International Journal of Education

The idea of combining number and shape is a core strategy for solving comprehensive problems of quadratic functions. It effectively connects geometric intuition and algebraic reasoning, and greatly improves problem-solving efficiency. Taking a comprehensive parabola problem from the Shanghai High School Entrance Examination as a starting point, this paper deeply explores the application of the number-shape combination idea in quadratic function problems. First, the study deconstructs the original problem, which is set against the background of parabola and integrates the geometric properties of isosceles right triangles to assess students' ability to use function graphs and geometric figures comprehensively. Through coordinate setting, equation establishment and classified discussion, geometric features are transformed into algebraic equations, and the rationality of solutions is verified by function graphs, fully reflecting the typical application characteristics of number-shape combination. On this basis, five types of variant problems are designed, including changing geometric figures, adjusting parameter ranges and introducing dynamic conditions, so as to verify the universality of the proposed thinking model. Finally, a four-step general method is refined: extracting graphic features, constructing algebraic models, solving through mathematical calculation, and optimizing by geometric verification, with emphasis on the key role of mutual translation between number and shape in each step. This study not only provides theoretical references for quadratic function teaching, but also demonstrates the practical value of the number-shape combination idea through concrete cases, which helps to cultivate students' core mathematical competencies such as intuitive imagination and logical reasoning. Further exploration can be made on the application of this idea in higher-dimensional mathematical problems in the future.

Combination of number and shape; Quadratic function; Parabola; Isosceles right triangle; Problem-solving strategy