High School - Gateway 3

The materials reviewed for Open Up High School Mathematics 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.

Each unit contains an Overview that provides teachers with information about what is taught and how it will be presented.  Each Overview begins with a discussion of where the students may have touched on the concepts in previous years, or in previous units of the course, and how the unit will build on that knowledge. It also provides a table that provides a summary of the mathematics in each lesson. 

Each lesson is presented in a consistent format in order to provide teacher guidance on how to present the student materials in a way that engages students and guides their mathematical development. Examples of how the instructional materials provide guidance on how to present the materials include:

• In Math 1, Lesson 9.6, students review standard representations for single variable numerical data, compare how representations represent the data in different ways, and create their own representations of data given specific characteristics of center, shape, and spread. The Launch portion of the lesson includes a Notice and Wonder activity which provides detailed narratives and prompts for teachers to facilitate students’ work. In the Explore portion of the lesson, teachers are encouraged to pair students up or place them in small groups to discuss the sketches they have created. Also in the Explore portion, under the heading Selecting and Sequencing Student Thinking, teachers are guided to select two students who have sketches that meet the given criteria but are very different. The Teacher Notes state “For instance, select a student for 4b who has a distribution with median of 5 with two equal quartiles on either side and another student whose distribution has a median of 5 that has two different sized quartiles.”  The Discuss portion of the lesson allows students to present their graphs. The directions for the teacher guide them through possible student examples and how to use those examples as a part of a full class discussion that will enable students to address key takeaways of the lesson. 

• In Math 2, Lesson 5.7, students practice writing proofs to show that conjectures are true. The Explore Narrative in the Teacher Lesson recommends that students work with a partner or in small groups. Teachers are reminded that “a variety of ways may be used to present the proof both in terms of the conceptual approaches they use (transformations, linear pairs, congruent triangle criteria) and in terms of the format they use to write their proofs (two-column, flow diagrams, narrative paragraphs, algebraic proof).” Additionally, teachers are reminded that they should select a variety of approaches when presenting work during the whole class discussion. The Monitor Student Thinking section in the Explore Narrative presents questions teachers can ask students if they notice that students are struggling with proof-writing. 

• In Math 3, Lesson 3.3, students compare operations with polynomials to operations with integers and use those comparisons to add and subtract polynomials algebraically. The Launch Narrative in the Teacher Lesson guides the teacher to help students see how numbers are structured as the sum of powers of 10 while polynomials are structured as sums of powers of x. The Connect Student Thinking portion of the Launch Narrative describes the types of examples the teacher should select from student work to share with the class in order to highlight different approaches to answering the problems. In the Discuss Narrative, the Pause and Record routine included in the Selecting, Sequencing, & Connecting Chart guides the teacher through the Think-Pair-Share routine to have students identify the three most important things to remember about subtracting polynomials. The three important takeaways are also provided for the teacher. 

Each Teacher Lesson includes Learning Goals (for the teacher), the Learning Focus (for the student), Standards for the Lesson, Materials (when necessary), Required Preparation, BLMs, Progression of Learning, and Purpose. Notes are provided for the teacher to anticipate, monitor, and connect student thinking. These notes also provide detailed information about how to be prepared for common student questions and to guide student work to ensure that key takeaways are reached and understood. There are narratives that break down the specific instructional strategies and what the students should be doing to get the most out of each task.

The Anticipate & Monitor Charts list solutions, misconceptions, and other possible ideas that students may have when completing the tasks. These charts also include follow up questions and suggestions of how to address misconceptions and to expand upon the solutions and other ideas offered by students.

The Selecting, Sequencing, & Connecting Charts provide some examples of how individual, small group, and full class discussions can be facilitated to use students’ answers and ideas to guide them toward reaching the goals of the lesson.

