The materials reviewed for Open Up High School Mathematics Traditional 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. “Some lessons are devoted to developing a concept, others to solidifying understanding, yet others to practicing mathematics.” For example, in Algebra 1, Unit 4, each lesson builds upon the previous to reinforce concepts:

• In Algebra 1, Lesson 4.1, students develop conceptual understanding by explaining each step in the process of solving an equation (A-REI.1).

• In Algebra 1, Lesson 4.2, students solidify their understanding by rearranging formulas to solve for a variable (N-Q.1, N-Q.2, A-REI.3, A-CED.4).

• In Algebra 1, Lesson 4.3, students practice solving literal equations (A-REI.1, A-REI.3, A-CED.4). 

The materials systematically support the development of conceptual understanding as students build on the previous day’s learning with each new lesson.

• N-RN.3: In Algebra 2, Lessons 3.5 and 3.6 develop the concept of irrational numbers. The students begin by plotting rational numbers on a number line and then move on to plotting irrational numbers on the number line. This activity helps students understand the similarities and differences between irrational and rational numbers. Once students establish this understanding in Lesson 3.5, the students develop their understanding of the properties of irrational numbers in Lesson 3.6. The materials address irrational numbers and their properties over two lessons, so students have the opportunity to develop a more thorough understanding of the mathematical concept.

• F-IF.7: In Algebra 1, Lesson 7.1, students develop an understanding of transformations on a graph and how a graph relates to a corresponding equation. Students explore the changes of a graph in relationship to the area of a square. By the end of this lesson, students identify the key features of the graph and how changes to a corresponding equation will change the graph. This development is continued in Algebra 1, Lesson 7.2, where “students write and graph quadratics with multiple transformations.”

• G-CO.10: In Geometry, Lessons 3.1, as students explore ways of knowing the triangle interior angles sum theorem - one based on experiments with specific triangles and the other based on a transformational argument - they consider the difference between making a conjecture based on experimentation versus reasoning about the validity of the conjecture using diagrams. In Lesson 3.2, students generate proofs for specific cases showing that the base angles of isosceles triangles are congruent; and in Lesson 3.3, students prove that the base angles of isosceles triangles are congruent for all cases.

• G-GPE.B: In Geometry, Lessons 7.1-7.3, students engage in a complete learning cycle. In Lesson 7.1, students find the distance between two points; and in Lesson 7.2, students prove that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals. In Lesson 7.3, students apply coordinate geometry to quadrilaterals to prove that a given quadrilateral is a parallelogram, a rectangle, a rhombus, or a square.

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Examples in the materials where students independently demonstrate procedural skills include, but are not limited to:

• N-CN.7: In Algebra 2, Lesson 3.5, Retrieval, Ready, Set, Go, students solve six quadratic equations, three of which have complex solutions (problems 28, 31, and 32). In Algebra 2, Lessons 3.6-3.8 and Lesson 4.5, students solve additional problems that have complex solutions.

• A-SSE.2: In Algebra 2, Lesson 2.3, students derive the product, quotient, and power rules for logarithms by graphing various logarithmic functions and generalizing the patterns found in corresponding equations. In Retrieval, Ready, Set, Go, problems 1-12, students practice rewriting radical expressions with fractional exponents and logarithmic expressions with radical exponents. In problems 15-16, students practice applying the rules of logarithms by identifying which of the given expressions are, in fact, equivalent, and showing why. In problems 17-22, students practice converting exponential equations into their logarithmic equivalents, and vice versa. In Algebra 2, Lesson 4.6, students use the given features of a polynomial to find other features, such as writing the function in factored form when given the graph of the function or its roots. Because only some information about the function is provided, students write the equation of the polynomial from scratch or write the equation in a different form than is given in order to find the other missing information. Students continue practicing these types of problems on their own in Retrieval, Ready, Set, Go, problems 10-15. In problems 1-8, students also practice rewriting rational expressions by dividing out common factors from the numerator and denominator. 

