2.3 Assignment Carnegie Teachers Answer Sheet

Alignment

The instructional materials reviewed for Grade 8 partially meet the expectations for alignment to the CCSSM. The instructional materials partially meet the expectations for Gateway 1 because they meet the expectations for focus on major work and do not meet the expectations for coherence. Since the materials partially meet the expectations for Gateway 1, evidence was collected in Gateway 2. The instructional materials meet the expectations for rigor and balance and do not meet the expectations for practice-content connections. 

GATEWAY ONE

Focus & Coherence

PARTIALLY MEETS EXPECTATIONS

The instructional materials reviewed for Course 3 partially meet the expectations for focus and coherence with the CCSSM. For focus, the instructional materials meet the criteria for summative assessment items on grade-level and delivered in a challenging and effective manner with most units having little or no above grade-level standards. Focus is also met in the time devoted to the major work of the grade with 78.0 percent of the days allocated in the timeline aligning to the major work. For coherence, supporting work is sometimes connected to the focus of the grade with some missed opportunities for natural connections to be made. The amount of content for one grade level is not viable for one school year and will have difficulty fostering coherence between the grades. Content from prior or future grades is clearly identified, but materials that relate grade level concepts to prior knowledge from earlier grades is not explicit. Overall, the materials are shaped by the CCSSM and incorporate some natural connections that will prepare a student for upcoming grades. The material does lack some consistency for grade-to-grade progressions, and content that is not on grade level or supports on grade-level learning is not explicit.

Focus

The instructional materials reviewed for Course 3 meet the expectations for focus within the grade. The materials reviewed for Course 3 do assess mostly grade-level content with some above grade-level topics, but if the future grade content was removed, it would not change the underlying structure of the assessments. The instructional materials do meet the expectations for majority of class time on the major clusters of this grade. In Grade 8, the materials have 78.0 percent of the days suggested for major work of this grade. Overall, the instructional materials meet the criteria for grade level assessments and also spend the majority of time in the major clusters of this grade.

Criterion 1a

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Materials do not assess topics before the grade level in which the topic should be introduced.

The post, chapter, and standardized assessments that are included in the Teacher's Resources and Assessments were reviewed for Course 3 and found to meet the expectations for instructional material that assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades is sometimes introduced, but students should not be held accountable on assessments for those future expectations. If the future grade content was removed, it would not change the underlying structure of the assessments. Overall, the instructional material in the summative assessment items reviewed in Course 3 addressed the grade-level content in a challenging and effective manner with most units having little or no above grade level standards addressed.

2/2

Indicator 1a

The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The post, chapter, and standardized assessments that are included in the Teacher's Resources and Assessments were reviewed for Course 3 and found to meet the expectations for instructional material that assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades is sometimes introduced, but students should not be held accountable on assessments for those future expectations. If the future grade content was removed, it would not change the underlying structure of the assessments. Overall, the instructional material in the summative assessment items reviewed in Course 3 addressed the grade-level content with most Units having little or no above grade level standards addressed.

Quality, on grade-level examples are:

  • Chapter 4, End of Chapter Test. Question 5a-e uses a real-world scenario to assess 8.F by having students create a graph from information given about a quiz and then explain the relationships between the slopes in terms of the context given.
  • Chapter 11, End of Chapter Test. Question 6a-d asks students to write and solve a system of equations based off a real-world scenario and interpret the solution in the context of the problem. Using context problems to assess 8.EE allows students to work with the MPs to persevere and allows for multiple entry points to a problem.

The following items are above grade and should not be assessed, but they can be removed without drastically changing the material:

  • Chapter 9 Post Test, Question #2 and End of Chapter Test #4, #5, and #6. Students are asked which Similarity Theorem applies. This is not a Grade 8 standard; it is first defined in high school.
  • Chapter 9 Post Test, Question #11. Students are asked to construct a perpendicular line through a given point not on the line.

Criterion 1b

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

4/4

Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Course 3 meet the expectations for spending the majority of class time on the major clusters of each grade. A chapter overview was found at the beginning of each chapter. This included the standards being taught in the lesson and a suggested pacing guide. Overall the instructional materials meet the criteria outlined in the CCSS publisher guidelines for the majority of class time on the major clusters of each grade.

To determine the three perspectives we evaluated: 1) the number of chapters devoted to major work, 2) the number of lessons devoted to major work, and 3) the number of days devoted to major work. It was decided that the number of days devoted to major work is the most reflective for this indicator because it specifically addresses the amount of class time spent on concepts and our conclusion was drawn through this data.

Evidence was determined from the Table of Contents pages FM-6 through FM-56 and the number of days suggested in each chapter overview found in the the Teacher Implementation Guide and written by the publisher.

  • Chapters – 12 out of 17 chapters, or approximately 70.58 percent of time spent on major work.
  • Lessons – 60 out of 78 lessons, or approximately 76.9 percent of time spent on major work.
  • Days – 64 out of 82 days, or approximately 78.0 percent of time spent on major work.

The major clusters of the grade are:

  • 8.EE.A - Work with radicals and integer exponents.
  • 8.EE.B - Understand the connections between proportional relationships, lines, and linear equations.
  • 8.EE.C - Analyze and solve linear equations and pairs of simultaneous linear equations.
  • 8.F.A - Define, evaluate, and compare functions.
  • 8.F.B - Use functions to model relationships between quantities.
  • 8.G.A - Understand congruence and similarity using physical models, transparencies, or geometry software.
  • 8.G.B - Understand and apply the Pythagorean Theorem.

Modules and Chapters that contain these Standards are:

  • Module 1 (Focus on Linear Equations and Functions): Chapter 1- 1.1, 1.2, 1.3, 1.4 (4 days); Chapter 2- 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7 (8 days).
  • Module 2 (Focus on Rate of Change and Multiple Representations of Linear Functions): Chapter 3- 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 (6 days); Chapter 4- 4.1, 4.2, 4.3, 4.4, 4.5 (6 days).
  • Module 3 (Focus on Pythagorean Theorem): Chapter 6- 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 (7 days).
  • Module 4 (Focus on Transformational Geometry): Chapter 7- 7.1, 7.2, 7.3, 7.4 (4 days); Chapter 8- 8.1, 8.2, 8.3, 8.4 (4 days); Chapter 9- 9.1, 9.2, 9.3, 9.4 (4 days).
  • Module 5 (Focus on Lines and Angle Relationships, Systems of Linear Equations and Functions, and Solving Linear Systems Algebraically): Chapter 10- 10.1, 10.2,10.3, 10.4, 10.5 (6 days); Chapter 11- 11.1, 11.2,11.3, 11.4 (4 days); Chapter 12- 12.1, 12.2,12.3, 12.4, 12.5 (5 days).
  • Module 6 (Focus on Properties of Exponents): Chapter 13- 13.1, 13.2, 13.3, 13.4, 13.5, 13.6 (6 days).

Students and teachers using the materials as designed will devote a majority of class time in Grade 8 to the major work of this grade. The instructional materials reviewed for Course 3 meet the expectations for majority of class time on the major clusters of the grade. For example, based on the pacing (one period = 50 minutes), 64 days out of 82 days total have 78.0 percent of the time spent directly on the major work of the grade.

Coherence

The instructional materials reviewed for Course 3 partially meet the expectations for being coherent and consistent with the standards. Supporting work is sometimes connected to the focus of the grade with some missed opportunities for natural connections to be made. The amount of content for one grade level is not viable for one school year, and the materials do not foster coherence between the grades. Content from prior or future grades is clearly identified, but materials that relate grade level concepts to prior knowledge from earlier grades is not explicit. Overall, the materials are shaped by the CCSSM and incorporate some natural connections that will prepare a student for upcoming grades. However, the material does lack some consistency for grade-to-grade progressions, and content that is not on grade level or supports on grade-level learning is not explicit.

Criterion 1c-1f

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

1/2

Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

  • The instructional materials reviewed for Course 3 partially meet the expectations for the non-major content enhancing focus and coherence simultaneously by engaging students in the major work of the grade. In some cases, the non-major work enhances and supports the major work of the grade level, while other areas could be stronger.

    Non-major clusters of the Grade 8 are:

    • 8.NS.A - Know that there are numbers that are not rational, and approximate them by rational numbers.
    • 8.SP.A - Investigate patterns of association in bivariate data.
    • 8.G.C - Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

    Evidence of non-major content enhancing focus and coherence and supporting a partially meet's score are:

    • 8.NS.A supports 8.G and 8.EE in Module 3 - Chapter 5, lesson 2 by having students work with Irrational Numbers so that they are able to fully understand the major work of the Pythagorean Theorem by working with square and cube roots.
    • 8.NS.A supports 8.G and 8.EE in Module 3 - Chapter 6, Lesson 1 by having students work with square and cube root symbols so they are able to fully comprehend and solve problems involving right triangles.
    • 8.SP.A supports 8.EE and 8.F in Module 7 - Chapter 15, Lessons 1-3 by having students work with grade level vocabulary supporting the major work and interpreting and analyzing scatter plots to determine the relationship between the two variables.

