### Alignment

The instructional materials reviewed for Eureka Traditional partially meet expectations for alignment to the CCSSM for high school. For focus and coherence, the series showed strengths in the following areas: attending to the full intent of the mathematical content contained in the standards, spending the majority of time on the content from CCSSM widely applicable as prerequisites, requiring students to engage in mathematics at a level of sophistication appropriate to high school, making meaningful connections in a single course and throughout the series, and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. For rigor and the mathematical practices, the series showed strengths in the following areas: supporting the intentional development of students' conceptual understanding, opportunities for students to develop procedural skills and fluencies, utilizing mathematical concepts and skills in engaging applications, and displaying a balance among the three aspects of rigor.

#### Focus & Coherence

MEETS EXPECTATIONS

**Criterion 1a-1f**

- 0
- 10
- 14
- 15
- 18

Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).

The instructional materials reviewed meet the expectations for focus and coherence. The materials attend to the full intent of the mathematical content contained in the high school standards for all students, but they partially attend to the full intent of the modeling process. The materials, when used as designed, allow students to spend the majority of their time on the WAPs, but due to extensions beyond the non-plus standards, the materials partially allow students to fully learn each standard. The materials do engage students in the mathematics at a level of sophistication appropriate to high school; make meaningful connections in a single course and throughout the series; and identify and build on knowledge from Grades 6-8 to the high school standards.

##### Indicator 1a

The materials focus on the high school standards.*

##### Indicator 1a.i

The materials attend to the full intent of the mathematical content contained in the high school standards for all students.

The materials meet the expectation for attending to the full intent of the mathematical content contained in the high school standards for all students. For standards that span across a course or series, the development of the standard within a course and across the series was examined to determine if all aspects of the standard were addressed. Some specific examples where the materials attend to all aspects of a standard either within a course or across the series are shown below.

F-IF.4: The materials fully develop F-IF.4 through the course of four modules and extending into a fifth module. In Algebra I, the standard is developed for linear, quadratic and exponential functions including modeling opportunities. The materials further develop this standard in Algebra II from the foundation provided through previous learning experiences in Algebra I. In Algebra II, this standard is extended to trigonometric and logarithmic functions. It would be helpful if the materials identified the intended standard at the lesson level rather than just at the topic level. (Note: student books do not identify standards at all.) Courses are organized into modules, topics and then lessons. F-IF.4 is addressed in multiple locations, shown below, across Algebra I and II.

- Algebra I, module 3: topics B and D
- Algebra I, module 4: topics A and B
- Algebra I, module 5: topics A and B
- Algebra II, module 2: F-IF.4 is not explicitly noted in the teacher guide, but numerous opportunities to identify key features from a table, graph or context are noted throughout this module.
- Algebra II, module 3: topic C

G-CO.1: This standard was fully developed in module 1, topics A and G, for all concepts listed in the standard. The module started with students completing constructions to develop precise definitions of angle, circle, perpendicular line, parallel line and line segment. Students are asked to recall, compare, and expand upon geometric definitions from previous grades (Grades 4-8). Then students are asked to make conjectures and write proofs. As students progress through the module they extend this knowledge in developing definitions of rotations, reflections, and translations. (G-CO.4).

A-SSE.2: This standard is fully developed over the span of three modules: in Algebra I, module 1, topic B and module 4, topic A, then in Algebra II, module 1, topics A and B.

- In the Algebra I modules, students use and develop the concepts of algebraic properties by rewriting expressions and looking at patterns, such as the distributive and associative properties. Then, in module 4, students use the properties of exponents to support the development of understanding and procedures for multiplication of polynomials and factoring.
- In Algebra II, students expand on these concepts from Algebra I and extend them to higher degree polynomial operations and algebraic manipulations. Each series of lessons begins by having students focus on rewriting polynomials or decomposing numbers to identify patterns and then expanding upon those patterns to construct new understandings.

For teachers utilizing these materials, note that some standards are addressed more fully in the teacher discussion notes, and routine use of the teacher discussion notes is essential. Otherwise, there is the potential for some standards to receive superficial treatment. For example, the discussion notes on page 16 of the teacher edition for Algebra II, module 2, lesson 1, suggest that the teacher have students review and explain their thinking and consider revisions to the methods used in completing exploratory challenge 1 as they gain greater understanding of F-IF.7 and F-TF.5. Without using these teacher notes, the students may not necessarily have the opportunity to engage with all aspects of these two standards.

There were a few assessments which tested material before it was actually addressed.

- In the Algebra I, module 1, mid-module assessment, A-SSE.1a and A-SSE.1b were not covered in the material but were on the assessment and a part of the rubric.
- In Algebra II, module 2, F-TF.3 and F-LE.2 were tested in the mid-year assessment but not covered until the second half of the module. These standards were re-assessed at the end of the module.
- In Algebra II, module 4, S-CP.3 was assessed in the mid-module assessment but not covered until the second half of the book.

##### Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.

The materials partially meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. The review identified multiple units and assessment items across the series where the modeling standards are addressed to varying levels. There are some modeling standards for which the opportunities for student engagement in the modeling process do not meet what is intended as described in the CCSSM for high school mathematics: “Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.” The CCSSM further describes modeling as having attributes such as choice, decision-making, creativity, estimation, drawing and validating conclusions, design and re-design, as well as reasoning and communicating.

In addressing the modeling standards, many lessons limited students to analyzing relationships presented by the teacher or materials, reflecting on results and making adjustments, if needed. Many lessons were structured with learning opportunities that were teacher-led and contained step-by-step instructions for students with minimal opportunities for creativity, estimation and student choice of math concepts and skills to combine and utilize for problem solving. Examples of modeling standards where the full intent of the modeling process was not met include the following:

A-CED.3 was addressed in Algebra I, module 1, topic C. The lessons start by leading students to solve problems using a specific method utilizing provided graphs, tables and fill-in-the-blank equations. The lessons are teacher-led and then move to more open-ended questions that ask students to utilize some components of the modeling process. The students experience the modeling process in a highly-structured manner by following predetermined steps disclosed by the teacher or the materials. If the early lessons of Algebra I, module 1, were amended by placing the tables, graphs and other modeling hints or directives in the teacher notes as facilitation guidelines, then students would have the opportunity to engage with the full intent the modeling process.

A-CED.4 was addressed in Algebra I, module 1, topic C, lesson 19. This lesson is constructed as a teacher-led lesson containing non-contextual problems, falling short of attending to the modeling process.

