CLASSROOM CAPSULES EDITOR Michael K. Kinyon Indiana University South Bend South Bend, IN 46634 Classroom Capsules consists primarily of short notes (1–3 pages) that convey new mathematical in- sights and effective teaching strategies for college mathematics instruction. Please submit manuscripts prepared according to the guidelines on the inside front cover to the Editor, Michael K. Kinyon, Indiana University South Bend, South Bend, IN 46634. A Non-Visual Counterexample in Elementary Geometry Marita Barabash (marita@macam.ac.il), Achva Academic College of Education, Beer- Tuvia, Israel Future math teachers, whether at the primary or secondary level, should enter their profession with a solid foundation in geometry. There are times when, in teaching elementary topics to such students, we have an opportunity to introduce more profound ideas at the same time. An example of this occurs in the context of area and perimeter. Every student needs to understand that there is no dependence whatsoever between a change in perimeter and a change in area—one can go up or down while the other remains constant. (Some theoretical background on this topic may be found in Courant and Robbins [1, Ch. VII], and in a more general discussion leading to the general isoperimetric problem and its dual in Jacobs [2, pp. 307–328].) In my experience, many students do not comprehend this as it contradicts their erroneous intuition that an increase in one of these quantities leads to an increase in the other. An exchange similar to the following takes place between me (Instructor) and my students nearly every year. Instructor: If two polygons are congruent, obviously they must have the same area and the same perimeter. Is the opposite true? That is, if two polygons have the same area and the same perimeter, do they have to be congruent? Students (after some work): No, there is quite a simple example. Take a trapezoid with a right angle and attach a right triangle in two different ways, like this: Figure 1. Instructor: Very good, so it’s not the case for quadrilaterals. But what about triangles? As you know, there are some ways in which triangles are special polygons. VOL. 36, NO. 5, NOVEMBER 2005 THE COLLEGE MATHEMATICS JOURNAL 397