High School - Gateway 1

The materials meet the expectation for attending to the full intent of the mathematical content contained in the high school standards for all students. For standards that span across a course or series, the development of the standard within a course and across the series was examined to determine if all aspects of the standard were addressed. Some specific examples where the materials attend to all aspects of a standard either within a course or across the series are shown below.

F-IF.4: The materials fully develop F-IF.4 through the course of four modules and extending into a fifth module. In Algebra I, the standard is developed for linear, quadratic and exponential functions including modeling opportunities. The materials further develop this standard in Algebra II from the foundation provided through previous learning experiences in Algebra I. In Algebra II, this standard is extended to trigonometric and logarithmic functions. It would be helpful if the materials identified the intended standard at the lesson level rather than just at the topic level. (Note: student books do not identify standards at all.) Courses are organized into modules, topics and then lessons. F-IF.4 is addressed in multiple locations, shown below, across Algebra I and II.

• Algebra I, module 3: topics B and D

• Algebra I, module 4: topics A and B

• Algebra I, module 5: topics A and B

• Algebra II, module 2: F-IF.4 is not explicitly noted in the teacher guide, but numerous opportunities to identify key features from a table, graph or context are noted throughout this module.

• Algebra II, module 3: topic C

G-CO.1: This standard was fully developed in module 1, topics A and G, for all concepts listed in the standard. The module started with students completing constructions to develop precise definitions of angle, circle, perpendicular line, parallel line and line segment. Students are asked to recall, compare, and expand upon geometric definitions from previous grades (Grades 4-8). Then students are asked to make conjectures and write proofs. As students progress through the module they extend this knowledge in developing definitions of rotations, reflections, and translations. (G-CO.4).

A-SSE.2: This standard is fully developed over the span of three modules: in Algebra I, module 1, topic B and module 4, topic A, then in Algebra II, module 1, topics A and B.

• In the Algebra I modules, students use and develop the concepts of algebraic properties by rewriting expressions and looking at patterns, such as the distributive and associative properties. Then, in module 4, students use the properties of exponents to support the development of understanding and procedures for multiplication of polynomials and factoring.

• In Algebra II, students expand on these concepts from Algebra I and extend them to higher degree polynomial operations and algebraic manipulations. Each series of lessons begins by having students focus on rewriting polynomials or decomposing numbers to identify patterns and then expanding upon those patterns to construct new understandings.

For teachers utilizing these materials, note that some standards are addressed more fully in the teacher discussion notes, and routine use of the teacher discussion notes is essential. Otherwise, there is the potential for some standards to receive superficial treatment. For example, the discussion notes on page 16 of the teacher edition for Algebra II, module 2, lesson 1, suggest that the teacher have students review and explain their thinking and consider revisions to the methods used in completing exploratory challenge 1 as they gain greater understanding of F-IF.7 and F-TF.5. Without using these teacher notes, the students may not necessarily have the opportunity to engage with all aspects of these two standards.

There were a few assessments which tested material before it was actually addressed.

• In the Algebra I, module 1, mid-module assessment, A-SSE.1a and A-SSE.1b were not covered in the material but were on the assessment and a part of the rubric.

• In Algebra II, module 2, F-TF.3 and F-LE.2 were tested in the mid-year assessment but not covered until the second half of the module. These standards were re-assessed at the end of the module.

• In Algebra II, module 4, S-CP.3 was assessed in the mid-module assessment but not covered until the second half of the book.

The materials meet the expectations for, when used as designed, allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs, and careers. Overall, the series does devote a majority of instructional time to the WAPs.

Algebra I contains five modules of instruction, and three of the modules are almost completely focused on the WAPs. Most of the WAPs are addressed in the Algebra I instructional materials, and the WAPs primarily addressed in the Algebra I materials are the non-plus standards from A-CED and F-IF.

• A-CED is addressed in topics C and D of module 1; topic D of module 3; topics A, B, and C of module 4; and topics A and B of module 5.

• F-IF is addressed in topics A, B, C and D of module 3; topics A, B, and C of module 4; and topics A and B of module 5.

All of the lessons within Geometry address standards from the Geometry category, and although topics were addressed that could be considered part of a 4th-year course, these lessons were not distracting from the time spent on the WAPs. The WAPs from the Geometry category are largely addressed in the first two modules of the Geometry materials, and the Geometry materials, as a whole, do spend a majority of time on the WAPs from the Geometry category.

• G-CO.A is a part of topics A, C and G of module 1.

• G-CO.B is a part of topics C, D and G of module 1.

• C-CO.9,10 are found in topics B, E and G of module 1.

• G-SRT.A,B,C are found in topics A, B, C, D and E of module 2.

In reviewing the Algebra II instructional materials, most of the lessons aligned to the WAPs. Although topics were addressed that could be considered part of a 4th-year course, these lessons were not distracting from the time spent on the WAPs. The WAPs are largely addressed in the first and third modules of the Algebra II materials, and the Algebra II materials, as a whole, do spend a majority of time on the WAPs.

• A-SSE is included in topics A and B of module 1 and topics D and E of module 3.

• F-IF is addressed in topic B of module 1; topics A and B of module 2; and topics A, C, D and E of module 3.