F-LE.1a, F-LE.1b and F-LE.1c were addressed in Algebra I, module 3, topic A or module 5, topics A and B. Students compared graphs and equations and wrote equations, but student actions were directed by the teacher and/or materials with little opportunity for student choice, creativity, design or re-design. Some modeling components were completed by the students, but some questions were exclusively in the teacher notes and not in the student activity. Without using the teacher notes, the students may not necessarily have the opportunity to engage in the full modeling process with these standards.

F-LE.4 was addressed in Algebra II, module 3-topics B, C and D. The exit tickets have modeling questions for students, but the actual lessons are teacher-led and scaffolded leaving little in the way of possibilities for multiple solution pathways.

There were lessons where the materials did attend to the full intent of the modeling process when applied to modeling standards. Two examples describing those lessons are shown below.

In Geometry, module 3, lesson 1 provides an exploration to find out how much paint is needed to cover an oval area where different approaches are used to determine the area of the figure. The students use the modeling process in order to determine an approach that is appropriate for working with an oval. Students compare and contrast different approaches to calculate the area of the figure and how much paint is needed. This exploration is used as the materials begin to lay the foundation for G-GMD.3 in topic B of the module.

In Algebra II, module 2, lessons 1 and 2 use physical models (a paper plate) and mathematical models (graphs) to represent and explore the real world situation of the height of a passenger car on a Ferris wheel. Students are prompted to create a sketch of the height of a passenger car on a Ferris wheel as it rotates four times. The way in which the challenge is presented allows students to make some decisions and assumptions as they begin to complete the challenge. Ensuing questions and discussion allow for additional student choices and re-designs in order to find a better model or to make the model meet new information, for example a specific diameter. The execution of the questions in the teacher notes and ensuing discussions help to provide the full intent of the modeling process in these lessons.

##### Indicator 1b

The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.

##### Indicator 1b.i

The materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

The materials meet the expectations for, when used as designed, allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs, and careers. Overall, the series does devote a majority of instructional time to the WAPs.

Algebra I contains five modules of instruction, and three of the modules are almost completely focused on the WAPs. Most of the WAPs are addressed in the Algebra I instructional materials, and the WAPs primarily addressed in the Algebra I materials are the non-plus standards from A-CED and F-IF.

- A-CED is addressed in topics C and D of module 1; topic D of module 3; topics A, B, and C of module 4; and topics A and B of module 5.
- F-IF is addressed in topics A, B, C and D of module 3; topics A, B, and C of module 4; and topics A and B of module 5.

All of the lessons within Geometry address standards from the Geometry category, and although topics were addressed that could be considered part of a 4th-year course, these lessons were not distracting from the time spent on the WAPs. The WAPs from the Geometry category are largely addressed in the first two modules of the Geometry materials, and the Geometry materials, as a whole, do spend a majority of time on the WAPs from the Geometry category.

- G-CO.A is a part of topics A, C and G of module 1.
- G-CO.B is a part of topics C, D and G of module 1.
- C-CO.9,10 are found in topics B, E and G of module 1.
- G-SRT.A,B,C are found in topics A, B, C, D and E of module 2.

In reviewing the Algebra II instructional materials, most of the lessons aligned to the WAPs. Although topics were addressed that could be considered part of a 4th-year course, these lessons were not distracting from the time spent on the WAPs. The WAPs are largely addressed in the first and third modules of the Algebra II materials, and the Algebra II materials, as a whole, do spend a majority of time on the WAPs.

- A-SSE is included in topics A and B of module 1 and topics D and E of module 3.
- F-IF is addressed in topic B of module 1; topics A and B of module 2; and topics A, C, D and E of module 3.
- F-BF.1 is included in topics A, B, C, D and E of module 3.

##### Indicator 1b.ii

The materials, when used as designed, allow students to fully learn each standard.

The materials partially meet the expectation for allowing students to fully learn each standard, when used as designed. Overall, the materials provide limited opportunities for students to engage with some standards, and some materials are distracting as they extend beyond the non-plus standards.

Standards with multiple components, such as A-REI.4 and A-REI.11, were examined carefully within the materials. There were instances within the series where standards with multiple components were addressed through a series of lessons or revisited across the series, building logically toward full depth of understanding for all students. However, for some standards, a series of lessons or revisiting a standard across the series did not occur. Examples of standards that have a limited number of opportunities for students to engage with the standard in a focused manner are:

- In Algebra I, module 3, lessons 8 through 10 focus on standards F-IF.1,2. These two standards include the formal definition of a function, the use and interpretation of function notation, and the graph of a function,
*f*, is the graph of the equation*y = f(x)*. These three lessons represent a limited opportunity for students to focus on and fully learn F-IF.1,2 as they pertain to all types of functions. - In Algebra II, module 1, a series of six lessons addresses standards within A-REI, culminating in a lesson on solving simple rational equations (A-REI.2). The series of lessons follows a logical progression through simplifying rational expressions; adding, subtracting, multiplying and dividing rational expressions; and solving rational equations in mathematical and real-world contexts. The culminating work in solving rational equations is addressed in one lesson. Though the series of lessons addresses the prerequisite skills students would need when solving rational equations, students would have a limited opportunity to fully learn A-REI.2 for rational equations.

There are lessons that include problems which are distracting to students as they extend beyond what is required by the non-plus standards. Examples of where the materials are distracting include, but are not limited to:

- In lesson 5 of module 2 in the Algebra I materials, students calculate the standard deviation for a set of data by hand, and in lesson 7 of the same module, students use the
*1.5 X (IQR)*method for determining outliers. These topics extend beyond what is required by standards in S-ID.A. - Set-builder notation and pseudocode are included in lessons 11 and 12 of module 3 in the Algebra I materials. These topics extend beyond what is required by standards in F-IF.A.
- In the middle of lesson 14 of module 3 in Algebra I, the teacher materials introduce and have students discuss geometric mean, but geometric mean is an extension beyond the standards in F-IF which are aligned to the lesson.
- Points of concurrencies are addressed by lesson 5 of module 1 in Geometry, and these are extensions beyond G-CO.1, G-CO.1.12 and G-CO.1.13, which are the standards aligned to the topic for this lesson.
- Toward the end of lesson 10 in module 4 of Geometry, the materials have students find the area of a hexagon using coordinates, but this figure is beyond what is in G-GPE.7, which includes finding the area of triangles and rectangles.
- Lesson 15 of module 4 in Geometry has students develop a formula for the distance that a point is from a line that is given by the equation
*y = mx + b*. The proof that is used to develop the formula extends beyond G-GPE.4 and G-GPE.6 which are the standards aligned to the lesson. - During lesson 39 of module 1 in Algebra II, the materials ask students to solve a fourth-degree polynomial that has complex solutions, and this example extends beyond N-CN.7 which includes solving quadratic equations that have complex solutions.
- Lesson 7 of module 2 in Algebra II addresses the secant, cosecant, and cotangent functions which are extensions beyond any of the non-plus standards in F-TF.
- Lessons 11-13 of module 3 in the Algebra II materials include various properties of logarithms, problems that involve rewriting logarithmic expressions from one form to another, and justifications of logarithmic properties for all possible values of the base in the logarithmic expression. These topics extend beyond what is required by F-LE.4.

