High School - Gateway 1

The instructional materials reviewed for the Discovering series meet the expectations for attending to the full intent of the mathematical content contained in the High School Standards for all students. Overall, the instructional materials address most of the non-plus standards, however, there are a few instances where all aspects of the non-plus standards are not addressed across the courses of the series.

The following are examples of standards that are fully addressed:

• The standards from F-BF are developed starting in the Discovering Algebra course with a study of recursive sequences and writing explicit expressions to represent the sequences. Work with recursive sequences is then expanded to working with exponential equations. In Discovering Algebra, the work in this domain concludes with students identifying and exploring the various function transformations. Work with the F-BF domain continues in the Discovering Advanced Algebra course with a further study of recursive sequences and transformations and a study of building inverse functions.
• F-IF.7: The materials introduce time-distance graphs in Discovering Algebra Lesson 3.4 by discussing the intercept, periods of non-movement, and speeding up and slowing down. The standard is further addressed through Discovering Advanced Algebra as students examine maximum, minimum, and zero values of quadratic and polynomial functions as well asymptotes and end behavior of rational functions.
• A-CED.3: In Discovering Advanced Algebra Lesson 2.5, students determine appropriate and reasonable constraints for application problems involving profit from two types of birdbaths. Students also determine if solutions are reasonable in various problems.
• G-CO.7: In Discovering Geometry Lesson 4.4, students create various triangles using constructions and GeoGebra when given select criteria (such as, one side and two angles must be the same) to generate congruence “shortcuts” for triangles such as ASA, AAS, SAS, and SSS. Students also reason and investigate as to why the shortcut “SSA” does not exist.

The following standards are partially addressed:

• G-SRT.1a: In Discovering Geometry, Coordinate Geometry 7, students are questioned if “...the corresponding sides are parallel? Explain.” when examining a dilated set of triangles. However, the materials do not address whether a line passing through the center of dilation remains unchanged in either Coordinate Geometry 7 or Discovering Geometry Lesson 7.1.
• G-GPE.5: In Discovering Geometry, Coordinate Geometry 5, Example A, students are directed to find the equation of a perpendicular line through a point to find perpendicular bisectors. In Discovering Geometry, Coordinate Geometry 11, students are provided with the parallel slope property and perpendicular slope property and are given problems in which to use them in proofs. However, the materials do not contain a proof of these two properties.
• S-IC.5: This standard is not identified in the Discovering Algebra, Discovering Geometry, or the Discovering Advanced Algebra correlation documents. However, examination of the materials reveal that students compare treatments in Discovering Advanced Algebra Lesson 9.1, but at no time do the materials contain simulations to decide if differences between parameters are significant.

Standard S-IC.6 is not addressed within the three courses of the series. The correlation document for Discovering Advanced Algebra suggests this standard is addressed in Lesson 9.4. Upon examination of the materials, no indication can be found where reports that were either publisher-created or student- generated based on data are to be evaluated.

The instructional materials reviewed for the Discovering series meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The full intent of the modeling process is used to address nearly all of the modeling standards by the instructional materials of the series. Throughout the series, there are a number of lessons and activities that contain a variety of components of the modeling process described in the CCSSM. Each course also contains a separate document titled Modeling Tasks, which contains a preface that accurately describes the modeling cycle and provides an example modeling problem with a possible solution.

While the Modeling Tasks contain the same preface for each course, the problems provided are different for each course. The Modeling Tasks do not indicate standards or topics, and most of the problems included in the Modeling Tasks allow students the opportunity to engage in all aspects of the modeling cycle. The Modeling Tasks address nearly all of the standards identified as modeling standards. The tasks are not explicitly aligned to standards or to lessons in the materials, so students are able to complete the task by referencing all of the mathematics they have learned, rather than being guided by the lesson or chapter in which the task appears. The Modeling Tasks also include a rubric that can be used to score the modeling problems.

Some examples of Modeling Tasks that address modeling standards and the full intent of the modeling process include:

• In Discovering Algebra, Get Your Hamburgers Here!, students determine how much to charge for hamburgers in order to maximize their profits. Students are provided some assumptions based on a survey given to current customers. Students make sense of the problem by creating a model to compute a variety of hamburger prices. Students use their model to interpret and validate their solutions. Students may need to adjust their model as needed to determine the best price for the hamburgers. Students must report their solutions by providing an explanation of their results.
• In Discovering Algebra, Who Doesn’t Love Honey?, students determine whether they can isolate half of the bees while they are still healthy in order to produce enough honey for the science fair. Students must make assumptions about how many bees the infection began with, formulate a way to determine the time when half of the bees would be infected, and determine if they would be able to save half the bees based on when the infection was discovered.
• In Discovering Geometry, Let’s Go Camping!, students design a tent that fits a set of constraints. Students create a sketch of their designs; however students are not provided direction in regards to the design or formulas used to solve the problem. Students create a model of their tents and create formulas to compute if the surface area and volume are within the given constraints. Students interpret and validate their solutions. Students prepare a report to present to their family which includes a list of pros and cons of each design.
• In Discovering Geometry, Country Boy Gas Solution, students determine new dimensions of a given propane gas tank in order to double its volume. Students formulate a model to find the volume of the described tank. Students determine how the tank can change and how the changes will affect the volume. Students use their model to check dimensions to meet the demands of the problem. Students report what the new dimensions are. In this task, the sample answers consider changing either the diameter or the length of the tank; students may also find a way to change both dimensions which would create a tank with double the volume.
• In Discovering Advanced Algebra, A Heavy Fish Tale, students develop a plan to create a method to determine the largest fish caught. Students create a model to determine the largest fish caught and calculate repeatedly to determine if the model is accurate (interpretation, validation).
• In Discovering Advanced Algebra, What Will You Have with Your Coffee?, students are given information about running a concession stand in a local hospital. Based on the given information and making assumptions about the number of people they would be serving and how many items they could sell, students present a sales strategy for the business. Students create a model for each product and use the model to compare the costs and profits for the products. Based on the results, students modify their sales strategy for the business.