Examples of how the instructional materials provide teacher guidance on how to plan for instruction include:

• In Math 1, Lesson 5.3, in which students graph the solution set for linear inequalities in two variables, teachers are provided with a detailed Materials list including graph paper, colored pencils, graphing calculators, and a link to a GeoGebra app. The Launch Narrative discusses the possibility that “[s]tudents may need direct instruction in how to access these tech tools and may benefit from a list of steps to be able to use the applet or software.”  

• In Math 2, Lesson 3.3, students develop fluency and flexibility in solving quadratic equations and determine the most efficient method for solving any quadratic equation. In the Required Preparation portion of the lesson, teachers are directed to work the task and consider given questions in order to Anticipate Student Thinking. In the Explore portion of the lesson, teachers are guided to provide sentence frames to support discussions (Math Language Routine 8: Discussion Supports).

The materials reviewed for Open Up High School Mathematics Integrated series 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 teacher edition contains thorough adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge. Each unit includes an overview of the content addressed in each lesson. The narrative is presented in adult language with a quick table reference of math concepts presented per lesson. In addition, each lesson’s Progression of Learning and Purpose sections describe specifically how lessons connect content throughout the learning cycles of multiple lessons. As a quick reference point, Open Up High School Math Dependency, a chart provided with the series, gives teachers the opportunity to see where a given lesson connects to another course in the series but not outside the scope of the current materials. Examples of course-level explanations include:

• In Teacher Notes, Purpose, there is information about a topic to be introduced. For example, in Math 2, Lesson 3.4, students are introduced to non-real solutions of a quadratic function. The Note in this lesson discusses a brief history of complex numbers including dates and names of mathematicians responsible for current notations.

• In Math 1, Unit 3, at the end of the Unit Overview, there is a paragraph on notation that explains the different ways to write interval notation and how interval notation connects to the set notation students have been using. 

• In Math 2, Lesson 5.10, detailed information is provided in Teacher Notes regarding the medians, angle bisectors, and the perpendicular bisectors of triangles and how these points of concurrency relate to balance, inscribing a circle, and circumscribing a circle.

For each course, the materials also provide adult-level explanations and examples for teachers to improve their own knowledge of concepts beyond the current course through a collection of essays titled Connections to Mathematics Beyond the Course. These essays are also directly connected to the lessons with which they are relevant, and examples include:

• In Math 1, Lesson 2.9 is connected to Rate of Change.

• In Math 2, Unit 1 is connected to First and Second Differences.

• In Math 3, Lessons 9.9 and 9.10 are connected to Confidence Intervals.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The teacher materials provide information to explain coherence across multiple courses and to enable teachers to make connections to prior and future content. Examples include but are not limited to:

• The Open Up High School Math Dependency Chart illustrates how units are connected in the series. For example, the chart shows that Math 3, Unit 6, Modeling Periodic Behavior depends on Math 3, Unit 1, Functions and their inverses. The document also includes Additional Dependencies. For example, Math 2, Unit 6 depends on Math 2, Unit 2 (algebra skills for working with quadratic expressions).

• In the Course Guide, the Standards Alignment for HS Integrated lists each lesson from the series that addresses each standard and also identifies which sections of each of the Ready, Set, Go problem sets in each lesson are aligned to which standards.

• At the course level, the Course Overview documents make general references to standards that are covered in the course and the units in which those standards will be covered. For example, the Course Overview for Math 2 explains how “[t]he major purpose of Math 2 is to  extend the mathematics that students learned in Math 1, including working with quadratic, piecewise and absolute value functions, using rigid transformations and triangle congruence criteria to prove geometric relationships, examining the geometry of circles, and using conditional probability to make and evaluate decisions.” Later in the Course Overview, the connection between Math 1, Units 1-3, and Math 2, Units 1-4, is described and then connected to the work with functions in Math 3.  

• The Unit Overview in each unit provides information about prior knowledge and where it was addressed. For example, the Unit Overview for Math 3, Unit 3, begins by identifying that the work with higher order polynomials in Math 3 connects to students’ understanding of linear and quadratic functions in Math 1 and Math 2. Towards the end of the overview, connections are made to Math 3, Units 4 and 8.