Time requirements for each component of the lessons are specifically noted throughout the teacher materials. The series provides a problem set at the end of every lesson, and class time for working on the problem sets is not included in the timings assigned to the lessons. The problem sets are intended to be utilized as homework assignments, and time for student questions or classroom discussion concerning the problem sets is also not included. The pacing for the course does include time for assessments with return for remediation or further applications.

##### Indicator 1c

The materials require students to engage in mathematics at a level of sophistication appropriate to high school.

The materials meet the expectation for requiring students to engage in mathematics at a level of sophistication appropriate to high school. Overall, the materials include problems for all students that are appropriate to high school, present contexts that motivate the mathematical content of the high school standards, and use types of numbers that are part of real-world scenarios.

In general, the series expects all students to engage with the materials through similar experiences at a level of sophistication appropriate to high school. There are no indications in the student materials of optional exercises or problem sets for students not ready for the course-level work, but there are some scaffolding notes in the teacher materials. For advanced learners, some lessons provide an extension problem, but when these extensions do exist, there is typically not more than one problem. For example, in Algebra I, module 1, lesson 9, the extension problem is: "Find a polynomial that, when multiplied by 2x^2 + 3x + 1, gives the answer 2x^3 + x^2 - 2x - 1." This is a reasonable extension problem as it stays on course level, and this is the only extension problem provided in the lesson.

Real-world contexts are used throughout the series to motivate the content of the high school standards. In some cases, the contexts are technical but do promote the mathematical content of the context. For example:

- In Algebra II, module 1, telescopes are used to assist in developing models for parabolas and to provide reasons why parabolic equations are needed.
- In Geometry, module 3 concludes with a lesson on 3-D printers which is a context that promotes Cavalieri’s principle and the volume of various three-dimensional objects.
- In Algebra I, module 5, students bring together multiple standards from throughout the course as they solve problems in settings that include fish populations in a lake, investing money, and the concentration of medicine in a patient’s blood.

The numbers included in the instructional materials are appropriate for high school. Types of numbers used within problems do not just consist of positive integers or integers, and answers to problems are not always integers. For example:

- Lesson 24 of module 4 in Algebra I includes a data set for a problem that has rational numbers in tenths and produces a solution that has rational numbers to the ten-thousandths place.
- Lesson 33 of module 2 in Geometry uses rational numbers to the hundredths place for lengths of sides and angle measurements that produce irrational results when working with trigonometric functions.
- Lesson 28 of module 3 in Algebra II uses rational numbers as data points, along with the number
*e*, when discussing Newton’s law of cooling.

##### Indicator 1d

The materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

The materials meet the expectation for fostering coherence through meaningful connections in a single course and throughout the series, where appropriate and where required by the standards.

Examples of connections that are in the series:

- Solving quadratic equations in Algebra I with solving rational equations in Algebra II;
- Factoring in Algebra I with radical expressions in Algebra II;
- Within Algebra I, creating equations that describe relationships with constructing linear and exponential models in modules 3 and 5;
- The Pythagorean theorem is used in both Geometry and Algebra II;
- Within Geometry, solving systems of equations with using coordinates to compute perimeters of polygons in topic A of module 4;
- Operations with radicals from Geometry are used again with radical equations in Algebra II; and
- Within Algebra II, representing data on two quantitative variables and modeling periodic phenomena with trigonometric functions in topic B of module 2.

The teacher materials communicate connections. Each module begins with an overview, which describes the standards that will be addressed, how those standards are connected to prior learning and how the work will help prepare students for subsequent lessons and courses. This information describes the intended flow of the module. The foundation and extension standards are shown along with new terms, familiar terms and symbols. The facilitator notes for each lesson are thorough and routinely make explicit to the teacher exactly how each example, discussion, activity, etc. connects to previous and subsequent learning. The teacher materials remind the teacher of the "big picture."

##### Indicator 1e

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.

The materials meet the expectation for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. When necessary, the materials reference prior knowledge in the teacher edition of the materials. Each unit across the series, especially for Algebra I and Geometry, identifies the "foundational standards" from Grades 6-8 that underlay the development of concepts in each module in each course. All of the Grades 6-8 standards that are identified as foundational standards in Algebra I and Geometry are standards that are built upon and extended to the high school standards in the high school series. The foundational standards are not explicitly taught within the lessons in the high school series, instead they serve as the basis for extending to the high school standards. These connections to the grades 6-8 standards are made explicitly for teachers, but not for students. The "foundational standards" in Algebra II are all from the Algebra I materials.

Some examples of where the materials connect standards from Grades 6-8 to the high school standards:

- Algebra I, module 2 is about statistics. The focus standards for this module address number line plots; shapes, centers, and spreads of distributions; categorical data; scatter plots, correlation, and linear regression equations. The Grades 6-8 standards that are identified as foundational for the focus standards include recognizing statistical questions; understanding the shape, center, and spread of a distribution; number line plots; summarizing numerical data sets; constructing scatter plots and estimating trend lines with linear equations. The discussion and work with scatter plots and regression equations do not duplicate what was addressed in middle school, instead the focus moves to to least-squares regression and deeper understanding of correlation.
- Geometry, module 2 lists familiar terms from previous courses in the teacher edition and notes connections to standards addressed in Grades 6-8 while also discussing how the standards progress in high school. The teacher notes detail how the material/lesson meets the intent of progression. In topic A, the teacher materials explain that, in Grades 6-8 (with a mention of a Grade 4 standard), the purpose was to for students to observe how dilations worked. The expectation in high school is to explain why dilations work.

