The use of dynamic geometry software in exploring and investigating geometrical tasks is important for teaching mathematics, especially when paired with appropriate investigative tasks that can inspire while illustrating various properties and hypotheses. In this paper, we exemplify this by examining a specific investigative task through the use of illustrative software, which was tested among pre- and in-service mathematics teachers. The teachers’ response to the use of the software in solving the task and developing a proof for a hypothesis is observed. We emphasize the contribution a stimulating investigative task has in improving the quality of teaching as part of a micro-investigation, and study the students’ willingness to utilize the technological tools.
The particular task is presented in detail. It entailed finding the relative area of a quadrilateral formed at the vertex of a triangle by pencils issuing from the other two vertices. The relative areas were found to be conserved even as the side lengths of the triangle changed and the mathematical explanation is given. However, the task also led to a surprising discovery in that all the relative areas of any polygon within the triangle were conserved when the line pencils cut segments of different lengths of the opposite sides. A mathematical explanation is given for this case as well. Based on their experience, students were asked to provide a proof for particular cases of number of lines in the pencil, followed by a generalized proof for any number of lines.