The materials clearly indicate how individual lessons or activities throughout the series are correlated to the CCSSM. Each lesson identifies the mathematical content standards as well as the relevant Standards for Mathematical Practice (SMP). Examples include:

• In Math 1, Lesson 6.3, the materials identify Focus Standards G-CO.4 and G-CO.5 and Supporting Standards G-CO.1, G-CO.2, and G-CO.6. The lesson also identifies MPs 3 and 7. The Exit Ticket identifies G-CO.4 as the standard to which the lesson should build. 

• In Math 2, Lesson 9.6, the materials identify Focus Standard G-GPE.2. The lesson also identifies MPs 3, 7, and 8. The Exit Ticket is also aligned to the Focus Standard for the lesson.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

The Open Up Math HS Course Guide provides detailed information regarding instructional approaches and research-based strategies used throughout the program. It begins with a discussion of the Comprehensive Mathematics Instructional Framework (CMI) stating “The CMI framework developed by the BYU-Cites public school partnership provides access to research-based principles and practices of teaching mathematics through problem solving and inquiry. This framework includes a teaching cycle and learning cycle connected within a continuum of mathematical understanding. The CMI Framework focuses task instructional implementation on the future—where the learning can go—through what is referred to as The Teaching Cycle. Also within the framework is careful attention to The Learning Cycle continuum of conceptual, procedural, and representational. By using the Teaching Cycle, teachers guide students through the Learning Cycle in order to help them progress along the Continuum of Mathematical Understanding.” Also included in the Course Guide are explanations of instructional routines and Mathematical Language Routines (MLR) used throughout the series. 

In each Teacher Lesson, specific information about how to employ various routines and strategies is provided. For example, in Math 1, Lesson 5.4, in the Launch Narrative, specific guidance is provided around Speaking: MLR 8 Discussion Supports. The guidance suggests that some students might benefit from sentence frames to prompt their thinking or support their thinking. These suggestions are specific to the lesson context. Any lesson throughout the series that includes a routine or an instructional routine will have explanations specific to the lesson in addition to the general explanations provided in the Course Guide. In Math 2, Lesson 6.3, as part of the Launch Narrative, the routine “Pause and Record” is described in the lesson context to help teachers understand how to implement the routine specifically for this lesson.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

Course documents provide an Integrated Materials List which lists all materials needed separated by course. In the unit overviews, information about digital tools and other materials that will be helpful for certain lessons are provided at the end of each overview. Each lesson also provides a list of materials needed specifically for that lesson.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for having assessment information included in the materials to indicate which standards are assessed.

The materials include multiple formal assessments for each unit including Self Assessments, Quick Quizzes, Unit Tests, and Performance Assessments. Formative assessments at the lesson level include Exit Tickets. Examples of how the content standards are consistently identified for all assessments, except Self Assessments, include:

• At the end of each Quick Quiz, there is a list of standards aligned to each item in the quiz. For example, in Math 1, Unit 9, the Quick Quiz for Lessons 9.1-9.5 lists the standards by item 1. S-ID.6, 2. S-ID.8, 3. S-ID.7, 4. S-ID.6a, 5. S-ID.6a, 6. S-ID.8, and 7. S-ID.9. 

• At the end of each Unit Test, a list of standards aligned to each item is provided. For example, in Math 2, the Unit 10 Test identifies that question 9 aligns to three different content standards: S-CP.1, S-CP.4, and S-CP.7. All items are listed and have one to three content standards listed for alignment. 

• The Teacher Notes of the Performance Assessments contains Core Standards Focus, which lists the standards addressed in the task. For example, in Math 3, Unit 5 Performance Assessment, the Teacher Notes lists G-MG.1-3, G-GMD.4, and G-SRT.9-11 as aligned to the task. 

