High School - Gateway 3

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

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

The Front Matter section includes:

• Guiding Principles of Carnegie Learning.

• Content Organization including Modules, Topics, and Pacing.

• Course standards overview chart.

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

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

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

• Guidance for implementing MATHia.

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

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

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

Evidence for materials including sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives to engage and guide student mathematical learning include: 

• In Math 1, Module 4, Topic 1, Lesson 1, Getting Started, Teacher’s Implementation Guide, the materials include annotations with suggestions on how to chunk the activity. The “Chunking the Activity” suggests the following: “ Read and discuss the introduction and situation, Group students to complete 1, Check-in and share, Group students to complete 2 and 3, and Share and summarize.” 

• In Math 2, Module 5, Topic 1, Lesson 1, Activity 3, Teacher’s Implementation Guide, students are introduced to the terms disjoint sets, intersecting sets, and union of sets. The materials include a Language Link to support English Language Learners. The Language Link defines the prefix of the terms to better understand the definition in mathematical context. For example, “The prefix dis- means to take the opposite. Disjoint means not joined.”

• In Math 3, Module 1, Topic 3, Lesson 4, Activity 3, Teacher’s Implementation Guide, students choose a set of functions whose product builds a quartic function with two imaginary zeros and a double zero. The materials provide differentiation strategies to support students who struggle and to challenge advanced learners to extend. For advanced learners the materials suggest to “challenge students to choose sets of functions whose product builds a quintic function with varying numbers of real and imaginary zeros.”

The materials reviewed for Carnegie Learning High School Math Solution Integrated meet expectations for containing adult-level explanations and examples of the more complex course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The materials provide an overview at the beginning of each module and for each topic within the module. The Module Overview provides an explanation for the naming of the module, research for why the module is included as part of the scope and sequence, connections to prior learning, and connections to future learning. The Topic Overview provides an explanation of how key topics are developed, an examination of the entry point for students to connect to prior learning, and identification of the importance of the topic for future learning. 

Examples of how the materials support teachers to develop their own knowledge of more complex, course-level concepts include:

• In Math 1, Module 3, Topic 1, Lesson 3, Teacher’s Implementation Guide, the materials provide an adult-level explanation of how the graph of a function changed horizontally and vertically based on a constant being added or subtracted inside or outside a function. The materials state, “You can describe transformations performed on any function f(x) using the transformation function g(x)=Af(B(x-C)+D where the D-value translates the function f(x)  vertically, the C-value translates f(x)  horizontally, the A-value vertically stretches or compresses f(x) .”

• In Math 2, Module 3, Topic 1, Topic Overview, the materials provide an adult- level explanation of how the key concepts of functions derived from linear relationships are developed. The materials explain the connection between solving and graphing absolute value equations and inequalities, analyzing linear piecewise and step functions, and deriving inverses of linear functions. Additionally, it explains to the teacher what students will be doing with each concept.

• In Math 3, Module 5, Module 5, Teacher’s Implementation Guide, Overview, the materials answer the question “When will students use knowledge from Relating Data and Decisions in future learning?”, it then goes on to talk about the collegiate paths that will require statistics. Additionally, it addresses how the module “provides students with the tools to collect, analyze, and draw conclusions from data.”

Examples of how the materials support teachers to develop their own knowledge beyond the current course: 

• In Math 1, Module 4, Module Overview, Connection to Future Learning, the material’s state, “This module supports future learning by deepening students’ proficiency with the statistical process, data displays, and numeric summaries of data. In later courses, students will use the mean and standard deviation of a data set to fit a normal distribution. They will use a normal distribution to estimate population percentages. Advanced studies in statistics will teach students methods to determine whether an observed relationship between two variables is statistically significant. Because statistics lies at the heart of important advances in the physical, economic, and political sciences, students will encounter the skills learned in this module in many higher education fields.” The materials then provide a Math Representation illustrating how a normal distribution can be drawn given a sample mean and standard deviation.

