The surface area of a sphere as a function of radius, "r," is a derivative of the volume of the sphere, and a few other families of solids with this property are known. Sometimes, the derivative relationship can be reached with a tricky choice of size parameter "r." This paper presents an activity for students that discovers…

When examining students' participation in these mathematics discussions, the focus is on their verbal contributions. However, students' nonverbal contributions--such as pointing, drawings, and models-- can be crucial resources for advancing the mathematical thinking and the collective activity of a classroom community (Johnson et al., 2023; Webb…

A series of course-specific MAP Growth Mathematics and Science subject tests were released successively starting in August 2017 to replace the older NWEA End-of-Course (EOC) tests. Different from the prior NWEA EOC tests taken only at the end of a course, these course-specific tests can be administered multiple times throughout the school year,…

The paper explores the division of a polygon into equal-area pieces using line segments originating at a common point. The mathematical background of the proposed method is very simple and belongs to secondary school geometry. Simple examples dividing a square into two, four or eight congruent pieces provide a starting point to discovering how to…

A Steiner chain is defined as the sequence of n circles that are all tangent to two given non-intersecting circles. A closed chain, in particular, is one in which every circle in the sequence is tangent to the previous and next circles of the chain. In a closed Steiner chain the first and the "n"th circles of the chain are also tangent…

In this work we describe a teaching proposal to calculate the eccentricity of the Moon's trajectory by applying a geometrical technique. The values of the ratios between the Earth-Moon distance and the diameter of the Moon at apogee and at perigee were calculated from a kinematic model associated with a geometrical technique of image analysis. The…

Inside the discipline, mathematical work consists of the interplay between stating and refining conjectures and attempting to prove those conjectures. However, the mathematical practices of conjecturing and proving are traditionally separated in high school geometry classrooms, despite some research showing that students can successfully navigate…

Most any students can explain the meaning of "a[superscript b]", for "a" [element-of] [set of real numbers] and for "b" [element-of] [set of integers]. And some students may be able to explain the meaning of "(a + bi)[superscript c]," for "a, b" [element-of] [set of real numbers] and for…

The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

These notes discuss several related problems in geometry that can be explored in a dynamic geometry environment. The problems involve an interesting property of hexagons.

This short note presents and discusses an interesting area partition result related to a parallelogram. It is, then, shown how proving the result, and understanding why the result is true based on the principle of conservation of the area of triangles with the same base and between the same parallel lines, leads to further generalizations to…

A key topic throughout school geometry is measurement--namely, distance, area, and volume. This article focuses on one key idea for finding and justifying the area of two-dimensional (2D) shapes: area-preserving transformations. Although especially pertinent to geometry teachers, this article highlights a vertical connection of ideas progressing…

This article discusses funky protractor tasks, which are designed to provide opportunities for students to reason about protractors and angle measure. These tasks afforded opportunities for students to think critically about angular measurement tools, which supported the teachers with valuable insights into students' developing conceptions of…

Ariadne is a touch-based program for the learning of homotopies of paths, without the use of formalism, by building mental models. Using Ariadne, the user can construct points, paths by dragging points and homotopies by dragging paths as well as compute winding numbers of paths, all on a variety of surfaces, through touch gestures. Ariadne…

Calculus, as commonly taught, describes certain operations on explicit functions, but science relies on experimental data, which is inherently discrete. In the face of this disparity, how can we help students transition from lower-division mathematics courses to upper-division coursework in other STEM disciplines? We discuss here our efforts to…