• The Exit Tickets included for each lesson identify Focus Standards which are addressed in the problem(s). For example, in Math 1, Lesson 5.3, the Exit Ticket is aligned to A-REI.12 as the Focus Standard. 

Examples of the materials identifying the Standards for Mathematical Practice (MPs) in many of the Performance Assessments include: 

• In Math 1, Unit 3, the Performance Assessment is aligned to F-IF.1-5 and MPs 1, 2, 3, 6, and 8.

• In Math 2, Unit 3, Performance Assessment, the Teacher Notes identify content standards A-SSE.3, A-REI.4, and F-IF.8 and MP7.

• In Math 3, the Unit 6 Performance Assessment aligns to F-TF.5 and MP4.

The materials reviewed for Open Up High School Mathematics 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.

The materials include an array of assessment types and opportunities to assess student understanding for each unit. Each lesson includes Exit Tickets and Retrieval, Ready, Set, Go problem sets. Each unit includes Self Assessments, Quick Quizzes, a Unit Test, and a Performance Assessment. 

Examples of different types of modalities used for student assessments include:

• In Math 1, Unit 6 Test, students graph, express transformations algebraically, and justify their answers. The Unit 6 Performance Assessment provides an example of students playing a game that includes an expectation of precise mathematics (MP6) as well as engagement in math practices by all participants. Specifically, the instructions state, “Following the end of the game, each player needs to write a justification describing how they know each quadrilateral they recorded on their recording sheet is actually a quadrilateral. The recording sheets will be turned in and graded for accuracy and completeness of the justifications. This step needs to be completed by each player, regardless of the number or type of quadrilaterals that were completed during the playing of the game.” 

• In Math 2, Unit 9, Performance Assessment, students work individually or in pairs to: write equations of different conic sections given specific information about each, describe the important features of the graphs, sketch the graphs, and find intersections of different graphs. Students prove the points of intersection they have identified are correct for problem 5.

Examples of different types of items used for student assessments and how they are used to measure student performance include:

• In Math 1, Unit 8, Quick Quiz 1 assesses student knowledge of the key features of quadrilaterals and the properties of parallel and perpendicular lines. In problems 1 and 2, students select four points that form a given quadrilateral then prove how they know the chosen points form the given quadrilateral. Problem 3 assesses students’ knowledge of the properties of parallel lines. The notes provided for teachers indicate that students are expected to use slope in their explanation of why the two lines are parallel (MP6). 

• In Math 2, Unit 10, Performance Assessment, students analyze the data from a 2-way table, using probability notation, a Venn diagram and a Tree diagram in order to determine if the test is useful to identify whether a patient has the condition. Students must state and justify their conclusion (MP3) based on the data. 

Examples of how assessments address complexity include:

• In Math 3, Unit 1 Test, problem 2 states “f(x) and g(x)are inverses of one another and drawn on the same graph with the same scale on both the horizontal and vertical axes. Which of the following would be true?” There are four multiple choice responses offered. While only one is true, the remaining three distractors test the student’s clear understanding of inverse functions and vocabulary related to transformations. If students choose any of the three distractors, the teacher would have additional insight into any misconceptions.

• In Math 2, Unit 3 Test, Problem 2, students write functions in standard form and factored form for functions which are given as graphs (one of the functions has no real solutions). Students utilize MP2 as they shift from a graphical to a symbolic representation of the functions, consider the units involved, and attend to the meanings of the root quantities that must be present in the factored form.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning course-level mathematics.

