Daniel Ness

Quadrilaterals and Bretschneider's Formula

We talk about a formula deduced by Bretschneider in the mid-nineteenth century that not only allows the computation of areas of planar quadrilaterals, regular and irregular, but also provides a more thorough understanding of the geometry of such figures. We present one version of Bretschneiders formula and talk about its scope of application and what it can say about the possible shapes of quadrilaterals.

Shoelace formula: Connecting the area of a polygon and the vector cross product

Understanding how one representation connects to another and how the essential ideas in that relationship are generalized can result in a mathematical theorem or a formula. In this article, we demonstrate this process by connecting a vector cross product in algebraic form to a geometric representation and applying a key mathematical idea from the relationship to prove the Shoelace theorem.

A new look at an old triangle counting problem

The correct answer to the problem in figure 1 is 15. There are nine triangles congruent to ABE, three triangles congruent to ACH, two triangles congruent to BGH, plus ADJ, for a total of 15 equilateral triangles. Co-author Sutcliffe recently encountered this problem on a MATHCOUNTS® poster titled “What number do the following have in common” in her high school classroom while student teaching, and the problem generated a great deal of excitement among her students. Of fifty-four students in three different classes, none was able to identify BGH and CEI as equilateral, but they all tried very hard to find them. The second source of excitement came from the handful of students who were eager to know how the solution generalizes to larger arrays.

Solving and graphing quadratics with symmetry and transformations

One of the central components of high school algebra is the study of quadratic functions and equations. The Common Core State Standards (CCSSI 2010) for Mathematics states that students should learn to solve quadratic equations through a variety of methods (CCSSM A-REI.4b) and use the information learned from those methods to sketch the graphs of quadratic (and other polynomial) functions (CCSSM A-APR.3)