• F-BF.3: In Algebra 1, Lesson 7.1, students complete scaffolded questions about the effect on the graph of f(x) by replacing it with f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative). In Retrieval, Ready, Set, Go, students complete problems to develop and demonstrate their procedural skill. In Lessons 7.2 and 8.3-8.4, students write equations and graph functions that have multiple transformations for quadratic and absolute value functions, respectively. Students continue to write equations and graph functions that have multiple transformations for cubic, rational, logarithmic, and trigonometric functions in Algebra 2, Units 2-8.

• G-GPE.7: In Geometry, Lesson 7.1, students determine the distance between two points on the coordinate plane. At times they are only required to calculate the distance vertically or horizontally. Other times, students are asked to determine the distance between two points on a diagonal. Students use the Pythagorean Theorem to determine the distance and then formalize that understanding into the distance formula. Students practice with the procedure and formalize their thinking (strategies) conceptually. The procedural use of the distance formula is reinforced in Geometry, Lesson 7.3, where students prove that figures are parallelograms by comparing the lengths of opposite sides.

• S-CP.3: In Geometry, Lessons 9.4-9.6, students connect conditional probability and general probability and come to understand the meaning of independence, using tests to determine if two events are independent, and using the Multiplication Rule to find probabilities for independent events. In Lesson 9.4, Retrieval, Ready, Set, Go, problems 9-14, students find the conditional probability in contextual situations. In Lesson 9.5, Retrieval, Ready, Set, Go, problems 7-10, students use formulas for P(A and B) and P(A|B) to determine independence; in problems 11-22, students find general and conditional probabilities. In Lesson 9.5, Retrieval, Ready, Set, Go, problems 5-15, students find general and conditional probabilities and determine independence within a range of contexts.

The materials reviewed for Open Up High School Mathematics Traditional 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. 

Lessons often begin with an engaging scenario that is either a direct, real-world application of the content for that day or a unique, novel problem for the students to solve. The applications within the series support students as they develop their conceptual understanding of the mathematics and subsequent acquisition of abstract notation and procedural skills. There are also scenarios that recur throughout the series. As a result, students contextualize many different mathematical ideas to the same scenario.

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

• A-CED.3: In Algebra 1, Lessons 5.1, 5.2, 5.4, and 5.5, and within a pet-sitting context, students use systems of equations and inequalities to build a business model, minimize cost, and maximize profit. 

• A.REI.3: In Algebra 1, Lesson 4.1, students “develop insights into how to extend the process of solving equations to multistep equations” associated with the actions of Elvira, a cafeteria manager. This storyline continues in Lesson 4.2, where students “apply the equation solving process developed in the previous task to solving literal equations and formulas” as Elvira continues to work to improve the efficiency of the cafeteria.

• F-IF.4,5: In Algebra 1, Lesson 3.2, students use tables and graphs to interpret key features of functions (domain and range, increasing and decreasing intervals, 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.

• F-IF.7: In Algebra 2, Lesson 5.1, students write, graph, and solve rational equations in the context of winning the lottery. Students compare different points on the equation and graph based on different ways of splitting the prize money. In Retrieval, Ready, Set, Go, problems 7-10, students interpret an equation and a graph to determine different ways of paying for a gift among friends.

• F-TF.5: In Algebra 2, Lesson 6.1, students will use right triangle trigonometry to develop a function for finding the height of a rider at any position on a stationary Ferris wheel with specified midline (center) and radius (amplitude). In Algebra 2, Lesson 6.2, students model the circular motion of a rider as the angle measures the amount of rotation around a moving Ferris wheel, creating a function that models the amplitude, period, and average height (or midline) of the rider.

• G-SRT.8: In Geometry, Lesson 4.10, students determine the height of a tree using angle of elevation and shadows. Students work within the same scenario to determine unknown angles of depression and elevation. In the Retrieval, Ready, Set, Go problem set, students work multiple real-world problems using trigonometric ratios to determine missing lengths and angles.

• S-ID.2: In Algebra 1, Lesson 9.7, students select the bridge design that will support the most weight upon calculating and comparing the mean and standard deviation of their weight-bearing data. In Algebra 1, Lesson 9.8, students calculate measures of center (mean and median) and spread (interquartile range and standard deviation) of test results of six classes; by analyzing these results, students determine which class performed the best on the test and wins free pizza. 