• F-BF.1 is included in topics A, B, C, D and E of module 3.

The materials meet the expectation for requiring students to engage in mathematics at a level of sophistication appropriate to high school. Overall, the materials include problems for all students that are appropriate to high school, present contexts that motivate the mathematical content of the high school standards, and use types of numbers that are part of real-world scenarios.

In general, the series expects all students to engage with the materials through similar experiences at a level of sophistication appropriate to high school. There are no indications in the student materials of optional exercises or problem sets for students not ready for the course-level work, but there are some scaffolding notes in the teacher materials. For advanced learners, some lessons provide an extension problem, but when these extensions do exist, there is typically not more than one problem. For example, in Algebra I, module 1, lesson 9, the extension problem is: "Find a polynomial that, when multiplied by 2x^2 + 3x + 1, gives the answer 2x^3 + x^2 - 2x - 1." This is a reasonable extension problem as it stays on course level, and this is the only extension problem provided in the lesson.

Real-world contexts are used throughout the series to motivate the content of the high school standards. In some cases, the contexts are technical but do promote the mathematical content of the context. For example:

• In Algebra II, module 1, telescopes are used to assist in developing models for parabolas and to provide reasons why parabolic equations are needed.

• In Geometry, module 3 concludes with a lesson on 3-D printers which is a context that promotes Cavalieri’s principle and the volume of various three-dimensional objects.

• In Algebra I, module 5, students bring together multiple standards from throughout the course as they solve problems in settings that include fish populations in a lake, investing money, and the concentration of medicine in a patient’s blood.

The numbers included in the instructional materials are appropriate for high school. Types of numbers used within problems do not just consist of positive integers or integers, and answers to problems are not always integers. For example:

• Lesson 24 of module 4 in Algebra I includes a data set for a problem that has rational numbers in tenths and produces a solution that has rational numbers to the ten-thousandths place.

• Lesson 33 of module 2 in Geometry uses rational numbers to the hundredths place for lengths of sides and angle measurements that produce irrational results when working with trigonometric functions.

• Lesson 28 of module 3 in Algebra II uses rational numbers as data points, along with the number e, when discussing Newton’s law of cooling.

The materials meet the expectation for fostering coherence through meaningful connections in a single course and throughout the series, where appropriate and where required by the standards.

Examples of connections that are in the series:

• Solving quadratic equations in Algebra I with solving rational equations in Algebra II;

• Factoring in Algebra I with radical expressions in Algebra II;

• Within Algebra I, creating equations that describe relationships with constructing linear and exponential models in modules 3 and 5;

• The Pythagorean theorem is used in both Geometry and Algebra II;

• Within Geometry, solving systems of equations with using coordinates to compute perimeters of polygons in topic A of module 4;

• Operations with radicals from Geometry are used again with radical equations in Algebra II; and

• Within Algebra II, representing data on two quantitative variables and modeling periodic phenomena with trigonometric functions in topic B of module 2.

The teacher materials communicate connections. Each module begins with an overview, which describes the standards that will be addressed, how those standards are connected to prior learning and how the work will help prepare students for subsequent lessons and courses. This information describes the intended flow of the module. The foundation and extension standards are shown along with new terms, familiar terms and symbols. The facilitator notes for each lesson are thorough and routinely make explicit to the teacher exactly how each example, discussion, activity, etc. connects to previous and subsequent learning. The teacher materials remind the teacher of the "big picture."

The materials meet the expectation for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. When necessary, the materials reference prior knowledge in the teacher edition of the materials. Each unit across the series, especially for Algebra I and Geometry, identifies the "foundational standards" from Grades 6-8 that underlay the development of concepts in each module in each course. All of the Grades 6-8 standards that are identified as foundational standards in Algebra I and Geometry are standards that are built upon and extended to the high school standards in the high school series. The foundational standards are not explicitly taught within the lessons in the high school series, instead they serve as the basis for extending to the high school standards. These connections to the grades 6-8 standards are made explicitly for teachers, but not for students. The "foundational standards" in Algebra II are all from the Algebra I materials.

Some examples of where the materials connect standards from Grades 6-8 to the high school standards:

• Algebra I, module 2 is about statistics. The focus standards for this module address number line plots; shapes, centers, and spreads of distributions; categorical data; scatter plots, correlation, and linear regression equations. The Grades 6-8 standards that are identified as foundational for the focus standards include recognizing statistical questions; understanding the shape, center, and spread of a distribution; number line plots; summarizing numerical data sets; constructing scatter plots and estimating trend lines with linear equations. The discussion and work with scatter plots and regression equations do not duplicate what was addressed in middle school, instead the focus moves to to least-squares regression and deeper understanding of correlation.

• Geometry, module 2 lists familiar terms from previous courses in the teacher edition and notes connections to standards addressed in Grades 6-8 while also discussing how the standards progress in high school. The teacher notes detail how the material/lesson meets the intent of progression. In topic A, the teacher materials explain that, in Grades 6-8 (with a mention of a Grade 4 standard), the purpose was to for students to observe how dilations worked. The expectation in high school is to explain why dilations work.