初等數學中的典型多項式問題之研究

本研究是為了建構多項式單元的學習需求，故以國中及高中教授的內容做為基本架構，編寫初等數學中的典型多項式問題之研究的教材，讓教師能藉由此教材提升教師具備基本之學習邏輯，以此邏輯架構下，提供多樣化的解題方向及技巧，培養洞察力之廣度、邏輯思考推理之深度、加強研究思考的習慣，並提昇教學興趣。大綱如下：一、初等數學中的典型多項式問題之研究的教材，內容包含多項式、多項式函數、連續觀念、勘根定理(中間值定理)、極值問題(最大值與最小值)、微分(切線斜率問題)、局部極值(局部最大值與局部最小值)及積分(面積概念)之編排順序，讓教師可以由淺入深，逐步了解多項式真正的內涵。二、防範定義及定理易犯錯之處，如未考量分母不等於零，根號內的算式須大於零，在在強調數學題型之定義及其性質之應用。三、定義數學運算式模型之建構，並以幾何圖形做主要架構，以連續觀念做幾何圖形之輔助，研討國中及高中數學之課程內容及理論基礎作主軸，幫助教師可以快速進入多項式主題，並避免因未知數”x,y,z”造成學習此課題單元之困擾。

關鍵字

多項式

並列摘要

This study is to construct the learning needs of polynomial units. The Classical Polynomial Problems in the Elementary Mathematics is based on the teaching materials from junior and senior high schools, and is expected to develop teachers’ learning logic. Teachers provide diverse methods and skills to solve mathematical problems, cultivate excellent intuition, improve the ability of logical reasoning, strengthen the habits of independent thinking, and promote teaching interests. Following are the outlines of this study: 1.The study of classical polynomial problems in the elementary mathematics includes Polynomial, Function of Polynomial, Continuous Concepts, the root of located theorem (intermediate value theorem), extrema (maximum value and minimum value ), differentiation( the problems of tangent slope ), local extrema (local maximum value and local minimum value), and integration (area concept). According to the order of the arrangement, teachers can understand the real meanings step by step. 2.Prevent from making mistakes easily, for example, that the denominator doesn’t equal to zero isn’t be considered, the formula in the root number should be bigger than zero or equal to zero. The definition and application of math problems should be emphasized strongly. 3.Definite the construction of mathematical operation model, regard geometric shapes as the main structure, take continuous concepts as assistance, discuss the math curriculum which are taught in junior and senior high schools. With the basic theory above, teachers can understand more about polynomial topic and avoid being confused with the lesson from unknown x, y, z.