High School - Gateway 3

The materials reviewed for Open Up High School Mathematics Traditional 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 lesson uses a consistent format to provide teacher guidance on how to present the student and/or ancillary materials in a way that engages students and guides their mathematical development. For example, the instructional materials, which are divided by section (e.g., Jump Start, Launch, Explore, Discuss, Exit Ticket), include Groupings; Routines; and Narratives. Examples of how the instructional materials provide teacher guidance on how to present the student and/or ancillary materials include:

• In Geometry, Lesson 2.4, where students justify triangle congruence criteria using rigid transformations, the materials for the Jump Start indicates a duration of 10 minutes and Individual and Whole Class groupings. Annotations about possible student misconceptions are also included. Specifically, the materials indicate “Students may rely upon it 'looks like they are congruent' as their justification. If so, draw a second triangle, ∆ABD, by sliding point A slightly left or right. Such a triangle may appear to be congruent, but only measuring sides and angles to verify they are the same length would verify that they are so.” The Launch Narrative suggests that teachers introduce Proof by Contradiction as a way to help students think about how their arguments support their claims. 

• In Algebra 2, Lesson 7.8 the instructional materials include Unit Circle and game cards Black Line Masters (BLMs) to support student engagement with the learning objectives as well as students’ understanding of the mathematical concepts through visual representations. Related annotations provide guidance for questions that prompt student thinking, narrative remarks to connect one student’s process to another student’s process, and tips for transitioning from the game card activity to the whole class discussion. For example, the Launch Narrative includes the note: “Some equations require students to use trigonometric identities as part of the solution process. This work is the ‘practice’ part of the task, so watch how adept students are at this work, and select particularly difficult work for the whole class discussion.” The Explore Narrative includes Think-Pair-Share Teacher Notes and suggestions for Selecting and Sequencing Student Thinking while making connections to the Whole Class discussion.

Each lesson uses a consistent format to provide teacher guidance on how to plan for instruction in a way that engages students and guides their mathematical development. For example, at the beginning of each lesson, teacher guidance includes Learning Goals (Teacher), Learning Focus (Student), Standards for the Lesson, Materials (when necessary), Required Preparation (which includes Anticipate Student Thinking), BLMs, Progression of Learning, and Purpose. In the course of the lesson, the materials also include Anticipate & Monitor and Selecting, Sequencing, & Connecting charts. Examples of how the instructional materials provide teacher guidance on how to plan for instruction include:

• In Algebra I, Lesson 5.8, which addresses A-REI.5, the materials include Anticipate & Monitor as well as Selecting, Sequencing, & Connecting charts so that teachers can plan to guide students’ understanding of the mathematical concepts both individually and as a whole class. The materials prompt teachers to listen for students making arguments in Monitoring Student Thinking. One example for teachers to listen for is: “For problem 1, since Carlos bought the same number of bags of Tabitha Tidbits, the difference in the total cost must be due to the two extra bags of Figaro Flakes.” In Connect Student Thinking, students generate a list of key points to demonstrate their understanding of underlying processes that helped them to solve each scenario. One possible key idea is: “If the number of one type of item is the same in both purchases, then the difference in total cost is due to the difference in the number of the second item purchased.”

• In Geometry, Lesson 8.5, the instructional materials include Required Preparation notes that suggest using a quick quiz to assess student understanding of the previous lessons and to forego the exit ticket. The materials also remind the teacher to make copies of the quick quiz prior to class. In addition, the instructional materials include timing guidance.

The materials reviewed for Open Up High School Mathematics Traditional 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 how the materials support teachers to develop their own knowledge of more complex, course-level concepts include:

• At the end of the Algebra 1, Unit 3 Overview, teachers learn how to write features of functions using interval notation. In addition, there is an explanation about how interval notation connects to set notation, which is the notation that students have been using in previous lessons. 

