High School - Gateway 3

The materials reviewed for Carnegie Learning High School Math Solution Traditional series series meets 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 Algebra I, Module 5, Topic 1, Lesson 2, Activity 3, Teacher’s Implementation Guide, the materials provide sets of questions to support student discourse within the lesson. Questions are identified by type such as Gathering, Probing, or Seeing Structure. A sample Seeing Structure question is, “Why does it make sense that pairs of symmetric points will have the same numeric rate of change with opposite signs?”. 

• In Geometry, Module 1, Topic 3, Lesson 2, Activity 1, Teacher’s Implementation Guide, students are introduced to slash markers and arc markers to represent congruent parts. An annotation in the material provides teachers with a differentiation strategy which suggests for students to, “use colored pencils to trace congruent sides and mark congruent angles.” The materials suggest by using colored pencils students may better understand how to interpret labeled diagrams. 

• In Algebra II, Module 2, Topic 1, Lesson 2, Activity 2, Teacher’s Implementation Guide, students factor binomials completely using the difference of squares. The materials indicate a common misconception would be for students to believe they cannot rewrite the function h(x) as a product of linear factors since they just concluded the function is factored completely; however, teachers are instructed to, “Remind them that h(x) is factored over the set of real numbers, but they can use complex numbers to rewrite h(x)as a product of linear factors.”

The materials reviewed for Carnegie Learning High School Math Solution Traditional 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 Algebra I, Module 4, Topic 1, Lesson 2, Teacher’s Implementation Guide, the materials provide an adult-level explanation on describing a data set using center, shape, and spread. The materials state, “The median is the better measure of central tendency, and the IQR is the better measure of spread to describe a skewed data set. The mean is the better measure of central tendency, and the standard deviation is the better measure of spread to use to describe a symmetric data set. You identify outliers in a data set using the formula Q1-(IQR\cdot1.5) to determine a lower fence and Q3+(IQR\cdot1.5) to determine an upper fence. Any value outside these is an outlier.”

• In Geometry, Module 3, Topic 2, Topic Overview, the materials provide an explanation of how key concepts of trigonometry are developed. Students will explore concepts of special right triangles and trigonometric ratios in order to solve problems involving right triangles. Students will also explore the complement angle relationships in right triangles and derive the Triangle Area Formula. A sample math representation is also provided to show how the key concepts will be used. 

• In Algebra II, Module 1, Topic 3, Lesson 5, Teacher’s Implementation Guide, the materials include an adult-level explanation for the average rate of change formula to provide an interpretation for application of polynomial functions. The materials state, “The formula for the average rate of change is \frac{f(b)-f(a)}{b-a} for an interval (a,b) . The expression b-a represents the change in the input of the function f. The expression f(b)-f(a)represents the change in the function f as the input changes from a to b.”

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

• In Algebra 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 Geometry, Module 3, 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 Algebra 2, 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 Traditional 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 Algebra I, Module 4, Topic 1, Topic Overview, the materials provide an overview of how key concepts for one-variable statistics are developed. The topic begins with a remainder of the statistical process from middle school and a connection to the modeling process. Building on these statistical data from models in middle school, students develop toward understanding the formal notation for mean, how to calculate standard deviation, and how to find outliers. 

• In Geometry, Module 4, Teacher’s Implementation Guide, Module Overview the materials provide a connection to prior and future learning. Students use simple and compound probabilities developed in middle school to make predictions which will develop into using formal and intuitive strategies to determine the probabilities of real-world events. 

• In Algebra II, Module 1, Teacher’s Implementation Guide, Module Overview the materials provide a connection to prior and future learning. Students previously learned the key characteristics of linear, exponential, and quadratic functions and will build on that foundation to use the key characteristics of polynomials to expand the inventory of functions that they know and with which they can model scenarios.

The materials reviewed for Carnegie Learning High School Math Solution Traditional series 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.” Finally, DEMONSTRATE is intended to, “Reflect on and evaluate what was learned.”