##### Indicator 1f

The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.

The plus standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college- and career-ready. These standards are identified in the teacher materials as extension standards at the beginning of the module. The materials do not, however, directly identify these standards at the lesson level. Plus standards within the series include:

- Geometry, module 2, lessons 31-33 -- G-SRT.9,10,11 are fully addressed by the lessons listed.
- Geometry, module 3, lessons 10-12 -- G-GMD.2 is fully addressed by the lessons listed.
- Geometry, module 5, lesson 11 -- G-C.4 is fully addressed by the lesson listed.
- Algebra II, module 1, lessons 24-25 -- A-APR.7 is not fully addressed by the lessons listed. Closure for the system of rational expressions under the operations of addition, subtraction, multiplication, and division is not addressed.
- Algebra II, module 1, lessons 39-40 -- N-CN.8,9 are fully addressed by the lessons listed.
- Algebra II, module 2, lessons 4, 9, 10, and 14 -- F-TF.3 is fully addressed by the lessons listed.
- Algebra II, module 2, lesson 17 -- F-TF.9 is fully addressed by the lessons listed.

In all instances, these standards were coherently connected to the non-plus standards. In many instances, lessons covering plus standards could be omitted, causing minimal issues in studying concepts found later in the series. For example, Geometry module 2 addresses G-SRT.9, G-SRT.10 and G-SRT.11 in lessons 31-33. These lessons could be used as extension lessons or skipped without disrupting the flow of the material.

### Rigor And Mathematical Practices

PARTIALLY MEETS EXPECTATIONS

#### Rigor and Balance

**MEETS EXPECTATIONS**

- 0
- 5
- 8 8

The materials meet the expectation for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to developing students’ conceptual understanding, developing students' procedural skill and fluency, and providing engaging applications. Within the materials, the three aspects of rigor are not always treated together and are not always treated separately, and the three aspects are balanced with respect to the standards being addressed.

**Criterion 2a-2d**

- 0
- 5
- 8 8

Rigor and Balance: 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; procedural skill and fluency; and engaging applications.

##### Indicator 2a

Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The materials meet the expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Exploratory challenge lessons tend to challenge student thinking and help students to build conceptual understanding through the use of activities and tasks. For example, lesson 5 of module 4 in Algebra I has students explore the zero-product property to begin to develop understanding of solving quadratic equations. Socratic lessons are set up to engage the class in discussions about mathematical problems and ideas, linking logical ideas together to formulate a description or summary of a big mathematical idea. These types of lessons also tend to assist students in building conceptual understanding. For example, lesson 5 of module 3 in Algebra I is a Socratic lesson that highlights similarities and differences between arithmetic sequences/ linear equations and geometric sequences/ exponential equations.

These materials offer regular opportunities to develop conceptual understanding in relationship to the development of procedural skill and fluency and work with applications. A few examples of the development of conceptual understanding related to specific standards are shown below:

- A-SSE.1 was addressed in Algebra I module 1 and module 4. In module 1, lesson 25, the materials used area models to help set up equations from problem situations in context. The lesson then moved from using area models to using variables to set up the equations. Various models were used to assist the students in identifying and interpreting complicated variable expressions. Later, in module 4, lesson 1, the materials again used area models, this time to assist the students in finding products of variable expressions. This work involved several different types of drawings, models and tables as tools for helping students to understand the relationships between terms and to interpret terms, factors, and coefficients.
- A-REI.1 was addressed in Algebra I, module 1 and Algebra II, module 1. In Algebra I, the emphasis in lessons 10 and 11 is on whether equations are true or false and using that concept to build up solution sets. lesson 12 has students verify that the addition and multiplication properties of equality work. They use the properties to rewrite equations to get true statements, and then the solution is easily recognizable. Lesson 13 has students explain properties that allow them to create the next step in solving the equation. The exit ticket requires that students work on their own for one equation and demonstrate that certain procedures either do not affect the solution set or that they can affect the solution set. Later, in Algebra II, module 1 extends these ideas to rational expressions in lessons 22 through 25 and to rational equations in lessons 26-27.
- G-SRT.1 was addressed in Geometry module 2, where topic A focuses on the parallel method and the ratio method to promote conceptual understanding of dilation. The students are given the opportunity to explore the concept geometrically and with an algebraic algorithm. Subsequent topics in this module reference both methods to aid in developing understanding of other concepts.
- Geometry module 2 also addresses G-SRT.6. Lesson 25 guides students to the idea that values of the ratios of the side lengths depend solely on a given acute angle in the right triangle before the trigonometric ratios are defined explicitly in lesson 26. In lesson 27, students examine the relationship between sine and cosine, discovering that the sine and cosine of complementary angles are equal. The lesson develops this concept through examples where students find that the sine and cosine of complementary angles are equal. The closing of the lesson reiterates and emphasizes this point.
- S-ID.3 is addressed at the beginning of Algebra I, module 2. In lesson 1, students recognize the first step in interpreting data is making sense of the context for the data. They practice connecting distributions to contexts. In lessons 4, 5, nd 7, all of the work is based within the context of different data sets. The students must interpret that larger deviations are the result of greater spread in the data and vice versa. In lesson 5 students interpret standard deviation as a typical distance from the mean, and then in lesson 7 they must interpret the interquartile range as a description of the variability in the data.
- Algebra I, module 4 requires students to build conceptual understanding through investigative exercises. These exercises rely on a variety of tools and strategies, including but not limited to using area models, comparing and contrasting transformations of quadratic functions utilizing the graphing calculator, looking for patterns in tables, looking for patterns in related expanded and factored forms of quadratic expressions, and discussion questions which constantly focus students on the meaning of key features of the graphs - both mathematically and contextually. Repeatedly students are asked to interpret key features of a graph in reference to the context of the problem.

##### Indicator 2b

Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The materials meet the expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

In a problem set lesson, the teacher and/or students work through a series of examples which are designed to sharpen procedural skill and reinforce conceptual understanding. Examples of problem set lessons that promote procedural skill and fluency are lesson 28 of module 2 in Geometry, solving problems using sine and cosine, and lesson 24 of module 3 in Algebra II, solving exponential equations.

