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• Author or Editor: Kenneth Chelst
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## Finding Meaning in the Quadratic Formula

Connecting the formula to the graphic representation of quadratic functions makes the mathematics meaningful to students.

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## Technology Tips: Linear Programming with More Than Two Decision Variables

In a 1999 article in Mathematics Teacher, we demonstrated how graphing systems of linear inequalities could be motivated using real-world linear programming problems (Edwards and Chelst 1999). At that time, the graphs were drawn by hand, and the corner-point principle was applied to find the optimal solution. However, that approach limits the number of decision variables to two, and problems with only two decision variables are often transparent and inauthentic.

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## Purchasing a Used Car Using Multiple Criteria Decision Making

A Multiple Criteria Decision Making process (MCDM) in which students assign a numerical value to each alternative in a situation, and compute the highest value in order to quantify the optimum choice. MCDM can be used in making decisions such as choosing the best car or right college. This lesson can be used across the mathematics curriculum.

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## Visualizing Complex Roots of a Quadratic Equation

A quadratic equation was the basis for activities involving both concrete and technological representations.

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Analyzing profit from building LEGO® pets allows students to solve an authentic problem, first concretely and then abstractly.

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## Mathematical Explorations: Exercise Away the Big Mac: Ratios, Rates, and Proportions in Context

An activity centers around an authentic context: the nutritional content of a Big Mac and how to exercise the calories away. The task, which focuses on ratios, percentages, rates, and proportion concepts, also includes activity sheets.

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## Activities for Students: Controlling Inventory: Real-World Mathematical Modeling

Amazon, Walmart, and other large-scale retailers owe their success partly to efficient inventory management. For such firms, holding too little inventory risks losing sales, whereas holding idle inventory wastes money. Therefore, profits hinge on the inventory level chosen. In this activity, students investigate a simplified inventory-control problem. Within this context, students develop tables, graphs, and algebraic representations to reach a decision. We have successfully completed this activity with students in both first- and second-year algebra.

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## Queueing Theory: A Rational Approach to the Problem of Waiting in Line

All of us have experienced the annoyance of having to wait in line: to pay for our purchases in stores, to place orders in fast-food restaurants, to purchase tickets for concerts or sporting events, to enter amusement-park rides, or to get off an airplane. As customers, we generally do not like these waits. The managers of the establishments at which we wait also do not like long wait times, since such waits may adversely affect revenues. Why, then, is waiting necessary? According to Gross and Harris (1998), the answer is relatively simple: The need for service is greater than the ability to provide service. Often, an organization can increase the amount of service that it can provide. However, such improvements are often costly. Management then wants to know how much improvement is possible and wants answers to such questions as, On average, how long does it take a customer to receive service? and What is the average number of people waiting in line for service? Queueing theory, the detailed mathematical analysis of customers waiting for service, attempts to answer such questions, and it succeeds in many situations. Those successes can frequently be translated into increased business profits or greater efficiency by government agencies. Queueing theory originated in response to telecommunications problems in Scandinavia and retains its connection with the telecommunications industry in solving Internet problems today.

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## Promote Systems of Linear Inequalities with Real-World Problems

Because operations researchers solve problems in the real world, operations-research-based problems have rich connections to the world in which students live and work. Drawing on such problem situations is one way in which teachers can let applications of mathematics drive instruction. We believe that doing so will better motivate students to learn the mathematics they encounter in the classroom.

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## Investigating Integer Restrictions in Linear Programming

Restricting variables to integer values can lead to interesting classroom dialogues.