    Examples of missed opportunities:

    • 8.G.C has no support of major work noted.
    • Supports of the major work are not often specifically called out as a support. Often times the connections are there, but a teacher would need to know the cohesiveness on their own to be able to make the connections for the students.

    Though the supporting standards have made some connections to major work, they are not specifically written as such, and the non-major clusters of this grade are taught in isolation and miss some opportunities to engage students in the major work of Grade 8, which is why this team supports a partially meets score.

    In chapter 15, slopes and intercepts are interpreted as constant rates of change and initial values in the data, which supports major work of Grade 8.
  • Opportunities to connect student-derived formulas for volume with nonlinear functions (8.F.5) were missed.
  • In chapter 13, 8.EE.A, which is major work, is supported by 8.G.C.9.

0/2

Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Course 3 do not meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades. Without including any assessment days, there are approximately 82 days of lessons in the materials. There needs to be additional material, other than assessment days, to ensure a students' grasp of all major work at this grade level. Overall, the amount of content that is designated for this grade level is short of the amount of material needed to make it truly viable for one school year.

  • According to the pacing guide, each period is 50 minutes in length and there is a suggested 82 days of lessons.
  • When pre-tests, mid-chapter tests and post-test assessments are also included in the pacing, this would add an additional 51 days. If all assessments are given during the course of the year, one extra day per assessment, the total would be 133 days.

The guiding focus taken for this indicator for our team was, "Will the non-major and major work of this material be enough to prepare a student for the next grade level?" With the amount of days, many of those days not focusing on major work, the non-major work days not often supporting the major work of the grade, it will require the teacher to make significant modifications to prepare the student for the next grade level and supports this indicator receiving a does not meet rating.

1/2

Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials reviewed for Course 3 partially meet the expectations for the material to be consistent with the progressions in the standards. Content from prior grade standards is clearly identified; however above grade-level standards are not clearly marked as such. There is ample practice for students to engage deeply with with the problems related to the Grade 8 standards, but no connections are explicitly made to prior or future content in the Teacher Implementation Guide or the student text.

Some examples of areas where identification of standards from lower grades is beneficial and supports a partially meets rating along with a an example of not meeting the full depth of the standard:

  • Lower grade-level material is clearly identified in the grade level outline found in the Teacher Implementation Guide on page FM-30. They are also identified and explained in the same resource at the beginning of each lesson.
  • Chapter 5, pages 281-310, titled "The Real Number System" starts with 7.NS.3, a below grade-level standard, as indicated in the pacing guide and in the chapter overview, Teacher Implementation Guide page 281A. This standard is included, as stated by publisher, to review the sets of natural numbers, whole numbers, integers, and rational numbers.
  • Occasionally, the lessons do not seem to go to the full depth of the standards.
    • Chapter 5 which is suppose to cover 8.NS.1 and 8.NS.2 does not cover 8.NS.2 as deeply as suggested in the standards. 8.NS.2 states, "Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of the expressions (e.g., pi squared)." The lesson and skills practice only asks the students to compare or place numbers on a number line to the hundredths place. This is not a common occurrence in the curriculum for this grade.

The instructional materials reviewed for Grade 8 partially meet the expectation of giving all students extensive work with grade-level problems. Overall, the materials do not consistently give students of varying abilities extensive work with grade-level problems.

Some examples of giving all students extensive work with grade-level problems, but not of varying abilities and supports a partially meets rating:

  • There is ample practice for each standard. Every lesson has guided practice with a script for the teacher to follow. This portion has the students conceptually developing the skill being taught and are given practice problems as well. Along with the guided practice are assignments. The number of assignments and number of problems varies per lesson. In addition there are skill practice pages to accompany each lesson as well. The number of skill pages also varies with each lesson.
  • The Teacher Implementation Guided does not list any lessons or ideas for differentiated instruction except when it talks about the Mathia Software product. No differentiated or extension lessons in the Student Text, Students Skills Practice book, or the Student Assignment book were found by the reviewers.

The instructional materials reviewed for Grade 8 do not meet the expectation of relating grade-level concepts explicitly to prior knowledge from earlier grades. Overall, no support materials were found that relate grade-level concepts explicitly to prior knowledge from earlier grades.

  • The Teacher Implementation Guide is a wrap around of the Student Text. In the margins of the Teacher Implementation Guide, the authors have reworded the question asked in the student text but these does not seem to add anything to the instruction. The margins also have steps for the teachers to follow, ways to groups students (i.e., "Have students complete questions 2 and 3 with a partner. Then share the responses with a class," page 5), and guiding questions to ask students. However, it does not clearly make connections between previous knowledge and new concepts. There are not any indicators that knowledge is being extended.
  • The warm-up sections for each lesson listed would be an ideal place to include connections to prior standards covered in this curriculum. For instance:
    • Chapter 10, lesson 4, when equations of perpendicular and parallel are introduced.
    • Chapter 12, lesson 2, students are asked to write a linear system to represent each graph, yet there are no discussion questions that could guide discussion to perpendicular and/or parallel linear equations.

1/2

Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials reviewed for Course 3 partially meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings.

  • The Chapter titles are clearly labeled and aligned to the standards without a need for much interpretation.
    • Chapter 2 - Linear Functions (8.F)
    • Chapter 4 - Multiple Representations of Linear Functions (8.EE)
    • Chapter 6 - Pythagorean Theorem (8.G.B)
    • Chapter 9 - Similarity (8.G.A)

The instructional materials do include some problems and activities that serve to connect two or more clusters in a domain. They include a few problems and activities that connect two or more domains in a grade, in cases where these connections are natural and important. However, overall the materials only partially foster coherence through connections in Course 3.

  • For the majority of the work, most standards were taught and covered within one Unit out of the entire series and not aligned with any other concept throughout the year.
  • There are no connections identified by publisher. However, there are connections within the Grade 8 standards that are just not noted or stated by the publisher.

Some examples of where connections were made and support a partially meets rating is:

  • Chapter 3, "Slope: Unit Rate of Change," lessons 1 and 5 connect 8.EE.B.5 and 8.F.A by having students determine the rate of change from graphs by using the formal definition of rate of change and using rise/run formula. Students will compare the rates of graphs, compare the steepness of four lines on the same graph and relate the steepness of the lines to the magnitudes of their rates of change.
  • Chapter 4, "Multiple Representations of Linear Functions," lessons 1 through 3 connect 8.EE.C.7.B and 8.F.A by having students, given linear equations written in standard form, complete tables by evaluating each equation and solving for the value of either x or y. The points are graphed and then used to calculate slope. Finally, the students convert the standard form linear equations into slope-intercept form.
  • Chapter 6, "Pythagorean Theorem," lesson 1 connects 8.NS.A and 8.EE.A.2 by having students determine the area of a larger square and the sum of the areas of the two smaller squares to prove they are equal. Students also use the Pythagorean Theorem to solve for the length of unknown sides of right triangles set in a variety of contexts. In lessons 2 through 6, 8.EE.A.2 and 8.G.B are connected by having students use the Pythagorean Theorem to determine that the diagonals in a rectangle and square are congruent along with the diagonals of a trapezoid only when the figure is isosceles.

GATEWAY TWO

Rigor And Mathematical Practices

PARTIALLY MEETS EXPECTATIONS

The materials reviewed for Course 3 partially meet the expectations for Gateway 2: Rigor and Mathematical Practices.

The materials reviewed for Course 3 meet expectations for rigor and balance by providing a balance of all three aspects of rigor throughout the lessons. The Grade 8 instructional materials reflect the balances in the standards and help teachers to help their students meet rigorous expectations. They do this by helping students develop conceptual understanding, procedural skill and fluency, and application. The materials weakest section, but still enough quality representations to receive the "meets" rating, is on conceptual understanding. Students are able to work in groups to develop understanding, but then sometimes the narrative, in the text, scaffolds the work in such a way that the students are just walking through that understanding step by step.

The materials reviewed for Course 3 do not meet the expectations for practice-content connections. The materials attempt to incorporate the MPs in each lesson but all instruction of the MPs happens at the beginning of the Teacher Implementation Guide and never directly links the standards to the lessons using the MP vocabulary. This makes it extremely difficult for a teacher to reliably use the materials to know when MPs are being carefully attended to. The materials incorporate questions in which students have to justify and explain their answers, but no teacher supports are given which creates a lack of lesson structures for which students would discover their own solution paths, present their arguments, and justify their conclusion. Vocabulary is presented and almost always incorporated meaningfully into the lesson.

Overall, the materials partially meet the expectations for Gateway 2 in rigor and mathematical practices.

