High School - Gateway 2

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. The lessons throughout the materials are generally well developed to allow students to build conceptual understanding of the key mathematical concepts. The few missed opportunities are noted below. 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 are asked to demonstrate their understanding in a variety of ways including within class discussion, within groups and/or pairs, and individually. 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 some specific ways the development of students' conceptual understanding is met:

F-IF.A: Function notation is consistently used and developed throughout the entire series. A sorting activity in Algebra 2, lesson 1.3 on pages 27-33 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 represents a function and which does not represent a function.

G-SRT.6: Geometry, lesson 9.1 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 section 9.2 on page 672.

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 Algebra 1 on pages 170 and 176. Slope and y-intercept are again interpreted in context of a given scenario and data set in Algebra 1 on pages 524-525. Another example (page 531) is given in which the y-intercept must be obtained through extrapolation and the students must determine whether the extrapolated y-intercept makes sense in terms of the context.

Some specific places where opportunities for students to fully develop conceptual understanding are partially met:

N-RN.1 - The relationship between rational exponents and radical notation is provided to students in Algebra 1, sections 5.5 and 14.3, and in Algebra 2, sections 9.4 and 9.5. 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 (Algebra 2, page 699) is utilized to group equivalent expressions that are written in non-simplified form. One question in this section (Algebra 2, page 707) shows three examples of student work and has the student determine whose work is correct. A similar question (Algebra 2, page 706) 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 the rationale for those rules.

A-REI.A: This cluster is addressed in Algebra 1 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 that the teacher ask about the process for solving and if there is more than one way, 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 standard is not fully developed in this lesson or in subsequent lessons in this course. In contrast, Algebra 2, Section 8.3 includes many opportunities to develop students' conceptual understanding of solving rational equations. A cut and paste sorting activity on page 631 is utilized to distinguish between the methods of cross-multiplying and using the least common denominator and when it is most advantageous to use each method. One question in this section (page 626) shows two examples of student work where each student uses a different method. Students are asked to compare and contrast the methods.

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 that are specified in the standard are not addressed in any of the courses.

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. 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, Geometry FM-22 to FM-30) as well as aligning the types of problems students will encounter to the MPs (for example, see Geometry FM-42 to FM-45). Although the material shows 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 mimics the examples given or that follow 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. Students are often asked to use or create definitions, use units appropriately when necessary, and expected to communicate understanding clearly in writing and/or orally.

Although many of the components, of the practice standards are included in the lessons, the use of the practice standards would be enhanced if the publisher identified which practice standard(s) are best emphasized in each lesson or group of lessons to provide focus and direction to the teacher.

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 standard is evident. It is stated that exemplar answers are provided but how to get students to get to those types of answers is not addressed. 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 uncover a (potentially new) 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 Algebra 1 and Algebra 2 materials, tables are utilized to consider the units involved in a problem (for example, Algebra 1 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, Algebra 1 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 sense making required.

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 (Algebra 1, 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.

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 often true. 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 Algebra 1, lesson 11.6 on page 664, 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 gives no answers and has many - students can look at the graph, the table, or calculate it all using the calculator. No connections are made for teachers or students about why this is, and therefore, MP4 is lacking in this lesson.

Lessons throughout the series prompt students to engage in scaffolded steps in the modeling process as required by MP5.

• A variety of tools are utilized to perform geometric constructions (i.e. compass, paper, pencil, rule, patty paper). Tools in the Algebra 1 and Algebra 2 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 includes, "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 (Algebra 1, pages 167 and 426).
• Many lessons within the series utilizes multiple representations to model a problem context. For example, an exponential growth problem discussed in Algebra 1, pages 348-349, represents the scenario in a table, graph, and equation. Questions in the textbook are included to identify relationships among the representations.

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, Algebra 1, 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.

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.

The materials require students to look for patterns, make generalizations, and explain the structure of expressions. Teacher-guided questions used during 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 (Algebra 1 section 12.5).
• "Why does this construction work?" is frequently asked of students in Chapter 1 of the Geometry textbook when students are making several constructions.
• The teacher guiding questions included in Algebra 1, 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?" "Area all curved graphs considered graphs of non functions?" "Are all linear graphs considered graphs of functions?"

Some lessons include a focus on seeing structure and generalizing (e.g., Algebra 1, lesson 11.4 "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 Algebra 1. However, most problems are typically 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 repeated practice or scaffolded examples rather than an intentional outcome of student discussion or problem-solving.