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Report Overview
Summary of Alignment & Usability: Carnegie Integrated | Math
High School
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Report for High School
Alignment Summary
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.
High School
Alignment (Gateway 1 & 2)
Usability (Gateway 3)
Overview of Gateway 1
Focus & Coherence
Gateway 1
v1.0
Criterion 1.1: Focus & Coherence
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
Indicator 1A.i
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.
Indicator 1A.ii
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
Indicator 1B.i
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.
Indicator 1B.ii
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.
Indicator 1C
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.
Indicator 1D
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.
Indicator 1E
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
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.
Overview of Gateway 2
Rigor & Mathematical Practices
Gateway 2
v1.0
Criterion 2.1: Rigor
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.
Indicator 2A
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.
Indicator 2B
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.
Indicator 2C
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).
Indicator 2D
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.
Criterion 2.2: Math Practices
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.
Indicator 2E
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.
Indicator 2F
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.
Indicator 2G
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).
Indicator 2H
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?"