• In Math 2, Module 2, Topic 2, Topic Overview, Connection to Future Learning, the materials state, “Trigonometry provides a bridge between geometry and algebra. Understanding the trigonometric ratios in terms of side length ratios prepares students to study trigonometric functions in the next course. They will use right triangles to build the unit circle. Unrolling the unit circle leads students to an understanding how these ratios form the basis for periodic functions. Trigonometry has applications across the STEM fields: in computer science, where angles are used to design computer programs; in physics, where a periodic function models a pendulum swing; in aviation, where angles of elevation and depression help to plot paths for aircraft; and in the engineering of bridges.” The materials then provide a Math Representation illustrating how an angle measure used as an input results in a real number output corresponding to coordinate points on the unit circle. 

• In Math 3, Module 4, Module Overview, Connection to Future Learning, the materials state, “Students in advanced courses will use trigonometric functions to model real-world scenarios involving circular motion. They will use radian measures extensively, which will lead to simple formulas for derivatives and integrals of periodic functions. Students who pursue post-secondary mathematics courses will use the periodic functions to determine the velocity and acceleration of objects in motion and learn that trigonometric functions are the building blocks for modeling any periodic phenomenon.” The materials then provide a Math Representation displaying how to highlight certain features of a sound wave by decomposing a periodic graph.

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Examples of materials providing correlation information for the mathematics standards addressed throughout the series include:

• Each Module within the courses contains a Module Teacher’s Implementation Guide Overview. The Module Teacher’s Implementation Guide Overview provides the standards for each topic as well as the standards for each MATHia workspace that is paired with each topic.

• Found under each topic’s Teacher Materials section, the Front Matter in the Teacher’s Implementation Guide provides a Standards Overview chart. The chart identifies lesson standards in green and spaced practice standards in gray. Each module with the Teacher’s Implementation Guide also has a standard overview represented as a dot matrix that identifies the standards addressed in each module, topic, and lesson. Additionally, each topic has a Topic Overview which lists the standards for each lesson. 

• Each topic also has a Topic Overview under the Teacher Materials section that identifies the standards in each lesson. A session log is also available that identifies the sessions MATHia will be utilized..

Module Overviews located at the beginning of each Module identifies specific course-level mathematics. The Topic Overview located at the beginning of each Topic identifies the role of the mathematics present within the Module. Examples of where explanations of the role of the specific course-level mathematics are present in the context of the series include: 

• In Math 1, Module 1, Teacher’s Implementation Guide, Module Overview, the materials provide connections to prior and future learning. Students use prior reasoning with middle school concepts of independent and dependent quantities, to help connect to the future learning of how their understanding of arithmetic sequences can be used to launch their study of linear functions. 

• In Math 2, Module 4, Topic 1, Topic Overview, the materials connect the students’ prior learning of factoring quadratic functions with real roots to their current learning of solving quadratic equations with complex roots. Materials provide connections to future learning by stating, “...the mechanics of factoring will be important in future courses when students determine the zeros of graphs of higher-order polynomials, to rewrite rational expressions, and to identify discontinuities in complex functions.”

• In Math 3, Module 1, Teacher’s Implementation Guide, Module Overview, the materials connect the students' prior learning of end behavior, symmetry, zeros, and the degree of the function to current learning extending to higher order polynomial functions to create sketches and represent scenarios.

The materials reviewed for Carnegie Learning High School Math Solution Integrated meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Front Matter in each course provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted, all examples are found in the Front Matter of the Teacher’s Implementation Guide.

Examples of the materials explaining the instructional approaches of the program include: 

• The Front Matter of the Teacher’s Implementation Guide includes the programs, “Guiding Principles.” The four guiding principles state, “All students are Capable Learners,”“Learning by Doing™,” “Learning Through Assessments,” and “Education is a Human Endeavor.”

• The program’s instructional approach is, “...based on a scientific understanding of how people learn, as well as an understanding of how to apply the science to the classroom.” There are three phases to the instructional approach: ENGAGE, DEVELOP, and DEMONSTRATE. The materials provide an explanation for each instructional approach. ENGAGE is intended to, “Activate student thinking by tapping into prior knowledge and real-world experiences.” DEVELOP is intended to, “Build a deep understanding of mathematics through a variety of activities.” FDEMONSTRATE is intended to, “Reflect on and evaluate what was learned.”

