Untitled Cross-Quadrant Positioning Method (CQPM): A Visualization Approach for Sum and Difference of Squares Based on a Cross Structure
Year: 2026
Type: preprint
Abstract: In elementary algebra, the sum of squares a²+b² and the difference of squares a²−b² are extremely common expressions, yet traditional textbooks present them only as linear formulas, lacking intuitive geometric correspondence. This paper proposes a novel "Cross-Quadrant Positioning Method" (CQPM), which fixes these four quantities at four specific endpoints (upper-right, upper-left, lower-left, lower-right) of a cross structure, establishing a unidirectional mapping from base numbers to results. Furthermore, it reveals the bidirectional reversible property of this quadruple under specific constraints—"knowing any two, the other two can be uniquely determined." Through verification across 10 classical application scenarios in algebra, number theory, trigonometry, complex numbers, and geometry, the method demonstrates strong educational and practical value.摘要在初等代数中，两数 a、b 的平方和 a²+b² 与平方差 a²−b² 是极其常见的表达式，但传统教科书仅以线性公式呈现，缺乏直观的几何对应关系。本文提出一种全新的“十字象限定位法”，将这四个量固定于一个十字结构的四个特定端点（右上、左上、左下、右下），构建了从底数到结果的单向映射，并进一步揭示了该四元组在特定约束下的“任意知二求二”双向可逆性质。通过代数、数论、三角、复数、几何等10个经典应用场景的验证，表明该方法具有较强的教育实用价值。
Source: Figshare
Author: yang
Subfield: Education
Field: Social Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7213198154
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