Below are examples of standards in which students are expected to develop procedural skills and fluencies and the places within the materials where these standards are addressed:

- A-SSE.2 is addressed in Algebra I, module 1, topic B; Algebra I, module 4, topics A and B; and Algebra II, module 1, topics A and B. In Algebra I, students use different properties of equality, along with the structure of expressions, to identify ways to rewrite mainly quadratic expressions, and in Algebra II, this skill is extended to polynomial expressions of degree 3 or higher.
- A-APR.1 is addressed in Algebra I, module 1, topic B and Algebra I, module 4, topic A. In the Algebra I materials, students develop the skills in this standard primarily by multiplying linear, binomial expressions and adding and subtracting like terms once the product has been found.
- A-REI.6 is addressed in Algebra I, module 1, topic C and Algebra II, module 1, topic C. In Algebra I, students are given multiple opportunities to solve systems of linear equations exactly and approximately, primarily with systems that consist of two equations in two variables, and in the Algebra II materials, students extend this skill by solving systems of linear equations in three variables.
- F-BF.3 is addressed in Algebra I, module 4, topic C and Algebra II, module 3, topic C. In Algebra I, students develop fluency with transformations using functions that are polynomial, radical, absolute value, and piece-wise, and in Algebra II, this fluency is further developed with functions that are exponential and logarithmic.
- G-GPE.4 is addressed in Geometry, module 4, topics B and D and module 5, topic D. In module 4, students use coordinates to prove simple geometric theorems primarily about lines, line segments, and polygons, and in module 5, students continue to develop the fluency of using coordinates to prove theorems as they work with the equations of circles.

Overall, the series does address the development of procedural skills and fluency, especially where called for in specific content standards and clusters. Below are two standards for which limited opportunities for students to develop procedural skills and fluencies exist:

- G-SRT.5 expects students to use congruence and similarity criteria in solving problems and proving relationships. Lesson 15 of module 2 in Geometry has students work with the angle-angle criterion as it pertains to G-SRT.5, and lesson 17 of module 2 has students work with the side-angle-side and side-side-side criterions. Opportunities for students to use these criterions are limited to these two lessons, and this standard is not directly addressed within the remaining lessons of the module.
- N-CN.2 expects students to develop skill and fluency in adding, subtracting, and multiplying complex numbers. The student outcomes for lesson 37 of module 1, in Algebra II, include students learning that complex numbers share many of the properties of real numbers including addition, subtraction, and multiplication. Lesson 37 has limited opportunities for students to develop fluency with operations on complex numbers, and the following lessons extend to equations that yield complex solutions and factoring with complex numbers.

##### Indicator 2c

Attention to Applications: The materials support 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 meet the expectations for supporting the intentional development of students' abilities to utilize mathematical concepts and skills in engaging applications.

The entire series regularly contains a variety of application problems in mathematical contexts and in real-world contexts. Some of these application problems are relatively simple while other problems are more complex. Some applications, for example those found in Algebra II, module 2, lessons 22-23, rise to the level of offering opportunities to engage in mathematical modeling.

Application opportunities are routinely found in each course, Descriptions of some of the application opportunities are given below:

Algebra I, module 3 contains multiple lessons having several different types of applications. For example, in lesson 21 there are applications that include fitting a function to data in order to work with minnow populations, a dog-walking business, and certificates of deposits. Lessons 22-24 go on to include problems related to invasive plants, Newton's law of cooling, and parking rates. All of this work was supported earlier in the module (lessons 13-16), where students explored linear, exponential and piecewise functions using both non-contextual data and data tied to a specific real world situation.

The end of Geometry module 2 contains several lessons which have a variety of applications. These include determining the distance to the moon (lesson 20), determining the heights of objects and the distance from objects (lesson 29), and determining heights and distances using trigonometry (lesson 34).

The entire Algebra II, module 4 book is constructed of a series of real-world applications and data, which are used to teach statistics and probability standards. Topic B deals with modeling data distributions. Throughout the module discussions are based on real data, such as determining the heights of dinosaurs from fossil remains and analyzing the fuel economy of a specific car over a 25-week period. Topic C, module 4 deals with sampling and sampling variability, and the data is either real-world or student-generated.

##### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

The series meets the expectations that the three aspects of rigor are not always treated together and are not always treated separately. In all three courses, the series makes a visible effort to develop conceptual understanding and to provide opportunities for students to develop procedural skill and engage in mathematical applications. The series utilizes four different lesson structures, problem set, modeling cycle,exploration and Socratic. Lesson types are determined by the requirements of the content. Exploratory challenge lessons tend to challenge student thinking and help students to build conceptual understanding through the use of activities and tasks. Socratic lessons are set up to engage the class in discussions about mathematical problems and ideas, linking logical ideas together to formulate a description or summary of a big mathematical idea. These types of lessons also tend to assist students in building conceptual understanding. In a problem set lesson the teacher and/or students work through a series of examples which are designed to sharpen procedural skill and reinforce conceptual understanding, The modeling cycle lessons do not always offer deep mathematical modeling opportunities but do tend to offer application problems which are built around a mathematical or a real-world context. Every lesson of the series offers an exit ticket which can be used by the teacher for formative assessment. The lessons also include a problem set which can be used for homework. The exit tickets and problems sets contain conceptual, procedural and/or application items, based on the content of the lesson. A problem set might contain items that tend toward one element of rigor or may contain a combination of the three elements of rigor, based on the content of the lesson.

An example of the use of all three elements of rigor can be seen in Algebra II, module 4, lessons 1-7. These lessons focus on A-APR.1, A-SSE.2, A-SSE.3a and A-CED.1. Early lessons rely on making connections between multiplication and factoring, with considerable use of area models and tables to develop understanding of factoring as the reverse process of multiplication, and to help students understand the structure of a polynomial expression. These early lessons contain exercises, examples, exit tickets and problem sets that include items tending toward procedural skill and conceptual understanding, though there are some simple application problems. By lessons 3 and 4, the materials are focused on advanced methods for factoring quadratic expressions. At this point, problem sets and exit tickets are more focused toward procedural skill, but are not completely void of conceptual base problems and applications. As the series of lessons build to developing an understanding of the zero-product property in lesson 5 and solving one-variable quadratic equations in lessons 6-7, the module work becomes more and more focused on applications. Lesson 7, is completely focused on application problems arising from situations modeled by quadratic equations in one variable.

