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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.

Boxplots are statistical representations for organizing and displaying data that are relatively easy to create with a five-number summary. However, boxplots are not as easy to understand, interpret, or connect with other statistical representations of the same data. We worked at two different schools with 259 middle school students who constructed and interpreted boxplots. We observed that even students who were able to create boxplots had difficulty interpreting data represented in a boxplot. After sharing specific difficulties that we observed students having, we discuss ways to help students to make sense of data presented in boxplots.

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.

A common activity--having students collect data to measure the radius and circumference of circular objects--can be given a new twist. These data can be viewed virtually using technology. A dynamic geometry environment coupled with the powerful capabilities of a spreadsheet can greatly enhance student learning. The activity in this article addresses these grades 6-8 tenets found in "Curriculum Focal Points," which state that students should be able to: (1) analyze tables to discern relationships; (2) use equations to describe relationships inherent in a table; (3) study areas and circumferences of circles; (4) identify the unit rate of the graph of a proportional relationship as the slope of the line; and (5) translate among verbal, tabular, graphical, and algebraic representations of functions. These suggestions can be explored in a technological environment where teachers and students use TI-Nspire[TM], GeoGebra, or The Geometer's Sketchpad[R]. Students collect data from a dynamic geometric construction and study the relationships among the radius of a circle, its circumference, and its area. Many technologies allow students to create and study dynamic geometric constructions by changing a part of the construction. The effects of that change in other parts of the same construction become clear for students to see. For instance, students can use a graphing calculator such as TI-Nspire with dynamic geometry features to grab and drag the radius of a circle to make it longer or shorter. However, in GeoGebra or Sketchpad, students would need to drag the endpoint of a radius. Students can also use the technology to quickly measure many circles, display these measurements, and observe the effect that changing the radius has on the circumference. These tools can also capture an approximation of the areas of the circles. A TI-Nspire-based activity sheet is included at the end of this article. (Contains 6 figures and 1 table.)

"I liked to grab things on the graphics and see it first hand, see what we were doing on the problem. It makes more sense." These words were from a student who completed a series of lessons on lines of best fit. She made this comment after comparing a hands-on approach using a strand of spaghetti with a high-tech approach using a calculator. To give students meaningful experiences while studying regression lines, the authors designed a learning environment that accommodates "Bruner's three forms of representation." Bruner advocated a sequence of learning that runs the gamut from using physical materials to using symbols to teach mathematical ideas effectively. This article presents a regression activity using the TI-Nspire[TM] calculator and spaghetti. (Contains 4 figures.)

Evolving technology has played an important part in a common quadratic-function lesson. Having been mentioned repeatedly in numerous reform documents, a recurring lesson has involved changing the parameters in f(x) = ax[superscript 2] + bx + c and studying the effects on the graph. In both NCTM Yearbooks and NCTM Standards documents, technology is discussed and explored as a way to enrich quadratic function tasks. However, these authors will show how advances in technology have enriched these activities. Using quadratic functions as an example, this article describes how the evolution of technology has altered the expectations of students' understanding and critical thinking. (Contains 9 figures.)

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.)

This article describes one way of embedding remediation in the context of studying more advanced mathematical concepts. In the process of doing so, ways to foster mathematical argumentation and proof are also introduced.

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)

Identifies big ideas for the informal study of algebra-related topics in the middle grades including notation, variable, function, and properties of number. Suggests meaningful ways to accomplish them. (YDS)

Presents activities to find Pythagorean triples using the TI-83, a graphing calculator. (KHR)

Describes how three experiments suitable for upper-elementary students can be used to foster classroom discourse in which students begin to explore probability. (KHR)

Describes an action-research collaboration between a middle school mathematics teacher and a mathematics teacher educator in which the teacher wrote a narrative description of the collaboration and the changes she made in her instructional practice as a result of the collaboration. Contains 34 references. (Author/ASK)

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

A model for conceptualizing teacher change was developed during a two-year study of mathematics teachers' implementation of an innovative curriculum. Based on constructivist views of teaching and learning, the model suggested that one way to promote change in teaching practice is to structure interactions among teachers to promote reflective thinking. (SM)

Describes an inquiry activity in which students explore the variables that influence the amount of time it takes a ball to roll down an inclined plane. Relates features of the activity to recommendations in the NCTM Standards. (MKR)

Several activities that are based on the ancient method of false position, also called false assumption, are presented in this article as a way to motivate students to find the solution of literal equations in beginning algebra.

Explores the effects of varying the coefficients in the general quadratic function using graphing calculators. (MKR)

Mathematics instructional programs in elementary school should include opportunities for students to understand and apply basic concepts related to chance and probability. The discussion draft of Principles and Standards for [...]