Rigor and Balance

The materials reviewed for Course 3 meet expectations for rigor and balance. The Grade 8 instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application. The materials weakest section, but still enough quality representations to receive the "meets" rating, is on conceptual understanding. Students are able to work in groups to develop understanding, but then sometimes the narrative, in the text, scaffolds the work in such a way that the students are just walking through that understanding step by step.

Criterion 2a-2d

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

2/2

Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials for Course 3 meet the expectations to develop conceptual understanding of key mathematical concepts, especially when called for in specific content standards or cluster headings. Overall the instructional materials present real-world situations and multiple visual examples as a way to develop conceptual understanding.

The materials in the Teachers Implementation Guide, student text, and student assignments were all used to gather evidence. In the Teachers Implementation Guide it states that they provide many opportunities for conceptual understanding, especially those in which the standard calls for it. Each chapter begins with an overview and in this overview is a column titled "highlights." This section summarizes what the students will be able to do after the lesson and was a guide to searching for the evidence below to justify the meets rating.

  • Chapter 3, Lesson 2: Students use visual representations and linear graphs to represent a situation from a given context.
    • Problem 1, page 164, gives students information about a school soccer team trip and then has follow-up questions that ask students to determine rate of speed during the trip. They are then asked create a concrete visual representation of the story based upon facts presented.
  • Chapter 4: Students interpret meanings of equations and analyze intervals. The lessons have students use tables and graphs to see the relationships of the function which correlates with standards in 8.F and 8.EE.
  • Chapter 6: Teaches the use of the Pythagorean Theorem (8.G). Lesson 1 has students use shapes to develop the Pythagorean Theorem. They use grid paper to show the relationship of the area of the triangles.
  • Chapter 7: Students explore transformations (8.G) with shapes in the coordinate plane. Many questions require students to explain their thinking.
  • Chapter 9: Students work with understanding dilation and similarity (8.G) by investigating and exploring shapes in the coordinate plane.

In the lessons listed below and others, the students are asked to explain their reasoning or explain why they believe the answer to be correct. The student assignment book allows individual students to show their understanding through the following questions in the listed lessons.

  • Lesson 1.2 - Why Doesn't This Work?
  • Lesson 2.1 - Patterns, Patterns, Patterns ...
  • Lesson 7.4 - Mirror, Mirror
  • Lesson 12.3 - Making Decisions
  • Lesson 14.2 - Piling On!

An area of concern for this review team was that many of the guided lessons walk students through entire procedures and do not allow them to explore or discover concepts on their own.

2/2

Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials for Course 3 meet the expectations to give attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, there are multiple opportunities for students to develop procedural skills and fluency which include various questioning strategies for students to explain procedural skills, and chances for students to apply procedural skills to new situations. Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency meeting the expectations for this indicator.

  • Based on where the lesson is in the timeline of the unit, the level of difficulty varies. As expected, it starts off with a low level of difficulty, and as the students gain more practice, the difficulty increases as the unit progresses.
  • Along with problems during the guided lessons, each lesson begins with a warm up that is procedural practice.
  • Some of the student assignments are procedural in nature as well as each lesson having suggested pages for students to complete in the student skill practice book.

Examples of fluency practice that justify the meets rating are:

  • Chapter 1, Lesson 1.1, when the students start by having to write the steps they follow to solve multi-step equations with one variable then they move onto just solving the problems.
  • Chapter 5, Lesson 5.1, students practice writing fractions as decimals.
  • Chapter 8, Lesson 8.2, students practice determining what transformation is present.

2/2

Indicator 2c

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

The instructional materials for Course 3 meet the expectation so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. Overall, the materials have multiple opportunities for real-world application.

There are a variety of single and multi-step contextual application problems found in the student text, students' assignment book and student skills practice.

  • Chapter 2, Lesson 2.6 has students calculate the cost for orders of T-shirts for various given values from a competitor with a different cost value, determine the amount of shirts that can be purchased, and create a table and graph to represent the situation.
  • Chapter 3, Lesson 3.3 asks the students to determine the rate of change based on a soccer tournament.
  • Chapter 11, Lesson 11.3 asks the students to apply what they know about rate of change and linear equations to solve real-world problems such as how much money a person will make if they work at an hourly rate. It also asks them to apply what they know about functions and graphing to demonstrate a variety of scenarios for a person to earn a certain amount of money.

2/2

Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials reviewed for Course 3 meet the expectation that the materials balance all three aspects of rigor with the three aspects not always combined together nor are they always separate. Overall, the majority of the lessons focus on procedural skills and fluency, but do balance that out with conceptual understanding and application problems. The material's weakest section, but still enough quality representations to receive the "meets" rating, is on conceptual understanding. Students are able to work in groups to develop understanding, but then sometimes the narrative, in the text, scaffolds the work in such a way that the students are just walking through that understanding step by step.

The student text, along with the wrap around Teacher Implementation Guide help guide the educator and student to the level of rigor needed to prepare the student for upcoming mathematics. An example of this is in Chapter 3, Analyzing Linear Equations.

  • The chapter begins with introducing students to finding two points on a line and then determining the rate of change.
  • Delving deeper into this skill, students are then asked, on page 143 problem 1, to determine the unit rate of four different cars using the same graph.
  • They are then asked to describe what the steepness of each line implies regarding their individual unit rate.
  • Then each lesson ends with a complex problem which delves deeper into student understanding of the standard.

Mathematical Practice-Content Connections

The materials reviewed for Course 3 do not meet the expectations for practice-content connections. The materials attempt to incorporate the MPs in each lesson. However, all instruction of the MPs happens at the beginning of the Teacher Implementation Guide and never directly links the standards to the lessons using the MP vocabulary. This makes it extremely difficult for a teacher to reliably use the materials to know when MPs are being carefully attended to. The materials incorporate questions in which students have to justify and explain their answers, but no teacher supports are given which creates a lack of lesson structures for which students would discover their own solution paths, present their arguments, and justify their conclusion. Vocabulary is presented and almost always incorporated meaningfully into the lesson.

Criterion 2e-2g

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

1/2

Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The Instructional materials reviewed for Course 3 partially meet the expectation for identifying and using MPs to enrich mathematics content within and throughout each grade. While each practice is represented in this book, they are not often used in a way that would promote or enrich the mathematics content, are not user-friendly and are under-identified in many of the units because they are not specifically stated.

  • MPs are described on pages FM-34 through FM-42 in the Teacher's Implementation Guide, along with a description of what it looks like in the student text.
  • On pages FM-45 through FM-55, there are more examples of how the practices are implemented throughout the series. This section also defines symbols that clue teachers and students that they should discuss to understand, think for yourself, work with your partner, and share with the class. It also defines that a "thumbs up" means a worked example is correct and a "thumbs down" means a worked example is incorrect.
  • Specifically, FM-45 through FM-49 attempt to define the academic terms analyze, explain your reasoning, represent, estimate, and describe. When these verbs appear in the series, they are suppose to correlate with the MPs listed. There is no further identification of MPs in the Teacher's Implementation Guide.
  • All eight MPs are evident throughout the materials, but it was very difficult to find them since they are not specifically marked and not all practices had identifying academic terms to label them.
  • The MPs could be used to enrich the mathematical content, but one would have to keep referencing the only guide to using them in the first few pages of the Teachers Implementation Guide, in order to have a better understanding of what practices to emphasize and how to use the problems to enrich them. Since all of the chapters have an overview section, this would be a place to identify the practices for each lesson and further encourage the use of these practices to enrich the mathematics content.

0/2

Indicator 2f

Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Course 3 do not meet the expectations for carefully attending to the full meaning of each practice standard. The publisher rarely attends to the full meaning of the practice standard and when they do, it is cumbersome to use since they are not specifically called out.

  • The lessons are set up for students to work in groups and then the teacher is to guide them through discussion of their findings. The graphs, tables, equations, etc., are usually created for the students, which doesn't allow them to have to model with mathematics.
  • There are sequential questions to lead the students through the process, so they are not having to make sense of the problem, or persevere, because the text book does it for them. With this guidance on how to complete the problem, there is usually only one way to solve the problem, not allowing for multiple entry points.
  • Students are not given the opportunity to choose tools to help them with the mathematics, the tools are provided.
  • Almost all of the lessons are designed for group work. Students are rarely asked to work independently, and given opportunity to compare.
  • There are a variety of questions for teachers to pose during each lesson, however there are no MPs indicated and no sample student responses given to aid a teacher who is unsure what MP is involved and how students may be thinking as related to that MP.
Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

1/2

Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The materials reviewed for Course 3 partially meet the expectation for appropriately prompting students to construct viable arguments and analyze the arguments of others. Each chapter provides opportunities for students to construct viable arguments and places to critique worked examples that are sometimes correct and sometimes wrong, however, each time they let the students know which argument is correct and which is incorrect by putting a thumbs up or a thumbs down with the problem. This directs students thinking and doesn't force them to go through the thought process of finding viable arguments to critique the reasoning of others.

