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. Although the materials generally allow students to build conceptual understanding of key mathematical concepts, there are missed opportunities. 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 demonstrate their understanding individually, in pairs, in small groups, and as a class. 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 specific standards for which the materials partially met the expectation for developing conceptual understanding:

• N-RN.1: This standard states the following: "Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents." The relationship between rational exponents and radical notation is provided to students in Integrated Math I, Lesson 5.5, Integrated Math II, Lesson 15.3, and Integrated Math III, Lessons 11.4 and 11.5. 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 (Integrated Math III, page 807) is utilized to group equivalent expressions that are written in non-simplified form. One question in this lesson (Integrated Math III, page 815) shows three examples of student work and has the student determine whose work is correct. A similar question (Integrated Math III, page 814) 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 or explain the rationale for those rules.
• N-RN.3: This standard is addressed through Lessons 15.1 and 15.2, but conceptual understanding is partially developed as students are not given opportunities to explain "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."
• A-REI.A: This cluster is addressed in Integrated Math I 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 asking students about solution paths, 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 cluster is not fully developed in this lesson or in subsequent lessons in this course.
• 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, which are specified in the standard, are not addressed in any of the courses.

The following are specific standards for which the materials met the expectation for developing conceptual understanding:

• F-IF.A: A sorting activity in Integrated Math III, Lesson 3.3 on pages 135-141, 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 represent a function and which do not represent a function.
• G-SRT.6: In Integrated Math II, Lesson 8.1 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 Lesson 8.2 on page 582.
• 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 Integrated Math I on pages 170 and 176. Slope and y-intercept are again interpreted in the context of a given scenario and data set in Integrated Math I on pages 524-525. In an example on page 531 of the Integrated Math I materials, the y-intercept must be obtained through extrapolation, and the materials ask students to determine whether the extrapolated y-intercept makes sense in terms of the context.

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. MPs are not explicitly identified throughout the series. Course/series scope and sequence charts do not include identification of MPs to chapters or lessons. 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, Integrated Math II FM-24 to FM-32) as well as aligning the types of problems students will encounter to the MPs (for example, see Integrated Math II FM-44 to FM-47). Although the materials show 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 follows the examples given or 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 even though it is not specifically identified in any lessons. Students are often asked to use or create definitions, students are expected to use units appropriately when necessary, and they are often expected to communicate understanding clearly in writing and/or orally.

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 practice is evident. Although exemplar answers are provided, teachers are not given guidance on how to get students to provide those types of answers. 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 examine a correct 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 Integrated Math I and Integrated Math III materials, tables are utilized to consider the units involved in a problem (for example, Integrated Math I 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, Integrated Math I, 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 explanations and critiques created by students.

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 (Integrated Math I, 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 true in some instances. 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 Integrated Math II, Lesson 12.6 on page 904, 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 is presented with no answers in the teachers' materials, but it has many - students can look at the graph, the table, or calculate them all using the calculator. No connections are made for teachers or students about why this is, and therefore, MP4 is lacking in this lesson.

An example of where connections among multiple representations are made is in Integrated Math I on pages 348-349. In this example, the scenario of an exponential growth problem is represented in a table, graph, and equation. Questions in the textbook are included to identify relationships among the representations.

Also for MP4, 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, Integrated Math I, 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.

For MP5, tools in the Integrated Math I and Integrated Math III 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 include "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 (Integrated Math I, pages 167 and 426). In the Integrated Math II materials, multiple tools are utilized to perform geometric constructions (i.e. compass, paper, pencil, rule, patty paper).

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.

Opportunities to develop MP7 are missed in the instructional materials. For example, in the Integrated Math II materials, Lesson 8.1 has students draw in vertical lines inside an existing right triangle so that trigonometric ratios can be developed and defined. Due to the lack of descriptions and teacher guidance for the MPs within lessons, the connection of students drawing auxiliary lines in order to solve a problem (MP7) is not made with the content and activities present in the lesson. Also in the Integrated Math II materials, Lesson 8.6 uses the technique of drawing auxiliary lines to solve problems and derive formulas, but the absence of descriptions and guidance for teachers or students does not support the intentional development of seeing structure (MP7).

Some lessons include a focus on seeing structure and generalizing (e.g., Integrated Math II, Lesson 12.4 “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 Integrated Math I. However, most problems are 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 scaffolded examples rather than an intentional outcome of student discussion or student calculations. An example of this can be found on pages 531-532 of Lesson 7.4 (problems 4 through 16) in the Integrated Math II materials where the problems represent scaffolded questions that lead students directly to the formula for the sum of the interior angles of an n-sided polygon. As students answer these questions, they are not given the opportunity to utilize MP7 or MP8 on their own.

Teacher-guided questions used during some 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 (Integrated Math II Lesson 13.5).
• "Why does this construction work?" is frequently asked of students in Chapter 12 of the Integrated Math I textbook or Chapter 1 of the Integrated Math II textbook when students are making several constructions.
• The guiding questions for teachers included in Integrated Math I, 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?" "Are all curved graphs considered graphs of non functions?" "Are all linear graphs considered graphs of functions?"