High School - Gateway 1

The materials partially attend to the full intent of the mathematical content contained in the high school standards for all students. In most cases, the standard was addressed, clearly aligned and fully developed throughout the instructional materials. However, there were several instances where the instructional materials addressed the standard(s) to which it was aligned, but a particular component of the standard was not evident. In only a few instances were standards not clearly incorporated into the materials.

The following standards were identified as being especially strong/exemplary in terms of attending to the mathematical content:

• F-IF.2: This standard is fully met. Function notation is consistently used throughout the entire series.
• G-CO.6: This standard, using geometric descriptions for transformations, is addressed extensively. Examples of lessons addressing this standard are found in the Geometry materials on pages 24, 52 and 523.
• G-GMD.4: This standard requires students to identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects. In Geometry, lessons 4.1, 4.4, 4.5 and 4.7 include rotations of rectangles, circles and triangles as well as the cross-sections of cylinders, spheres, cubes, pyramids and cones. Additionally, the lessons include applications of the volume formulas for cylinders and cones.

The following standards were identified as not being fully attended to in the instructional materials:

• N-RN.1: The material provides an explanation of how the definition of the meaning of rational exponents follows from extending the properties of integer exponents but does not provide students the opportunity to explain themselves (Algebra 1 and Algebra 2).
• N-Q.1: Units are attended to repetitively throughout the instructional materials, especially in Chapters 1 and 2 of Algebra 1, where this standard is addressed. However, the portion of this standard regarding interpreting the scale and the origin in graphs and data displays is not called out in the problems. The lessons begin with a table of values and then use that data to create a graph (sections 2.1, 2.2, 2.6). There were not student or teacher prompts that provided an opportunity for discussion, nor were there explanations or clarification about how the data in the table was used to create a scale for the graph.
• N-Q.3: Determining level of accuracy is addressed on a basic level in the Algebra 1 materials in ways such as, "It isn't possible to have .3 boxes of popcorn so we must round up to the next box." Opportunities were not found that allowed students to consider the level of accuracy needed in ways such as, “Should we measure in feet or inches or sixteenths of inches?” or “Are integers or decimals two or three decimal places appropriate for the solution to a system?”
• A-REI.11: This standard requires students to "explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately." This is addressed only for linear and quadratic equations. Polynomial, rational, absolute value, exponential, and logarithmic functions that are specified in the standard are not addressed in any of the instructional materials.
• G-CO.2: The instructional materials provide students with the opportunity to represent transformations in the plane in a variety of ways. Students translate by moving the individual points of an angle (Geometry, page 52). Students also cut out the copy of a polygon and move it around the coordinate plane (Geometry, page 515). The materials do not address "describing transformations and functions." The materials have students analyze what happens to the x-values of the points of a figure and the y-value of points in a figure (Geometry, pages 517-518), but the materials do not have them write the transformation as a function.
• G-CO.13: Inscribed equilateral triangle, square, and regular hexagon are not included.
• G-GPE.5: Slope criteria is addressed in Geometry, lessons 3.1-3, but not as proofs.
• G-CO.4: Several sections of materials throughout the Geometry materials focus on experimenting with transformations in the coordinate plane. Materials provide ample opportunities to translate, reflect, and rotate line segments and figures in sections 1.2, 1.4, 1.5, and 7.1 of Geometry. However, evidence was not found in which the definitions of rotations and reflections are developed in terms of angles, circles, perpendicular lines, and parallel lines as outlined in this standard.
• 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 this procedure was not appropriate. There were no options for determining the area under the normal curve other than the text materials providing the percent for each interval.
• S-IC.4 and S-IC.5: Evidence was not found of the use of simulation to attend to the standard.
• S-CP.1: The instructional materials attend to the mathematical intent of the standard in that the terms, sample space and outcome, are consistently used throughout the probability sections of the materials. There is discussion of disjoint, intersecting, and complementary events (Geometry, sections 14.1, 14.2). However, there is no formal use of the term "union" used in the materials. Additionally, this standard is aligned to lessons beyond where the publisher shows in the alignment documents.

The following standards were identified as NOT ALIGNED to the mathematical content:

• N-RN.3: This standard is not addressed, although Algebra 1, lesson 14.1 is related. There is a statement that operations of rational numbers is closed; however, no explanation is provided. Furthermore, there was no evidence regarding the operations of adding and multiplying of irrational numbers as closed operations.
• G-MG.2: This standard was not identified in the publisher materials as being addressed in the materials nor was evidence of this standard found during the review process.

Additional notes about alignment:

• G-CO.9: This standard requires students to prove theorems about lines and angles. There are lessons throughout Chapter 2 of Geometry about writing proofs, lessons about lines and angles, and tasks that allow students to develop their understanding and then write proofs. However, one concern about this particular standard is that the standard is aligned to lessons where the standard is not specifically addressed. For instance, Geometry, lessons 2.1 and 2.3 contains prerequisite skills in the lesson that lead up to the standard rather than addressing the actual standard that is aligned to a given lesson. These lessons are not actually aligned to any high school standards and are only aligned to G-CO.9 in the alignment guide provided by the publisher.