  • The questions following 'thumbs up' are usually comparing how two different people solved it correctly, but differently, which does allow for a student to then describe why they are both right.
  • For 'thumbs down' situations, students are asked to find the errors that were made.
  • In the Teacher's Implementation Guide, page FM-45, there are icons that direct student questioning that provide norms for what students are to do when they see the icons throughout the text.
  • In the Teacher's Implementation Guide, page FM-48 students learn specifically how to explain their reasoning which is identified as SMP3.
  • In the Teacher's Implementation Guide, pages FM-52 to FM-55, there is a "Who's correct" option that, if used, better allows students to form their own opinions and arguments for the work done.

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Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The materials reviewed for Course 3 do not meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, there is not enough guidance in the teacher materials to direct teachers on questioning strategies, setting up scenarios where students experiment with mathematics and based on those experiments construct and present ideas, examples of higher level questions and suggested activities that lead students to construct viable arguments and analyze the arguments of others.

In the Teacher's Implementation Guide, page FM-37 and pages FM-43 through FM-55 do provide teachers with instruction on how to get students to construct viable arguments and critique the work of others, while each lesson also has a "Share Phase" section in the margins that poses questions teachers are supposed to ask for discussion; however, many of the questions being asked are closed ended questions and do not promote discourse. Also, there are no suggestions for how students should share or report out their thinking, and the lessons are written in the same format all the way through the series which does not promote students to think about mathematics in different ways.

As the wrap around teachers edition was reviewed, the publisher did not specifically address potential teacher moves regarding constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics. The vast majority of the time, the areas that addressed MPs were merely closed ended questions added to the practice section of the lessons. Teachers are not given any specific examples on how to address this practice in their daily lessons.

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Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The materials reviewed for Course 3 meet the expectation for attending to the specialized language of mathematics. Overall, there are several examples of the mathematical language being introduced and appropriately reinforced throughout the unit.

  • In the Teacher Implementation Guide, page FM-40, there is an explanation of MP6.
    • It states that students should communicate clearly and use clear definitions in discussions and writing.
    • It emphasizes that students should label things to clarify their work.
    • This section also states that the answers provided in the Teacher's Implementation Guide are exemplars of student responses and model precision appropriately.
  • Along with information about a vocabulary section in the skill practice for each lesson. The book also states that each lesson provides opportunities for students to communicate precisely in writing and when sharing their solutions.
  • Each chapter has key terms listed in Lesson 1. The words are then defined somewhere in the lesson and written in bold font.
    • The terms are not in bold or referenced after that first lesson.
    • The terms are listed again in the chapter summary, but it does not define or use them in any way.
    • Lesson 1 in the student skills practice book, has students work with the vocabulary and the answers in the teacher's guide provide the precise answers written in the language of mathematics.

GATEWAY THREE

Usability

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The instructional materials reviewed for the Integrated series from Carnegie partially meet expectations for alignment to the CCSSM for high school. The materials partially meet the expectations for focus and coherence as they: partially attend to the full intent of the mathematical content contained in the high school standards for all students, do not attend to the full intent of the modeling process when applied to the modeling standards, partially allow students to fully learn each standard, partially make meaningful connections in a single course and throughout the series, and partially identify and build on knowledge from Grades 6-8 to the High School Standards. The materials also partially meet the expectations for rigor and the MPs as they partially support the intentional development of conceptual understanding and do not meet the expectations for meaningfully connecting the MPs to the standards for mathematical content.

Criterion 1a-1f

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 partially meet the expectation for attending to the shifts of focus and coherence. The instructional materials reviewed in this series focus the students' time on the Widely Applicable Prerequisites for a range of college majors, postsecondary programs, and careers. The focus is diminished as there are some standards which are not fully developed throughout the series due to aspects that are never addressed or specific methods/content identified in the standards that are not addressed throughout the series. Also, there is a lack of coherence across materials as the connections between standards, clusters, domains, and conceptual categories called for in the standards are not identified for teachers and students which leaves the content disconnected and a series of topics to be covered. There was also a lack of connection to the Grade 6-8 standards with clear guidance about how the high school work built upon the work from the middle school grades. There were many lessons of content that were repeated in courses. These lessons were identified with different standards, but the content was largely the same, if not identical. This becomes distracting and misleading for students and teachers.

Indicator 1a

The materials focus on the high school standards.*

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Indicator 1a.i

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

The instructional materials reviewed for the series partially meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. In general, the series included the majority of all of the non-plus standards, but there were some instances where the full intent of non-plus standards was not met.

For G-MG.2, evidence was not found anywhere throughout the series, and the standard was not identified in the materials.

The following standards are examples identified as having been fully met in this series in the conceptual categories and domains listed:

  • A-SSE.1: In instances where this standard is listed (Integrated Math I Chapters 2, 3, 5; Integrated Math II Chapters 12, 13; and Integrated Math III Chapters 3, 5, 8), the use of real world applications and varied types of expressions (linear, quadratic, power, exponential) that are used throughout the series clearly meets the standard.
  • G-MD.4: In instances where this standard is listed (Integrated Math II, Lessons 11.1-11.5, 11.7), the use of rotating and stacking two-dimensional figures to create three-dimensional solids, opportunities for informal argument of the derivation of formulas for volume of a cone, pyramid, and sphere, and opportunities to explore cross sections of solids clearly meets the standard.

The following standards are identified as having been partially met in this series in the conceptual categories and domains listed. In general, many of the standards that are partially met earn that classification due to the lack of student opportunity to engage in certain aspects stated in the standards.

  • N-Q.2: In instances where this standard is listed in Integrated Math I (Lessons 1.1, 3.1, 5.6), students are not provided opportunities to independently identify quantities to represent the context; rather students are provided with pre-labeled tables or graphs with pre-determined numbers making the quantities that they represent obvious to the student.
  • F-BF.3: This standard appears in Integrated Math I, Lesson 5.3, Integrated Math II, Lesson 12.7 and Integrated Math III, Lessons 4.2-4.4, 5.3, 9.2-9.3, 11.3, 12.3, 12.5 and 14.6. When students are asked to identify the effect on the graph of replacing f(x) by f(kx) for specific values of k (both positive and negative) any problem that involves a negative uses k=-1. Students are not given graphs and asked to find the value of k.
  • G-CO.9: In four lessons where this standard was identified within Integrated Math II (Lessons 2.1, 2.2, 7.4 and 7.5), students were never asked to construct a proof about lines and angles.
  • G-CO.10: The proof of the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; or the medians of a triangle meet at a point was found in Integrated Math II, Lesson 4.3, Problem 6. However, this standard was not referenced for teachers and students.
  • G-CO.12: Geometric constructions largely rely on compass and straight-edge techniques, with a few references to patty paper. The other methods of strings, reflective devices, and dynamic geometric software, etc., are not present in the materials.
  • G-GMD.1: Application of Cavalieri's Principle was present in Integrated Math II, Lesson 11.3. Students did not have opportunities to use dissection arguments and informal limit arguments for the circumference of a circle, area of a circle and volume of a cone.
  • S-ID.4: This standard requires students to: “use mean and standard deviation to fit it to a normal distribution and to estimate population percentages. Recognize there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve.” Problems were not found that included data sets where the normal distribution did not apply.
  • S-IC.4 and S-IC.5: There was not evidence of the use of simulation as stated in the standards.

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Indicator 1a.ii

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

The instructional materials reviewed for the series do not meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. Throughout the series, there are a number of lessons that contain a variety of components of the modeling process described in the CCSSM. Students are provided scaffolding questions to help guide them through the process of modeling an equation and reasoning from that model. However, throughout the series, students do not have an opportunity to authentically engage in the modeling process by gathering their own data, organizing it, creating multiple representations of it and interpreting the representations.

A few examples of when and how components of the modeling process are not fully attended to include:

  • Integrated Math I and Integrated Math II materials provide multiple opportunities to interpret features of graphs and tables, yet lack all the steps included in the modeling process to meet the full intent of the modeling standard which the standard F-IF.4 requires. Students do not have the opportunity to develop their own solution strategies due to the presence of scaffolded questions or identify variables and formulate a model by creating tabular models due to the presence of predetermined graphs with scales and some given equations.
  • In Integrated Math I, Lesson 2.3: Modeling Linear Inequalities, students model the profit of selling popcorn. In this lesson, students are provided the inequality already graphed and shown how to solve the problem algebraically. In every case, students are provided pre-made graphs with the scale already selected, partially completed tables and step-by-step directions.
  • In Integrated Math I, Lesson 4.3, the modeling standard A-SSE.1a is listed, however, the formulas are given and not interpreted through modeling. The modeling standards F-BF.1, F-BF.2 and F-LE.1,F-LE.2 are also listed, however, no opportunities are provided for students to build their understanding of the development of recursive and explicit formulas for sequences.
  • In Integrated Math I Lesson 11.3, People, Tea, and Carbon Dioxide, all problems are intended to provide students with an opportunity to engage in modeling. Students are asked to compute, interpret and report their work through a series of scaffolded steps. However, not all six steps of the modeling process are included. For instance, students never chose a model to use – they are given a table to fill in and an equation to use to find the values.
  • In Integrated Math II, Lessons 7.1 and 7.3 - 7.5, the modeling standard G-SRT.8 is listed. Students, however, are not provided any opportunities to attend to the modeling process using this standard.
  • In Integrated Math II, Lesson 11.6, for G-GMD.3, problems provided within the teacher implementation guide and student materials allow for students to think more critically about the use of formulas for irregular shapes as well as how different formulas compare and making a decision based on that comparison. Examples include approximating the volume of a vase and finding about how many cubic feet of hot air a typical hot-air balloon holds, and the guiding questions, such as "Is there more than one correct strategy to approximate the volume of the vase?" and "How do you decide which strategy will product a more accurate result?" However, students are not provided opportunities to develop their own solution strategies and develop ways to analyze the results.