#### Mathematical Practice-Content Connections

**PARTIALLY MEETS EXPECTATIONS**

- 0
- 4 4
- 8

The MPs are used to enrich the mathematical content in some instances, and the identification of the MPs is inconsistent within the courses and across the series. Throughout the courses, there is little instructional guidance given to the teacher concerning strategies for promoting the MPs aligned to lessons, and the teacher is inconsistently prompted to encourage or look for specific student behaviors indicative of a particular MP. There is little, if any, guidance given for how the student use of the MPs should be deepening across the series, and there is also not any guidance or support for students to help them develop their understanding or use of the MPs.

In each module's overview section, focus MPs are identified for the module. A brief and general explanation of how the focus MPs enrich the content of the module is also provided, but the focus MPs are not aligned to any topics or lessons in the module overviews. In the materials for Algebra I, MPs are not mentioned in any of the topic overviews, but MPs are mentioned in some of the topic overviews in the materials for Geometry and Algebra II. When the MPs are mentioned in the topic overviews in Geometry and Algebra II, they are aligned to individual lessons, and a description is given for how the MPs enrich the content of those lessons.

**Criterion 2e-2h**

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- 4 4
- 8

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

##### Indicator 2e

The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for this series partially meet the expectations for supporting the intentional development of overarching mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the MPs. The materials do engage students in MP1 and MP6 throughout the materials, and there are not any instances where these two MPs are treated separately from the content standards. Overall, however, there are instances when the materials do not sufficiently support the intentional development of MP1 and MP6 by not accurately attending to the intent of these two MPs and by not fully supporting the instructional implementation of the MPs.

The following are examples that do not meet the intent of MP1 and MP6 or are not connected to content:

- Throughout the series, portions of lessons cite MP1, but often what is labeled is a place where students are asked to solve a problem but have been given a prescribed formula or steps to solve the problem in a previous example. The directions will even tell the teacher/student to use the steps already given. An example is Geometry module 2, topic A, lesson 3, Example 1. The context changes very little, and the main difference in the problems are numbers.
- For MP1, in Algebra II module 3 lesson 9 on page 132 of the teacher's edition, students are asked to figure out why social security numbers are 9 digits and how many digits long do phone numbers need to be to meet demand. In the previous example, students are shown how to use logarithms to figure out how many digits for ID numbers of a certain length. While the context changed, the work needed to be done is exactly the same just with larger numbers.
- For MP6, in Algebra I module 2, topic D, lesson 16, students work with residual graphs. However, the materials walk students through the graph and do not require them to attend to precision. Although the materials themselves attend to precision, there is no work for the students to develop this Standard for Mathematical Practice.

The following are ways in which the materials do not fully support the instructional implementation of the MP1 and MP6:

- At the lesson level, MPs are identified in three ways in the teacher materials across the series: in Lesson Notes, within the lesson itself, and with a blue box in the margin of the lesson. Across the series, the MPs are usually identified with a blue box in the margin of the lesson, and when the blue box is used, there is little description or guidance as to how the identified portion of the lesson exemplifies the noted MP. Examples of blue MP boxes include the following:

- For MP1, the blue box found on page 54 of Algebra I module 4 lesson 4 states, “This question is open-ended with multiple correct answers. Students may question how to begin and should persevere in solving.” There is no other guidance for teachers on integrating MP1 or description of how the question exemplifies MP1.
- For MP1, the blue box found on page 219 of Algebra II module 1 lesson 20 is drawn around four questions that teachers can ask students during a whole-class problem, but there is no guidance for teachers on when to ask the questions or if all or only some of the questions should be asked.
- For MP6, the blue box on page 377 of Geometry module 2 lesson 24 states, “Ask students to summarize the steps of the proof in writing or with a partner.” There is no other guidance for teachers on integrating MP6 or description of how the proof exemplifies MP6.

- When the MPs are mentioned in the Lesson Notes, there is typically a brief description as to how the MP will generally be exemplified in the lesson, but these brief descriptions are not necessarily connected to specific portions of the lesson. Examples of this characteristic of the materials include the following:

- For MP1, the Lesson Notes on page 109 of Geometry module 1 lesson 13 state, “Additionally, students develop in their ability to persist through challenging problems (MP.1).” There is no connection to portions of this lesson, or following lessons, to indicate where or how students develop their ability to persist.
- For MP6, the Lesson Notes on page 250 of Algebra I module 4 lesson 23 state, “Throughout this lesson, students...report their results accurately and with an appropriate level of precision.” There is no connection to any portions of the lesson for MP6, and MP6 is not directly referenced at any other point in the lesson.
- For MP6, the Lesson Notes on page 369 of Algebra II module 3 lesson 23 state, “In the main activity in this lesson, students work in pairs to gather their own data, plot it (MP.6), and… .” There is no connection to any particular part of the main activity, and MP6 is not directly referenced at any other point in the lesson.

The following are examples that meet the intent of MP1 and MP6, are connected to content, and engage students in these two MPs:

- For MP6, in Geometry module 3, the students are working with volume, cross sections, and areas of three-dimensional figures. They adjust formulas and work with various units to arrive at precise answers. Students are given the opportunity to develop their use and understanding of MP6 from the first to the last lesson of the module.
- For MP1, in Algebra II module 3 lesson 1 on page 15 of the teacher’s edition, students are given a chance to solve problems and preserve by brainstorming ideas to explore questions that arose from the opening problem, coming up with plans of actions, and sharing ideas. Students get the chance to develop ideas about a problem without having seen an example first and decide what information they need, and then, students work with classmates to consolidate ideas and make revisions to their original ideas.

##### Indicator 2f

The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for this series partially meet the expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the MPs. The materials do engage students in MP2 and MP3 throughout the materials, and there are not any instances where these two MPs are treated separately from the content standards. Overall, however, there are instances when the materials do not sufficiently support the intentional development of MP2 and MP3 by not accurately attending to the intent of these two MPs and by not fully supporting the instructional implementation of the MPs.