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

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

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

Examples of materials including and referencing research based strategies include:

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

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

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

  • In Algebra I, Teacher’s Implementation Guide, Module 5 Overview, the materials cite research from Focus in High School Mathematics: Reasoning and Sense Making, NCTM, pg. 41 by stating, “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 Geometry, Teacher’s Implementation Guide, Module 4 Overview, the materials cite research from Douglas H. Clements and Michael T. Battista, “Geometry and Spatial Reasoning,” Handbook of Research on Mathematics Teaching and Learning, 2004 by stating, “School geometry is the study of those spatial objects, relationships, and transformations that have been formalized ... Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum.”

  • In Algebra II, 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. by stating, “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 Traditional 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 Traditional 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 gauges, “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 provide 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 assessments include: 

• In Algebra I, Module 3, Topic 1, Assessment Overview, the materials identify “The Real Number System, Interpreting Functions, Building Functions, and Linear, Quadratic, and Exponential Models” as the standard domains for the assessment. The materials then provide a specific standard for each question in the Pre-test, Post-test, End of Topic Test, Standardized Test, and the Performance Task.

• In Geometry, Module 4, Topic 1, Lesson 1, the Talk the Talk assesses the standards G-C.1, G-C.5, G-GMD.1 by having students determine the length of each steel arc connecting one passenger car to the next on a ferris wheel. 

• In Algebra II, Module 2 Teacher’s Implementation Guide Overview, the nine MATHia sessions aligned with Topic 3, assesses A-APR.6, A-CED.1, A-REI.2, and F-IF.7d (+).

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 Algebra I, Module 5, Topic 2, Lesson 1, Activity 5, the summary in the Teacher’s Implementation Guide states, “You can multiply polynomials using the area model or Distributive Property. The product of two polynomials is always a polynomial.” The Habits of Mind listed are, “Model with mathematics. Use appropriate tools strategically.” Within the activity, the Questions to Support Discourse states, “Why does each model include multiplication twice? How can you tell when a result is in general form? How is the multiplication problem different than s(100-2s) ? How did you modify the area model to multiply a trinomial and binomial? Explain how you used the Distributive Property to calculate each product. How can you tell the degree of the polynomial from its factors? Can you tell the number of terms in a product from its factors? Explain your thinking.”

• In Geometry, Module 3, Topic 2, Lesson 2, Activity 3, the summary in the Teacher’s Implementation Guide states, “If you are using congruent reference angles in similar triangles, the tangent ratio of the reference angles is constant.” 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 can you represent an unknown dilation factor? Why? How is the expression representing tan x^0 in the first triangle different than the expression representing tan x^0 in the second triangle? Why will your proportion always be true? 

• In Algebra II, Module 5, Topic 1, Lesson 3, Activity 2, the summary in the Teacher’s Implementation Guide states, “A percentile is a data value for which a certain percent of the data is below the data value in a normal distribution.” 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, “Is it possible to be at the 100th percentile? Explain. Why does it make sense that the mean coincides with the 50th percentile in a normal distribution? What is the relationship between percentiles and the mean? Explain why a z-score of 1.28 makes sense for the 90th percentile on the normal curve. How did you use technology to determine the value that corresponds to the 20th percentile? How can you tell that your response is reasonable?”

The materials reviewed for Carnegie Learning High School Math Solution Traditional 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 Traditional 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 Algebra I, Module 2, Topic 2, the End of Topic Test Form A develops the full intent of the standard A.REI.3. Problem 2 states, “Determine whether the equation has one solution, no solution, or infinite solutions. a)4(x+5)+4=\frac{1}{2}(8x+48) b)5(x-2)-x=5x+2 c)2(4-x)=-2(1+x)-2 d)3(8x-4)=6(4x-1)-6.”

• In Geometry, Module 1, Topic 3, MATHia Software Workspace, Triangle Congruence Theorems, Using Triangle Congruence develops the full intent of standard G.CO.7, 8. Students are instructed to, “Analyze the worked example and then complete each two-column proof to prove the given statement.” Students are given two proofs where they must use various theorems, postulates and definitions to prove triangle congruence. 

• In Algebra II, Module 3, Topic 1, the Performance Task develops the full intent of the standards F.IF.4, F.IF.5, and F.IF.7b. Students are given the following scenario,”Doctors sometimes need to calculate the body surface area of their patient when they are determining the specific dosage of a medicine they are administering. The formula for body surface area is B=\sqrt{\frac{H\cdot W}{3131}}where B is the body surface area in square meters, H is the height of the patient in inches, and W is the weight of the patient in pounds.” Then students answer a series of questions using the formula for body surface area.