Notably, the “formulate” part of the modeling provided in the CCSSM is consistently lacking in the lessons provided in the material(s). The CCSSM states that students should be “formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables. There was no evidence that students were ever required to formulate a process for solving any problems or work through the modeling process on their own.

A few examples of standards for which no part of the modeling process attended to:

  • No aspect of the modeling process is addressed within the Integrated Math II, Lesson 3.1, problem 4, for the standard G-MG.2.
  • In Integrated Math I, Lessons 12.1, 12.2, 12.4, 14.1, 14.2, 14.3, 14. 4 and Integrated Math II, Lesson 4.3, the modeling standard G-GPE.7 is listed, however, students are not provided any opportunities to attend to the modeling process using this standard. Four of these lessons are duplicated within Integrated Math II and the modeling standard G-GPE.7 is not listed in these Lessons 1.2-1.4 and 17.1 as it was indicated within Integrated I.
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.

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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 for this series, when used as designed, meet the expectation for allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, post-secondary programs, and careers. Examples include, but are not limited to:

The Algebra standards are included throughout the series and are seen as a focus.

  • A-SSE evidence is found in Integrated I Chapters 2-5, Integrated II Chapters 12 and 13; and Integrated III Chapters 3-6, 8-10.
  • A-CED evidence is found in Integrated I Chapters 1-3, 5 and 7; Integrated II Chapters 12-14 and 16; and Integrated III Chapters 3, 4, 6-10 and 14.
  • A-REI evidence is found in Integrated I Chapters 1-3, 5-7; Integrated II Chapters 13-15; and Integrated III Chapters 3, 6, 7, 9-11 and 14.

The F-IF standards are included throughout the series and are seen as a focus.

  • Evidence is found in Integrated I Chapters 1-5, 11 and 16; Integrated II Chapters 12, 14, and 16; and Integrated III Chapters 3-7, 9, 11, 12, 14 and 15.
  • A variety of functions are interpreted and analyzed. Integrated Math I focuses on linear, quadratic, and exponential, while Integrated Math III focuses on quadratic, polynomial, exponential, logarithmic, rational, and trigonometric.
  • Within the series, students graph functions and identify/analyze key features of those functions.

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Indicator 1b.ii

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

The instructional materials reviewed for this series partially meet the expectation that the materials allow students to fully learn each standard. The materials for the series, when used as designed, would not enable students to fully learn some aspects of the non-plus standards.

All non-plus standards, other than G-MG.2, are referenced at least once.

There are several examples of when the materials would not enable students to fully learn some aspects of the non-plus standards:

  • N-Q.1: In Integrated Math I, Lessons 2.1, 2.2 and 2.6, Student Assignments and Skills Practice both heavily favor identifying the independent and dependent quantities and their units with the use of a table and have very few problems identifying these with graphs. Students are not provided adequate opportunities to choose their own scales or origins in the student textbooks and assignment books. There is one blank graph made available in the student textbook for Lesson 2.6, page 139 and in the Skills Practice book, pages 302-304, problems 8-12, the grid given has no markings of scale or unit.
  • N-RN.1: In Integrated Math I, Lesson 5.5, students are not asked to explain how rational exponents are an extension of the properties of integer exponents. Students are provided one example where they are given an equation, given the substitution that results in the rule for the fractional notation of a radical number and asked to apply it.
  • N-RN.2: In Integrated Math I, Lessons 5.5 and 5.6, the majority of material content aligns with 8.EE, simplifying integer negative exponents, and the only high school appropriate work is provided on page 342, explaining and extending the properties of exponents to rational exponents. (Note, this lesson contains typographical errors.) In Integrated Math II, Lesson 13.6, this standard is listed, however, the student work involves rewriting radicals and does not provide opportunities to work with rational exponents. In Integrated Math II, Lesson 15.3, this standard is listed, however, students are provided one review problem within this lesson related to this standard. The rest of Lesson 15.3 addresses N-CN.1.
  • G-CO.2: In Integrated Math I, Chapter 12, the materials did not provide students opportunities to represent transformations in the plane using geometry software. Also, transformations were not described as functions that take points in the plane as inputs and give other points as outputs.
  • G-SRT.8: In Integrated Math II, Chapter 7, several lessons require the use of trigonometric ratios which have not yet been introduced to students. Introduction to trigonometry occurs in Chapter 8. Problems solved using trigonometric ratios are found in Lesson 7.1, Problem 3, Number 4; Lesson 7.4, Problem 4, Number 4; and Lesson 7.5, Problem 2, Number 10. In Integrated Math II, Chapter 7, Lessons 7.1, 7.3 - 7.5, this standard is listed, however, one problem (7.3 problem 5) is provided for students to use the Pythagorean Theorem. Furthermore, in the Chapter 7 Skills Practice for the same lessons, students are not provided opportunities to use trigonometric ratios and/or the Pythagorean Theorem to solve right triangles in applied problems.
  • F-IF.4,5 and 7; F-BF.1 and 4; F-LE.1 and 2: Integrated Math I includes Chapter 16 on logic. This chapter does not provide students with opportunities to learn any of the function standards identified. This additional material distracts student learning from the high school standards.
  • G-MG.3: In Integrated Math II, Lesson 10.4 when problems were provided for extension work, students are asked to solve problems for linear velocity and angular velocity. This additional material distracts student learning from the high school standards.

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Indicator 1c

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

The instructional materials reviewed in this series fully meet the expectation that students engage in mathematics at a level of sophistication appropriate to high school. Materials meet the depth of the non-plus standards. When used as designed, all students are given extensive work with non-plus standards. Following are several examples of when students are given extensive work with the non-plus standards:

  • In Integrated Math I, Lesson 8.4, for standard S-ID.2, students revisit data sets previously used and have the opportunity to use the formula and technology to compute the standard deviation of each data set. In the last activity, students compare measure of center (median and mean) and the measures of spread (IQR and standard deviation) with respect to their sensitivity to outliers.
  • In Integrated Math II, Lesson 15.5, for standard N-CN.7, the lesson begins with quadratic functions having one, two or no x-intercepts graphed on a coordinate plane. Students list the key characteristics of each graph. Students rewrite a quadratic function with imaginary zeros written in standard form to factored form and then to vertex form.
  • In Integrated Math III, Lesson 10.2, for standards A-SSE.2 and A-APR.7, students use the structure of an expression to identify ways to simplify rational expressions and list their restrictions for the variables.
  • Throughout the series, for standards A-REI.10, 11, and 12, students are provided extensive opportunities to represent and solve equations and inequalities both algebraically and graphically. Problems involve real world scenarios and students are instructed on how to use several graphing calculator strategies.

The materials provide students with opportunities to engage in real-world problems throughout the courses. The students engage in problems that use number values that represent real-life values - fractions, decimals and integers. Solutions to problems also are typical of real-life situations, and the context of most of the scenarios are relevant to high school students.

Individual standards are given more instructional time than the whole clusters. There are only a few opportunities for non-plus standard work in the Number and Quantity category, but many opportunities on Algebra and Functions. N-RN is found within four chapters throughout the entire series in eight lessons. N-Q is found within four chapters within Integrated Math I in eight lessons. In contrast, F-BF.1 is found within eleven chapters throughout the entire series in over 20 lessons.

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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 instructional materials reviewed partially meet the expectation that the materials are mathematically coherent through meaningful connections in a single course and throughout the series. There are two conceptual categories which are not coherently connected to other conceptual categories across the series.