The following are examples that do not meet the intent of MP2 and MP3 or are not connected to content:

- Algebra I module 1, lesson 12 labels the Closing with MP2. The Closing has students using different properties of equality to create new equations that have the same solution set as an original equation, but none of the equations involve units or contexts, which means that students are not reasoning quantitatively during the Closing. By not reasoning quantitatively, the full intent of MP2 is not met. Also, in lesson 14 of module 1, Exercise 5 is labeled with MP2, but the exercise does not reach the full intent of MP2 because students are not given the opportunity to reason quantitatively.
- In Algebra I module 5, lesson 2 part of the Opening Exercise is labeled with MP3. During the exercise, students are asked to make conjectures and support them with evidence. In the rest of the lesson, students are not asked to revisit their conjectures nor are they asked to critique other students’ conjectures. By making conjectures and not determining if they are viable, students have not reached the full intent of MP3.
- MP3 is inconsistently identified in Geometry module 1. Lessons 22-27 have students generating arguments to show that triangles are congruent by different methods, including indicating where the triangles cannot be proven congruent, but there is no identification of MP3 anywhere in those six lessons. Lessons 10 and 11 are about proofs of parallel lines and angles that are congruent in relation to them, and there is no identification of MP3 in the teacher’s materials.

The following are ways in which the materials do not fully support the instructional implementation of the MP2 and MP3:

- At the lesson level, MPs are identified in three ways in the teacher materials across the series: in Lesson Notes, within the lesson itself, and with a blue box in the margin of the lesson. Across the series, the MPs are usually identified with a blue box in the margin of the lesson, and when the blue box is used, there is little description or guidance as to how the identified portion of the lesson exemplifies the noted MP. Examples of blue MP boxes include the following:

- For MP2: The blue box found on page 11 of Algebra I module 2 lesson 1 is drawn around 3 bullets that teachers should use as they review different types of graphs. There is no other guidance for teachers on integrating MP2 or description of how these bullets exemplify MP2.
- For MP2: The blue box found on page 58 of Algebra II module 1 lesson 4 is drawn around problem 8 of the problem set, but there is no guidance for teachers on how to emphasize MP2 or how the problem exemplifies MP2.
- For MP3: The blue box on page 36 of Geometry module 2 lesson 2 is drawn around a portion of Exercise 6, but there is no other guidance for teachers on integrating MP3 or description of how the Exercise exemplifies MP3.

- When the MPs are mentioned in the Lesson Notes, there is typically a brief description as to how the MP will generally be exemplified in the lesson, but these brief descriptions are not necessarily connected to specific portions of the lesson. Examples of this characteristic of the materials include the following:

- For MP2: The Lesson Notes on page 466 of Geometry module 2 lesson 31 state, “Students carefully connect the meanings of formulas to the diagrams they represent (MP.2 and 7).” There is no connection to any particular part of the lesson, and although MP2 is referenced again for the Discussion portion of the lesson, there is no description within that portion as to how it exemplifies MP2.
- For MP3: The Lesson Notes on page 224 of Algebra I module 3 lesson 17 state, “In the Exploratory Challenge, consider highlighting MP.3 by asking students to make a conjecture about the effect of
*k*.” There is no connection to any portion of the lesson for MP3, and MP3 is not directly referenced at any other point in the lesson. - For MP3: The Lesson Notes on page 293 of Algebra II module 2 lesson 17 state, “The lesson highlights MP.3 and MP.8, as students look for patterns in repeated calculations and construct arguments about the patterns they find.” There is no connection to any particular part of the lesson, and MP3 is not directly referenced at any other point in the lesson.

The following are examples that meet the intent of MP2 and MP3, are connected to content, and engage students in these two MPs:

- Algebra II, module 4 addresses trigonometry within the unit circle. Lessons 1 and 2 in this module meet the intent of MP2 with connection to content by using a physical model of a paper plate for a Ferris wheel to assist students to relate to periodic functions. Students are reasoning both abstractly and quantitatively as they relate the height of a car and the distance the car has traveled over time.
- Module 4 of Algebra II also has lessons that meet the intent of MP3 with connection to content. In lessons 15-17, students are asked to construct valid arguments to extend trigonometric identities to the full range of inputs. In addition, lesson 12 displays graphs which have been identified by fictitious students with a function, and the students are asked to identify which fictitious student is correct and explain why.

##### Indicator 2g

The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for this series partially meet the expectations for supporting the intentional development of modeling and using appropriate tools (MPs 4 and 5), in connection to the high school content standards, as required by the MPs. The materials do engage students in MP4 and MP5 throughout the materials, and there are not any instances where these two MPs are treated separately from the content standards. Overall, however, there are instances when the materials do not sufficiently support the intentional development of MP4 and MP5 by not accurately attending to the intent of these two MPs and by not fully supporting the instructional implementation of the MPs.

The following are examples that do not meet the intent of MP4 and MP5 or are not connected to content:

- Algebra I module 3, lesson 23 states in the Lesson Notes that MP4 is a focus of the lesson, but in the Opening Exercise, students are given the mathematical model that they will use throughout the lesson. In the remainder of the lesson, students evaluate the model for different values of the parameters in it and create graphs for given values of the parameters, but these student actions do not meet the intent of MP4. In the lesson, students do not create a model on their own, nor do they make any assumptions in their calculations or ever revise the model that is given to them.
- The study of Geometry contains routine opportunities to use tools to develop understanding and skill and to engage in applications. However, modules 3, 4, and 5 in Geometry do not mention MP5. When tools are used in the series, there are times when an explicit instruction to use a specific tool or set of tools is given, for example, Exercise 1 in lesson 15 of module 2. Rarely in any module across the series is there an intentionally designed opportunity for students to choose the most appropriate tools from a selection of available tools.
- The Discussion after the Opening Exercise of lesson 2 in module 1 of Geometry is labeled with MP5. At the beginning of the Opening Exercise, students are directed which tools to use, and the Discussion is focused on the importance of students describing objects using correct terminology. The Discussion would be more appropriately labeled with MP6 than with MP5.

The following are ways in which the materials do not fully support the instructional implementation of the MP4 and MP5:

- At the lesson level, MPs are identified in three ways in the teacher materials across the series: in Lesson Notes, within the lesson itself, and with a blue box in the margin of the lesson. Across the series, the MPs are usually identified with a blue box in the margin of the lesson, and when the blue box is used, there is little description or guidance as to how the identified portion of the lesson exemplifies the noted MP. Examples of blue MP boxes include the following:

- For MP4: The blue box found on page 181 of Algebra I module 4 lesson 16 states, “In the activity above, students model the situation using tables and graphs. Then, they conclude that the graphs of the equations can move up or down by adding or subtracting a constant outside the parentheses.” There is no other guidance for teachers on integrating MP4 or description of exactly how the activity exemplifies MP4.
- For MP4: The blue box found on page 130 of Algebra II module 3 lesson 9 is drawn around parts c and d of the Exploratory Challenge, but there is no guidance or description for teachers on how these two parts of the challenge exemplify MP4.
- For MP5: The blue box on page 13 of Geometry module 1 lesson 1 is drawn around Example 1. Example 1 starts with “You need a compass and a straightedge,” but since this is not allowing students to choose appropriate tools and no other guidance or description is provided for teachers, this example does not exemplify MP5.

