High School - Gateway 1

The instructional materials reviewed for the Big Ideas Traditional series partially meet the expectation that materials attend to the full intent of the mathematical content contained in the high school standards for all students. For this Indicator, the materials were examined, guided by included correlation documents: Common Core State Standards for Mathematical Content Correlated to Algebra 1, Geometry, and Algebra 2 and a course specific table at the beginning of each book which listed the standards addressed in each lesson. Overall, most of the non-plus standards are included in the materials; however, some aspects of the non-plus standards have not been completely addressed by the instructional materials. Additionally at least two standards are completely omitted.

• G-CO.3: In Geometry, Lesson 4.2, students are asked about lines of symmetry but are not directly asked to describe the reflections that carry a specific polygon onto itself as called for in the standard. Lesson 4.3 does ask students to describe rotations that map a figure onto itself, page 195, problem 20, and to select angles of rotational symmetry for a given regular polygon, page 195, problems 21 - 24.
• S-IC.4: In Algebra 2, Lesson 11.5, students are given a version of the Margin of Error formula and use it, but they do not develop the concept using simulations as required by the standard.
• S-IC.5: In Algebra 2, Lesson 11.6, problems 3-4 and 7-9 do "use data from a randomized experiment to compare two treatments," yet no evidence was found requiring students to "use simulations to decide if differences between parameters are significant."
• S-CP.5: In Geometry, Lesson 12.2, students are asked about a variety of everyday situations and whether they are independent (problems 3 - 10). However, there is no evidence of students connecting the concepts of conditional probability and independence.
• N-Q.2 and N-Q.3: No evidence was found where students had to define their own quantities or determine the appropriate level of accuracy of quantities. These standards are indicated to be present throughout sections of Algebra 1 and Algebra 2 but are not noted as a primary focus in any lesson as stated on page xxxii in both the Algebra 1 and Algebra 2 Teaching Editions. Upon examining the identified sections in Algebra 1 and Algebra 2, no evidence of N-Q.2 or N-Q.3 was found to be incorporated into the lessons.

The instructional materials reviewed for Big Ideas Traditional 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. Overall, the lessons are presented in a way that will allow students to fully learn many of the standards. Throughout the series, students are not spending time on content previously learned but are constantly moving forward. Students, however, are frequently not given the opportunity to develop their own definitions, and where the standards expect students to prove or develop a concept, the materials often provide students the information.

The following standards were included in the materials, yet were not presented in a way that would allow students to fully learn that standard.

• S-ID.4: In Algebra 2, Lesson 11.1 includes practice on using "the mean and standard deviation of a set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate." The materials make use of calculators and tables to estimate areas under the normal curve; however, they do not make use of spreadsheets as called for in the standard.
• N-RN.3: In Algebra 1, Lesson 4.1, problems 99 and 100, students do a limited number of calculations with pre-selected numbers then use that information in problem 100 to answer if this is always, sometimes, or never true. This is the only place where the standard is located. With this limited amount of exposure to the standard, students may not reach the full depth of understanding.
• G-MG.1: Students are frequently provided or told the exact geometric shape to use to solve a problem; the material does not offer students a chance to select for themselves the appropriate shape to model a situation. Throughout the series only one problem was located that asked students to “draw an object (or part of an object) that can be modeled” by a certain shape (problem 40 on page 237 in the Geometry materials).
• G-GPE.2: In Algebra 2, Lesson 2.3 the derivation of a parabola is given to students on pages 68-69. Students are not given the opportunity to derive the equation of a parabola given a focus and directrix for themselves.
• G-C.5: In Geometry, lesson 11.1, page 597, students are provided with the derivation of the fact that the length of the arc intercepted by an angle is proportional to the radius. Students are not given the opportunity to complete this derivation for themselves.
• G-CO.1: Throughout the geometry materials, students are not provided the opportunity to develop their own definitions as required by the standard; instead, the definitions were given to them. The textbook does tell students that there are undefined terms in geometry (with the exception of distance around a circular arc) and point, line, and plane are mentioned to be “undefined terms;” however, the textbook does not build the definitions through this notion.
• A-APR.1: Students add, subtract and multiply polynomials, but the “understanding” aspect of this standard is not developed. In Algebra 2, Lesson 4.2 the closed nature of polynomials is explicitly stated; however, students are not required to demonstrate understanding of the concept.
• A-SSE.1a: Problem 49 of lesson 3.2 in Algebra 1 is the only problem that addresses this standard. The materials list various other lessons in both Algebra 1 and 2 as addressing this standard; however, evidence of meeting the full depth of this standard was not found.
• A-REI.4a: In Algebra 1, page 515, students are given the derivation of the quadratic formula and then work with a partner to provide the justification for the given steps. Students are never given the opportunity to derive the formula as stated in the standard.
• G-CO.10: The standard calls for students to “prove theorems about triangles;” however, many proofs are simply provided by the materials throughout the geometry materials. For example, proof that the measures of interior angles of a triangle add to 180 degrees is provided as an example on page 234, proof that the base angles of isosceles triangles are congruent is provided as an example on page 252, and proof that the medians of a triangle meet at a point is offered as a theorem on page 320 with the proof provided online.
• G-C.1: Proof that all circles are similar is provided for the students on page 541 of the Geometry textbook.
• S-CP.2: There were four problems—Geometry, lesson 12.2, problems 3-6 with an identical chapter in Algebra 2—throughout the series that fully addressed this standard Geometry.
• S-CP.6: Problem 23 in lesson 12.2 of the Geometry textbook is the one problem which addresses the standard.

