The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Every unit attends to the learning cycle, interweaving aspects of mathematical proficiency. Most of the lessons across the series are exploratory in nature and encourage students to develop understanding through questioning and various activities. Concepts build over many lessons within and between courses in the series. Examples include:

• N-RN.A: In Math 1, Unit 2, a contextual situation offers students the opportunity to understand how values of a dependent variable can exist on the intervals between the whole number values of the independent variable for a continuously increasing exponential function. Next, students examine the role of positive and negative integer exponents and begin to understand the need for rational exponents. Students further develop their conceptual understanding by verifying that the properties of integer exponents remain true for rational exponents.

• A-APR.B: In Math 3, Unit 3, students develop an understanding of multiplicity and a deeper understanding of the relationship between the degree and the number of roots of a polynomial. Then, students use their background knowledge of quadratic functions and end behavior to extend their understanding to higher-order polynomials. The polynomials in this unit are factorable and allow students opportunities to solidify their understanding of end behavior, the Fundamental Theorem of Algebra, the multiplicity of a given root, and what the multiplicity would look like graphically. Finally, students extend their understanding of the Fundamental Theorem of Algebra and the nature of roots by applying the Remainder Theorem.

• A-REI.A,B: Math 1, Unit 4 builds students’ conceptual knowledge by first introducing multivariable linear equations and then having students express given relationships in equivalent forms. Students engage with inequalities as they encounter the contextual need for inequalities. Students consider the differences and similarities between solving inequalities and solving equations, including that inequalities produce a range of solutions, the inequality symbol must be changed when multiplying or dividing by a negative number, and the reflexive property is true only for equations.

• G-GPE.1: Math 2, Lesson 9.1, students cut out triangles and pin them to a coordinate plane to build a unit circle effectively developing their understanding of the relationship between the Pythagorean Theorem and the equation of a circle at the origin. Students connect their geometric understanding of circles as the set of all points equidistant from a center to the equation of a circle. This task focuses on a circle (constructed of right triangles) with a radius of 6 inches in order to focus on the Pythagorean theorem and use it to generate the equation of a circle centered at the origin. After constructing a circle at the origin, students consider how the equation would change if the center of the circle is translated. 

• G-GPE.5: In Math 1, Lesson 8.3, students prove that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals. The proofs use the ideas of slope triangles, rotations, and translations and are preceded by a specific case that demonstrates the idea before students are asked to follow the logic using variables and thinking more generally.

• G-GPE.6: In Math 1, Lesson 8.2, students use similar triangles and proportionality to find the point on a line segment that partitions the segment in a given ratio. Students first find the midpoint of a segment using two possible strategies and use similar triangles to find segments in ratios other than 1:1. The formula for finding the midpoint of a segment is formalized during the discussion. The discussion can also be extended to derive a formula for finding the point that partitions a segment in any given ratio.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills and students independently demonstrate procedural skills throughout the series. Examples include: 

• N-CN.7: In Math 2, Lesson 3.4, students write 10 equations in vertex form, standard form and factored form. Five of these equations have complex roots. In Math 2, Lesson 3.5, students find complex roots of quadratic equations in problems 11, 13, and 14. In Math 2, Lesson 3.5 Retrieval Problem 2 has a complex solution and Go Problems 28, 31, and 32 have complex solutions. Additional practice is provided in the RRSG problem sets for Lessons 3.6-3.8. 

• A.APR.2: In Math 3, Lesson 3.5 students explore the relationship between the remainder and factor in problems 3-8 and make conjectures based on their work. In Math 3, Lesson 3.5 RRSG, students practice this concept in the Set Problems 10-20.

• A-CED.4: In Math 1, Lesson 4.2 students apply the equation solving process to solve literal equations and formulas. There are multiple opportunities to solve a formula for a given variable throughout the lesson and several practice examples in the RRSG exercises. Students are presented with multiple opportunities to practice this standard in the RRSG throughout the remainder of Math 1, Unit 4. Additionally, practice for this standard is also found in the RRSG exercises of Math 1, Lessons 5.5, 8.3, and 9.5. The standard is revisited in Math 2, Lessons 4.4 and 4.6.