The materials reviewed for Carnegie Learning High School Math Solution Traditional 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 Algebra I, Module 2, Topic 1, Lesson 4, Activity 3, the materials state, “Ensure students are familiar with terms such as bank account, account balance, and deposit. Ask students to share definitions for these terms in their own words, and then clarify as needed.”

  • In Geometry, Module 3, Topic 2, Lesson 2, Activity 4, the materials state, “Ensure students understand the term inverse in reference to inverse tangent. Connect to students’ prior experience solving equations with inverse operations such as addition and subtraction. Help students see that inverse tangent is the inverse operation of tangent.”

  • In Algebra II, Module 1, Topic 1, Lesson 6, Activity 1, the materials state, “Discuss the meaning of the terms real and imaginary as opposites in everyday life to help students connect why you apply the term imaginary numbers to non-real numbers.” 

• 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 Algebra I, Module 1, Topic 1, Lesson 3, Activity 1, the materials suggest to support struggling learners by having teachers, “Suggest students lay their pencil down on top of the arrow and move it horizontally across the graph to apply the vertical line test.”

  • In Geometry, Module 5, Topic 2, Lesson 2, Activity 2, the materials suggest to support all learners as students share and summarize Questions 1 by having teachers, “Stress the connection between the two meanings of 6, as the desired outcome of P(A) and the total outcome in the conditional probability P(B|A). 

    • Have students use shading in the first table in Question 1 to demonstrate the two meanings of 6. 

    • Discuss why this occurs regardless of the values so that students understand the proof in general terms in the worked example.”

  • In Algebra II, Module 1, Topic 3, Lesson 1, Activity 3, the materials suggest to support students who struggle as they work on Question 4, teachers’ should, “Encourage students to use patty paper to test whether the function is even, odd, or neither. Remind students to trace the graph and the x- and y-axis before they perform the reflections.”

• 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 Traditional 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 Algebra I, Module 2, Topic 2, Lesson 2, Activity 3, students are given the following question, “The formula for the area of a trapezoid is A=\frac{1}{2}h(b_1+b_2), where h is the height and b_1 and b_2 are the lengths of each base. (a.) Rewrite the area formula to solve for the height. (b) Use your formula to determine the height of a trapezoid with an area of 24 cubic centimeters and base lengths of 9 cm and 7 cm.” The materials suggests extending the question for advanced learners by having, “students solve for b_1 and b_2 and compare it to the process they used to solve for h.”

• In Geometry, Module 4, Topic 1, Lesson 1, Activity 1, the students are instructed to, “Summarize the proof to show that circles are similar.” The materials suggest extending the statement for advanced learners by asking, “students to formally prove that all squares are similar.”

• In Algebra II, Module 1, Topic 2, Lesson 2, Activity 2, the materials suggest extending Activity 2 for advanced learners by having, “students explore multiplicity with zeros of degree-3 functions. Have students predict the shape of the graph of f(x)=(x+3)(x+3)(x+3). Use graphing technology to explore multiple zeros of functions.”

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 Algebra I, Module 2, Topic 1, Lesson 3, Assignment, a graph is provided showing the graphed function of f(x) and g(x). An optional “Stretch” question to extend learning asks students to, “Consider the graphs of the functions f(x) and g(x). 1) Write an equation for each function in general form. 2) Write an equation for g(x) in terms of f(x).”

• In Geometry, Module 4, Topic 1, Lesson 4, Assignment, an optional “Stretch” question to extend learning gives the following scenario and question, “ 1) Lake Erie, the smallest of the Great Lakes by volume, still holds an impressive 116 cubic miles of water. Suppose you start today dumping out the entire volume of Lake Erie using a cone cup. A typical cone cup has a diameter of 2\frac{3}{4} inches and a height of 4 inches. About how long would it take you to empty the lake if you could dump out one cup per second? 2) Lake Erie has an average depth of 62 feet. Suppose the volume of Lake Erie were contained in a cylinder. What would be the radius of the cylinder?”.