The instructional materials reviewed for the Discovering series, when used as designed, meet expectations for spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

In Discovering Algebra, students spend a majority of their time working with WAPs from the Number and Quantity, Algebra, Functions, and Statistics and Probability conceptual categories. For example:

• In Lesson 8.4, students factor quadratic equations to determine zeros using the Zero Product Property (A-SSE.3a). Students also factor quadratic equations from general form to vertex and/or factored form. Students check their work by confirming the locations of zeros as a result of factoring via the graphing calculator.
• In Lesson 4.1, students use "secret codes" in which each input of the code has only one output to introduce functions and vocabulary such as domain and range (F-IF.1). In Lesson 7.1, students encounter function notation and its relationship to the graph.

The Discovering Geometry course focuses on the widely acceptable prerequisites in the Geometry conceptual category. For example:

• In Lesson 12.2, students solve word problems using trigonometric functions (G-SRT.8). In Example 1 of the lesson, students use the angle of elevation to determine the distance a sailboat is located from a lighthouse.
• In Lessons 1.1, 1.2, and 1.3, students learn and use precise definitions for the terms such as line segment, angle, parallel lines, and perpendicular lines (G-CO.1). Students practice labeling each of these and answer questions about each of them.

During Discovering Advanced Algebra, students spend a majority of their time working with widely acceptable prerequisites from Number and Quantity, Statistics and Probability, Algebra, and Functions:

• In Lesson 4.6, students graph logarithmic functions by looking at how different transformations change each function (F-IF.7e). In Lesson 7.5, students graph trigonometric functions by looking at how different transformations change the period, midline, and amplitude.
• In Lesson 9.1, students compare and contrast various types of studies such as experimental, observational, and surveys. Students also make predictions based on sample data from a population in Exercise 7 (S-IC.1).

The instructional materials reviewed for the Discovering series, when used as designed, meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials provide students with opportunities to engage in real-world problems throughout the series. Students engage in problems that use number values that represent real-life values--fractions, decimals, and integers. The context of most of the scenarios are relevant to high school students.

Examples of where the materials require students to engage in mathematics at a level of sophistication appropriate to high school include:

• In Discovering Algebra, Lesson 4.1, students apply ratios and proportions from middle school to finding the rate of change of a function. Students write the rate of change in terms of unit rates that use compound units.
• In Discovering Algebra, Lesson 8.1, students solve quadratic equations that have various types of real numbers including terminating decimals, irrational numbers, and integers. Students also work with age-appropriate contexts including the height of a falling baseball, the height of a model rocket, and the height of an arrow being shot from ground level.
• In Discovering Advanced Algebra, Lesson 4.1, students use exponential equations that model both growth and decay. Students use various types of real numbers including decimals and integers and age-appropriate context such as population growth, growth of plants, and the price of a used automobile.
• In Discovering Advanced Algebra, Lesson 8.2, students apply key takeaways of basic statistics and probability from middle school to find the probability of compound events such as the probability that two students will be successful, and the probability of getting a number of questions correct in a row on a true/false test when just guessing.
• In Discovering Geometry, Lesson 11.4, students use rational numbers in fraction and decimal form. There are also operations with radicals and numerical manipulations where students leave “pi” in the answer.
• In Discovering Algebra, each chapter begins with a “Refreshing Your Skills” section which often allows for practice with varying number types. Discovering Algebra Lesson 4.0, Exercise 2 includes repeating decimals and radicals. In Lesson 6.0, students convert decimals to percents, and students review scientific notation in Lesson 6.4, which is reviewed throughout the exercises thereafter. Negative exponents are presented in Lesson 6.6 and used thereafter.
• In Discovering Advanced Algebra, students encounter the same variety of number types with the addition of complex number operations in Lesson 5.4. Complex numbers are then used in later work.

The instructional materials reviewed for the Discovering series meet expectations for explicitly identifying and building on knowledge from grades 6-8 to the High School Standards.

In Discovering Algebra, standards from grades 6-8 are explicitly identified in the teacher assistance portal on the left side of the page in the online teacher manual. Standards from Grades 6-8 are listed and aligned to lessons in the Discovering Geometry Correlation Guide. There are no standards from Grades 6-8 listed or aligned to lessons in the Discovering Advanced Algebra Correlation Guide nor in the lessons.

Some examples where the materials explicitly identify content from Grades 6-8, make connections between Grades 6-8 and high school concepts, and allow students to extend their previous knowledge include:

• Discovering Algebra, Chapter 0, Lesson 0.4 explicitly identifies 6.NS, 7.NS, 7.EE.1, 8.EE, and 8.F.1 and builds upon them to introduce F-BF.1a. Through the context of operations with signed numbers, students look for patterns in order to determine an explicit expression, a recursive process, or steps for calculation.
• In Discovering Algebra, Lesson 4.1, 8.F.1 is explicitly identified and built upon to address F-IF.1. Students examine functions through the use of “secret codes” and determine that a function is a rule (8.F.1) in order to understand that one element in the domain of a function corresponds to exactly one element of the range (F-IF.1).
• In Discovering Geometry, Lesson 7.3, students solve problems using scale drawings of geometric figures (7.G.1) while simultaneously solving problems using similar triangles (G-SRT.4).
• In Discovering Geometry, Lesson 2.5, students use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve problems (7.G.5). This knowledge is extended as students prove that vertical angles are congruent (G-CO.9) in Investigation 1.