A Study of Teaching for Readiness to Introduce Square Roots: With the Aim of Constructing Dynamic Images of Irrational Numbers

Abstract

　　Seven companies publish mathematics textbooks for use in junior high schools in Japan. Introducing square roots in mathematics class, all of them discuss the relationship between the area and the side of a square. For example, there is a description of “If the area of a square is 2cm², how many centimeters is the length of one side of a square?” Hitotsumatsu (1990) casts doubt upon the description of “The solution of x² - 2 = 0 is + √2 and - √2 ” (p.7).

　　The aim of this paper is to develop a new approach to introduce square roots in teaching for readiness. That lies hidden in Square Dotty Grid and Isometric Grid. Our idea is based on the figures of “If a segment is multiplied by a and one more time running, then it is multiplied by 2 and 5 on Square Dotty Grid, 3 and 7 on Isometric Grid” We carried out classwork study and realized that the approach will succeed. 67 second-grade students in junior high school made three squares or three rhombuses using a diagonal’ length of a square or a rhombus, and responded to a question in worksheets during class. As a result of the survey, their notes gave us three suggestions. First, there is a possibility that students consider with the ratio in length between each one side of three squares or three rhombuses. Second, these teaching materials provide learning opportunities of inductive, analogical and deductive inference for students. Third, students may notice irrationalness themselves.