Practical problems that use mathematical concepts are among the highlights of any mathematics class, for better and for worse. Teachers are thrilled to show applications of new theoretical ideas, whereas most students dread "word problems." This article presents a sequence of three activities designed to get students to think about…

Research shows that students often struggle with understanding empirical sampling distributions. Using hands-on and technology models and simulations of problems generated by real data help students begin to make connections between repeated sampling, sample size, distribution, variation, and center. A task to assist teachers in implementing…

The advent of dynamic geometry software has changed the way students draw, construct, and measure by using virtual tools instead of or along with physical tools. Use of technology in general and of dynamic geometry in particular has gained traction in mathematics education, as evidenced in the Common Core State Standards for Mathematics (CCSSI…

The Gale-Shapley stable marriage theorem is a fascinating piece of twentieth-century mathematics that has many practical applications--from labor markets to school admissions--yet is accessible to secondary school mathematics students. David Gale and Lloyd Shapley were both mathematicians and economists who published their work on the Stable…

The process of prime factor splicing to generate home primes raises opportunity for conjecture and exploration. The notion of "home primes" is relatively new in the chronicle of mathematics. Heleen (1996-97) first described a procedure called "prime factor splicing" (PFS). The exploration of home primes is interesting and…

In line with the Common Core and Standards for Mathematical Practice that portray a classroom where students are engaged in problem-solving experiences, and where various tools and arguments are employed to grow their strategic thinking, this article is the story of such a student-initiated problem. A seemingly simple question was posed by…

Educational games provide a fun introduction to new material and a review of mathematical algorithms. Specifically, games can be designed to assist students in developing mathematical skills as an incidental consequence of the game-playing process. The programs presented in this article are adaptations of board games or television shows that…

If students are going to develop reasoning and thinking skills, use their mathematical knowledge, and recognize the relevance of mathematics in their lives, they need to experience mathematics in meaningful ways. Only then will their mathematical skills be transferrable to all other parts of their lives. To promote such flexible mathematical…

What if Stephen Douglas instead of Abraham Lincoln had won the U.S. presidential election of 1860? What if John F. Kennedy had not carried some of the eight states he won by 2 percentage points or fewer in 1960? What if six hundred more people in Florida had voted for Al Gore in 2000? And what if, in that same year, the U.S. House of…

The 2012 U.S. presidential election is the perfect opportunity to present a timely civics lesson on how a U.S. president is elected. More important, it offers opportunities for students to reason mathematically about election issues--for example, about how much time and resources the candidates should invest in particular states. The results of…

Every two years in the United States, districts in each state elect representatives to the U.S. House of Representatives. The district boundaries are not permanent; rather, they are redrawn every ten years in a process known as redistricting. Mathematics is useful in understanding this important and often contentious process. Redistricting is a…

Students frequently have difficulty determining whether a given real-life situation is best modeled as a linear relationship or as an exponential relationship. One root of such difficulty is the lack of deep understanding of the very concept of "rate of change." The authors will provide a lesson that allows students to reveal their misconceptions…

By working through well-designed tasks, students can expand their thinking about mathematical ideas and their approaches to solving mathematical problems. They can come to see the value of looking at tasks from different perspectives and of using different representations. This article discusses four tasks that encourage high school students and…

Certain dates stand out in history--October 12, 1492; July 4, 1776; and May 8, 1945, to name a few. Will December 21, 2012, become such a date? The popular media have seized on 12/21/12 to make apocalyptical prognostications, some venturing so far as to predict the end of the world. Scholars reject such predictions. But major archeological finds…

Getting students to think about the relationships between area and perimeter beyond the formulas for these measurements is never easy. An interesting, nonroutine, and accessible problem that will stimulate such thoughts is the Lattice Octagon problem. A "lattice polygon" is a polygon whose vertices are points of a regularly spaced array.…