The materials reviewed for Open Up High School Mathematics Traditional 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.

Throughout the materials students engage in each of the aspects of rigor in a cycle: each unit contains at least one Developing Understanding lesson to build conceptual understanding in students, at least one Solidifying Understanding lesson to build on that conceptual knowledge, and at least one Practicing Understanding lesson. Real-world applications are often incorporated in these cycles. Within each lesson there are Retrieval, Ready, Set, Go problems that spiral procedural skills for students. Examples include:

• In Algebra 1, Lesson 2.1 (a Developing Understanding lesson), 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 multiple representations. In Algebra 1, Lesson 2.2 (a Solidifying Understanding lesson), students discern when it is appropriate to represent a situation with a discrete or continuous model, thus deepening their conceptual understanding. Within this lesson, students also practice modeling with mathematics by connecting the type of change (linear or exponential) with the nature of that change (discrete or continuous), thus supporting student development of procedural skill and fluency. Throughout both of these lessons, students learn the concepts and procedures through their application to real-world contexts (e.g., medicine metabolized within a dog’s bloodstream and pool filling/draining). (F-IF.3, F-IF.5, F-BF.1, F.LE.1, F.LE.2)

• In Algebra 1, Lessons 8.5-8.7, students work with inverse functions and engage in all three aspects of rigor. The emphasis of this learning cycle, students work with inverse functions. In Lesson 8.5 (a Developing Understanding lesson), students compare the two different ways of modeling the same real-world context (two friends who ride bikes for exercise) to see characteristics of inverse functions, such as: the graphs are reflections over y = x, the inputs and outputs in the tables are reversed, and the equations have reciprocal slopes with the variables reversed. In Lesson 8.6 (a Solidifying Understanding lesson), students find inverse functions from equations and graphs. In the culminating lesson of the unit, Lesson 8.7 (a Practice Understanding lesson), students combine the work of piecewise functions, inverse functions, and absolute value functions using tables, graphs, and equations. (F-LE.2, F-LE.3, F-LE.5, F.BF.2, F-IF.5, F-IF.7, F-IF.9, A-REI.10)

• In Geometry, Lesson 6.4 (a Develop Understanding lesson), students use the formulas for arc length and area of a sector, developed in Lesson 6.3, and proportional reasoning to calculate the ratio of arc length to radius. Students use this ratio to define a constant of proportionality for any given central angle that intercepts arcs of concentric circles and develop radians as a way to measure an angle. Students conduct this work as they help Madison design her circular garden. In the Retrieval, Ready, Set, Go problem set, students practice the procedures by answering problems related to area, circumference, and the arc length of circles. (G.C.5)

• In Geometry, Lesson 7.2 (a Solidifying Understanding lesson), students use the slope formula and determine the relationship between parallel and perpendicular lines. Students engage with procedures by finding the slope between two points. Students then think conceptually when comparing the slopes of two lines to determine if the lines are parallel, perpendicular, or neither. Students also use rotations and transformations to hypothesize about the relationships between the slopes of lines that have undergone a transformation. (G.GPE.B, G.GPE.5)

• In Algebra 2, Lesson 5.2 (a Solidifying Understanding lesson), students engage with the same function f(x) = 1/x, but with a primary focus on the procedures of graph transformations and writing equations from graphs, without incorporating application problems. The lesson begins with practice problems that activate students’ prior knowledge about transformations of functions addressed in earlier lessons in the course. The lesson then revisits a graph of the function f(x) = 1/x and students name the asymptotes and anchor points. Later, given a set of problems containing either a graph or a description of a function that is a transformation of f(x) = 1/x, students write the equation for each. In yet another problem set in the lesson, students match a given equation to one of the given phrases that describes a transformation from y = 1/x. (F-IF.7d+, A-CED.2) 

The materials reviewed for Open Up High School Mathematics Traditional  series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6) in connection to the high school content standards.