The following standards provide students thorough exposure through multiple experiences to fully learn each standard.

• G-C0.4: Students are given several explorations throughout Chapter 4 of the Geometry materials through which they can develop the ideas of transformations before the mathematical definition is provided for them.
• G-CO.12: Students are given opportunity to explore constructions using “a variety of tools” (i.e., compass, straightedge, paper folding, and dynamic software) as called for in the standard.
• A-CED: The entire Creating Equations cluster is thoroughly developed throughout the algebra 1 and algebra 2 materials.

The materials offer additional resources to help all students fully learn each standard.

• Each section begins with two or three exploration activities that offer students an opportunity to engage with the content before formal presentation of the terms, definitions, facts, theorems, or procedures. These explorations help students with content mastery by allowing them to "play" with the mathematics and familiarize themselves with the concepts in a seemingly informal way. Many of the explorations present content from different perspectives. For example, in Algebra 1, Lesson 1.4, students explore the concept of absolute value first by considering it as an equation and looking for values that make it true, then as a number line, and finally numerically using a spreadsheet.
• The end of each section offers suggestions for students who may need extra help as well as for students who may need additional challenge problems. The Resources by Chapter book provides on grade-level, additional problems for struggling students and extension problems for students who need a challenge. The extension problems focus on moving the learner forward. For example, in Algebra 1, section 5.2, the lesson dealt with solving systems of equations algebraically. The extension asks students to consider how to solve a system of three equations.
• "Laurie's Notes" in every chapter and lesson provide guidance to teachers in presenting lessons, which in turn could help all students better learn all aspects of standards. Furthermore, the beginning of each chapter includes "Scaffolding in the Classroom" notes in the margin. Each lesson also includes "Differentiated Instruction" and "English Language Learner" boxes providing strategies for teachers to use in order to reach all learners. "Assignment Guide and Homework Check" boxes included before student exercises break down what problems teachers could use for Basic/Average/Advanced assignments and homework checks. After each Chapter Test in the book, teachers are given ideas on what materials to use if students need help or if students got the info. Students are given opportunities to use paper and pencil, graphing calculators, and dynamic software.

The Instructional materials reviewed for Big Ideas Traditional Series partially meet the expectation that all students engage in mathematics at a level of sophistication appropriate to high school. Overall, the materials offer real-life and relevant situations to high school students; however, problems often involve integers and avoid more complex solutions.

• The materials present a majority of problems with integer values. In Algebra 1, Chapters 3 and 4 deal with linear functions, both graphing them and then writing equations. Students do not see non-integer y-intercepts until they are introduced to the linear regression feature of the calculator in Lesson 4.5. The majority of y-intercepts are integer values. Likewise, Chapter 5 (solving systems of equations) primarily contains solutions that consist of integer values. Algebra 2, Lesson 1.4 extends the ideas of solving a system of equations to three variables but still has a large majority of solutions that are integer values and rarely involves non-integer values. Similarly, the Geometry textbook remains focused on integer values much like that of the Algebra 1 and Algebra 2 textbooks.
• The materials offer students problems frequently based on real-life and relevant situations.
  • Geometry, page 198, problem 19 connects the Tetris game to rigid transformations.
  • Geometry, page 608, problem 37 connects pizza with circle/sector area and reasoning.
  • Algebra 1, in the Chapter 6 Performance Task "The New Car," students weigh the many costs of a car when deciding which is the best buy.
  • In the Algebra 2, Chapter 2 Performance Task "Accident Reconstruction," students analyze car speed and braking distance.
• Daily journal entries, either online or utilizing the Student Journal book, are age appropriate for high school students.
• One worksheet per lesson provides problems for students working below course level as well as students working above level. These enrichment and extension worksheets are similar to that of the chapter.

