國中資優數學之完全平方數的探討

The Study of Gifted Mathematics in Some Courses of Complete Square Number in Junior High School

余雅惠(Ya-Hui Yu)

指導教授 ： 王牧民

中原大學/理學院/應用數學研究所/碩士(2013年)

https://doi.org/10.6840/CYCU.2013.00016

摘要

本研究旨在解決國中數學資優生的特殊學習需求在普通常態班級中常被忽視的問題。故以國中數學領域課綱範圍為基礎，編寫國中資優數學之完全平方數的教材，讓教師能藉由教材，使資優生能體會到數學問題的本質，並提供給學生多樣化的解題方法，培養學生創造思考的能力，提升學習興趣。研究結果如下：
一、 國中資優數學之完全平方數的教材，能提供學生不同的解題策略且在驗證答案的過程中，培養學生多元思考的精神。
二、 研究者所編製的國中資優數學之完全平方數教材，最後歸納出的解題技法，能符合研究目的。本研究歸納出的完全平方數解題技法如下：
（1）平方數的奇偶性。
（2）平方數的餘數關係。
（3）連續兩整數的平方之間不可能有另一個平方數。
（4）以十進位制分析完全平方數。
（5）利用因式分解，解不定方程式。

關鍵字

資優數學 ； 完全平方數

並列摘要

This study aims to solve the problem of the mathematically gifted junior high school students’ special learning needs, which is usually neglected in regular classes. Based on Mathematics Learning Area in Grade 1-9 Curriculum, teaching materials on the complete square number for these students were designed to make them understand the essence of mathematics questions. In addition, the teaching materials provided the students with various ways to solve mathematics questions, developed students' creative thinking ability, and further increased their interest in learning mathematics.
The results are as follows:
1.The teaching materials in the present study provided students with different strategies
and techniques to solve mathematics questions. Furthermore, in the process of answer validation, students’ diversified thoughts were built.
2.The teaching materials were summarized by the researcher and the solving techniques
in them met the purpose of the present study. The techniques are listed below:
(1)The parity of the square number.
(2)The remainder relationship of the square number.
(3)It’s impossible to have another square number between the squares of two
consecutive integers.
(4)Adopt the decimal system to analyze the complete square number.
(5)Adopt the factorisation to solve the Diophantine equation.

並列關鍵字

complete square number ； gifted mathematics

參考文獻

許介彥(2003)。同餘的基本概念。科學教育月刊，261，23-25。

劉貞宜(2000)。數學資優生的解題歷程分析。國立臺灣師範大學特殊教育研究所論文。

Groen, G & Resnick, L.B.(1977).Can preschool children invent addition algorithms? Journal of educational psychology,69,645-652.

Renzulli,J S. (1978).What makes giftedness：Reexamining a definition.Phi Delta Kappan,60,180-184.

中文部分