• In the Teacher Notes, there is an Anticipate & Monitor Chart that gives explanations to guide student thinking. Within these explanations, teachers are able to improve their own knowledge of the subject. In Geometry, Lesson 2.4, teachers are given an explanation in the Teacher Notes with the transformations that can prove two triangles congruent using SSS criteria, including a diagram. 

• In Algebra 2, Lesson 5.3, the Selecting, Sequencing, & Connecting chart offers multiple possibilities for how students might respond to Explore question 9, which asks them to draw conclusions about the degree of the numerator and denominator in a rational function, the location of asymptotes for the rational function, and the function’s intercepts. In unpacking a solution in which a student claims that the x-intercept can be found by setting the function equal to 0, the teacher notes offer sample follow-up questions that would allow students to test that proposed solution, then offers additional clarification about why those questions are relevant in the context of that solution. Using adult language, the teacher note explains that although the student’s solution is correct, setting just the numerator equal to zero would be a more efficient approach to the problem.

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 Algebra 1, Lesson 2.9 is connected to Rate of Change.

• In Geometry, Lessons 6.1 and 6.2 are connected to Limits, and Lessons 8.2 and 8.3 are connected to Riemann Sums and the Definite Integral.

• In Algebra 2, Lessons 6.4, 6.5, and 7.1 are connected to Parametrically-defined curves.

The materials reviewed for Open Up High School Mathematics Traditional 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 allow coherence across multiple course levels and to allow a teacher to make prior connections and teach for connections to future content. Examples include but are not limited to:

• In the Course Guide, the Standards Alignment for AGA lessons indicates where the standards can be found within the lessons and Retrieval Ready, Set, Go problem sets. 

• The Course Overview describes the purposeful design of each course. For example, the Algebra 2 Course Overview explains how the course is designed to build upon student understanding of functions developed in Algebra 1 and how the work in the first three units reviews students’ learning of linear, quadratic, and exponential functions, thus preparing them to engage with other functions throughout the course. The Course Overview highlights that the major purpose of Algebra 2 is for students to use and apply the accumulation of learning they have gained from their previous courses, in that students build on their understanding of right triangle trigonometry from the Geometry course, as well as their knowledge of functions and how to manipulate them from Algebra 1.

• The Progression of Learning section of the instructional materials situates the intended learning outcomes of the current lesson within the continuum of past and future learning. For example, in Geometry, Lesson 9.1, the materials indicate that the lesson builds on students’ prior experience with using tree diagrams to find probabilities from Grade 7 (7.SP.6; 7.SP.7b; 7.SP.8a-c) to introduce conditional probability (S.CP.5, S.CP.6). The materials also indicate Supporting Standards (S.CP.3, S.CP.4) that will be a focus in ensuing lessons. In addition, Algebra 2, Lesson 1.1 explains how the lesson revisits and builds upon concepts related to inverse functions, which were introduced in Algebra 1, Unit 8. It further explains how the lesson fits into the unit as a whole: how students will later explore the inverse of an exponential function, leading to the concept of logarithms which are explored in more depth in the following unit.

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 Geometry, Lesson 1.4, the materials identify Focus Standards G.CO.1, G.CO.2, G.CO.4 and Supporting Standards G.CO.5 and G.CO.6 related to the content. The lesson also identifies the MPs 3, 6, and 7. In the Exit ticket, G.CO.4 is identified as the content standard to which the lesson should build.

• In Algebra 1, Lesson 1.9, the Selecting, Sequencing, and Connecting chart includes a sample solution indicating that a student might determine that there are two possible common ratios when there is an even number or jumps of one possible common ratio when there is an odd number of jumps. The materials include a teacher note to emphasize MP3 by asking the whole class “What do you think about this claim?” and encouraging students to test the claim with a couple of problems.

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

The Open Up High School Math Course Guide is replete with information that explains the instructional approaches of the program as well as identifies research-based strategies that are used in the design. Specific examples include, but are not limited to:

• In “About These Materials,” the materials describe the student engagement, the classroom experience, and teacher role that one can expect when implementing a problem-based mathematics curriculum.