• “Introduction to Blended Learning”, explains how MATHbook and MATHia are designed to be used simultaneously to support student learning. Students will “Learn Together” using the MATHbook approximately 60% of the time and “Learn Individually” using MATHia 40% of the time.” 

• The Front Matter of the Teacher’s Implementation Guide provides a rationale for the sequence of the modules, topics, and lessons within the course and series.

• “Comprehensive Assessment” includes checking student readiness using the MATHia Ready Check Assessments and the MATHbook Getting Ready resources, monitoring learning by question to support discourse, and measuring performance using pre-tests, post-tests, end of topic tests, standardized tests, and performance tasks. 

Examples of materials including and referencing research based strategies include:

• In the Front Matter of each course in the Teacher’s Implementation Guide, the materials state, “The embedded strategies, tools, and guidance provided in these instructional resources are informed by books like Adding It Up, How People Learn, and Principles to Action.”

• In the Front Matter of each course in the Teacher’s Implementation Guide, the materials state, “MATHia has its basis in the ACT-R (Adaptive Control of Thought-Rational) theory of human knowledge and cognitive performance, developed by John Anderson - one of the founders of Carnegie Learning (Anderson et. al., 2004; Anderson, 2007).”

• Each Module Overview includes a section on “The Research Shows…” citing research related to a strategy, tool, or content matter within the module. Examples include:

  • In Math 1, Teacher’s Implementation Guide, Module 2 Overview, the materials cite research from Progressions for the Common Core State Standards in Mathematics (draft). High School, Algebra., pg. 7 to state, “The Algebra category in high school is very closely allied with the Functions category…The separation of algebra and functions…is not intended to indicate a preference between these two approaches. It is, however, intended to specify the difference as mathematical concepts between expression and equation on the one hand and functions on the other. Students often enter college level mathematics courses apparently conflating all three of these.”

  • In Math 2, Teacher’s Implementation Guide, Module 4 Overview, the materials cite research from Focus in High School Mathematics: Reasoning and Sense Making, NCTM pg. 41 to state, “Functions appear in most branches of mathematics and provide a consistent way of making connections between and among topics. Students’ continuing development of the concept of functions must be rooted in reasoning, and likewise functions are an important tool for reasoning. Thus, developing procedural fluency in using functions is a significant goal of high school mathematics.” 

  • In Math 3, Teacher’s Implementation Guide, Module 2 Overview, the materials cite research from Progressions for the Common Core State Standards in Mathematics (draft), HS Algebra, pg. 4-5. to state, “Seeing structure in expressions entails a dynamic view of an algebraic expression, in which potential rearrangements and manipulations are ever-present. An important skill for college readiness is the ability to try possible manipulations mentally without having to carry them out, and to see which ones might be fruitful and which not.”

• The materials of each course provides a link to a website referencing more extensive research on the research-based strategies incorporated in the program.

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Examples of where materials include a comprehensive list of supplies needed to support the instructional activities include:

• The online materials for each course provides a “Course Materials List” located in the General section of the Teacher Materials. The list contains the supplies needed for each Module.

• In each course, the Front Matter in the Teacher’s Implementation Guide provides Module pages in the Table of Contents. The Module pages specify materials needed for each module in the right corner of the page. 

• The list of materials is also provided in the Topic Overview included at the beginning of each topic.

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials state, “Assessment is an arc and not a one-time event. It is a regular part of the instructional cycle. Ongoing formative assessment underlies the entire learning experience driving real-time adjustments, next steps, insights, and measurements. Check Readiness > Monitor Learning > Measure Performance.” The materials identify the following as assessments: 

• Check Readiness

  • Module Readiness is in the MATHia Readycheck Assessment and measure, “student readiness of concepts and skills that are prerequisite for any upcoming content. The scoring guide informs student instructional needs.” The MATHbook Getting ready reviews prior experiences with mathematical content that will be built upon in the module. 