Examples of where students make sense of problems and persevere in solving them include, but are not limited to:

• In Algebra 1, Lesson 3.1, students make sense of a story context to make a graph of a function without any knowledge of shape or scale. Working within the context of a pool’s water level over time, students persevere in making sense of increasing and decreasing intervals, minimum and maximum values, and domain and range. 

• In Geometry, Lesson 2.1, students construct two shapes: a rhombus and a square. Within the materials, teachers are prompted to provide students with enough time to explore the constructions fully. Within this lesson, students make sense of the constructions and persevere through the process of making the construction.

• In Algebra 2, Lesson 1.1, students make sense of two different views of a problem situation, each using a different dependent and independent variable. Because students who attempt to model the problem using the same variables in both cases will likely encounter confusion, students have an opportunity to persevere and in doing so see the importance of defining and making consistent use of variables as well as highlighting features of a relationship between a function and its inverse.

Examples of where students attend to precision include, but are not limited to:

• In Algebra 1, Lesson 2.2, students identify the domains of two sequences. As students “connect the type of change---either linear with a constant rate of change or exponential with a constant change factor, with the nature of the change, either discrete or continuous”---they further conceptualize that an arithmetic sequence is a linear function with a domain that is limited to the natural numbers and that a geometric sequence is an exponential function with a domain of the natural numbers. In Lesson 2.2, problems 6-11, students discuss a situation that is not clearly discrete or continuous and attend to precision as they make mathematical arguments about modeling choices.

• In Geometry, Lesson 1.1, students attend to precision in their language for transformations. Students use precise definitions for each of the transformations so the final image is a “unique figure, rather than an ill-defined sketch.” The materials prompt students to see how precision is needed when defining geometric relationships to make sure that images are well defined. 

• In Algebra 2, Lesson 8.3, “students naturally attend to precision as they attempt to refine the parameters of the equations they write to model the data given in graphs and tables.” Students become aware of how small changes in some parameters of certain equations result in large differences in results, while other parameters may be less sensitive to small changes. 

The materials reviewed for Open Up High School Mathematics Traditional series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. Throughout the materials, there are many opportunities for students to critique the reasoning of others and to reason abstractly.

Examples where students reason abstractly and quantitatively include, but are not limited to:

• In Algebra 1, Lesson 4.4, students are given a statement and determine which of the two expressions represent a larger value. In addition, students reason abstractly about an expression, compare it to another expression, and explain their reasoning.

• In Algebra 1, Lesson 7.1, students reason abstractly by relating the numeric results in a table to the graphs and explaining how the graph is transformed. Students examine the abstract relationships between the different representations (table, graph, and function) and how a change in one form impacts a change in the other.

• In Geometry, Lesson 6.2, students reason about how an infinite process might converge on a unique value and examine the case of how an inscribed regular polygon with more and more sides converges on the shape of a circle. This limiting process provides an informal proof for the circumference and area of a circle.

• In Algebra 2, Lesson 4.2, students identify the characteristics of and graph the basic cubic function. Students come to 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 problem set, students reason quantitatively by substituting in values to compare different power functions. Students reason abstractly by making generalizations based on their knowledge of exponents.

Examples where students construct viable arguments and critique the reasoning of others include, but are not limited to:

• In Algebra 1, Lesson 3.2, students interpret two representations (a table and a graph) and determine if Sierra’s statements are correct. Within the lesson, students analyze the situations, justify their reasoning, and communicate their conclusions to others. Students listen to the reasoning of others and decide if they make sense.

• In Geometry, Lesson 4.8, students consider examples of right triangles and explain why their specified ratios are the same, or nearly the same. Students identify and justify that having one acute angle congruent in all of the triangles creates a set of similar right triangles. Students use this observation to hypothesize the trigonometric relationships for all right triangles. Students determine how the ratios are related and construct an argument for what they believe about the trigonometric ratios.