  • The majority of the Geometry standards are isolated within Integrated Math II and are not connected to other conceptual categories such as Algebra and Functions. In Integrated Math I, Chapter 14 contains unique geometry material as Chapters 12 and 13 are duplicated content within the Integrated II series. Students do not have the opportunity to work with Geometry standards at all within Integrated Math III. For example, students are not given the opportunity to connect creating polynomial equations, A-APR.3, to volume formulas in G-GMD.3. The publishers missed the opportunity to purposefully connect Geometry standards to Algebra through the use of creating equations A.CED. There is no chapter within the series where geometric coordinates, G-GPE.7, is referenced along with equations, A-REI, as suggested on page 74 of the CCSSM. Integrated Math II, Lesson 17.3, is a missed opportunity for the problems to require students to connect these standards.
  • The majority of the Statistics and Probability standards are isolated within Integrated Math I and III and are not connected to other conceptual categories such as Algebra and Functions. The publishers missed the opportunity to purposefully connect S-ID and S-IC with Algebra and Functions.

Examples of connections between standards within a single course include:

  • Integrated Math I demonstrates strong connections between the conceptual categories of Algebra and Functions. The materials connect linear functions, exponential functions, arithmetic and geometric sequences, and recursive and explicit representations. Chapters 1 and 2 introduces students to the concepts of functions and linear functions. In Chapter 4 students begin to work with arithmetic and geometric sequences. The relationship between arithmetic sequences and linear functions and some geometric sequences and exponential functions is developed. Students use recursive and explicit formulas to connect these concepts.
  • Integrated Math II Chapters 7 and 8 connect the idea of using trigonometric ratios, G-SRT.C, as a way of analyzing quadrilaterals and proving their properties, G-CO.B, to aid in problem solving.
  • Integrated Math III, Lesson 3.4 connects the Algebra category to Geometry through modeling scenarios in an engineering-based problem. While Geometry standards from G-MG are not explicitly identified, students engage in problem solving and critical analysis of a figure modeled geometrically and define it algebraically through equations and functions to reason about and justify their solutions.

Examples of connections within a single course that are not adequately developed include:

  • Integrated Math I, Lessons 9.1 - 9.5, addressing S-ID.6, 7, 8 and 9, has students use the calculator to produce a regression line, use this line to make some predictions, and then asked to find the equations of lines between pairs of points in a data set to determine which lines best "matches" the data. There are several missed opportunities to connect to the function standards in domains F-IF, F-BF and F-LE.
  • Integrated Math I, Lesson 11.4, Choosing the Best Function to Model Data lists only the function standards. There are missed opportunities to connect to the statistics standards in S-ID.
  • Integrated Math I, Chapter 13 addressing G-CO.6-8 missed opportunities within the Student Assignments and Skills Practice sections to identify which rigid motion created the pairs of triangles in problems where both triangles are given.
  • Integrated Math II, Lesson 1.2 (duplicated from Integrated Math I, Lesson 12.1) addressing G-CO.2, translating line segments, is a missed opportunity to connect transformations to functions F-BF.3 covered in Lesson 5.3.
  • Integrated Math II, Lesson 1.2 (duplicated from Integrated Math I, Lesson 12.1) addressing G-CO.4 is a missed opportunity to connect the rotation of a line segment around its endpoint to the creation of a circle (Problem 3), as a way of developing a definition for a circle, as called for in the standard.
  • In Integrated Math II, Lesson 10.1 which addresses G-C.3, the Skills Practice has students determine one angle of a quadrilateral given its opposite angle (Quad-Opp angle theorem). There is a missed opportunity to connect to G-C.2 and G-CO.11.

Additionally, lessons are renamed and nearly identical across the series but are not indicated as review or repeats to students or teachers. Lessons that are repeated with minor alterations, such as a few of the graphics changed and a few additional problems added, do not explicitly connect this repetition (nor do they point out this repetition) to the original lesson in which the content was developed. Those lessons include:

  • G-CO.A, B, D Integrated Math I, Lessons 12.1, 12.2, 12.3, and 12.5 and Integrated Math II, Lessons 1.2-1.5.
  • G-GPE.B Integrated Math I, Lesson 12.4 and Integrated Math II, Lesson 17.1.
  • G-CO.A, B, D Integrated Math I, Lessons 13.1 - 13.6 and Integrated Math II, Lessons 5.1-5.6; Lesson 5.7 is the only new lesson in Integrated Math II.
  • G-GPE.B Integrated Math I, Lessons 15.1-15.2 and Integrated Math II, Lessons 17.2-17.3.
  • G-GPE.B Integrated Math II, Lesson 15.4 and Integrated Math III, Lesson 4.6.
  • G-CO.9 Integrated Math I, Lesson 16.1 and Integrated Math II, Lesson 2.1.

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Indicator 1e

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

In the instructional materials reviewed for this series, content from Grades 6 to 8 is present but not clearly identified and/or does not fully support the progressions of the high school standards. Connections between the non-plus standards and how those standards are built upon from Grades 6-8 is not clearly articulated for teachers.


In the teacher’s or student's materials, there is no reference to 6-8 CCSSM throughout the series. Course Content Maps downloaded from online Integrated Math I (2012) and Integrated Math II, III (2013) sometimes have information in a column titled “Access Prior Knowledge” that references middle school standards. Repeatedly, the series for high school introduces and/or develops a 6-8 standard, but instead of identifying it as such and clearly making the connection, the series introduces it as a high school standard.


Examples of how lessons connect to middle school content include:

  • G-MD.4: Students in middle school calculate area and perimeter of two-dimensional shapes and calculate volume and surface area for three-dimensional shapes, 6.G.A. 7.G.B, 8.G.9. In high school, students use these skills in a more sophisticated fashion through the use of application problems. The connection between two-dimensional and three-dimensional figures culminates with the topics of cross sections and diagonals in three dimensions.
  • A-REI.5-7; A-REI.11-12: Students in middle school analyze and solve pairs of simultaneous linear equations, 8.EE.8. In high school, students connect this standard to solving a linear equation algebraically and graphically and extending this to solving and graphing systems of linear inequalities.

Here are some examples where the materials do not correctly identify content from Grades 6-8 in an appropriate way for high school:

  • Integrated Math I, Lessons 2.1-2.2, 3.2 and 3.4: In the material with F-IF referenced, student problems found on pages 75, 84 ("Talk the Talk"), 174 and 186 are aligned with 8.F, using functions to model relationships between quantities.
  • Integrated Math I, Lesson 5.5: In the material with N-RN.1 referenced, student problems found on pages 338-342, are aligned with 8.EE.A, radicals and integer exponents.
  • Integrated Math II, Lesson 15.1: In the material with N-RN.3 referenced, problem 3 is aligned with 8.EE and 8.NS, translating between decimal and fraction notation, particularly when the decimals are repeating.

Here are some examples where the materials fail to reference standards from Grades 6-8 for the purpose of building on students’ previous knowledge:

  • Integrated Math II, Lessons 13.1-13.3: In the content on pages 952-956; 963; 972-973, the properties of real numbers (commutative, associative, distributive, etc.) are presented as applying the properties of operations to generate equivalent expressions without mention to build upon students’ prior knowledge of 6.EE.3.
  • Integrated Math II, Lessons 15.1 and 15.2, standard N-RN.3 is listed but the material covered in these lessons has students work with content below high school level, defining sets of numbers, determining which sets of equations can be solved and writing repeating decimals as fractions. Students are walked through a history of numbers and given definitions and rules all along the way. Students are told that an irrational number has an infinite, non-repeating decimal form but no explanation is given. On page 1088, students are asked to simplify expressions and identify the property. The expressions the students are to simplify contain whole number coefficients aligned with 6.EE.
  • Integrated Math III, Lessons 2.1-2.2: In the lessons on “Sample Surveys, Observational Studies, and Experiments” and “Sampling Methods and Randomization,” students are introduced to sampling and making inferences without mention of the standard addressed in 7.SP.
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.

Some of the plus standards, when included, are clearly indicated in the Teacher Implementation Guides located in the Chapter overviews. The inclusion of the plus standards follows logically in progression with the material. Lessons including the plus standards could be omitted without interrupting meaning, or the understanding for the student.

Integrated Math II, Lesson 15.4, and Integrated Math III, Lesson 4.6, are in fact the same; however, both follow logical ordering in their respective materials. The lesson includes standards, N-CN.3 and N-CN.8, working with complex numbers as an extension of learning both quadratics and the real number system. Students are able to practice finding complex conjugates, N.CN.3, and rewriting/extending polynomial identities, N.CN.8, throughout student materials.


Integrated Math III, Chapter 9, walks students through identifying zeroes, asymptotes, end behavior and factorization in order to graph rational functions, F-IF.7d, which students practice throughout their materials. The teacher material which supports the students assignment in Lesson 9.5 suggests using a graphical approach to solve one of the problems, but the problem could be solved without needing to graph. Thus, it is not dependent on 9.1-9.4.

Integrated Math III, Chapter 13, could not be omitted in it's entirety, as it is the only location where students develop skills in F-LE.4, which has a natural connection with F-BF.5.