• In “Design Principles,” the materials highlight the Comprehensive Mathematics Instructional Framework (CMI), which provides access to research-based principles and practices of teaching mathematics through problem solving and inquiry. It also refers to the Teaching Cycle and the Learning Cycle. “By using the Teaching Cycle, teachers guide students through the Learning Cycle in order to help them progress along the Continuum of Mathematical Understanding.”

• In “The Five Practices,” the materials state “Every lesson follows the framework for organizing task-based instruction as described in The 5 Practices for Orchestrating Productive Mathematics Discussions (Smith M., & Stein M.K., NCTM 2018), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014).”

• In “Routines,” explanations for the implementation of two routines are highlighted.

• In “Instructional Routines,” the what? and why? of four routines (e.g., Notice and Wonder) are included.

• In “Supporting All Students,” the materials state that “Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students.” The Design Principles and Design Elements for All Students as well as support for English Language Learners are explained in detail.

• Eight Mathematical Language Routines (MLRs), selected because they are effective and practical for simultaneously learning mathematical practices, content, and language, are explained in detail. Sources are cited.

• “Supports for Students with Disabilities” offers additional strategies for teachers to meet the individual needs of a diverse group of learners. The supports for students with disabilities were developed using the three principles of Universal Design for Learning (http://udlguidelines.cast.org/): Engagement, Representation, and Action and Expression.

• In “Assessment and Analysis,” the materials describe the “wide variety of assessment resources are provided as a tool to assist teachers.”

Teacher Edition: 5 Practice Charts, which is separate from the Course Guide, provides guidance particular to the Launch; Explore; Anticipate & Monitor Chart; and Selecting, Sequencing & Connecting chart.

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

The “AGA Materials List” specifies the materials list for each course. The document specifies physical materials (e.g., highlighters for Algebra 1, compasses for Geometry, and Dry Spaghetti for Algebra 2) as well as suggests digital tools (e.g., GeoGebra for Algebra 1 and Geometry as well as Desmos activities for Algebra 2).

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

Explicit assessment resources include Quick Quizzes, Self Assessments, Unit Tests, and Performance Assessments. Lesson-based, formative assessments include Exit Tickets and listening to students as they work on the tasks. The materials identify the content standards for all types of assessments, and the Standards for Mathematical Practice (MPs) are identified for many of the Performance Assessments.

Examples of the materials identifying the MPs include:

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

• In Geometry, seven of the nine Unit Performance Assessments are aligned to content standards and practices, and in Units 7 and 9, the Performance Assessments are aligned to content standards.

• In Algebra 2, the Unit 6 Performance Assessment aligns to F-TF.5 and MP4, and the Units 8 and 10 Performance Assessments also align to MP4. The Unit 7 Performance Assessment aligns to MP8.

Throughout the series, the Quick Quizzes, Unit Tests, and Exit Tickets align to the content standards. Examples include:

• Quick Quizzes, which assess student learning over a cluster of lessons within a unit, indicate a content standard alignment at the problem level. For example, in Algebra 2, Quick Quiz 2.1-2.2, Problem 8, students estimate the value of a logarithmic expression that does not have an integer value. The materials clearly indicate that the question aligns to F-BF.5+, as students show that they understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

• Unit Tests consistently identify the content standards that are assessed across the courses. For example, in Algebra 1, Unit 9 Test includes the content standard alignment at the problem-level.

• Exit Tickets are lesson-based assessment opportunities that identify the content standards. The Course Guide indicates that across the series, “every Exit Ticket includes a short narrative that describes it, any focus or supporting standards, and solutions.” For example, in Algebra 2, Lesson 1.1, the Exit Ticket identifies F-BF.4 as the Focus Standard, which aligns to the task in that students are asked to find the inverse of each function given: one each in a table format, an equation, and a graph.

