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Volume 1: Algebraic Modeling

Algebraic modeling skills are important for improving decision making in businesses, governmental agencies and even on a personal level. This curriculum provides an enormously wide array of contexts that involve the development and interpretation of algebraic models. However, computational skills are deemphasized, as such tasks are primarily handled using a spreadsheet. Throughout the curriculum, we demonstrate mathematics as a tool for exploring options and determining a preferred solution and not simply a process for getting the one correct answer.

Chapter 1 uses basic algebraic skills taught in middle school and Algebra 1 to explore decisions that involve multiple criteria. The decision contexts in this chapter are 1) select a cellphone plan, 2) pick a college to attend, and 3) purchase a used car. The student learns how to frame a decision using criteria and measures, and then to determine the preferred choice by calculating a weighted sum. This chapter stands alone, is not prerequisite for subsequent chapters, and can be taught at any high school grade level. This chapter leads naturally to the development of an individual or team project.

Chapters 2 through 6 build on the concepts of systems of linear equations and inequalities that are taught in all algebra 2 courses. However, algebra 2 textbooks generally limit their application problems to two variables, the limit for an easy graphical representation. However, two decision variables and a couple of constraints cannot demonstrate the true potential of this modeling process. In this course students are taught how to use EXCEL Solver to tackle decisions that involve many decision variables and constraints. Equally as important, students are taught to explore the robustness of the optimal solution and answer “what if” questions. Chapter 2 can be easily incorporated into an Algebra 2 course to enrich the modeling domain. Chapter 5 explores the impact of restricting the domain of the variables to integer values. Lastly, Chapter 6 presents a new modeling concept, binary decision variables. This concept is critical to modeling a collection of yes and no decisions such as which projects should be taken on or who should be assigned to a particular team. There are literally thousands of published applications that can form the basis for a student project.

Chapters 7 and 8 are stand-alone chapters which do not depend on the earlier ones. Their level of mathematical and conceptual sophistication is appropriate for any high school mathematics course. Chapter 7 introduces the common problem of deciding where to locate a facility. It presents algorithms for finding the optimal location which can be counterintuitive.

Chapter 8 introduces the mathematical models that used to study the phenomenon of waiting in line. There are relatively few broadly relevant applications of non-linear functions in high school mathematics curriculum. Mathematical models of waiting in line provide a rich array of decision contexts that utilize non-linear functions to calculate critical performance measures. Queues is the British word for waiting lines, and they are an ever-present element of all societies. The chapter’s contexts explores waiting in line to buy tickets to a show, the impact of merging small rural post offices and alternative layouts of airport security screening stations. The nonlinear functions presented in this chapter include higher order polynomials as well complex exponential functions.

Sample Activities: