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While tutoring his granddaughter in second-year algebra recently, the second author lamented that every textbook he could find expresses the quadratic formula as probably the most common form of the formula. What troubled him is that this form hides the meaning of the various components of the equation. Indeed, the meaning was obscured by the simplification that arose from combining two fractions. When a contextual problem is associated with the mathematics, meaning obscured due to a simplification is often critical. Sometimes, it is critical even in the absence of any context. The authors think this is particularly true in the case of the quadratic formula. In this article, they explain the current approach of quadratic formula and propose a new way of connecting the formula to the graphic representation of quadratic functions that makes the mathematics meaningful to students. They believe the approach we have proposed above is consistent with the visions of mathematics teaching and learning articulated in both "Principles and Standards for School Mathematics" (NCTM 2000) and the Common Core State Standards for Mathematics (CCSSI 2010). When students engage in activities such as those we have described, they will be using algebraic symbols to represent, analyze, and understand a mathematical situation, as suggested in "Principles and Standards." Moreover, they will be reasoning abstractly and quantitatively, making use of structure, attending to precision, and, in particular, using an appropriate tool strategically, as the Common Core State Standards recommend.

Linear programming (LP) is an application of graphing linear systems that appears in many Algebra 2 textbooks. Although not explicitly mentioned in the Common Core State Standards for Mathematics, linear programming blends seamlessly into modeling with mathematics, the fourth Standard for Mathematical Practice (CCSSI 2010, p. 7). In solving a linear programming problem, we always seek to find the optimal solution, which might be a maximum or a minimum depending on the nature of the problem. When the variables are restricted to integer values, as often happens in the real world, the problem is then an example of integer linear programming (ILP). In this article, we show why this distinction matters and how it might provide an interesting classroom investigation.

In mathematics, students should represent, model, and work with such real-world situations as those found in the physical world, the public policy realm, and society (CCSSI 2010). Additionally, students need to make decisions and be flexible enough to improve their decisions after analyzing realistic situations. The LEGO® Pets activity does just that. Modeling with the Common Core's Standards for Mathematical Practice is a process that begins with identifying variables, analyzing relationships, and formulating a model, then moves into interpreting, validating, and reporting on the results (CCSSI 2010). To bring this modeling process to life in classrooms, teachers need sources of authentic problems. But how can teachers determine the authenticity of a problem that will allow students to model and analyze a situation mathematically? Herrington and Oliver (2000) and Lombardi (2007) have developed frameworks for authentic learning situations. They indicate that authentic problems share a number of characteristics. In particular, they (1) have clear connections to the real world; (2) require significant investments of time and intellect; (3) use information from multiple sources and often require taking more than one perspective; (4) allow for student collaboration; and (5) require not only finding a solution but also interpreting that solution in the context of the problem. In this article, the authors describe how they developed the LEGO® Pets activity (activity sheet included at the end of the article) in which students model a real-world problem situation, first concretely and then abstractly. They taught the activity in three sixth-grade classes in an affluent suburban district as well as in one sixth-grade and one seventh-grade class in a much less affluent suburban district. Approximately thirty students were in each class. The students in all five classes were motivated, interested, and able to work through all the questions posed. Sometimes they needed no scaffolding to answer the questions; at other times, minimal scaffolding; and a few times, extensive scaffolding.

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. This activity has been successfully completed with students in both first- and second-year algebra.

When studying mathematics, students often ask the age-old question, "When will I ever use this in my future?" The activities described in this article demonstrate for students a process that brings the power of mathematical reasoning to bear on a difficult decision involving multiple criteria that is sure to resonate with the interests of many of them: choosing the right used car to purchase. This objective is met by introducing the concept of Multiple Criteria Decision Making (MCDM), as well as a structured method for identifying a best alternative in such situations. This lesson easily demonstrates the breadth of mathematics applications and possible avenues for non-traditional problem solving through mathematics. (Contains 9 tables.)

Discusses students' exploration of a particular rational function in the context of people waiting in line for service. The concepts of domain, range, and asymptotes are also developed in that context as is the effect of changes in input variables on function outputs. (Author/NB)

Uses everyday situations that students experience, in order to attract and hold student attention and promote understanding of linear inequalities. (ASK)

The article focuses on the benefit of building LEGO bricks to develop child ability on how to solve algebraic problems. Among the benefits discussed include student collaboration, require significant investments of intellect and time, and uses information from multiple sources on more than one perspective.

The article offers suggestions on using graphing systems technology in classroom activities for teaching and learning of linear programming problems with two or more variables. It discusses use of Solver software for translating algebraic formulation of two variables into a spreadsheet formulation. It mentions solutions for linear programming problems with four variables. It also mentions that these classroom activities connected with Common Core State Standards for Mathematics.

This paper quantifies key issues with regard to concurrent engineering through the use of risk and decision analysis techniques that enable us to better understand, structure, and manage the design process. In concurrent engineering, the information structure of a design process does not usually imply the execution patterns of the corresponding design tasks. On the contrary, this gap between the information structure and execution patterns is the essence of concurrent engineering and its basic advantage over traditional sequential design. In this paper, we relate the structure of information flow in a design process to three different execution strategies: sequential, partial overlapping, and concurrent. The risks of excessive task iterations or redesigns associated with each execution pattern are probabilistically modeled. Risk and decision analysis methodology is used to determine the best execution strategy and the optimal overlapping policy for a set of activities given their information structure. Applying this theoretical framework to a real-world design application of an automotive cylinder block suggested a potential 18% reduction in development cycle time. Index Terms - Concurrent engineering, design process structure, information requirements, partial overlapping, probabilistic modeling, risk and decision analysis.

The article describes an activity program for the seventh-grade level which presents a realistic context to help students think proportionally.

The article evaluates an inventory-control problem by implementing the modeling cycle described by the 2010 Common Core State Standards (CCSSI), with special focus on the time and number of video games consoles orders. It discusses the Economic Order Quantity, an inventory model that was developed by Ford W. Harris. The model works on assumptions such as ordering of product on exhaustion of current inventory only, fixed ordering cost irrespective of items ordered, and instant product delivery.