Teaching Research On Cosine And Sine Theorem Based On HPM Perspective

Triangle is the basic figure of plane geometry,and the relationship between sides and angles is one of the most important relations of triangle.The law of cosine and the law of sine are two important theorems to describe the relationship between sides and angles of triangle,and they are powerful tools for the transition from qualitative analysis of triangles in junior high school to quantitative analysis of triangles in senior high school.In high school mathematics classroom teaching,the law of cosine and the law of sine are relatively "studious" content,because students will memorize the two theorems as two sets of formulas in trigonometric function,and then apply to solve problems.However,if we examine the history of trigonometry,we will find that astronomical observation,calendar calculation and the development of navigation prompted people to study the sphere,which led to spherical trigonometry.The law of cosines and the law of sines were the products of people's study of spherical trigonometry.Later,plane trigonometry was produced because of the need of plane mapping.Therefore,whether teachers or students,just because the solution of triangle and trigonometric functions have the word "triangle",they will finally simplify the solution of triangle into two groups of trigonometric formula teaching and memorization,covering the origin of the law of cosine and sine.Based on mathematics teaching idea of HPM,this study examines the basic status of triangle teaching high school solution: First of all,according to a senior student in solving the problems exposed by the triangle "solution" on the topic,further through the study,class view,first-line teachers about solving triangle teaching literature,undertake to the student interviews,analysis and find solutions for the problems existing in the triangle teaching.Secondly,it combs the origin and development of the law of cosine and the law of sine,refines the important thought of the development of the two theorems,and puts forward the suggestion of redesigning the teaching of the two theorems.Coincides with a new round of curriculum reform,the use of new textbooks,this paper from various versions of the new textbooks on the law of cosines,the law of sine content arrangement of the similarities and differences,further to provide ideas for the design of teaching cases.Finally,based on the HPM theory and combining the history of the law of cosines and the law of sine,this study redesigned the teaching content of the two theorems,and carried out teaching experiments and evaluations.The new teaching design starts from the history of mathematics and establishes a new teaching starting point.It can make students realize that the essence of the law of cosine and sine is the quantitative expression of the relationship between sides and angles of a triangle.The law of cosines is the extension of Pythagorean theorem;The ratio of each side of a triangle to its diagonal sine is related to the outer circle of a triangle.Rich historical materials in mathematics are an important source of teaching representation.By infiltrating the origin and development of mathematical concepts and theorems into teaching,it can solve the cognitive conflicts that students encounter when learning new knowledge and make them feel the reality of mathematics.