The materials partially meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Although most non-plus standards were addressed, multiple standards were only partially addressed.

For some standards, the materials only addressed certain aspects of the standard. For example:

• N-Q.1,2: Throughout the series, students are always provided the units to use in their problems. When contexts are used, students are given the appropriate units, and in most problems, they are given the units to use in their solution process. Examples of this include:
  • In exercises involving area and volume, the unit of measure is consistent so that the student does not have to choose to convert units and decide which unit of measure to use for reporting solutions.
  • In other real-life problems, units could be used to gain understanding, as in problem 38 of Lesson 1.5 or problem 22 in Lesson 3.3 (both in Integrated Mathematics I), but students are not given opportunities to use units to solve problems that would require a series of conversions.
• For G-CO.3, the materials do not address the reflection of an object onto itself. In Integrated Mathematics I, 11.2, the materials ask students about lines of symmetry but do not ask students to describe reflections that carry a polygon onto itself. Lesson 11.3 does ask students to describe rotations that map a figure onto itself in problem 20 and to select angles of rotation symmetry for a given regular polygon in problems 21-24, which addresses the portion of the standard concerning rotations.
• S-IC.4 requires students to develop a margin of error through the use of simulation models for random sampling. In Integrated Mathematics III, Lesson 10.5, students calculate a sample mean and sample proportion and use those to estimate the population parameters. Students learn a formula for calculating a margin of error but do not “develop a margin of error through the use of simulation models for random sampling.”
• S-IC.5: In Integrated Mathematics III, Lesson 10.6, problems 3-4 and 7-9 do "use data from a randomized experiment to compare two treatments," and although problems 7-9 have students use simulations to calculate the differences between the means of two groups and draw a conclusion, the conclusion that is drawn does not include whether or not the differences are significant.
• For S-CP.5, students are given the formula for conditional probability, and then they are directed to use and apply the formula. When asked to explain the concept of conditional probability, they are asked to explain in terms of dependence and not independence. Students are not prompted to explain or make the connection between conditional probability and independence.

The instructional materials reviewed for Big Ideas Integrated Series partially meet the expectations that the materials, when used as designed, provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics. The materials for the series, when used as designed, would not enable students to fully learn some of the non-plus standards.

There were a number of examples where the materials would not enable students to fully learn a particular standard. Specific examples are shown below:

• G-CO.1: This standard asks students to know geometric definitions based on "undefined notions of point, line, distance along a line, and distance around a circular arc." Across the series, this standard was addressed by telling students there are undefined notions in geometry (with the exception of distance around a circular arc), and the geometric definitions were not developed based on the undefined notions.
• N-RN.3: The standard asks students to "explain why the sum or product of two rational numbers is rational; 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." In Algebra 1, Lesson 4.1, problem 99 has students perform a limited number of calculations with pre-selected numbers, and then, students use that information in problem 100 to answer if this is always, sometimes, or never true. These are the two problems that offer opportunities to address N-RN.3.
• Additionally, it is important to note that there were a number of standards, for example, A-REI.5 (Integrated Mathematics I, 5.3), G-SRT.7 (Integrated Mathematics II, 9.5) and S-ID.5 (Integrated Mathematics I, 7.5), which were addressed in one lesson throughout the series.

For standards that require students to derive, prove or explain, the materials often provide a derivation, proof or explanation rather than providing students with the opportunity to show their own understanding. Examples are shown below:

• N-RN.1 requires students to explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponent to those values. In Integrated Mathematics II, Lesson 1.5, an explanation was provided to students, with no opportunity for student explanations.
• G-C.5 calls for students to derive the fact that the length of the arc intercepted by an angle is proportional to the radius. In Integrated Mathematics II, Lessons 11.1-11.2, the derivation of arc length and the formula for area of sectors is provided for students. Students are not required to engage in constructing a derivation.
• G-GPE.2 requires students to derive the equation of a parabola given a focus and directrix. In Integrated Mathematics II, Lesson 3.6, the derivation is given to students, with no opportunity for active engagement or input by the student.

For some standards, the materials do provide sufficient opportunities for students to fully learn the standards. Examples where the materials provide sufficient opportunities for students to fully learn a standard include the following:

• F-IF.2: The materials introduce function notation to students in Integrated Mathematics I during Lesson 3.3, providing a description of function notation and how to use it to evaluate functions for specific input values and practice in using function notation when solving familiar problems in mathematical contexts and real-world contexts. Skill in utilizing function notation is continually promoted throughout the remainder of the series with the regularity of using function notation increasing from Integrated Mathematics I through Integrated Mathematics III. For example, by the end of Integrated Mathematics I, students build fluency in seeing f(4) = -2 as (4, -2) . In Integrated Mathematics II and III, basic skills extend to more complex uses of function notation, such as seeing Δy/Δx as or describing function transformations using f(x) notation.
• F-BF.3: In Integrated Mathematics I, students develop understanding of transformations with linear and exponential functions. In Integrated Mathematics II, students begin working with absolute value and quadratic function transformations. In section 3.4 of this course, students learn how to identify even and odd functions. There is a specific example to address identifying even and odd functions from the rules, and there is a study tip to expose students to identifying even and odd functions from a graph. There are practice problems to address both. Students continue to practice and develop understanding of transformations of functions in Integrated Mathematics III. In earlier chapters of this course, students work with transformations on function types previously learned in the first two courses. Students then develop understanding of transformations with new functions (polynomials, radical, exponential and logarithm, rational, and trigonometric) as they learn them.