• In Algebra II, Module 2, Topic 1, Lesson 2, Assignment, an optional “Stretch” question to extend learning asks students to, “Consider the function f(x)=3x^3-4x^2-17x+6. 1) Determine the values of a_0 and a_n for this polynomial function. 2) Determine the values of p, or the factors of a_0. 3) Determine the values of q, or the factors of a_n. 4) Determine the possible rational zeros of the function. 5) Check all the possible rational zeros to determine whether any of them are roots of the function f(x).”

The materials reviewed for Carnegie Learning High School Math Solution Traditional 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 include: 

• 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 Algebra I, Module 3, Topic 2, Lesson 4, Activity 2, the Language Link provided in the Teacher’s Implementation Guide states, “Support students by providing them with the following options in place of writing an article: Record a commercial or create a poster.”

  • In Geometry, Module 1, Topic 2, Lesson 3, Activity 3, the Language Link provided in the Teacher’s Implementation Guide states, “Ensure students understand the meaning of counterexample in the directions. Define counter as to go against or follow the opposite direction. A counterexample is an example that disproves a statement. Provide an example, such as ‘All triangles are isosceles.’ Discuss why a scalene triangle is a counterexample.”

  • In Algebra II, Module 1, Topic 2, Lesson 1, Activity 1, the language link provided in the Teacher’s Implementation Guide states, “Remind students to refer to the Academic Glossary…to review the definition of predict, describe and related phases. Suggest they ask themselves these questions: Does my reasoning make sense? Is my solution close to my estimate? How should I organize my thoughts? Did I consider the context of the situation?”

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

• 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 Traditional 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 Algebra I, Module 2, Topic 4, Lesson 1, Activity 2, students use a graphing calculator to compare absolute value parent functions, vertical and horizontal shifts, stretches/compressions, and reflections. Students record comparisons to draw conclusions about the coordinates resulting from the translations and transformations.

• In Geometry, Module 2, Topic 2, Lesson 2, Getting Started, students are given parallel lines cut by a transversal creating corresponding angles, Angle A and Angle B. Students determine corresponding angles are congruent by using patty paper to translate one angle onto its corresponding angle. Students describe how to use patty paper and vertical angles to show alternate interior angles are congruent. Later in the lesson, students formalize work in the Getting Started by proving the Corresponding Angles Theorem and the Alternate Interior Angles Theorem.

• In Algebra II, Module 1, Topic 2, Lesson 1, Activity 1, students are given a scenario about a civil engineer tasked with rebuilding a storm drain system in a city. Students use paper to model the design of a drain in the system. Students can use their model to complete a table relating the height of the drain to the width of the drain. Then, students must define a function representing the cross-sectional area of the drain in relation to the height of the drain. Finally, students use graphing technology to sketch a graph of the drain to determine the dimensions yielding the greatest cross-sectional area.

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

• In Algebra I, Module 2, Topic 1, MATHia Software Workspaces, Transforming Linear Functions, Exploring Graphs of Linear Functions Workspace, students are given a coordinate plane where f(x) and g(x) are graphed and g(x) is written as f(x)+D. Students use sliders to change the value of D and observe changes to the graph of g(x). Finally, at the end of the workspace, students choose an answer to complete the following statement, “The linear function of g(x)=f(x)+D is…”

• In Geometry, Module 3, Topic 2, MATHia Software Workspaces, Trigonometric Ratios, Relating Sines and Cosines of Complementary Angles, students are given an Explore Tool to investigate trigonometric ratio relationships. To use the tool, students can enter an angle measure in degrees and press enter to discover trigonometric values of sine, cosine, and tangent. Students are prompted to calculate the sin35\degree and the cosine of the complement of the 35\degree angle. Students are also prompted to calculate cos72\degree and the sine of the complement of the 72\degree angle. Finally, students make a conjecture by filling in the statement, “The sine of an angle is equal to the _____ of its complement, and the cosine of an angle is equal to the _____ of its complement.”

• In Algebra II, Module 4, Topic 1, MATHia Software Workspaces, Trigonometric Relationships, Understanding the Unit Circle, students watch a short animation video sharing key properties of the unit circle. Students answer questions related to properties of the unit circle.