Implementing the Assessment Standards for School Mathematics: Assessing Justification and Proof in Geometry Classes Taught Using Dynamic Software

Abstract

Consider the midpoints of all the chords that can be drawn from a given point, say, A, on a circle (see fig. 1). Can anything special be found about these midpoints? Using dynamic geometry software, students can trace the locus of these midpoints by dragging the chord AB from point B. That is, they can use the computer mouse to select and hold point 8 as it is moved around the circle. The computer displays a dynamic chord with a fixed endpoint A and traces the path of the midpoints. The small blue dots shown in figure 2 represent the midpoints of the chords generated as point B is dragged around the circle. Figure 2 suggests that these midpoints lie on a circle. Is this observation true? How can we be sure? When presented with this task, a high school student answered, “It forms … it forms a circle! The midpoints … the midpoints when you move it around form a smaUer circle inside the big circle!” When the student was asked to justify his answer, he said, “I can see it before me, and it does form a circle. I have evidence for it.”