The instructional materials reviewed for Big Ideas Traditional Series partially meet the expectations that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the standards. Overall, Chapter Summaries are provided at the beginning of each chapter that indicate what content the students should already be familiar with from previous grades and courses. The Common Core Progression offered at the beginning of each chapter indicates skills learned in previous courses, however does not make connections to specific standards.

• Notes for teachers are abundant in the margins and before each lesson; however, they focus mainly on identifying common errors, teacher’s actions, and ideas for next steps for students. Connections within a course and across a series are not made explicit for the teacher.
• Each course is presented in a traditional progression. Chapters are presented as isolated concepts or skills with little to no connection between mathematical concepts made explicit.
  • For example, Algebra 1: Lessons 8.1-8.4 each focus on graphing various forms of quadratic functions (8.1 - "Graphing f(x) = ax^2", 8.2 - "Graphing f(x) = ax^2 + c", 8.3 - "Graphing f(x) = ax^2+bx+c", and 8.4 - "Graphing f(x) = a(x-h)^2 +k"), however not until Lesson 9.2 do students discuss solving a quadratic function by graphing. Lesson 9.2 makes no reference to concepts learned in Lessons 8.1 - 8.4.
• “Common Core Progression” boxes located at the beginning of each chapter indicate general connections to previous courses; however, no standards are cited, simply skills and concepts.
  • For example, the "Common Core Progression" box at the beginning of Chapter 3 in Algebra 1 lists multiple skill focused statements not rooted in the standards in order to show connection among standards within and across courses.
    • "Identify linear functions, using graphs, tables, and equations."
    • "Use function notation to evaluate, interpret, and graph functions."
    • "Find the slope of a line and use it to write a linear equation in slope-intercept form."
    • "Solve real-life problems using function notation, linear equations, slopes, and y-intercepts."
    • "Translate, reflect, stretch, and shrink graphs of linear and absolute functions, and combine transformations of graphs of linear and absolute functions."

The instructional materials reviewed for Big Ideas Traditional Series partially meet the expectations that the materials explicitly identify and build on knowledge from grades 6-8 to the high school standards. Chapter Summaries and the Common Core Progression boxes indicate skills and procedures students should already be familiar with from previous mathematics classes; however, they do not explicitly identify the middle grades standards which they are referencing. Occasionally throughout the materials, explicit connections to middle grade standards are made in the Teacher Edition, specifically in “Laurie’s Notes” and the “Maintaining Mathematical Proficiency” sections. In some lessons, topics that are middle school standards are presented as if they are high school standards being presented for the first time.

• “Maintaining Mathematical Proficiency” sections are included throughout the materials and clearly state that these are below grade-level problems that have been included to help students retain previously learned skills necessary for further growth in mathematics. The Teacher Edition explicitly references these middle school standards.
• Connections to middle grades are typically mentioned for teachers in notes for the “Maintaining Mathematical Proficiency” sections at the beginning of each chapter. Explicit middle school standards are not referenced in these sections, but rather basic skills such as “Finding x-intercepts” and “The Distance Formula” (Algebra 2, page T-45). These notes are not included for the students.
• At the beginning of each chapter, the Teacher Edition does identify skills, not standards, from grades 6-8 that students should already be proficient at performing. Individual lessons, however, do not reference these connections.
• Multiple lessons present middle grades standards as if for the first time without identifying the standards are middle school standards but rather labeling them as high school standards. For example:
  
  • Algebra 1, Lesson 1.1 is about solving one-step, one variable equations, which is 8.EE.7, but is labeled as A-CED.1, A-REI.1, and A-REI.3. There is no mention in Lesson 1.1 that this is a concept that was previously learned and will be built upon.
  • Algebra 1, Lesson 6.1 is a re-teach of the rules of exponents, 8.EE.1, but does not indicate this. The rules of exponents are extended to rational exponents in Algebra 2, Lesson 5.2.
  • Algebra 1, Chapter 5 presents systems of equations to students as though the material is new, not a review of standards introduced in Grade 8, 8.EE.8.
  • The Overview of Algebra 1, lesson 3.1 explicitly states “Students have prior knowledge of functions from Grade 8 (8.F.1 - 8.F.5). Their understanding may be limited to discrete functions.” However, the lesson then proceeds as though the students have no understanding of the material.

Middle grades standards are rarely explicitly noted throughout the materials. Specific reference to middle grade standards are only found in the Teacher Edition in "Laurie’s Notes" and the Maintaining Mathematical Proficiency sections.