This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax[squared] + bx + c = 0, a [not equal to] 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the…

In this classroom note, the authors present a method to solve variable coefficients ordinary differential equations of the form p(x)y([squared])(x) + q(x)y([superscript 1])(x) + r(x)y(x) = 0. They propose an iterative method as an alternate method to solve the above equation. This iterative method is accessible to an undergraduate student studying…

In this paper, the properties of tangential and cyclic polygons proposed by Lopez-Real are proved rigorously using the theory of circulant matrices. In particular, the concepts of slippable tangential polygons and conformable cyclic polygons are defined. It is shown that an n-sided tangential (or cyclic) polygon P[subscript n] with n even is…

The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…

We sometimes teach our students a method of finding all integral triples that satisfy the Pythagorean Theorem x[squared]+y[squared]=z[squared]. These are called Pythagorean triples. In this paper, we show how to solve the equation x[squared]+ky[squared]=z[squared], where again, all variables are integers.

For most teachers, more important than supplies and a spacious classroom is a happy learning environment in which each child feels welcome and safe. That is why it is so dismaying that according to one recent study, 43% of students worry about going to the restroom for fear of encountering a bully. The same study reported that a child is bullied…

This article discusses divisibility tests for any prime number.

This article discusses the art of storytelling as a means of introducing new mathematics topics. Telling a mathematically based story can be a break from the routine and can serve as literary mnemonics full of mental imagery that helps students recall in problem-solving steps. The authors provide a certain story that has been tried by several…

The understanding and a liking towards mathematics can be very effectively developed in students by allowing them to find out the solutions for any basic problem or simulations, which are basically mathematical reenactments of nearly or completely hypothetical situations. The nontransitive relation of Efron's dice or the assignment of numbers in a…

Many school principals say that the toughest part of the day is recess. That's because recess is when most trouble starts. When the author asked one principal recently about recess, she promptly rattled off a list of headaches, such as "the teasing, the fights, the bullying, the injuries, the referrals." In this article, the author describes…

To better understand how teachers can capitalize on the power of story problems, Jacobs and Ambrose analyzed teacher-student conversations in problem-solving interviews. They identified eight categories of teacher moves that, when timed properly, were productive in advancing mathematical conversations. (Contains 2 tables.)

Learning mathematics has traditionally been thought of as a sequential progression. Children learn to count to 10, then to 20, and then to 100. They learn to add without regrouping and then with regrouping. The authors teach addition before multiplication and the two-times table before the six-times table. They usually teach division as a separate…

The number of students pursuing undergraduate degrees in mathematics is decreasing. Research reveals students who pursue mathematics majors complained about inadequate high school preparation in terms of disciplinary content or depth, conceptual grasp, or study skills. Unfortunately, the decrease in the number of students studying advanced…

Teachers who attempt to use inquiry-based, student-centered instructional tasks face challenges that go beyond identifying well-designed tasks and setting them up appropriately in the classroom. Because solution paths are usually not specified for these kinds of tasks, students tend to approach them in unique and sometimes unanticipated ways.…

The article proposes that individuals who acquire certain psychological support skills may experience accelerated learning and enhanced performance in many domains. In support of this proposal, we present evidence that these skills enhance learning and performance, that they are domain-general in that they can be applied in a variety of domains,…