• G-GPE.7: In Math 1, Lesson 8.1, students calculate the length of ribbon needed to create a specific pattern. Students then find the perimeter of a hexagon in a different pattern. In RRSG Set Problems 9-12 students find the perimeter of two triangles and two quadrilaterals. In Math 2, Lesson 9.3 RRSG Retrieval Problem 1 and Ready Problems 1-4 students also find the perimeter of polygons using ordered pairs in the coordinate plane.

• S-CP.3: In Math 2, Unit 10, students investigate conditional probabilities in a wide variety of different contexts and using different models such as tree diagrams, Venn diagrams, two-way tables, and formulas. Students then use what they know about conditional probabilities to determine whether events are independent. There are many opportunities for students to practice in the student tasks and Retrieval Ready, Set, Go (RRSG) problem sets.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. The materials use real-world situations in which students can apply mathematical concepts. In situations where a real-world context is not immediately appropriate, the materials begin with abstract situations (graphs, dot models, etc.) and build to the application of the concept in a real-world situation in a later task. Every lesson involves a task, and every task is a real-world situation or a mathematical model that builds to a real-world situation.

The series includes numerous applications across the series, and examples of select standards that specifically relate to applications include, but are not limited to:

• A-CED.3: In Math 1, Lessons 5.1 and 5.2, the pet-sitting problem uses systems of equations and inequalities to build a business model, minimize costs, and maximize profit.

• F-IF.4,5: In Math 1, Lesson 3.2, students use tables and graphs to interpret key features of functions (domain and range, where function is increasing/decreasing, x and y intercepts, rates of change, discrete vs. continuous) while analyzing the characteristics of a float moving down a river. Students interpret water depth, river speed, and distance traveled using the function skills they are developing.

• F-BF.1: In Math 2, Lesson 1.2, students develop a mathematical model for the number of squares in the logo for size n. Students are encouraged to use as many representations as possible for their mathematical model.

• F-TF.5: In Math 3, Lesson 6.2, students use the Ferris wheel to determine how high someone will be after 2 seconds, after observing that the Ferris wheel makes one complete rotation counterclockwise every 20 seconds. Students are continuing the work from a previous task in Math 3, Lesson 6.1. Students then determine elapsed time since passing a specific position. Students generate a general formula for finding the height of a rider during a specific time interval and are then asked how they might find the height of the rider for other time intervals.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Overall, the three aspects are balanced with respect to the standards being addressed. The instructional materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations by giving appropriate attention to developing students’ conceptual understanding, developing procedural skill and fluency, and providing engaging applications.

The materials engage students in each of the aspects of rigor in a pattern that repeats itself throughout the materials. Each unit contains Developing Understanding (conceptual understanding), Solidifying Understanding, Practicing Understanding tasks, in addition to the Retrieval Ready, Set, Go (procedural skill) problem sets.

For example, Math 1, Lesson 2.1, a Developing Understanding Task, focuses on conceptual understanding as students build upon their experiences with arithmetic and geometric sequences and extend to the broader class of linear and exponential functions with continuous domains. Students compare these types of functions using various representations (table, graph, and equation). In Math 2, Lesson 2.2, a Solidifying Understanding Task, students discern when it is appropriate to represent a situation with a discrete or continuous model, thus deepening conceptual understanding. This task also has students practice modeling with mathematics by connecting the type of change (linear or exponential) with the nature of that change (discrete or continuous) which develops students’ procedural skill and fluency. Throughout both tasks, problems are presented to students within real-world contexts (medicine metabolized within a dog’s bloodstream, library re-shelving efficiency, e-book download rate, savings accounts, pool filling, pool draining, etc), so students learn the mathematical concepts and procedures through the application of the mathematics.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of overarching mathematical practices (MPs 1 and 6), in connection to the high school content standards. Overall, MP1 and MP6 are used to enrich the mathematical content and are not treated as individual mathematical practices. Throughout the materials, students are expected to make sense of problems and persevere in solving them while attending to precision. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples where students make sense of problems and persevere in solving them include:

• Math 1, Lesson 3.1: Students are able to make sense of creating graphs given a situation. Students are already familiar with graphing rate of change and continuous and non-continuous situations. This task addresses domain and step functions. Students persevere in creating graphs by analyzing what is happening during each interval of time on their graph. 