The series is inconsistent in differentiating between plus and non-plus standards through introduction or description of the lesson. There are several lessons within the series that are not clearly identified as plus standards:

  • Integrated Math II, G-SRT.9, G-SRT.10, and G-SRT.11 in Lesson 8.6
  • Integrated Math II, G-GMD.2 in Lessons 11.3
  • Integrated Math II, F-BF.4b in Lessons 16.3
  • Integrated Math II, S-CP.8 in Lessons 19.3 and 19.5
  • Integrated Math II, S-CP.9 in Lessons 20.3 and 20.4
  • Integrated Math II, S-MD.6 and S-MD.7 in Lessons 20.5
  • Integrated Math III, A-APR.5 in Lesson 6.7
  • Integrated Math III, F-BF.1c and F-BF.4b in Lessons 14.1, F-IF.7d in Lessons 14.2 and 14.4

The plus standards are never identified within the student materials.

The following plus standards identified as addressed within the materials did not reach the full depth of the standards due to the lack of student opportunity to engage in certain aspects stated in the standards:

  • Integrated Math II, G-SRT.9, G-SRT.10, and G-SRT.11 in Lesson 8.6: Proper depth is not accessible for students for any of these standards. These standards are condensed into one lesson and suggested to be covered in one day within the timeline.

The instructional materials reviewed meet the expectation for rigor and balance. The aspects of rigor are balanced throughout the lessons, chapters and courses, and the lessons are often developed in a way to allow students to engage in relevant mathematics and develop their understanding. Many lessons begin with an application of the mathematical concept addressed in the lesson. Fluency is developed throughout the problems in the lessons and specifically through the work in the Student Skills Practice Book. A concern is that many lessons are scaffolded in such a way that students are guided through a solution path or given properties to use that are not fully developed by the students. This step-by-step process diminishes the rigor of those lessons and inhibits the development of conceptual understanding.

Criterion 2a-2d

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.

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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 partially meet the expectation for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. Although the materials generally allow students to build conceptual understanding of key mathematical concepts, there are missed opportunities. The lessons, practice, and assessments allow for students to develop and demonstrate their understanding through a variety of methods including models, constructions, and application problems. The materials often provide students with opportunities to justify, explain and critique the reasoning of others; however, sometimes steps for solving problems are scaffolded in a way that restricts alternate ways of approaching a problem and therefore diminishes the cognitive demand of the lesson (see N-RN.1 below). Students demonstrate their understanding individually, in pairs, in small groups, and as a class. The materials generally provide some opportunities for students to build their understanding from simpler problems and numbers to more complex situations and numbers.

The following are specific standards for which the materials partially met the expectation for developing conceptual understanding:

  • N-RN.1: This standard states the following: "Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents." The relationship between rational exponents and radical notation is provided to students in Integrated Math I, Lesson 5.5, Integrated Math II, Lesson 15.3, and Integrated Math III, Lessons 11.4 and 11.5. Although there are several opportunities with equivalent and simplified expressions, students are shown the rules and are expected to use them. For example, there are no connections for the product property between the exponent and repeated multiplication that would allow students to deepen their understanding of the properties rather than just repeat a rote process. A cut and paste grouping activity (Integrated Math III, page 807) is utilized to group equivalent expressions that are written in non-simplified form. One question in this lesson (Integrated Math III, page 815) shows three examples of student work and has the student determine whose work is correct. A similar question (Integrated Math III, page 814) shows three different methods for simplifying an expression (all methods are correct; one uses radical notation while the other two use rational exponent notation), and students need to identify similarities and differences among the methods and explain in writing why all three are correct. Although the variety of activities are included, the activities only require students to apply rules that are given, not develop or explain the rationale for those rules.
  • N-RN.3: This standard is addressed through Lessons 15.1 and 15.2, but conceptual understanding is partially developed as students are not given opportunities to explain "that the sum of a rational number and an irrational number is irrational and that the product of a nonzero rational number and an irrational number is irrational."
  • A-REI.A: This cluster is addressed in Integrated Math I Lesson 2.1, but not in a way such that students are required to justify the solution process. Students only have to solve problems and show work. The teacher notes suggest asking students about solution paths, but the justification or construction of a viable argument is not required by the prompts provided. Additionally, this lesson includes these problems as a portion of the lesson but not the emphasis of the lesson; therefore, this cluster is not fully developed in this lesson or in subsequent lessons in this course.
  • A-REI.11: This standard is thoroughly addressed only for linear and quadratic equations, and rational functions are addressed in only one example. Polynomial, absolute value, exponential and logarithmic functions, which are specified in the standard, are not addressed in any of the courses.

The following are specific standards for which the materials met the expectation for developing conceptual understanding:

  • F-IF.A: A sorting activity in Integrated Math III, Lesson 3.3 on pages 135-141, provides students with the opportunity to analyze relations (represented in an equation, table, graph, or scenario) and sort them into equivalent relations. As a follow up, students are asked to determine which of the equivalent relations represent a function and which do not represent a function.
  • G-SRT.6: In Integrated Math II, Lesson 8.1 features an exploration with ratios as an introduction to the trigonometric ratios of sine, cosine, and tangent. Students are expected to calculate ratios of sides in given triangles (concrete) and generalize these findings to overarching questions near the conclusion of the exploration (i.e., “Is each ratio the same for any right triangle with a congruent reference angle? As a reference angle measure increases, what happens to each ratio?"). This concept is extended in Lesson 8.2 on page 582.
  • S-ID.7: Students have many opportunities to develop their conceptual understanding of slope and intercept in the context of the data. The material repeatedly uses charts to break down functions into their components that the student must interpret in context and then draw conclusions about. Some examples of this are included in Integrated Math I on pages 170 and 176. Slope and y-intercept are again interpreted in the context of a given scenario and data set in Integrated Math I on pages 524-525. In an example on page 531 of the Integrated Math I materials, the y-intercept must be obtained through extrapolation, and the materials ask students to determine whether the extrapolated y-intercept makes sense in terms of the context.

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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 expectation for providing many intentional opportunities for students to develop procedural skills and fluency. The lessons begin with a "Warm Up" problem that often review the procedure from a previous lesson or lessons. Within the lessons, students are provided with opportunities to develop procedures for solving problems that begin to develop fluency. The lessons provide students with a variety of practice experiences - some problems are completed with the whole class, others with partners and some independent. Each classroom lesson ends with a "Check for Students' Understanding" that is often furthering the development of procedural skills learned in the lesson. The materials also include a Student Skills Practice workbook and a Student Assignments workbook. Both of these workbooks continue to develop procedural fluency by providing significant opportunities for students to practice independently. The Student Skills Practice that accompanies each course in the series primarily focuses on developing fluency of mathematical procedures.

Some highlights of strong development of procedural skills and fluency include:

  • A-APR.1 - Students are provided several opportunities to practice adding, subtracting, and multiplying polynomials within Integrated Math II, Lessons 13.1 and 13.2 to enhance student fluency in conducting this skill.
  • A-SSE.2: The instructional materials provide multiple opportunities for building fluency with factoring (Integrated Math II, Lessons 13.4, 13.5; Integrated Math III, Lesson 6.2).
  • F-BF.3 - Materials strongly emphasize transformations of functions, and this is evident in the amount of practice the materials provide. For several types of functions (quadratic, radical, rational, exponential, logarithmic), students practice graphing a transformed function, write in words how f(x) is transformed to g(x), write transformed functions in terms of other graphed functions (example problems in Integrated Math III, page 333 in the Student Skills Practice), and use a table to show how a reference point from a parent function is mapped to a new point as a result of a transformation.
  • G-GPE.4 - Materials provide several opportunities to use the distance formula and slope formula to classify quadrilaterals on the coordinate plane. Multiple types of quadrilaterals are discussed in the materials.
  • G-GPE.5 - Materials provide several opportunities in Integrated Math I, Lesson 12.4, to determine whether two lines are parallel or perpendicular given an equation or a graph with plotted points. Students also write an equation of a line passing through a given point that is parallel/perpendicular to a given line. Furthermore, in Integrated Math I, Lesson 15.2 uses information about the slope of parallel and perpendicular lines to classify quadrilaterals on the coordinate plane.

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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 expectation of 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. Students work with mathematical concepts within a real-world context. Sometimes contextual situations are used to introduce a concept at the beginning of a lesson while other times contextual situations are used as an extension of conceptual understanding. Single-step and multi-step contextual problems are used throughout all series materials and are intended to be utilized in different class settings (individual, small group, whole group).

Additional considerations related to real-world applications:

  • When students are given a mathematical object within a provided context, the materials have students decompose the object into its individual terms in which students need to identify the appropriate unit, contextual meaning, and mathematical meaning. For an example, see the table on page 78 in Integrated Math I.
  • Statistical concepts are taught within contextual settings requiring students to interpret data and make sense of their conclusions. For example, measures of central tendency are compared when analyzing the dot plots for the heights of players on two basketball teams. Polls and voting are used to provide context to teaching how to make inferences from population samples.