The materials reviewed for Open Up High School Mathematics 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.

The Open Up High School Mathematics Traditional series includes an array of assessment types and opportunities to test learning 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 Algebra 1, Unit 7, the Performance Assessment assesses students’ ability to identify the key features needed to write a quadratic function in standard, vertex, and factored forms. In this task, students write a step-by-step process. 

• In Geometry, Unit 1 Test, students graph, express transformations algebraically, and justify their answers. The Unit 1 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 Algebra 2, the Unit 8 Test includes questions that require responses in a table format, in a graph, and as written (verbal) explanations. In Algebra 2, Unit 8 Performance Assessment, students sketch graphs, write equations to model the frequency of notes being played, and write verbal explanations as to why some notes sound better than others when played together.

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

• In Algebra 1, Unit 3 Performance Assessment, students match functions in various representations using a card sort. Specifically, students work individually or in pairs to organize the cards so that each set of three cards all describe the exact same relationship. The cards contain representations of functions that include tables; descriptions of domain, range, and other defining characteristics; graphs; function notation; sequences; equations; and real-world context.

• In Geometry, Unit 1, the Quick Quiz, students respond to six problems where one correct answer is anticipated. Students agree or disagree with conjectures and justify their response. In the Performance Assessment for this unit, students justify why certain shapes are quadrilaterals. In the Unit Test, students respond to a variety of problem types (e.g., algebra-based, discussion, constructed response, and justification) from previous work in the unit.

Unit Tests in the materials address complexity through question type as well as the way the question is worded to prompt student thinking. Examples of how assessments address complexity include the following examples from the Algebra 2, Unit 3 Test:

• Given the equations of functions f(x), g(x), and h(x), students select, from a list, all statements that are true about the closure rules for those functions (A-APR.1). Students engage in MP3 as they analyze the situation and apply examples and counterexamples to generalized statements.

• Given the graphs of functions f(x) and g(x) (one of which has no real solutions), students write the equations of both functions in standard form and in factored form (N-CN.8+, F-BF.1). 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.

• Provided a statement that (a+bi) and (a-bi) are complex, conjugate pairs, students name at least four things they know about complex, conjugate pairs (N-CN.1, N-CN.2). Although not explicitly stated in the materials, this question requires students to utilize MP2 as they apply quantitative and abstract reasoning when considering how the expressions are related, how they could be manipulated, or what they might result in.

• Given an expression showing division between two polynomials, students write two equivalent expressions using multiplication statements. (A-APR.6)

The materials reviewed for Open Up High School Mathematics Traditional 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 Algebra 1, Lesson 9.1, Launch, the materials indicate support for engagement of SWD with an emphasis on Social-Emotional Functioning, Attention, and Organization. Teachers provide access by recruiting interest as students select one of seven data sets. This improves task initiation and encourages student connection to learning, and teachers separate the task into more manageable parts in an attempt to support students with additional processing time. This strategy also occurs in Algebra 2, Lesson 1.4, Explore, as students find the inverse of given functions by selecting 4 or 5 problems.

• In Algebra 1, Lesson 9.6, Launch, the materials provide the SWD support, Representation: Internalize Comprehension, where teachers encourage students to write ideas shared by other students on top of the corresponding graphs, highlight/circle the important features discussed, and provide specific things within representations for students to look for. The Narrative notes that differentiating instruction in this manner supports accessibility to Visual Spatial Processing and Conceptual Processing.

• In Geometry, Lesson 2.1, Explore, the materials provide the SWD support, Action and Expression: Conceptual Processing, Fine Motor Skills, which suggests providing access to GeoGebra or Sketchpad and notes that “technology tools can eliminate barriers and allow students to more successfully take part in the learning.” The Narrative indicates that this means of differentiating instruction supports accessibility for Conceptual Processing and Fine Motor Skills.

