直線群と包絡線について

概要

論文の詳細を見る
The following is in the exercises of mathematics for high-school students. Ex.1. As the real number t changes, illustrate the range which contains the straight lines y=tx+2t^2-1. Ex.2. As the real number t changes in the region -1≦t≦1, illustrate the range which contains the straight lines y=tx+2t^2-1. Ex.3. As the real number t changes in the region 0≦t≦1, illustrate the range which contains the straight lines y=3(t^2-1)x-2t^3. These exercises are difficult for high-school students. In order to let them understand a graphical meaning, it is necessary to consider the concept of the envelope for a family of straight lines. In this paper, we will try to persuade high-school students that each line of a family of straight lines is the tangent of the boundary of the domain obtained by the discriminant D≧0. First, we explain that the straight lines y=tx+2t^2-1 in Ex.1 have the contact point of the boundary of the domain obtained by the discriminant D≧0. Next, by using this idea, we explain how to find the domain which contains the straight lines in Ex.2. As for Ex.3, they can not use the discriminant D, so we need to explain the method by using the envelope and another method. Finally, we report the result that we actually made our students solve these exercises in mathematics classes of Kure College of Technology.
全国数学教育学会の論文