The Open Up HS Math Course Guide provides guidance on strategies and accommodations for special populations outlining best practices to support all students, as well as students with disabilities (SWD), English Language Learners, and students in need of enrichment. The Open Up HS Math Course Guide states “The philosophical stance that guided the creation of these materials is the belief that with proper structures, accommodations, and supports, all children can learn mathematics. Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students. While the suggested supports are designed for students with disabilities, they are also appropriate for many students who struggle to access rigorous, course-level content. Teachers should use their professional judgment about which supports to use and when, based on their knowledge of the individual needs of students in their classroom.” Examples of where and how the materials provide specific strategies and supports for differentiating instruction to meet the needs of students in special populations include:

• In Math 1, Lesson 4.6, students solve linear inequalities. The Launch Narrative in the Teacher Notes states “Representation: Language and Symbols: Encourage students to use the Takeaway section of their notebook that contains previously learned facts about inequalities including examples of how to read statements with inequalities correctly (for example, 1>x can be read as “1 is greater than x” or “x is less than 1”). Providing this resource to students can support them in decoding the symbols as well as recalling information from long-term memory. Therefore, students can focus using their short-term memory which is where problem-solving and computation takes place.”

• In Math 1, Lesson 9.2, students solidify understanding of correlation coefficients and develop linear models for data. In Explore, Teacher Notes, the Students with Disabilities (SWD) Support is identified as Action and Expression: Organization; Memory. In the Explore Narrative, this support is explained, and teachers provide students with a graphic organizer to assist students in staying organized and better reflect on the specific task. 

• In Math 2, Lesson 2.6, students factor trinomials using rectangular area models and the distributive property. In Explore, Teacher Notes, the SWD Supports are listed as Representation: Visual-Spatial Processing; Attention, Engagement: Attention; Organization; Social-emotional functioning. The listed supports are explained in the Explore Narrative. For Representation, teachers allow students to manipulate algebra tiles or use Desmos to assist students in visualizing a new concept and develop greater understanding. Teachers also encourage students to create the area diagrams without the tiles or technology. The Engagement support suggests students choose two or three problems on which to focus, which can improve task initiation.

• In Math 3, Lesson 8.3, teachers support students with disabilities as students choose the method they find most useful for understanding the graph of a bungee jump (time vs height). The options include a mental model, a graph on paper, or a graph with technology.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for providing extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity.

Each course in the series includes Enrichment lessons identified with an “E.” Based on the information provided in the Open Up HS Math Course Guide, these lessons “align primarily with CCSSM (+) standards and/or engage in mathematics that goes beyond the expectations of the standards (^). The content is not required for students to engage in the full set of non-enrichment lessons. Some non-enrichment lessons may include (+) standards and /or (^) if the content is related to the mathematics of the lesson and can be explored simultaneously with the non-plus standards of the lesson. Enrichment opportunities are distributed throughout the curriculum as natural extensions of the mathematics of the units. Consequently, the mathematical ideas of the Enrichment lessons are accessible to all students.” The use of these lessons is left to the discretion of the teacher; the guidance does not indicate whether the lessons should be used for the whole class or for specific groups of students. However, some units that include Extension lessons also include a second copy of the Unit Test which includes additional problems that address the Enrichment lessons. Examples of lessons that enrich and/or extend the learning of the course-level mathematics include:

• In Math 1, Lessons 4.7-4.9, students use matrices as a means to organize and manipulate data. This is the first time students have been introduced to the concept of a matrix. Students build, add, scale and multiply matrices over the three lessons. The data presented for use is very similar to the cafeteria information from previous (non-enrichment) lessons in the Unit. These lessons cover the N-VM.C+ standards. There are two versions of the Unit 4 Test. The Unit 4 Test E includes 8 additional questions (for a total of 18 questions) which align to the standards in Lessons 4.7-4.9.

• In Math 2, Lesson 3.8, students work with complex numbers as points and vectors as they justify operations with complex numbers. Students bring an understanding of a vector from Math 1 and will take the understanding of complex numbers to Math 3 (roots of polynomial functions). 

• In Math 3, Unit 4 is focused on rational functions and expressions and addresses F-IF.7d+ and A-APR.7+, along with additional non-plus supporting standards. Each lesson in the unit is labeled as an Enrichment lesson. This unit extends student understanding of polynomials by using them to create rational functions. Students extend transformations from quadratic functions in Math 2 and logarithmic and polynomial functions in Math 3 to rational functions. Finally students develop their own method to solve a rational equation.

