High School - Gateway 1

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for attending to the full intent of the mathematical content in the high school standards for all students. Overall, all of the standards are addressed during the course of this high school series.

All aspects of non-plus standards are addressed by the instructional materials and assessments. The following examples demonstrate the development of the aspects of the standards leading to the full intent:

• N-RN.1 (concerned with rational exponents and radicals) is developed in Algebra I Concept 7.3 and Algebra II Concept 3.1, moving from guided investigation of rewriting expressions with rational exponents and performing operations with rational exponents to solving equations with radicals and using properties of exponents.
• A-CED.2 (creating equations to solve problems) is a component of numerous lessons, including Algebra I Concepts 1.1, 1.2, 1.3 (simple expressions and equations); 2.1 (linear equations and expressions); and 11.1 (quadratics) and Algebra II Concepts 2.2 (exponential growth and decay); 6.3 (polynomials); 7.2 and 8.2 (rational equations).
• F-IF.2 (using function notation) is directly addressed in two lessons, Algebra I Concepts 3.1 and 4.1 and then built upon throughout the series in lessons that include functions.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation that they attend to the full intent of the modeling process when applied to the modeling standards. Overall, all modeling standards are addressed with the full intent of the modeling process in the series.

Modeling tasks are included regularly and frequently throughout the series. Apply tasks are included for every Concept and generally require students to engage in the full modeling process. These tasks typically involve real-world applications and require students gather, integrate, apply data in context, analyze, consider and validate results, and draw conclusions.

• Algebra I Concept 2.1 Apply 2 asks students to use a map and a limited amount of information on the speed and time spent driving each day for two drivers, one starting on the east coast and one starting on the west coast, to find when and where they might meet (A-CED.1, A-REI.3). Students are given information needed to understand the problem context but must develop and formulate the modeling process to complete the task.
• Geometry Concept 10.2 Apply 1 is a task about designing a parade balloon. Students conduct research to determine what information is important to find the amount of helium it would take to fill the balloon they design and the number of handlers it would require to control it. Students choose appropriate units and level of accuracy for calculations and must make sense of their calculations along the way. (G-GMD.7, G-MG.1)
• Algebra II Concept 2.2 Apply 1 asks students to research the Yellowstone National Park Wolf Project. Students review data about the number of wolves in the park since the beginning of the program in 1995, analyze the data to find a time period that can be modeled by an exponential growth or decay function, and write a paragraph to explain their analysis and give a comparison of the populations for one year before and one year after the time period selected. (F-IF.8b, F-LE.2, F-CED.1, F-LE.1b, F-LE.5)
• Algebra II Concept 7.2 Apply 3 requires students to use an app and workout information to explain how to change the next workout so that the average burn rate of calories increases eases by 5% over the average burn rate for the previous workout. (A-CED.1, A-CED.2, A-REI.2)

In addition, lessons consistently include application tasks that address modeling but provide students guidance in working through the modeling cycle.

• Algebra I Concept 2.1 Investigation 5 includes a task having students use equations to find the best deal for printing posters. Students are given costs by setup fees and the charge per poster printed for two companies and then asked to write equations modeling the cost of each company. They complete a table of costs for given numbers of posters and provide justification for which deal is best. (A-REI.11)
• Algebra I Concept 1.2 Investigation 4 asks students to make a conjecture about the relationship between the number of vertices in the polygon and the total number of diagonals. They attempt to write a mathematical model and share their ideas before being prompted to analyze data that will help them revise their original conjectures. To make sense of their models, this investigation guides them to organize their information in a table. Students share their results and discuss the validity of their expressions as well as how they might modify them. (A-SSE.1, A-SSE.1a)
• Geometry 1.1 Apply 2 asks students to choose a photograph that includes at least seven given geometric objects. They explain why the objects in the photo are the geometric objects identified, including the use mathematical definitions (G-CO.1, G-MG.1). Students must make decisions, provide evidence, and support the conclusions.
• Algebra II Lesson 9.3 Apply 2 asks students to research biorhythms; calculate their own physical, emotional, and intellectual biorhythm cycles; write and graph a function to represent each cycle; and interpret the points of intersection in the context. (F-TF.5)

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation that students spend the majority of their time on the content from CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, post-secondary programs, and careers.