• Monitor Learning

  • MATHia contains LiveLab where teachers can monitor student work for “real-time recommendations on how to support student progress.” The MATHbook contains Lesson Overview listing learning goals, review questions, and making connections to prior learning. The MATHbook contains Questions to Support Discourse for each activity to assess, “students’ sense-making and reasoning, to gauge what they know, and generate evidence of student learning.” The MATHbook also contains Talk the Talk tasks to allow students to reflect on their learning from the lesson and profice teachers with information on whether students can demonstrate the learning outcomes. 

• Measure Performance

  • MATHis provide Skill Reports monitor skill proficiency of students in mastery workspaces, Standard Reports provide an overview of students’ proficiency on specific standards, and Predictive Analytics allow teachers to monitor student progress to predict students’ year-end outcomes.

• MATHbook contains Summative Assessments in the form of Pretest, Post-test, End of Topic Test, Standardized Test, and Performance Tasks. 

Examples of how the materials consistently identify the standards for assessment include: 

• In Math 1, Module 5, Topic 2, Lesson 2, the Talk the Talk assesses G-CO.2 and G-CO.4 when students describe how to distinguish a transformation as isometric or non-isometric, write a function to describe a translation, and compare and contrast geometric translation functions versus algebraic equations. 

• In Math 2, Module 3, Topic 3, Lesson 4, the Questions to Support Discourse assess A-SSE.1a, A.CED.4, F-IF.9, and F-LE.3 by having students answer questions after completing activities comparing key characteristics of functions and the average rate of change of functions. 

• In Math 3, Module 2, Topic 2, Assessment Overview, the materials identify “Arithmetic with Polynomials and Rational Expressions and Interpreting Categorical and Quantitative Data” as the standard domains for the assessments.  The materials then provide a specific standard for each question in the pre-test, post-test, End of Topic Test, Standardized Test, and the Performance Task.

Standards for Mathematical Practice are referred to as habits of mind within the materials. The habits of mind are only identified within the activities in the MATHbook.  Within the activities the Questions to Support Discourse are used to assess the activities. Examples include:

• In Math 1, Module 2, Topic 3, Lesson 4, Activity 2, the summary in the Teacher’s Implementation Guide states, “You can solve a system of inequalities graphically or by using a combination of graphing and algebraic methods.” The Habits of Mind listed are, “Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others.” Within the activity, the Questions to Support Discourse states, “How many regions do you create when graphing two intersecting lines? How can you tell from the graph whether the points in each region satisfy both constraints, one constraint, or no constraints in the situation? How will the results from your substitution demonstrate whether a point is a solution to both equations? Is the intersection point to this system of inequalities included in the solution? Why not? How can you use the fact that the lines are solid or dashed to identify whether to include the intersection point in the solution? Is an intersection point formed by two solid lines always part of the solution to the inequalities? Part of the solution to the problem situation? For a system of inequalities to have no solution, do the lines have to be parallel? Explain. If a system of inequalities includes parallel lines, is there always no solution? Explain your thinking.”

• In Math 2, Module 3, Topic 1, Lesson 5, Activity 2, the summary in the Teacher’s Implementation Guide states, “The graph of the inverse of a function is a reflection of that function across the line y=x.” The Habits of Mind listed are, “Model with mathematics. Use appropriate tools strategically.” Within the activity, the Questions to Support Discourse states, “What is the relationship between the slope of the original graph and the slope of the inverse graph? How is this strategy related to writing an inverse equation? Why does it make sense that the inverse of a function reverses the x- and y-coordinates?”

• In Math 3, Module 2, Topic 1, Lesson 1, Activity 2, the summary in the Teacher’s Implementation Guide states, “Additional methods to factor polynomials include chunking, grouping, and treating degree-4 trinomials similar to quadratics.” The Habits of Mind listed are, “Look for and make use of structure. Look for and express regularity in repeated reasoning.” Within the activity, the Questions to Support Discourse states, “What is the benefit of chunking? How can you recognize when factoring by chunking is an option? How are the signs of the factors and the signs of its equivalent trinomial related? Explain how the factors create that product. What features do you need to recognize in a trinomial to know that it is a perfect square trinomial? How can you recognize when factoring by grouping is an option? What is the difference between factoring by grouping and chunking?”