The instructional materials reviewed partially meet the expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, yet do not vary the types of real numbers being used.

Throughout the series, many examples, exercises, and problems include real numbers that, in many instances, provide limited opportunities for students to use decimal and fractional constants, coefficients and solutions.

• In Integrated Mathematics I, the chapters that address linear equations, linear inequalities and geometric concepts avoid fractions, decimals and irrational numbers. Chapter 1 focuses on solving linear equations, and in Lesson 1.1, students solve one-step equations in examples and practice problems that use values integer coefficients, numbers that divide or multiply easily to give integer results, or fractions that have the same denominator.
• In Chapter 11 of Integrated Mathematics II, the figures in the exercises addressing area, surface area, and volume typically have dimensions that are whole numbers. When calculating area, surface area, or volume, the solutions are typically whole numbers unless the figure involves a circle or finding the area of a regular polygon. In those exercises where a missing dimension needs to be found, the solution is typically a whole number except for figures that involve a circle or finding the area of a regular polygon.
• In Lesson 6.5 of Integrated Mathematics III, students solve rational equations, and for most of the equations, the solutions are integers or simple rational numbers. This also occurs when students solve radical equation and inequalities in Lesson 4.4 and exponential and logarithmic equations in lesson 5.5.

Contexts include a variety of topics that help engage learners with different interests at a level appropriate to high school. Contexts include, but are not limited to: sports, animals, income, profit, investments, driving cars, and population density. Opportunities for experience with key take-aways from Grades 6-8 were prevalent, with the following seen more strongly than others:

• Applying ratios and proportional relationships
• Applying basic function concepts; e.g., by interpreting the features of a graph in the context of an applied problem
• Applying concepts and skills of geometric measurement; e.g., when analyzing a diagram or schematic
• Applying concepts and skills of basic statistics and probability (see 6-8.SP)

There were limited opportunities to apply the following key-takeaways:

• Performing rational number arithmetic fluently
• Applying percentages and unit conversions; e.g., in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/mL, kg/m3, acre-feet, etc.)

The instructional materials reviewed partially meet the expectations for fostering coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the Standards. While there is some evidence of mathematical connections within courses and across the series, overall the connections among standards within and between courses are not clearly shown for teachers. Even when solid connections exist within the mathematics, those connections may not be utilized to enhance the students' learning and understanding of the mathematics.

Some of the lessons and chapters are not connected to other lessons and chapters within the course/series where connections would be appropriate. For example:

• In Integrated Mathematics I, Chapter 7 primarily addresses analyzing and displaying univariate data, but there are no connections made to the analysis and display of bivariate data presented in lessons 4.4 and 4.5 of the same course. There was also no indication for teachers as to if or how the content of Chapter 7 would connect to future courses in the series.
• In Integrated Mathematics II, there is no connection made between geometric probability addressed in Lesson 11.2 and Chapter 5, Probability.
• In Integrated Mathematics III, the summary for Chapter 1, Geometric Modeling, indicates general concepts from the first two courses in the series that students will use in the chapter, but there are no other direct references to either Chapter 11, Circumference, Area, and Volume, of Integrated Mathematics II or Lesson 8.4, Perimeter and Area in the Coordinate Plane, of the first course.
• In Integrated Mathematics III, the summary for Chapter 10, Data Analysis and Statistics, indicates general concepts from the first two courses in the series that students will use in the chapter, but there are no other direct references to either Chapter 5, Probability, of Integrated Mathematics II or Chapter 7, Data Analysis and Displays, of Integrated Mathematics I.

The following examples include characteristics of the instructional materials reviewed that may promote mathematical connections within and between courses but do not clearly articulate those connections.

• The Teacher Edition includes overviews of the sections in each chapter as well as a chapter summary. Both the overviews and the chapter summaries indicate general concepts and skills teachers can expect their students to know from middle school and/or previous courses, but neither of these sections provide direct connections to previous courses.
• The student materials occasionally include “remember” arrows (as on pages 254 and 255 of Integrated Mathematics II) or phrases like “Previously, you…” that contain information about a previously learned concept. For example, in Lesson 2.6 of Integrated Mathematics III, the materials state, "Previously, you used transformations to graph quadratic functions in vertex form. You can also use the axis of symmetry and the vertex to graph quadratic functions written in vertex form". However, these aids do not foster direct connections within and among courses of the series.
• The connections between standards are listed in the Correlation to the Common Core State Standards Table of Contents documents for each course. However, the connections are not explained or emphasized for the teachers. These documents are not included in the Teacher Editions.
• There are "Connections to Algebra" and "Connections to Geometry" symbols and notes included in the lesson examples that make students and teachers aware that this is a concept that can be connected to other concepts in the course or series; however, these are used only occasionally. Some of these connections are very specific, as on page 390 of Integrated Mathematics I that states "In this step, you are applying the Substitution Property of Equality that you learned about in Section 1.1." Others are general or vague, as on page 405 of Integrated Mathematics I, where it states, "In this exploration, you expand your work on perimeter and area into the coordinate plane".