• In Algebra 2, Lesson 6.4, students work in partners to apply their understanding about sinusoidal functions to the construction of their own Ferris wheel in terms of radius, height, period, and direction of rotation, along with its graph and equation. Students then share their responses with another pair and decide if they agree with the connections they each have made between the three representations of each of their Ferris wheels: the description, the graph, and the function equation. Students are given sentence frames to provide support for constructing viable arguments and critique the reasoning of others; the sentence frames allow students to explain the connections in the representations of their Ferris wheel and provide supporting evidence for their claims.

The materials reviewed for Open Up High School Mathematics Traditional 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. 

Examples where students engage in modeling include, but are not limited to:

• In Algebra 1, Lesson 5.3, students manipulate a system of equations to model the constraints of setting up a cat and dog boarding business. They determine the best use of space to provide maximum profit. Students also have to understand what terms in their expressions are related to the different constraints. From this, they derive various forms of the equations to determine maximum profit.

• In Geometry, Lesson 6.1, students analyze a plan to build a regular, hexagonal gazebo. In the plan, there are several statements with which students must agree or disagree and then design their own gazebo.

• In Algebra 2, Lesson 8.3, students model two real-world behaviors: a dampened oscillation (the up and down motion of the bungee jumper) and Newton’s Law of Heating. To do so, students draw on mathematics they know well, such as how to model periodic behavior and exponential growth and decay, along with their emerging understanding of combining functions to model more complex behavior. Students use the graphing calculator to test and refine their assumptions until they have an accurate model that fits the data given in the tables and graphs provided in the lesson.

Examples where students choose appropriate tools strategically include, but are not limited to:

• In Algebra 1, Lesson 2.8, students compare linear and exponential growth related to two small companies. They are encouraged to use a calculator or spreadsheet to determine if this growth is continuous or discrete.

• In Geometry, Lesson 2.2, students use the circle as a tool to create congruent line segments. Students also consider transformations as tools to think about congruence when creating mappings.

• In Algebra 2, Lesson 2.5, students use various tools, such as tables, graphs, and technology, to find missing values for exponential functions. Because some problems require the use of a calculator whereas others do not, students 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 Traditional series meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards.

Examples where students look for and make use of structure include, but are not limited to:

• In Algebra 1, Lesson 8.3, students recognize that the operation of absolute value leads to a graph with two distinct sections. They relate the two sections of the graph to their experiences with piecewise functions and write equations in both piecewise and absolute value forms. Students use the structure of the graph to write piecewise functions and relate the structure of piecewise function rules to equivalent absolute value functions. 

• In Geometry, Lesson 1.6, students uncover the structure of regular polygons through experimenting with rotational symmetry and line symmetry. They notice that different types of line segments serve as lines of symmetry in regular polygons with an even number of sides versus those with an odd number of sides. 

• In Algebra 2, Lesson 3.4, students gain insight into the different types of roots for quadratic functions by making connections across multiple representations. Students connect information included in a table with the vertex and factored forms of a quadratic equation. Students use graphs to explain why roots of a quadratic function might be integers, non-integer rational numbers, irrational numbers, or non-real numbers.

Examples where students look for and make use of repeated reasoning include, but are not limited to:

• In Algebra 1, Lesson 8.6, students use the reasoning of how a function and its inverse are related to observe how they can write the inverse function by paying attention to the operations in the original function and the order in which they are performed. Students become familiar with characteristics of an inverse function and how to restrict are prompted to see that when finding an inverse you can sometimes just “undo” the operations in the opposite order of the original function.

• In Geometry, Lesson 1.2, students make a conjecture about the relationship between the slopes of perpendicular lines based on the collection of data obtained in four experiments. They also provide an informal justification as to why the conjecture is true by examining the rotation of the legs of a right triangle and relating this to the rise and run that determines slopes of the hypotenuse of a right triangle before and after rotation.

• In Algebra 2, Lesson 4.1, students apply previous experiences with functions and their representations to reason throughout this lesson. Students reason about the different rates of change and create recursive forms of equations as they seek generalizations. For example, students understand that a cubic function has a first difference that is quadratic, a second difference that is linear, and a third difference that is constant. Students generalize this pattern for all polynomial functions.

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 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 ____.”