High School - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 75% |
|---|---|
Criterion 2.1: Rigor | 8 / 8 |
Criterion 2.2: Math Practices | 4 / 8 |
Criterion 2.1: Rigor
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.
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.
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.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
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.
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).