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for including an assessment system that provides multiple opportunities throughout the series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Answer keys are provided to determine students’ learning and reports provide teachers’ guidance on interpreting student performance. Suggestions for follow-up are provided through LiveLab, which alerts the educator to students who may need additional supports in specific skills, and the Skills Practice which provides suggestions on how students can re-engage with specific skills. 

Examples include:

• In the MATHia Group Skills Report, teachers can view each student’s skill mastery progress organized by module, unit, and workspace. The materials state “For each skill, a student can be in one of the following categories: Proficient: The student has a greater than or equal to 95% probability of understanding and correctly executing that skill. Near Proficient: The students has a 70%-94% probability of understanding and correctly executing that skill. Remediation Suggested: The students has a <70% probability of understanding and correctly executing that skill. In Progress: The student is currently completing problems that address this skill. Not Started: The student has not encountered workspaces that address this skill.” In the digital material, Help center, Math, LiveLab, At Risk Student Alert in LiveLab, the MATHia Report states, “The At-Risk Student Alert tells a teacher when a student is at risk of not mastering a workspace, as he/she is struggling with the understanding of a specific math concept. The warning will appear as a life preserver icon next to the student's current status on the main Class Dashboard. Click to the Student Details screen to review which workspace he/she is struggling with and specific math skills covered in the workspace to better understand how to provide targeted remediation for this student. You can review the skills in the Mastery Progress section of the Student Dashboard to help you provide that targeted remediation.”

• Summative Assessments are provided in the form of Pre- and Post Tests, End of Topic Tests, and Standardized Tests. The materials provide answer keys with the correct answers for each of the summative assessments. Performance Tasks provide a sample student solution and a scoring rubric to interpret student performance. The materials provide Skills Practice located in the Additional Facilitation Notes at the end of each lesson. The materials state, “After working through MATHbook lessons and MATHia workspaces, some students may need to re-engage with specific skills. You can use the Skills Practice problem sets to support small group remediation.”

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices across the series. Assessments include opportunities for students to demonstrate the full intent of course-level standards and practices across the series. 

The Summative Assessment Suite provides opportunities for students to demonstrate understanding of the standards. The End of Topic Test assesses the full range of standards addressed in the topic using short-answer and open ended questions. Standardized Tests include multiple-choice and multiple-select questions. The Performance Task given for each topic provides open-ended questions allowing students to demonstrate learning of standards and mathematical practices. 

MATHia provides formative assessment data on standards aligned to each topic using the following item types: Grapher tool, Solver Tool, Interactive diagrams, Interactive Worksheets, Sorting Tools, and short-answer questions. 

Examples include: 

• In Math 1, Module 4, Topic 1, the Performance Task develops the full intent of the standards S.ID.2 and S.ID.3. Problem 3 asks, “Analyze both box-and-whisker plots. a) Compare the shape of the two box-and-whisker plots. b) Are there any outliers in either data set? Explain your reasoning. c)Which measure of central tendency - mean, median, or mode - is the most appropriate measure of center for each data set? Explain your reasoning.”

• In Math 2, Module 4, Topic 1, MATHia Software Workspaces, Adding, Subtracting, and Multiplying Polynomials, Adding Polynomials develops the full intent of the standards A.SSE.1a, and A.APR.1. Students add the polynomials 7x^2+9-3x and -7-x^2+2x. Students choose to perform one of the following actions, “Combine Like Terms, Factor Quadratic, Rewrite Fractions, Distribute, Perform Multiplication, or Rewrite Signs.” After choosing the correct action, students enter the solution after performing the selected action. 