Across the series there are problems that require higher levels of complexity and/or extend the mathematics beyond the scope of the standards, these problems are expected of all students. Examples include:

• In Math 2, Lesson 10.4, students calculate probabilities based on data from a two-way table. Students then write and verify conjectures to develop the definition of a conditional probability. 

• The Performance Assessments provide opportunities for students to engage in grade-level content at a higher level of complexity as students often explain and/or justify their thinking as part of the solution to the task. For example, in Math 3, Unit 9, Performance Assessment, Problem 6, students use a simulation to see if there is evidence to suggest that athletes from 2012 were faster than normal. There are multiple explanations students can provide to justify their reasoning, and different simulations can provide different data. 

• Each lesson provides at least one problem in a section titled Ready for More? which extends the mathematics of the lesson. For example, in Math 1, Lesson 4.6, students practice solving linear inequalities. The Ready for More? extends the mathematics of the lesson when students use reasoning skills to solve two absolute value inequalities. While the Open Up Math HS Materials Guidance indicates that “fast finishers should be encouraged to work on these extensions,” there is no specific guidance around students who demonstrate a need for an extension of the mathematics in the lesson.

The materials reviewed for Open Up High School Mathematics 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 Open Up HS Math Course Guide includes a section titled Supporting English Language Learners which states “Embedded within the curriculum are instructional supports and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). Therefore, while these instructional supports and practices can and should be used to support all students learning mathematics, they are crucial to meeting the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English.”

The Course Guide also indicates that the implementation of Mathematical Language Routines (MLRs) will support ELL students. The Course Guide states “The mathematical language routines (MLRs) were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language.” The eight MLRs included in the materials are: MLR1 Stronger and Clearer Each Time; MLR2 Collect and Display; MLR3 Clarify, Critique, Correct; MLR4 Information Gap; MLR5 Co-Craft Questions and Problems; MLR6 Three Reads; MLR7 Compare and Connect; and MLR8 Discussion Supports.

Examples of the materials utilizing the MLRs and additional supports for EL students based on the language demands of the lesson and examples of appropriate support and accommodations for EL students that will support their regular and active participation in learning mathematics include:

• In Math 1, Lesson 2.1, there are two opportunities for students to Pause and Record, which is an intentional break in the lesson where students record insights from the teacher to formalize concepts and describe it with the appropriate vocabulary and notation in order to develop their vocabulary and knowledge of mathematical concepts. During the Discuss portion of the lesson, students first Pause and Record vocabulary around the domain of a function and arithmetic sequences. The second opportunity for Pause and Record allows students to record takeaways from the lesson concerning geometric sequences, and discrete and continuous functions.

• In Math 1, Lesson 6.2, the Launch Narrative describes MLR2 Collect and Display. Teachers circulate throughout the classroom listening to students’ discussions and recording key words and/or phrases related to justifying whether lines are parallel. The words and/or phrases are then displayed for the class so students can refer to, build on, or make connections with the concepts during future discussions.

• In Math 2, Lesson 5.2, MLR 8 Discussion Supports is utilized. Teachers provide sentence frames for students to prompt their thinking or provide support to explain their thinking. After students have time to develop their thoughts, they share their thoughts with a partner rehearsing what they will share with the whole group. Rehearsing allows students to clarify their thinking further.

• In Math 2, Lesson 10.2, the Launch narrative describes MLR5 Co-Craft Questions. Teachers present a scenario with data without presenting the problems from the lesson. Students then craft one or two mathematical questions that could be asked about the data provided. Students share their questions with partners. Some students share their questions with the class. This allows students to spend some additional time making sense of the data provided and understanding what information is missing which will, in turn, increase their ability to answer the questions posed in the lesson. 

