High School - Gateway 1

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.

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.

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.

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.