• In Geometry, Lesson 3.4, Launch, the materials indicate the SWD support, Engagement: Attention; Organization; Social-emotional functioning, and the Narrative specifies that teachers “invite students to select two or four classmates’ arguments when following the diagram and making the two column proof.”

• In Algebra 2, Lesson 1.1, Launch, the materials provide the SWD support, Representation: Access for Perception, which suggests presenting the contextual task both visually and auditorily to increase sense-making, improve comprehension, and encourage students to annotate the task.

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

The materials provide opportunities for students to investigate course-level content 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 (^).” As indicated in the Course Guide, the content of the enrichment lessons is not required for all students, although the mathematical ideas are accessible to all students. The enrichment tasks are available to use for all students at a teacher’s discretion.

Examples of opportunities specific to extending students' learning of the course-level content, include:

• In Algebra 1, Unit 5, every lesson has at least one Ready for More? that extends the mathematics of the unit: Systems of Equations and Inequalities. Lessons 5.11E and 5.12E are enrichment lessons that extend the work of the unit to N-VM.6(+). In addition, the materials include a Self Assessment that complements the two enrichment lessons.

• In Geometry, Lesson 7.12 includes guidance for the instructor that relates to the progression of learning. In Ready for More?, the materials provide the teacher with one additional practice problem that is intended to deepen or extend the mathematics of the lesson. The Course Materials Guidance indicates that fast finishers should be encouraged to work on these extensions.

• The third learning cycle of Algebra 2, Unit 7, is a set of enrichment lessons extending the work of the unit to include additional trigonometric identities and their applications to changing the form of trigonometric expressions and solving trigonometric equations (Lessons 7.7E to 7.10E). The final enrichment lesson introduces students to the polar form of complex numbers and uses polar form to extend the arithmetic of complex numbers to finding roots of complex numbers.

• All of Algebra 2, Unit 10 is an enrichment unit, with each lesson addressing almost exclusively plus standards (with non-plus supporting standards).

Examples of opportunities for students to engage in course-level content at a higher level of complexity.

• In Algebra 1, Lesson 8.5, where students engage with F-BF.4a and F-BF.4c(+), Problems 6 and 7 involve quadratic functions whose domains must be restricted so that its inverse will be a function. As the notion of restricted domains and conditions under which functions are invertible is a topic of Algebra 2, teachers are encouraged to amplify the possibility that an inverse of a function may not be a function.

• In Geometry, Lesson 7.12E, all of the focus standards are plus standards: N-VM.1-5. Appropriately, students represent quantities that have magnitude and direction using vectors, examine the arithmetic of vectors, and sketch vectors in the coordinate plane. The materials provide teachers with a chart that outlines the Selecting, Sequencing, and Connecting Whole Class Questions & Connections for students. Teachers are encouraged to consider the different tasks/skills for which students might be prepared as doing so allows the teacher to extend the task complexity when appropriate. There are examples of real-world applications of vectors that can be introduced when students show readiness.

• In Algebra 2, Lesson 8.4, students are presented a scenario at an amusement park in which the output of one function becomes the input of another. The task allows students to approach working the problems either by decomposing the scenario into its component functions and working with them individually and in sequence or by composing a single function that accommodates the sequence of computations. Teachers are instructed to monitor student thinking, as the students who create the composition function first (before constructing a table of values) are navigating the task via the plus standard F-BF.1c(+), a higher level of complexity compared to the other standards addressed in the lesson.

The materials reviewed for Open Up High School Mathematics 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 provide general guidance about how to support English Language Learners (ELLs) in the Course Guide and indicate specific Math Language Routines (MLRs) within some phases of some of the lessons. The materials indicate that the curriculum builds on foundational principles for supporting language development for all students. The Supporting English Language Learners section “aims to provide guidance to help teachers recognize and support students’ language development in the context of mathematical sense-making. More specifically, the materials indicate “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).” The supports and practices “are crucial to meeting the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English.”