When used as designed, the instructional materials reviewed partially meet the expectation for allowing students to fully learn each standard.

The following are examples of where the instructional materials provide opportunities for students to fully learn non-plus standards.

A-SSE.1: The materials give students many opportunities to develop a deep understanding of these standards. Algebra 1 addresses these standards throughout with some specific examples in lessons 2.1, 2.2, 3.2, 3.4 and 5.1. Algebra 2 also continually makes reference to the components of expressions and equations and asks students to comment on the connection to the context of the problem.

The following are examples of where the instructional materials provide opportunities for students to partially learn non-plus standards.

A-CED.4: In Algebra 1, lesson 3.3, students re-write the temperature conversion formulas to solve for Celsius and Fahrenheit. This is the one example that was identified as addressing the aspect of the standard where students are required to rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

A-REI.1: The instructional materials provided insufficient opportunities for students to explain each step in solving a simple equation. Algebra 1, section 2.1 featured four examples in the textbook that required students to solve an equation and justify reasoning, although the "justifying" was overlooked as these steps were not included in the teacher edition. Justifications must be written out for a solved equation in Algebra 1, section 14.2. This section, however, is not aligned to this standard (it is aligned to N-RN.3), and the focus of this lesson is on the properties of real numbers rather than solving equations.

N-Q.1: Students have sufficient opportunities to interpret units and scale in the context of a variety of problems; however, the lessons always provide the units and scale. To fully meet the standard, students should be provided the opportunity to decide on units and scale.

N-RN.1: Algebra 1, Lesson 5.5 gives justification for properties of rational exponents but does not build to student explanations. The properties are shown with an example, and then students are expected to use them. These are not properties students are expected to know from prior grades, so the examples/explanations are not adequate and do not attend to developing conceptual understanding of the properties of rational exponents.

N-RN.2: Rewriting the radicals and rational exponents using the properties is included in Algebra 1, page 344; however, there are limited instances for student practice or development of the rules and few numbers without variables are included.

F-LE.3: Algebra 1, lesson 5.1, addresses this standard when comparing simple and compound interest. Algebra 2, lesson 5.4, also addresses this standard. However, the quantity and direct computations of problems in these lessons is not sufficient for mastery of this standard.

F-LE.4: The emphasis of the lessons aligned to this standard was on using the change of base formula. The components of the equation were not addressed, nor was base 2, which is specified in the standard.

A-SSE.3b: Algebra 1, lesson 2.7 primarily uses completing the square as a method to find zeros of a quadratic equation (ample opportunities for practice in textbook, skills practice, and assignment books). Two examples in the material (page 778) use completing the square to identify the vertex of a quadratic. The instructional materials do not use the language of maximum or minimum.

A-APR.6: No lessons ask students to use long division in the way specified by the standard -- that is, to recognize that division allows an equivalent form of the expression. The process of division is included, but the connection between the process and the zeros is not clear.

The materials partially meet the expectation for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. Sometimes the material reviews some Grade 6-8 concepts, but not in a distracting or excessive way; rather, the material is presented in a manner that refreshes students and prepares them for the new concept.

Grades 6-8 standards are not clearly identified in teacher and student materials. However, there is a column labeled "Access Prior Knowledge" in the alignment documents that sometimes specified a middle school CCSSM to which the lesson is aligned. There are not very many explicit connections that link the lessons to prior learning for the teachers.

Examples of how lessons connect to middle school content include:

• Students' exposure to calculating area and perimeter of two-dimensional shapes and calculating volume and surface area for three-dimensional shapes is built upon at the high school level through the use of application problems. The connection between two-dimensional figures and three-dimensional figures is a focus through the study of cross sections. Additionally, students learn how to calculate area and perimeter of a two-dimensional figure on a coordinate plane.
• Cluster 7.RP: Ratios and proportional relationships is built upon at the high school level when learning about similarity of figures and related proportionality theorems (i.e., triangle proportionality theorem, proportional side theorem, proportional segments theorem, and triangle midsegment theorem).
• The heading "Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers" under cluster 7.NS builds the foundation for the high school standard of A-APR.7 in which students rewrite rational expressions and perform operations with rational expressions.
• N-RN.2 closely aligns with 8.EE.1 in the Grade 8 standards. Materials require students to apply exponent rules in Algebra 1 (section 5.5) and Algebra 2 (section 9.4), but this does not go beyond the expectations for the middle school standard.
• 8.EE.8 ("Analyze and solve pairs of simultaneous linear equations") connects to standards A-REI.5, A-REI.6, A-REI.7, A-REI.11, and A-REI.12 in which students solve a linear equation algebraically and graphically and extend this to solving and graphing systems of linear inequalities.
• In Grade 8, students learn how to utilize the Pythagorean theorem to determine unknown lengths of sides of right triangles and calculate distances in a coordinate plane. At the high school level, students are expected to prove the theorem and apply the Pythagorean theorem and its converse in the context of real world problems. Furthermore, the Pythagorean theorem serves as an introduction to 45-45-90 and 30-60-90 triangles.