• In Math 3, Lesson 1.3, the Launch Narrative describes MLR6 Three Reads. Students will read the scenario, including problems 1 and 2, three times. The first read is to be able to tell the story without mathematical details. The second read is to find the mathematical details (both what they know and what they want to know). The third read is to find an idea on how to get started with solving the problems. This can help students make sense of the problem and decipher language as they discuss the scenarios with partners after each read. 

• In Math 3, Lesson 9.11, students use MLR1 Stronger and Clearer Each Time. Students develop a written first draft as a response to a prompt, meet with two to three partners where each partner gets a turn to be a speaker and listener, and students write a second draft of their response revising their thinking based on the conversations with their partners. Students defend their position on whether the data presented provides evidence that students perform better on tests when listening to music.

The materials reviewed for Open Up High School Mathematics 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.

The series uses physical and/or virtual manipulatives to help students develop understanding of mathematical concepts. Some manipulatives---such as colored pencils, scientific calculators, and Desmos---are used routinely by students at their discretion to support their learning and explain their understanding. Examples of the manipulatives and how they are used to help students develop understanding of a concept include:

• In Math 1, Lesson 1.1, students use colored pencils to shade cubes in ways that represent sequences. In Lesson 4.8E, students use colored pencils or markers to highlight how the elements from each of the matrices connect in matrix multiplication.

• In Math 1, Unit 6, each lesson uses a physical or virtual manipulative. In Lessons 6.1-6.6, Black Line Masters (BLMs) support students as they develop definitions of geometric transformations, use geometric descriptions to transform figures, and specify sequences of transformations that map one figure onto another.

• In Math 2, Lesson 6.1, students use a rubber band stretcher to develop the concept of dilations. There is also a GeoGebra app provided that replicates the activity. 

• The materials list for Math 2, Lesson 8.8 indicates two GeoGebra apps that are intended to support students develop their conceptual understanding of Cavalieri’s Principle. 

• In Math 3, Lesson 5.1 recommends a host of manipulatives to assist students as they explore cross sections of 3-D geometric solids. The materials list includes play-doh and dental floss for slicing solids, transparent 3-D figures to which water can be added, a sealed jar containing a colored liquid that can be tilted to illustrate possible cross sections, and flashlights for creating shadows of objects.

• In Math 3, Lesson 7.4, an alternative graphing activity is provided for Explore question 4, where students cut varying lengths of spaghetti to represent line segments for specific angles of rotation, then glue those pieces to a large graph. This allows students to create a physical model of the line segment used to represent the value of the tangent for a particular angle of rotation.

• In Math 3, Lesson 7.8E, the BLMs include images of the polar and coordinate planes as well as location and angle specification cards to help students as they engage with proofs and applications of trigonometric identities.

• In Math 3, Lesson 9.9, Launch, students find a margin of error and a plausible interval for a sample proportion using a simulation where student pairs are given a bag of dark and light colored beans representing artifacts more than 1000 years old and artifacts less than 1000 years old, respectively. Using physical manipulatives for this simulation helps students develop conceptual understanding around creating an interval that is likely to include the population proportion.

Examples of how manipulatives are connected to written methods include:

• In Math 1, Lesson 1.1, students “draw multiple diagrams with the checkerboard pattern such as a 3 ⨉ 3, 4 ⨉ 4, 7 ⨉ 7, etc., or use manipulatives to see patterns as the checkerboard increases or decreases.” Students then turn to a partner to use the following prompt to explain what they notice about the pattern: “When I looked at the diagram, I noticed _________________ and so I ____________________.” Students then “create numeric expressions that exemplify their process and require students to connect their thinking to the visual representation of the tiles.”

• As part of the alternative graphing activity in Math 3, Lesson 7.4, Explore task, question 4, students draw the unit circle on a large sheet of paper so that they can indicate how the line segment is defined in that context as well. The activity then allows students to practice using appropriate tools strategically (MP 5) by prompting them to refer back to their unit circle using sentence frames such as: “I used the unit circle as a tool to think about ____ by ____,” and “I used the unit circle to calculate ____.”