The materials continue that “the framework for supporting English language learners in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” The materials identify where both student successes and challenges may be rooted in misconceptions in content versus language demands, through learning and assessment. The Course Guide describes how MLRs have been incorporated throughout the series because they are effective and practical for simultaneously learning mathematical practices, content, and 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 Algebra 1, Lesson 3.2, the Discuss Narrative provides guidance on how to implement the MLR2 Collect and Display. Teachers are instructed to listen and collect the language students use to describe strategies for finding key features of functions.Teachers create a two-column table with the headings “table” and “graph” to record student language in the appropriate column for strategies identifying range in each representation. Students may then “borrow” language from the display as needed.

• In Algebra I, Lesson 4.4, Discuss, Problems 2-5, students share their explanations for each of these problems using models from the Jump Start and create arguments as to why the properties of inequalities can be generalized to all real numbers. Students speak to the entire group and engage in question and answer with the teacher and their peers. In addition, students paraphrase the properties of inequalities.

• In Geometry, Lesson 3.3, the Jump Start Narrative includes general guidance about MLR2 Collect and Display. The Launch Narrative includes general guidance about implementing MLR6 Three Reads and indicates that “students who would benefit from additional support may follow along as the teacher or a single student reads these paragraphs.”

• In Geometry, Lesson 5.3, students engage in content related to inscribed angles and polygons as well as circumscribed polygons. Using MLR7 Compare and Connect, students consider individually the work of other students before verbally describing not only their visual representations of different inscribed and circumscribed polygons but also the visual representation of others work.

• In Algebra 2, Lesson 6.1, Discuss, the MLR3 Clarify, Critique, Correct routine provides students with a structured opportunity to analyze, reflect on, and improve their written work by correcting errors and clarifying meaning. The teacher selects one of the student’s descriptions of the procedure for finding the distance a point on a circle is above or below the center of a circle from problem 5, then asks a series of Revise and Refine prompts one at a time so that students have an opportunity to offer suggestions following each prompt. This revision process continues until students have a shared statement that can be recorded in the takeaways chart in their lesson materials.

• In Algebra 2, Lesson 8.2, Launch, students engage in MLR6 Three Reads, which supports reading comprehension of the context in a word problem. Students sketch a graph of the path of a rider on proposed thrill rides at a local theme park. Three Reads enables students to focus on comprehending the situation by reading only about the context in the first read, then adding the proposals in the second read, then identifying key mathematical features of the story and reading for those in the third read, with the goal of devising a strategy for starting on the problems. Students also engage in several writing exercises because the task has them explain their reasoning about the shape of each graph.

• In Algebra 2, Lesson 8.6, Explore, students engage in MLR8 Discussion Supports. Students use sentence frames to prompt their thinking or to provide support for explaining their thinking. To prompt students to reason abstractly and quantitatively (MP2), students are provided sentence frames such as: “I had to think about __ in order to ...”, “It made sense to me to __ when I ....” Students share their answers with a partner and are encouraged to rehearse what they will say when they share with the full group. Teachers are provided a note that rehearsing provides opportunities for students to clarify their thinking.

• In Algebra 2, Lesson 9.11, Launch, students engage in MLR1 Stronger and Clearer Each Time. Students formulate a written response, share verbally, then revise their written response helps students to synthesize their understanding of the data and supports sense-making.

The materials reviewed for Open Up High School Mathematics 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.

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 Algebra 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 Geometry, Unit 1, each lesson uses a physical or virtual manipulative. In Lessons 1.1-1.6, 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.

• The materials list for Geometry, Lesson 6.8 indicates two GeoGebra apps that are intended to support students develop their conceptual understanding of Cavalieri’s Principle. 

• In Geometry, Lesson 8.1 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 Algebra 2, 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 Algebra 2, 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 Algebra 2, 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 Algebra 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 Algebra 2, 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 ____.”