The following are examples within the series materials that do promote and foster coherence:

• At the beginning of each chapter across the series, there is a "Maintaining Mathematical Proficiency" activity that reviews important concepts and skills from previous grades or courses. These activities help students connect prior learning to new learning in the chapter they are beginning.
• Students work with area, surface area, and volume in middle school, and these geometric concepts are reviewed in the last chapter of Integrated Mathematics II. The chapter extends this review work to include the area of regular polygons and surface area and volume of a variety of 3D figures not included in the middle school standards.
• In Integrated Mathematics I, students develop linear and exponential equations and functions. Then in Integrated Mathematics II, they extend their algebraic reasoning and understanding of functions with quadratic, absolute value, and polynomial functions. Finally, in Integrated Mathematics III, students study polynomials, radicals, logarithmic, rational, and trigonometric functions. In each of the three courses, students' ability to interpret the structure (A-SSE), perform arithmetic (A-APR), create equations (A-CED), reason with equations and inequalities (A-REI), and interpret functions (F-IF) builds upon the previous course. Students are given the opportunities to augment their work with many function types in these domains across the series.

The materials partially meet the expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. Overall, content from Grades 6-8 is present but is not explicitly identified and does not always fully support the progressions of the high school standards. There is some support for making connections between standards from Grades 6-8 and high school as seen in Laurie's Notes and Maintaining Mathematical Proficiency.

The following are examples of where the materials do not explicitly identify and/or build on standards from Grades 6-8:

• The Chapter Summary pages for all courses have a What Your Students Have Learned…Will Learn section, and throughout the series, this section includes references to topics or concepts students learned in middle school. However, when referencing these concepts from Grades 6-8, specific standards are not explicitly identified, and there are not clear connections between standards from Grades 6-8 and high school.
• Lessons 1.1-1.3 and 2.1-2.4 of Integrated Mathematics I are identified in the CCSSM document as addressing A-CED.1 and A-REI.3. These lessons address solving linear equations and inequalities, which aligns to standards from 8.EE.C, but there are no standards identified from Grades 6-8.
• Lessons 5.1-5.4 of Integrated Mathematics I address solving systems of linear equations graphically and symbolically for systems having one solution, no solutions and infinitely many solutions, which aligns to standards from 8.EE.C. The teacher materials do indicate that the concepts are being repeated from Grades 6-8, but no specific standards are identified. Also, the teacher materials do provide some pacing suggestions for these lessons, but there is no guidance for how to build on these concepts in order to enhance the learning for high school students.
• The Chapter 7 Summary of Integrated Mathematics I indicates the construction and interpretation of box-and-whisker plots as new learning rather than prior learning from middle school, but the overview for Lesson 7.2 states that “students should be familiar with representing data using box-and-whisker plots from middle school.” There are no standards from Grades 6-8 identified.
• Chapter 11 of Integrated Mathematics II mainly addresses area, volume, and surface area, which are concepts that have their origins in standards from Grades 6-8. However, there are no standards from Grades 6-8 identified, and the connections between the high school concepts and the concepts for Grades 6-8 are only described in the teacher materials.

The Maintaining Mathematical Proficiency lessons at the beginning of each chapter connect to concepts from Grades 6-8 as this section reviews concepts or skills that will be needed throughout the chapter. Most of the work with middle school concepts or skills in these lessons is intended to reinforce course-level standards, but identification of the concepts or skills from Grades 6-8 is inconsistent. For example, in Chapter 1 of Integrated Mathematics 1, adding, subtracting, multiplying, and dividing integers is specifically noted as Grade 7 work, but in Chapter 2 of the same course, graphing numbers on a number line and comparing real numbers are not identified as work from a previous grade.

An example of the materials attending to the progressions of standards that are referenced in the progression documents is in Chapter 11 of Integrated Mathematics 1. The publisher describes concepts from middle school that relate to what students will be learning, and the middle school concepts include the types of transformations students are familiar with and their understanding of congruency. In the introduction to Lesson 11.4, teachers are told, "In Grade 8, the concept of congruency was introduced. Students should understand that a two-dimensional figure is congruent to another when the second can be obtained from the first by a sequence of rotations reflections and translations." In Lesson 11.4, the understanding of congruency through rigid motions is continued, and then, students develop triangle congruency theorems through rigid motions in Chapter 12.