- 學位論文
國中資優數學之同餘的探討
The Study of Gifted Mathematics in Some Courses of Congruences in Junior High School
摘要
摘 要 本研究旨在探究國中資優數學中利用同餘概念解題所需具備的相關知識與解題思維，據以編製國中資優數學之同餘教材。 因此本研究根據研究目的，提出以下結論： 一、在九年一貫的數學課程綱要中，並未將同餘概念納入其中，故研究者選擇先由同餘基本性質出發，接著藉由同餘常見的應用，讓學生熟悉同餘式的操作，而同餘的常用定理則可擴充學生對於分析同餘問題的背景知識，讓學生對於同餘問題的思考方向有一基本的脈絡可循。 二、如上所述，本研究整理同餘的教材為「同餘基本性質」、「同餘常見的應用」與「同餘常見的定理」，主要內容如下： （一）同餘基本性質：求餘數是同餘的基本問題，藉由同餘基本性質的介紹與試題演練可知利用餘數為1或 是化簡同餘式求餘數的重要步驟。 （二）同餘常見的應用：同餘的概念可將整數做分類，可做為題目的討論方向與依據，由整數的分類進而可以分析整數為平方數或立方數的特殊類型，而不定方程可視為綜合演練，如何找到適合的模數簡化問題為解題的關鍵。 （三）同餘常見定理：討論中國剩餘定理、威爾森定理、費馬小定理與歐拉定理等同餘的常用定理，可用來更快找到某些特殊類型餘數為1的狀況，也可用來求解同餘方程式。
並列摘要
Abstract This study investigated the knowledge and skills required for solving math problems based on the concept of congruence in junior high school gifted math to provide some suggestions on compilation of learning materials about congruence for mathematically gifted students. Based on the objectives of this study, the following conclusions were proposed. 1. The concept of congruence is not covered in the Guideline for Math in Grade 1-9 Integrated Curriculum. Hence, the author proposed that instruction should start with basic properties of congruence and then the applications of congruence to familiarize students with congruence expressions. Further, common theorems of congruence should be introduced to enrich students’ background knowledge for analyzing congruence problems, allowing them to have a systematic understanding of congruence. 2. As mentioned above, the author suggested the learning materials about congruence should consist of three major topics, including “basic properties of congruence”, “common applications of congruence”, and “common theorems of congruence”. The content of each topic is as follows: (1) Basic properties of congruence: Finding the remainder is a basic problem of congruence. By learning the basic properties of congruence and how to solve some examples, students can understand that using remainder 1 or to simplify congruence expressions is an important step. (2) Common applications of congruence: The concept of congruence can be applied to classify integers. Through classification of integers, students can analyze if an integer is a square number or a cube number. Indeterminate equations can be viewed as general exercises. How to find an appropriate modulus to simplify a congruence problem is a key to solve the problem. (3) Common theorems of congruence: Common congruence theorems, including Chinese Remainder Theorem, Wilson’s Theorem, Fermat’s Little Theorem, and Euler Theorem, should be introduced. These theorems can help students identify some special problems with 1 as the remainder and solve congruent equations faster.