• Math 2, Lesson 4.1: Students write piecewise linear functions and interpret piecewise functions presented in algebraic form. Throughout the task, students use the piecewise-defined function to make sense of the problem to identify key features of the function including average rates of change, domain, and other relative information connecting the context to the function. 

• Math 3, Lesson 7.2: Students interpret a given function to answer questions about high tide, low tide, and the time between tide events. Students make sense of the problem by using multiple representations such as graphs, tables, the unit circle, and the meaning of the parameters of a periodic equation to answer these questions. 

Examples where students attend to precision include:

• Math 1, Lesson 5.5: Students represent constraints in the context of a pet sitting business with inequalities and with systems of inequalities. Students find the point of intersection and must interpret its meaning in the context of cats and dogs. Students must attend to the language in the constraints. Students must recognize that time is measured in both minutes and hours in the constraints which require them to attend to the units they choose to use. 

• Math 2, Lesson 7.1: Students attend to the precision of language by using correct mathematical vocabulary when describing and illustrating their process for finding the center of rotation of a figure consisting of several image/pre-image pairs of points.

• Math 3, Lesson 5.1: Students visualize two-dimensional cross sections of three-dimensional objects and draw the cross sections of those objects. Students are encouraged to use precise language as they work through the task in order to precisely identify components of the three-dimensional objects used.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. Overall, MP2 and MP3 are used to enrich the mathematical content found in the materials, and these practices are not treated as isolated experiences for the students. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples where students reason abstractly and quantitatively include:

• In Math 1, Lesson 5.1, students solve systems of linear equations using a variety of strategies. Then students use the solution to make decisions based on a context. Students decontextualize the contexts when working with a system of equations to represent pricing plans for each company. Students must attend to units and the meaning of operations used in the equations. Students then contextualize the solution to verify it makes sense in the context. Students reason abstractly and quantitatively which company to hire by using a table, a graph, and/or Algebra.

• In Math 2, Lesson 1.4, students write a function for a given context. Students must reason quantitatively to generate the function based on a numeric pattern. Students are required to consider two parameters, perimeter and area, and they must write the function for the area using one variable. Students will decontextualize the model in order to identify the key features and then relate the key features to the story context. 

• In Math 3, Lesson 3.2, students identify the characteristics and graph the basic cubic function. Students will understand that the same transformations they used to graph quadratic functions can be applied to cubic functions. Students reason abstractly and quantitatively as they compare the rates of change and end behavior of quadratic and cubic functions. In the Retrieval, Ready, Set, Go practice set, students reason quantitatively by substituting in values to compare different power functions. They reason abstractly by making generalizations based on their knowledge of exponents.

Examples where students construct viable arguments and critique the reasoning of others:

• In Math 1, Lesson 3.2, students explain why they either agree or disagree with each observation Sierra made. Students listen to the reasoning of others and decide whether the reasoning makes sense. Students also justify or explain flaws in Sierra’s observations. 

• In Math 2, Lesson 6.3, students read through Mia and Mason’s conjectures about similar polygons and decide which they believe are true. Students are also presented “explanations” from either Mia or Mason and must write an argument deciding whether they agree. 

• In Math 3, Lesson 3.9, students find patterns in the end behavior of functions and describe the end behavior of functions using the appropriate notation. Throughout the task students will use prior knowledge, make conjectures, and develop a series of statements to describe the relationship between expressions as the input values approach various quantities. Students justify conclusions and respond to others by listening, asking clarifying questions, and commenting on the reasoning of others. 

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. Overall, MP4 and MP5 are used to enrich the mathematical content and are not treated as individual practices. Throughout the materials, students model with mathematics and use tools strategically. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples where students engage in modeling with mathematics include:

• In Math 1, Lesson 5.4, students use systems of equations, tables and graphs to model the start-up costs of a new business and the space available to board cats and dogs. Students write equations for the constraints in different forms and then identify what information each form reveals in the context of the situation. 