- When the MPs are mentioned in the Lesson Notes, there is typically a brief description as to how the MP will generally be exemplified in the lesson, but these brief descriptions are not necessarily connected to specific portions of the lesson. Examples of this characteristic of the materials include the following:

- For MP4: The Lesson Notes on page 72 of Geometry module 4 lesson 7 state, “This lesson focuses on MP.4 because students work extensively to model robot behavior using coordinates.” There is no connection to any portions of the lesson for MP4, and MP4 is not directly referenced at any other point in the lesson.
- For MP5: In the Algebra I materials, the Lesson Notes section is not utilized to highlight MP5 in order to connect MP5 to any portion of the lesson or give a brief description as to how MP5 is exemplified in any of the Algebra I lessons.
- For MP5: The Lesson Notes on page 339 of Algebra II module 1 lesson 31 state, “The standards MP.5 … and MP.8 … are also addressed.” There is no connection to any particular part of the lesson, and MP5 is not directly referenced at any other point in the lesson.

The following are examples that meet the intent of MP4 and MP5, are connected to content, and engage students in these two MPs:

- In Geometry module 3, lesson 1 provides a framework for students to engage in MP4 while exploring the area of an oval using known polygons. This learning experience is tightly connected to the informal limits argument aspect of G-GMD.1. In the lesson, students are presented with a problem, and then they need to formulate how to model the oval with known polygons, compute the areas of the known polygons, interpret their answer, and revise as needed before reporting an approximate answer for the area of the oval.
- In Algebra II module 2 lesson 13, the Opening Exercise requires students to create a scatter plot for a set of data, and then in Example 1, students have to determine a function that models the data set. In the exercise and in the example, students have the choice to create the plot and function using technology or manually. Since students have the choice as to what tools they will use, the Opening Exercise and Example 1 meet the intent of MP5.

##### Indicator 2h

The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for this series partially meet the expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the MPs. The materials do engage students in MP7 and MP8 throughout the materials, and there are not any instances where these two MPs are treated separately from the content standards. Overall, however, there are instances when the materials do not sufficiently support the intentional development of MP7 and MP8 by not accurately attending to the intent of these two MPs and by not fully supporting the instructional implementation of the MPs.

The following are examples that do not meet the intent of MP7 and MP8 or are not connected to content:

- In Geometry module 2, the Opening Exercise of lesson 31 has students use trigonometry to determine the area of a triangle, but in the exercise, the materials do not address MP7 when drawing an auxiliary altitude in the triangle. By not addressing the auxiliary altitude with MP7, the materials miss an opportunity to develop students’ ability to look for and make use of structure.
- In Algebra I module 1 lesson 7, part of Exercise 8 is labeled with MP8, but in the exercise, calculations for the associative property of addition are given to the students. Students are not given the opportunity to perform repeated calculations for the associative property of multiplication. By not allowing students to perform some of the repeated calculations on their own and express the regularity in them, the intent of MP8 is not met in this exercise.

The following are ways in which the materials do not fully support the instructional implementation of the MP7 and MP8:

- At the lesson level, MPs are identified in three ways in the teacher materials across the series: in Lesson Notes, within the lesson itself, and with a blue box in the margin of the lesson. Across the series, the MPs are usually identified with a blue box in the margin of the lesson, and when the blue box is used, there is little description or guidance as to how the identified portion of the lesson exemplifies the noted MP. Examples of blue MP boxes include the following:

- For MP7: The blue box found on page 206 of Algebra I module 1 lesson 17 is drawn around Exercise 1 under Classwork, but there is no guidance for teachers on how to emphasize MP7 or how the problem exemplifies MP7.
- For MP8: The blue box found on page 58 of Algebra II module 1 lesson 4 is drawn around problem 8 of the problem set, but there is no guidance for teachers on how to emphasize MP8 or how the problem exemplifies MP8.
- For MP8: The blue box on page 78 of Geometry module 1 lesson 9 is drawn around a portion of Exercise 1, but there is no guidance for teachers on how to emphasize MP8 or how the specific portion of Exercise 1 exemplifies MP8.

- When the MPs are mentioned in the Lesson Notes, there is typically a brief description as to how the MP will generally be exemplified in the lesson, but these brief descriptions are not necessarily connected to specific portions of the lesson. Examples of this characteristic of the materials include the following:

- For MP7: The Lesson Notes on page 92 of Geometry module 5 lesson 8 state, “This lesson highlights MP.7 as students study different circle relationships and draw auxiliary lines and segments.” There is no connection to any particular part of the lesson, and although MP7 is referenced again within the Opening Exercise, there is no description with the exercise as to how it exemplifies MP7.
- For MP7: The Lesson Notes on page 28 of Algebra II module 1 lesson 2 state, “This lesson begins to address … and provides opportunities for students to practice MP.7 and MP.8.” There is no connection to any particular part of the lesson, and although MP7 is referenced again with Example 1, there is no description with the example as to how it exemplifies MP7.
- For MP8: The Lesson Notes on page 224 of Algebra I module 3 lesson 17 state, “This challenge also calls on students to employ MP.8, as they generalize the effect of
*k*through repeated graphing.” Although there is a connection to the Exploratory Challenge in the lesson for MP8, MP8 is not directly referenced at any other point in the lesson nor is there a description as to how MP8 is exemplified in any part of the lesson.

The following are examples that meet the intent of MP7 and MP8, are connected to content, and engage students in these two MPs:

- In lessons 11-14 of module 4 in Algebra I, students examine the method of completing the square, culminating in the use of this method to derive the quadratic formula. Throughout this series of lessons, students use the structure of quadratic equations to rewrite them in completed square form. The use of MP7 is tightly connected to content standards A-SSE.1-3 and A-REI.4.
- In Algebra II module 1, lesson 2, students meet the intent of MP8 as they develop their understanding of multiplication of polynomials (A-SSE.2) and develop some polynomial identities (A-APR.4). In Example 2 of the lesson, students express regularity in repeated reasoning to arrive at a generalized result when multiplying (x - 1)(x^n + x^(n-1) + … + x + 1).

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