The following examples show that the standards/clusters specified in the Publisher’s Criteria as WAPs are addressed across the entire series:

• N-RN and N-Q: Algebra I Units 7, 9, 10 and Algebra II Unit 3
• A-SSE, A-REI, A-APR, A-CED: Algebra I Units 1, 2, 4, 5, 7-11 and Algebra II Units 2-8
• F-IF, F-BF.1, and F-LE.1: Algebra I Units 3, 4, 7, 8, 10, and 11, Geometry Unit 7, and Algebra II Units 1-4, 6, and 9
• G-CO.1, G-CO.9, G-CO.10, G-SRT.B, and G-SRT.C: Geometry Units 1, 3, 4, 5, 6, and 8
• S-ID.2, S-ID.7, and S-IC.1: Algebra I Units 6, 11

The tasks aligned to WAPs from Grades 6-8 were included as a way to support student learning. For example, Algebra I Unit 5 begins with tasks that map to 8.EE.8 (solving pairs of linear equations) before moving on to the related high school content. This example is repeated across the Algebra 1 and Algebra 2 courses as students work with increasingly complex applications of systems for various parent functions.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for requiring students to engage in mathematics at a level of sophistication appropriate to high school.

Tasks are frequently presented in engaging contexts and vary in the types of numbers used.

• The tasks in Algebra I Unit 3.1 (addressing F-IF.1, F-IF.2, and F-IF.9) focus on understanding and interpreting functions. In Investigation 1, students use a telescope to determine the relationship between the distance of an object and the diameter of the field of view so that they can see advantages of representing a relationship between two variables algebraically. In Investigation 3, students determine whether or not different relations are functions (e.g., a planetary orbit and the area of a circle). In Investigation 4, students work with partners to view different functions and determine whether or not they are functions they know. Through extensions and applications students research the dietary intake of astronauts to determine the effects of space travel on the human body as well as daily caloric intake to find the basal metabolic rate, the weight of a million dollars, and the basics needed to calculate wind chill.
• In Geometry Unit 4.2, students research Da Vinci’s “Flower of Life” to find common structures and how to recreate their own versions of similar patterns. They further analyze abstract art to justify their own construction of triangles and angles with respect to midpoints and angle bisectors. As an extension, students construct a Wheel of Theodorus where they broaden their knowledge of angles and bisectors as well as the Pythagorean Theorem. Students also have three application tasks where they learn to copy an image using only a straightedge and a compass, create geometric figures in harmony, and use the geometry tool to explore “bank shots” with respect to the layout of a basketball court.
• In Algebra II Unit 5.1, students must research gas mileage, retail prices, and gas mileage to find the better deal, thereby allowing students to work with values that are real and appropriate for high school.

In addition, topics from grades 6-8 are included in a way that supports meeting high school expectations. For example, Algebra I Unit 2.1 (equations and inequalities) includes problems aligned with 8.EE.7.b (solving linear equations with rational number coefficients) but spends the bulk of the tasks on high school standards (including A-CED.3, A-CED.1, and A-REI.11).

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for fostering coherence and making meaningful connections in a single course and throughout the series.

All Concepts include a progressions and standards section for teachers. “Reach Back Standards” describe the groundwork needed to be prepared for concepts in the Standards Covered while “Reach Ahead Standards” identify connections to future mathematics.

The materials feature coherency within a course.

• In Geometry, there are connections among similarity, right triangle trigonometry, transformations, and circles. Students begin developing ideas of similarity in Unit 5 of the Geometry course. In Unit 6, students use similarity to develop an understanding that the ratios of sides in similar right triangles allows them to define the trigonometric ratios of angles. In Unit 7, students prove why all circles are similar using similarity transformations. The instructional materials make the connection clear to students by stating, “Recall that any two figures are similar if there is a combined transformation of a dilation and a rigid transformation that maps one to the other.” Students are prompted to use an exploration tool that allows them to determine that all circles are similar using their understanding of similarity transformations.
• In Algebra II, connections are made between inverse functions and the relationship between exponential and logarithmic functions. The concepts of inverse functions are learned in Concept 1.2, then in Unit 2 Concept 2.3 students connect to their prior learning of inverses to develop understanding of logarithms. The connection is pointed out to the teachers through instructional notes, “Before students start the activity, have them discuss what they recall about a function and its inverse—numerically, graphically, and algebraically,” and to the students by instructing them to “Use the Hands-On Activity: Exponential to Logarithmic to explore the relationship between an exponential function and its inverse.”
• In Algebra I Concept 3.2, linear functions and arithmetic sequences receive extensive development, including recursive and explicit aspects of sequences, along with exponential functions and geometric sequences. These concepts reappear in Algebra II when students model situations with either arithmetic or geometric sequences and extend their understanding of sequences to attend to deriving and using the formula for the sum of a finite geometric series.