• In Math 3, Module 1, Topic 1, the End of Topic Test Form A develops the full intent of standards A.CED.1 and F.BF.1a. Problem 13 states, “A basketball player stands near the middle of the court and throws the ball toward the basket. The path of the ball is a parabola. The ball leaves the player’s hands at a height of 6 feet above the ground. The ball travels to a maximum height of 12 feet when it is a horizontal distance of 18 feet from the player’s hand. Write a function to represent the height of the ball in terms of its distance from the player’s hands.”

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

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

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

  • In Math 1, Module 4, Topic 1, Lesson 2, Activity 1, the materials state, “Ensure students understand the key terms outlier, lower fence, and upper fence by connecting the terms to their common meanings outside of math class.”

  • In Math 2, Module 1, Topic 1, Lesson 1, Activity 1, the materials state, “Ensure students understand the meaning of semicircle by helping them connect to other uses of the prefix semi-. Have students compare this use of semi- with other uses such as semi-truck, semi-final, or semi-formal.”

  • In Math 3, Module 5, Topic 1, Lesson 2, Activity 3, the materials state, “Ensure students understand the terms concave downward and concave upward. Help students connect back to when they used the terms to describe parabolas to help them understand their meaning for the normal curve.”

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

  • In Math 1, Module 3, Topic 2, Lesson 2, Getting Started, the materials suggest supporting students who struggle as they work on Question 1 by having teachers, “Suggest students use graphing technology to graph the functions, then match them with the correct graph.”

  • In Math 2, Module 1, Topic 1, Lesson 1, Activity 1, the materials suggest supporting all students throughout the lesson by having students, “...draw three columns on a piece of paper and label the first column with term, the second column with definition, and the third column with picture. As students encounter new terms in the lesson, encourage them to add the term, definition, and a drawing to their table.”

  • In Math 3, Module 3, Topic 1, Lesson 2, Activity 1, the materials suggest supporting students who struggle as they work on Question 1 by having students “...modify the existing table to create a table for the inverse function. This may help students solidify their understanding of the relationship between the x-values and y-values of a function and its inverse.”

• The MATHia User Guide Implementation Tools state MATHia uses technology to,”...differentiate to create a personalized learning path for each student.” MATHia supports all students through “Step-by-Step” guided worked examples, “On-Demand Hints” providing multi-level hints, and “Just-in-Time Hints” to help correct common errors.

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for providing extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity. At the end of each lesson, the Teacher’s Implementation Guide has “Additional Facilitation Notes” to assist teachers in supporting all students, especially struggling students and advanced learners. The “Additional Facilitation Notes” include differentiation strategies, common student misconceptions, and suggestions to extend specific activities. 

Examples of suggestions in the Teacher’s Implementation Guide to extend student learning to provide opportunities for advanced students to investigate grade-level content at a higher level of complexity include:

• In Math 1, Module 3, Topic 1, Lesson 3, Activity 3, students are given the following question, “Use the table to compare the ordered pairs of the graphs of m(x) and n(x) to the ordered pairs of the graph of the basic function h(x). What do you notice?” The materials suggest to extend the question for advanced learning by having, “students investigate and explain why reflecting h(x) causes it to change from an increasing function to a decreasing function. Ask students whether this will hold true for all increasing exponential functions. Have them explain what they think would happen when they reflect a decreasing exponential function.”

• In Math 2, Module 1, Topic 2, Lesson 3, Getting Started, the materials suggest extending Question 2 for advanced learners by having students “extend the activity to quadrilaterals. What is the sum of the four interior angles of all quadrilaterals?”

• In Math 3, Module 3, Topic 1, Lesson 2, Activity 3, students compare domain, range, x-intercepts, and y-intercept of the functions c(x)=x^3 and c^{-1}(x)=\sqrt[3]{x}. The materials suggest extending for advanced learners by having students “investigate whether different transformations of c(x)=x^3 impact the inverse function.”