As noted previously, these applications are often given with extensive scaffolding, which could detract from the full depth of the standard being met, especially in regards to the modeling standards (see indicator 1aii).

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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 materials meet the expectation of providing balance among conceptual understanding, procedural skill and fluency, and application. No one aspect of rigor dominates problems/questions in the materials. In many lessons throughout the series, students are required to use multiple representations and written explanations to support their work and justify their thinking in order to demonstrate their understanding of procedures, skills, and concepts. The lessons generally provide opportunities for students to develop conceptual understanding - often through an initial application of a real-world concept - and are followed by opportunities for students to develop fluency through the Student Skills Practice sections.

The instructional materials reviewed partially meet the expectation for connections between the MPs and the standards for mathematical content. The instructional materials do not provide specific information for aligning the MPs to the Standards for Mathematical Content or to specific lessons. General information about the MPs is given at the beginning of each course within the teacher guides, but ongoing information for students or teachers is lacking. There are several components of the MPs within most lessons. However, teachers or students are not told which to focus on within the lessons because they are not specifically addressed/identified. An intentional structure for consistently addressing the MPs throughout the lessons would enhance the implementation of the MPs and benefit students and teachers.

Criterion 2e-2h

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

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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 partially meet the expectation of supporting the intentional development of overarching, mathematical practices (MP1 and MP6), in connection to the high school content standards, as required by the MPs. MPs are not explicitly identified throughout the series. Course/series scope and sequence charts do not include identification of MPs to chapters or lessons. A very brief overview of the MPs and how they are generally addressed throughout the series is included at the beginning of each course textbook (for example, Integrated Math II FM-24 to FM-32) as well as aligning the types of problems students will encounter to the MPs (for example, see Integrated Math II FM-44 to FM-47). Although the materials show an example of each MP, no notation/justification is given for why or how that particular example relates to the identified MP.

For MP1, the introductory Supporting the Practice section in the teacher materials states that a key component is for students to make sense of problems and develop strategies for solving problems. Student development of strategies is not evident in the majority of lessons other than students creating a pathway to a solution that follows the examples given or a scaffolded process that is provided for students. These support structures reduce the level of sense making required to fully address this practice standard. If the scaffolded and/or repetitive structure was abandoned, students would have the opportunity to make their own sense of problems and develop their own methods for solving them.

MP6 is addressed throughout the materials even though it is not specifically identified in any lessons. Students are often asked to use or create definitions, students are expected to use units appropriately when necessary, and they are often expected to communicate understanding clearly in writing and/or orally.

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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 partially meet the expectation of supporting the intentional development of reasoning and explaining, MP2 and MP3, in connection to the high school content standards, as required by the MPs.

For MP2, the overview says that this standard is addressed throughout the lessons because lessons often begin with real-world application and transition to mathematical representations. Although this may be a part of attending to MP2, this is not the entirety of the standard. For MP3, students often do construct viable arguments and do critique the reasoning of others. However, no additional support for helping teachers or students develop this practice is evident. Although exemplar answers are provided, teachers are not given guidance on how to get students to provide those types of answers. Some examples of how the materials align to components of MP2 and MP3 include:

  • "Thumbs Up" problems embedded throughout series materials provide opportunities for students to examine a correct solution pathway and analyze the approach as they try to make sense of another student's work.
  • "Thumbs Down" problems embedded throughout series materials provide opportunities for students to analyze an incorrect solution pathway and explain the flaw in the reasoning that was provided.
  • "Who's Correct" problems embedded throughout the series provide opportunities for students to analyze several solution pathways and decide whether they make sense. If a solution pathway is incorrect, students are asked to explain the flaw in the reasoning that was provided.
  • In Integrated Math I and Integrated Math III materials, tables are utilized to consider the units involved in a problem (for example, Integrated Math I textbook page 89). These tables provide the opportunity for students to attend to the meaning of quantities in an attempt to relate the contextual meaning and mathematical meaning of the provided scenario.

Problems frequently ask students to explain their reasoning. For example, Integrated Math I, Lesson 2.1 includes, “What is the slope of this graph? Explain how you know," but extensive use of scaffolding for problems reduces the depth of explanations and critiques created by students.

The material encourages students to decontextualize problems, often requiring them to come up with a verbal model or a picture of the problem and then put the mathematical measurements back in to find the answer. The material consistently provides opportunities for students to define the variables in the context of the problem and also define the terms of more complicated expressions within the context of the problem (Integrated Math I, page 185).

The material consistently poses problems that require students to examine simulated student work, determine if they were correct or not, and defend their answers with solid mathematical reasoning.

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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 partially meet the expectation of supporting the intentional development of modeling and using tools, MP4 and MP5, in connection to the high school content standards, as required by the MPs.

For MP4, the information states that the materials provide opportunities for students to create and use multiple representations, and this is true in some instances. However, there are not often specific instructions for teachers on how to make connections or get the connections from the discussion or even which connections to emphasize. For instance, in Integrated Math II, Lesson 12.6 on page 904, students have a table, a graph and a set of characteristics to identify. The guiding questions only call out characteristics of the problem and of using a calculator and do not make connections between the representations. The connections between the ways the zeros are represented is critical – in a table and on a graph. One question is "how do you use a graphing calculator to determine the x-intercepts?" This question is presented with no answers in the teachers' materials, but it has many - students can look at the graph, the table, or calculate them all using the calculator. No connections are made for teachers or students about why this is, and therefore, MP4 is lacking in this lesson.

An example of where connections among multiple representations are made is in Integrated Math I on pages 348-349. In this example, the scenario of an exponential growth problem is represented in a table, graph, and equation. Questions in the textbook are included to identify relationships among the representations.

Also for MP4, many lessons include mathematical models of real-world situations, but models are typically provided so that students are not asked to develop models themselves. For example, Integrated Math I, Lesson 2.1 includes a situation modeling the change in altitude of a plane but gives tables for students to complete and tells them to use one of the tables to draw a graph.

For MP5, tools in the Integrated Math I and Integrated Math III materials are primarily limited to paper, pencil, calculator and/or graphing calculator. Students rarely have opportunities to choose an appropriate tool to use to solve a problem. Materials often include "Use your calculator to…" within directions. Many lessons demonstrate the steps of using a graphing calculator and then provide students with opportunities to use the results to help find solutions to problems (Integrated Math I, pages 167 and 426). In the Integrated Math II materials, multiple tools are utilized to perform geometric constructions (i.e. compass, paper, pencil, rule, patty paper).

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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 partially meet the expectation of supporting the intentional development of seeing structure and generalizing, MP7 and MP8, in connection to the high school content standards, as required by the MPs.

Opportunities to develop MP7 are missed in the instructional materials. For example, in the Integrated Math II materials, Lesson 8.1 has students draw in vertical lines inside an existing right triangle so that trigonometric ratios can be developed and defined. Due to the lack of descriptions and teacher guidance for the MPs within lessons, the connection of students drawing auxiliary lines in order to solve a problem (MP7) is not made with the content and activities present in the lesson. Also in the Integrated Math II materials, Lesson 8.6 uses the technique of drawing auxiliary lines to solve problems and derive formulas, but the absence of descriptions and guidance for teachers or students does not support the intentional development of seeing structure (MP7).

Some lessons include a focus on seeing structure and generalizing (e.g., Integrated Math II, Lesson 12.4 “Factored Form of a Quadratic Function”). Instructional materials frequently summarize a lesson by having students compare several problems and identify similarities as on page 219 of Integrated Math I. However, most problems are scaffolded and provide students with a solution process which limits the students’ need to use structure and generalize. Students might be using repeated reasoning and structure to solve problems, but this is a byproduct of scaffolded examples rather than an intentional outcome of student discussion or student calculations. An example of this can be found on pages 531-532 of Lesson 7.4 (problems 4 through 16) in the Integrated Math II materials where the problems represent scaffolded questions that lead students directly to the formula for the sum of the interior angles of an n-sided polygon. As students answer these questions, they are not given the opportunity to utilize MP7 or MP8 on their own.

Teacher-guided questions used during some class discussions prompt students to look for structure and make generalizations. For example:

  • "How is the difference of two squares similar to the difference of two cubes? How is the difference of two squares different from the difference of two cubes" is asked during a lesson on factoring (Integrated Math II Lesson 13.5).
  • "Why does this construction work?" is frequently asked of students in Chapter 12 of the Integrated Math I textbook or Chapter 1 of the Integrated Math II textbook when students are making several constructions.
  • The guiding questions for teachers included in Integrated Math I, Lesson 1.2 are used to assist students in generalizing their findings after completing a sorting activity of graphs into a function group and a non-function group. Questions include: "Did all the graphs fit into one of the two groups? Can a graph be neither?" "What do graphs of non-functions look like?" "What do graphs of functions look like?" "Are all curved graphs considered graphs of non functions?" "Are all linear graphs considered graphs of functions?"

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