In addition, the materials feature coherency throughout the series.

• Connections are made for quadratics among the Algebra I and Algebra II courses. Students learn to solve quadratics in Algebra I, and then Algebra II Concept 3.2 connects that learning to the idea of non-real solutions and imaginary numbers. The student materials make the connection clear to students in Investigation 2, “Remember when you solved the equation x^2+25=0? You realized that there was no real number solution because the square root of –25 is not defined in the real number set. Now that you know the complex number set, try solving it again to find all the possible solutions.”
• In Geometry, in Unit 2 students connect to their use of function notation (F-IF.1, F-IF.2) and utilize the notation to represent transformations as functions.
• In Algebra II, in Unit 9 connections are made to students’ prior learning about units of measure, circles, similarity, and right triangle trigonometry in the Geometry Course. In Concept 9.1 Investigations 1-3, connections are made for students between the degree measure of the angle, the circumference of the circle, arc length, the radius, and radian measure. Then, in Investigation 4, Converting between Degrees and Radians, the instructional materials specifically connect the idea of converting between degrees and radians as a way of converting between units as students have done before. The investigation begins with the statement, “Think about how you convert between familiar units, such as feet to miles or days to hours. You can convert radians and degrees in a similar manner.” Connections continue to be made to students’ prior learning in Geometry as the unit progresses into further study of the unit circle and trigonometric functions. Students connect special right triangles and the Pythagorean Theorem to make sense of the structure of the unit circle, which leads to connections within the Algebra II course itself when students connect the unit circle to the graphs of the sine and cosine functions in Concept 9.

The instructional materials reviewed for the High School Discovery Traditional series meet the expectation for explicitly identifying and building on knowledge from grades 6-8 to the high school standards.

Content from grades 6-8 is clearly identified in each Concept’s instructional notes, called “Progressions and Standards,” in this series. Standards from grades 6-8 are identified for the teacher in the “Reach Back Standards” section. This section provides a detailed progression of student learning from their 6-8 experiences.

The grades 6-8 standards are consistently integrated into the tasks in a way that supports learning the high school standards.

• In Geometry Unit 2, Geometric Transformations, the instructional materials identify how Concept 1 connects to 7.G.2 (geometric figures) and 8.G.1 (rotations, reflections, and translation), building off these concepts to address G-CO.2, G-CO.3, G-CO.4, and G-CO.5 (transformations in the Euclidean and coordinate planes).
• Algebra I Concept 3.1, Understand and Interpret Functions, integrates a review of related grade 8 standards (8.F.1-4) about functions and then builds on that work to address high school standards F-IF.1 and 2.
• Geometry Concept 2.2 connects earlier exploration of congruence in terms of rigid motion and use of coordinates to describe the effect of a rigid motion (8.G.2, 8.G.3) to express congruence in terms of rigid motion (G-CO.6) and use rigid motions to show triangles are congruent and establish the SSS, SAS, and ASA triangle congruence criteria (G-CO.7, G-CO.8).
• Algebra II Unit 10 builds upon students’ previous understanding of using random samples to draw inferences about a population (7.SP.1, 2) and prior experience with determining probabilities (7.SP.5, 6, 7, 8) to develop ways to describe their sample spaces (S-CP.1) and to understand ideas of independence (including two-way frequency tables) and conditional probabilities (S-CP.2-5).

In addition, the materials frequently include brief references to previous grades 6-8 and high school work that help students connect their learning.

• Algebra I Concept 6.1 Investigation 1 has a note reminding teachers that “students created histograms, dot plots, and box plots and used terms such as cluster, peak, gap, symmetry, and skew to describe displays” in middle school (6.SP.4) and gives a brief task to check students’ understanding of these topics.
• Geometry Concept 4.1 includes a brief reference to earlier work about the sum of the angles in a triangle (8.G.5, though it is not explicitly identified in the materials) before leading students through a formal proof of the triangle sum theorem (G-CO.10).