• In Math 2, Lesson 10.3, students use Venn diagrams to model the situation, analyze the data, and write various probability statements (unions, intersections, and complements) and then apply the Addition Rule and interpret the answer in terms of the model.

• In Math 3, Lesson 6.1 students develop ways of thinking about the location of points around a circle, which become fundamental in their understanding of trigonometric functions, radian measure, and the unit circle. Students develop expressions to model the height of a rider at a particular angle. Students apply mathematics they know (right triangle trigonometry) to model motion around a Ferris wheel based on angles of rotation. Students use points to model the location of the rider.

Examples where students choose appropriate tools strategically include:

• In Math 1, Lesson 9.1 students strategically use graphing technology to interpret the correlation coefficient of a linear fit. Using technology, students also alter the data to see the effect on the scatter plot and correlation coefficient. 

• In Math 2, Lesson 3.1, students solve quadratic equations graphically and using different algebraic techniques and make connections between solving quadratic equations and graphing quadratic functions. Throughout the student tasks, students have options for which method they would like to use. Sometimes the problems leave it up to the student to choose which method they would like to use, and specific methods are called out to be used in other problems. Students solve quadratic equations by factoring, completing the square, using inspection, graphing, and using symmetry. Students will have opportunities to select appropriate tools to use while solving and graphing. They can hand sketch graphs or construct graphs using graphing calculators or other digital platforms.

• In Math 3, Lesson 2.5, students solve exponential equations which would require the use of logarithms using tables and graphs. In the last part of the task, students solve systems of linear and exponential equations using a method of their choice. Students may choose to use calculators or other technology with base 10 logarithmic and exponential functions to complete the problems. Students are encouraged to make appropriate decisions about using technology, like finding exact values for log expressions without relying on a calculator when they can.

The materials reviewed for Open Up High School Mathematics Integrated series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. Overall, MP7 and MP8 are used to enrich the mathematical content, and these practices are not treated as isolated experiences for the students. There is an increasing expectation that these practices will lead students to experience the full intent of the standards. 

The materials frequently take a task from a previous course and add a new contextual layer to the mathematics, such as Math 1, Lesson 5.2, “Pet Sitters” and Math 3, Lesson 1.1, “Brutus Bites Back.” Students are constantly extending the structures used when solving problems that build on one another and, as a result, are able to solve increasingly complex problems. In the instructional materials repeated reasoning based on similar structures allows for increasingly complex mathematical concepts to be developed from simpler ones.

Examples where students look for and make use of structure include:

• In Math 1, Lesson 2.1, builds upon students’ previous experiences with arithmetic and geometric sequences to extend to the broader class of linear and exponential functions with continuous domains. Students use tables, graphs, and equations to create mathematical models for contextual situations. Students continue to define linear and exponential functions by their patterns of growth. Students repeatedly use similarities and differences between situations to define the appropriate expression structure. 

• In Math 2, Lesson 4.3, students learn how to graph, write, and create linear absolute value functions by looking at structure and making sense of piecewise defined functions. They connect prior understandings of transformations, domain, linear functions, and piecewise functions and share strategies for how to go from one representation to another in order to graph and write equations for absolute value piecewise functions. 

• In Math 3, Lesson 3.7, students use their background knowledge of quadratic functions and end behavior to extend their understanding of polynomials in general. The polynomials in this task are easily factorable and allow students opportunities to solidify their understanding of end behavior, the Fundamental Theorem of Algebra, and the multiplicity of a given root, and what that would look like graphically. 

Examples where students look for and express regularity in repeated reasoning:

• In Math 1, Lesson 5.9, students practice solving systems of linear equations by obtaining systems of equivalent equations. By the end of the task, through repeated practice, students develop a procedure for solving a system of equations by elimination. 

• In Math 2, Lesson 3.2, students work with specific examples of quadratic functions in order to identify a process for locating the x-intercepts relative to the axis of symmetry. Students use the repeated reasoning after working the specific examples to develop the quadratic formula.

• In Math 3, Lesson 6.6, students calculate the x and y coordinates for stakes placed on concentric circles as well as the arc length on each circle placed around an archeological site. In problems 3 and 4 students look for patterns in the tables created and the processes used to calculate points and identify what the repeated patterns imply. 

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 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 ____.”