By the time they reach middle school, all students have been taught to add fractions. However, not all have "learned" to add fractions. The common mistake in adding fractions is to report that a/b + c/d is equal to (a + c)/(b + d). It is certainly necessary to correct this mistake when a student makes it. However, this occasion also…

"Mathematical Lens" uses photographs as a springboard for mathematical inquiry and appears in every issue of "Mathematics Teacher." Recently while dismantling an old wooden post-and-rail fence, Roger Turton noticed something very interesting when he piled up the posts and rails together in the shape of a prism. The total number…

Today, the Common Core State Standards for Mathematics (CCSSI 2010) expect students in as early as eighth grade to be knowledgeable about irrational numbers. Yet a common tendency in classrooms and on standardized tests is to avoid rational and irrational solutions to problems in favor of integer solutions, which are easier for students to…

The absolute value learning objective in high school mathematics requires students to solve far more complex absolute value equations and inequalities. When absolute value problems become more complex, students often do not have sufficient conceptual understanding to make any sense of what is happening mathematically. The authors suggest that the…

Different representations of rational numbers are considered and students are lead through activities that explore patterns in base ten and other bases. With this students are encouraged to solve problems and investigate situations designed to foster flexible thinking about rational numbers.

Suggests that good number sense is fundamental for success in estimation, approximation, and problem solving. Further, large-number concepts are appropriate for development in upper elementary school, high school, and beyond. Presents examples to enhance a sense of large numbers in middle and high school students. (PK)

A history of perfect numbers is presented, which briefly covers the 27 values known at this time. (MP)

Conjectures about triangular arrangements of nine digits are stated and proved. (DT)

The purpose of this article is to present a collection of problems that allow students to investigate magic squares and Latin squares, formulate their own conjectures about these mathematical objects, look for arguments supporting or disproving their conjectures, and finally establish and prove mathematical assertions. Each problem is completed…

The author observes that checking the closure property is redundant for most number systems and therefore hard for students to understand. He defines several systems which are not closed, develops two concepts related to closure, and provides many related examples. (SD)

An unusual way of using the long multiplication algorithm to solve problems is presented. It is conceptually harder, since it involves negative numbers but is easier to perform once mastered, since the size of the multiplication table required is smaller than the standard one. (MP)

A number for which the number of digits categorizes the number is called a star number. A set of star numbers having a designated property is called a constellation. Discusses nature and cardinality of constellations made up of star square, star prime, star abundant, and star deficient numbers. Presents five related problems for exploration. (MDH)