An article titled “Using the Assignment Stretch with Advanced Learners'' located in Help Center, Math, Teaching Strategies states, “Each Assignment includes a Stretch that provides an optional extension for advanced learners that stretch them beyond the explicit expectation of the standards. The Teacher’s Implementation Guide provides suggestions for chunking the assignment for each lesson, including the Stretch. These suggestions consider the content addressed in each session and recommend corresponding Practice, Stretch, and Mixed Practice questions. To ensure that advanced learners are not doing more work than their classmates, consider substituting the Stretch for Mixed Practice questions. When there are no Mixed Practice questions aligned with the Stretch, substitute the Stretch for the Journal or Practice questions.” 

Examples of the materials including “Stretch” questions to extend leaning of the course-level topic/concept include:

• In Math 1, Module 2, Topic 3, Lesson 5, Assignment, an optional “Stretch” question to extend learning gives the following scenario and questions. “Isla sells baked goods from her home kitchen. She offers decorated cookies for $15 per dozen and cupcakes for $13 per dozen. It takes her an hour to decorate a dozen cookies, but only 20 minutes to decorate a dozen cupcakes. She would like to make at least $300 per week and not put in more than 20 hours of work per week. 1) Create a system of linear inequalities that fits the situation and graph them… 4) What is the least amount of time she could work and still earn $300? What baked goods would she make?”

• In Math 2, Module 2, Topic 3, Lesson 3, Assignment, an optional “Stretch” question to extend learning gives the following scenario and questions. “The penny is a commonly used coin in the U.S. monetary system. A penny has a diameter of 19 millimeters and a thickness of 1.27 millimeters. The volume of a penny is 360 cubic millimeters. Suppose you stack 10 pennies on top of each other to form a cylinder. !) What is the height of the stack of pennies? ... 4) Multiply the area of a penny by the height of the stack of pennies. Compare your product to the answer from part(b). How do the two answers compare? What does this tell you about calculating the volume of a cylinder?”

• In Math 3, Module 1, Topic 1, Lesson 3, Assignment, an optional “Stretch” question to extend learning has students analyze three patterned figures. Students are then asked to “1)Write the function b(n) to represent the number of blue blocks in Figure n. 2) Write the function w(n) to represent the number of white block in Figure n. 3) Write the function g(n) to represent the number of green blocks in Figure n. 4) You can represent total number of blocks in Design n by the function t(n)=(n+\frac{5}{2})^2-\frac{17}{4}. Use the functions you wrote to show that t(n)=b(n)+w(n)+g(n).”

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. The materials state, “Everyone is an English language learner. Whether it’s learning the language itself or the specialized, academic language of mathematics, students consistently use various strategies to make sense of the world.”Additionally, the materials state that the design and recommended implementation of MATHbook and MATHia provides students with the structure to address all four domains of language (listening, speaking, reading and writing).

In the digital materials, Help Center, Math, Teaching Strategies, an article titled “Supporting ELL Students” provides strategies used in the materials aligning to best practices. The following are strategies the article states are used in the materials: “Text-to-speech and Google Translate make the text accessible in MATHia. Throughout the text, students demonstrate that vocabulary can have multiple meanings. Comprehensive monitoring of student progress via MATHia. The adaptive nature of MATHia. Worked examples in the software and textbook provide a way to explain and model the thought process in which problems can be solved; this also applies to the step-by-step problem in MATHia. Clear learning goals are stated in each lesson, and the narrative statement at the beginning of each lesson provides an opportunity for students to anticipate how the new information will connect to previous learning.”

The MATHbook allows students to “...highlight, annotate, and even write words in their native language.” The materials also provide graphic organizers to show relationships between concepts and terms. The materials also suggest, “Grouping students provides structured opportunities for ELL to practice speaking in English. Pairing an ELL with more proficient English speakers allows opportunities to communicate their thinking in a low-stress way.” Although MATHbook provides strategies/supports for students to read, write, and/or speak in a language other than English, the strategies/supports are only available in English and Spanish. 

General strategies/supports the materials provide in MATHbook for students who read, write, and/or speak in a language other than English: 

• The Academic Glossary provides definitions, ask yourself prompts, and related phrases for terms that will help students think, reason, and communicate ideas. The materials state, “There is strong evidence backing the importance of teaching academic vocabulary to students acquiring English as a second language.” 

