are encouraged to engage students in mathematical conversations to address student errors and misconceptions with phrases such as, “Remind the students…, Discuss with students…, Point out that….”
• MATHia software provides a solution pathway to common student misconceptions. “Like a human tutor, MATHia re-phrases questions, re-directs the student, and hones in on the parts of the problem that are proving difficult for the student. Hints are customized to address the individual student, understanding that there are often multiple ways to do the math correctly.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The materials provide several opportunities for ongoing review and practice:

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

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet the expectation for assessments clearly denoting which standards are being emphasized. The series offers several types of assessments, print and digital:

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

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. Materials include some guidance for teachers to interpret student performance. Answer keys are provided for all assessments. Performance Tasks include a detailed scoring rubric for teachers to use when interpreting student performance; however, no other assessment provides guidance for teachers about scoring student performance. MATHia reports provide teachers with detailed information about student performance in relation to progress on standards and suggestions on the skills that require additional support. Teachers can monitor students working in MATHia and view in-the-moment guidance that indicates to teachers which students need additional support. The materials also offer teachers an APSLE (Adaptive Personalized Learning Score) report which is a predictor for year-end summative assessments. Videos within MyPL explain this report in more detail while outlining the research and models behind the report.

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

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

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

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

A primary strategy for meeting the needs of all learners in this program is MATHia software. MATHia differentiates the learning experience for every learner, adapting the amount of support based on the students answers and path through each problem. This level of support is similar to a one-on-one tutored experience, where the software is adapting based on everything the student is doing.

Most lessons provide “Differentiation strategies,” “Questions to ask,” and a “Misconception” section. Most of the suggestions and the questions included in the “Questions to ask” section are intended for all students rather than geared toward helping students who struggle or challenging students ready to go deeper. For example, in Module 4, Topic 1, Lesson 2.1, Questions to ask: “Are natural numbers and counting numbers the same set of numbers? What is an example of a situation that would require you to use the set of natural numbers? What is an additive identity? How is the set of natural numbers different from the set of whole numbers? How is the set of natural numbers similar to the set of whole numbers? What is an example of a situation that would require you to use the set of whole numbers?”

However, in the “Differentiation strategies” section, suggestions are limited but more specific. For example, in Module 4, Topic 2, Lesson 4.2: “For students who struggle, have them concentrate on the structure of each problem first: Circle - Rectangle and Triangle + Semicircle. Provide a list of formulas for reference.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

• Each Topic Overview includes a section called, “What is the entry point for students?” Introduction to Functions Overview states: “Throughout elementary school, students described and explained features of patterns (e.g., 4.OA.5). They have also formed ordered pairs with terms of two sequences and compared the terms (5.OA.3). Therefore, sequences are used as the entry point for Introduction to Functions. Students are introduced to term, a key vocabulary word that will later be used to connect sequences to functions. They analyze a variety of sequences, write the sequences, predict next terms, and describe the patterns. This may be a departure from prior experiences when students were given the pattern and determined the terms. Students then compare the types of patterns in the sequences analyzed, searching for similarities in the pattern descriptions. Later, students will connect the term numbers and term values as the inputs and outputs, respectively, of a function.”
• Some application tasks, particularly the Performance Task, allow for multiple solution strategies or representations. For example, in Module 1, Topic 1, Lesson 6.3, Talk the Talk 2, students describe a transformation created from a sequence of at least two transformations and use coordinates to justify the answer.
• Some assessment questions allow for multiple entry points. For example, in Module 1, Topic 1, Post-test question 9, students explain how to determine the coordinates of the vertices of the image without using a graph.
• Lesson activities provide limited opportunities for students to create their own solution paths since strategies are often provided.

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

ELL Tips are specifically cited throughout the materials. For example, “ELL Tip: Instruct students to read and take notes on congruent line segments and congruent angles. Provide students with a concept map to support this stage in their note-taking.” (M1-86)

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

Some suggestions are specific to the lesson, but don’t necessarily further knowledge, such as in Module 2, Topic 4, Lesson 1.1: “Differentiation strategy: In order to have all students participate, rotate which students participate in the different-sized human chains.”

There are differentiation suggestions that do not include a rationale as to how they would provide support. For example, in Module 3, Topic 2, Lesson 1.3: “Provide students with a set of possible answers so they can check their work as they solve the equations.” It is unclear how this suggestion would support a student’s understanding - if the answer they calculate isn’t given, they still won’t know where they went wrong. In addition, it creates an opportunity for “guess and check” substitution rather than solving the equation.

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

• “Confusion may arise in Question 2 because the vertices to the images are not labeled. To support students who struggle, suggest they use visual tactics by looking at general side lengths and angle measures to determine corresponding angles and/or insert labels for the vertices of the images.” (M1-53F)
• “To extend the activity, discuss the meaning of these phrases: a scale factor of two, an image scaled to 200 percent, and a 100 percent increase.” (M1-109H)
• “Students may have a hard time visualizing y=x since it does not contain any numbers. Have students rewrite the equation as y=1x+0 and plot points on the line provided to make sense of the equation." (M2-53D)
• “As an alternative to using pasta as a movable line, students could draw a line on a sheet of patty paper and use the drawn line as their movable line.” (M2-289C)
• “Offer multiplying by -1 as an alternative to Sandy’s method.” (M3-7E)
• “To support students who struggle, draw right triangles with different orientations and have students practice labeling the legs and hypotenuse. Have them draw an arrow from the right angle to the hypotenuse before labeling each triangle.” (M4-55D)
• “To support students who struggle, model for students how to deal with larger exponents when they arise by writing a parallel problem using smaller numbers to create the rule.” (M5-7I).
• “To assist students struggling to visualize the cross-sections of the different cylinders, use clay and wire to create the cross-sections.” (M5-85J)

The instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for providing a balanced portrayal of various demographic and personal characteristics.

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