• The Glossary provides written definition and visual examples for mathematic specific vocabulary. The MATHbook glossary is only available in English and Spanish.

• The materials provide Language Links throughout lessons to support student language. Examples include:

  • In Math 1, Module 1, Topic 1, Lesson 3, Activity 1, the language link provided in the Teacher’s Implementation Guide states, “Ensure the students understand the meaning of non-function. The prefix non- means not or no. Therefore, a non-function is a relation that is not a function. Follow up with additional words with the prefix non-, such as nonsmoking, nonstop, and nonfat.”

  • In Math 2, Module 3, Topic 1, Lesson 2, Getting Started, the Language Link provided in the Teacher’s Implementation Guide states, “Ensure students understand the meaning of the term inspection. Relate it to other terms located in the Academic Glossary, such as analyze and examine. Ask students to provide examples of things that get inspected to help make sense of the meaning.”

  • In Math 3, Module 1, Topic 2, Lesson 5, Activity 1, the Language Link provided in the Teacher’s Implementation Guide states, “Support students with the definition, situation, and worked example by engaging with smaller pieces of the text. For example, have some students focus on relative maximum and others on the relative minimum. Then discuss both before moving to the situation.”

General strategies/supports the materials provide in MATHia for students who read, write, and/or speak in a language other than English: 

• The Glossary provides written definitions and visual examples for mathematic specific vocabulary. The MATHia glossary is only available in English and Spanish.

• MATHia Software Workspaces are available in English and Spanish. Students can use the text-to-speech feature in MATHia to hear the problems read aloud in several languages while customizing the speed and pitch at which the voice reads. Additionally, students are able to change the problems to all languages available within Google Translate.

• All MATHia videos are fully closed captioned and are available in English and Spanish.

The materials reviewed for Carnegie Learning High School Math Solution Integrated series meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

• In Math 1, Module 1, Topic 2, Lesson 2, Getting Started, students are given 13 cutouts of different sequences and are instructed to sort the sequences based on common characteristics. In Activity 1, students learn an arithmetic sequence has a common difference and a geometric sequence has a common ratio. Students revisit the 13 cutouts and identify the common differences for all the arithmetic sequences and the common ratios for the geometric sequences. Students also identify sequences that are neither arithmetic or geometric. 

• In Math 2, Module 4, Topic 2, Lesson 2, Activity 2, students use graphing technology to analyze positive intervals and negative intervals of quadratic equations to connect solutions to quadratic inequalities.

• In Math 3, Module 1, Topic 2, Lesson 1, Activity 1, students explore three-dimensional solids created by rectangles, discs, and isosceles triangles. Students are instructed to, “Draw the shape on an index card. Cut out the shape and tape it along the center to a pencil below the eraser, as shown. Hold the eraser and rest the pencil on its tip. Rotate the shape by spinning the pencil. You can get the same effect by putting the lower portion of the pencil between your hands and rolling the pencil by moving your hands back and forth.”

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

• In Math 1, Module 1, Topic 2, MATHia Software Workspaces, Recognizing Patterns and Sequences, Graphs of sequences, students are provided definitions and examples of arithmetic sequences and geometric sequences. Students complete a drag and drop activity sorting nine sequences by classifying them as arithmetic, geometric, or neither. 

• In Math 2, Module 3, Topic 1, MATHia Software Workspaces, Defining Absolute Functions and Transformations, Horizontally Translating Absolute Value Functions, student are instructed to translate the parent function f(x)=|x|. Then, students can choose to reflect, dilate, horizontally or vertically translate the graph and type in a value representing the specific transformation. The graph provided on the page will display the transformation the student has typed. Once the graph is correct, students enter the equation of the transformed graph.

• In Math 3, Module 1, Topic 1, MATHia Software Workspaces, Observing Patterns, Comparing Familiar Function Representation, students are provided with examples of linear, quadratic, and exponential functions. Students complete a drag and drop activity sorting twelve functions into the classification of linear, quadratic, exponential, or none of these.