Exploring Geometry through Research Methods and Transformations

1. Geometry: Teaching and Learning LESSON 11

2. PART ONE Geometry • 1. The basic research methods geometry research the nature of the graphics, and spatial graphic differences, their properties are so numerous and each are not identical, therefore, the methods to study graphic nature is diverse. In terms of elementary geometry, the graphic method can be divided into "experiment" "deductive" “analyzing "" transformation" 4 "class.

3. 1 Experimental Method • So-called experimental method, namely through the observation, measurement, experiment and so on means to gain knowledge of plane or space graphics. • For example, we want to know: how much is the sum of angles of a triangle? We can use a protractor to measure, add the each angular, or we can tear and put the three angles of the triangle, flat out on the table, and then use a protractor to measure. • Though the geometric result using the method of experimental is imprecise, but it can let students experience, to discover, explore graphic nature of the process, so in the lower grades, experimental method for learning graphic nature is the main methods.

4. 2 Deductive Method • So-called deductive method, that is, an axiomatic method is employed to set up the theory of system, through the pure logic reasoning prove to study and understanding the nature of the graphical method. Today, axiomatic method has become an important way of thinking in modern mathematics.

5. 3 Analytical Method • So-called analytical method, it is by introducing a vector, establish a coordinate system, then make the geometric position relationship and measure quantification . It a method of studying graphic nature. • For example, using the analytical method, the plane "dot" corresponds to a pair of real number, "line" corresponds to a binary equation, "plane" corresponding to a ternary equation, and so on.

6. 4 Transformation Method • So-called transformation method, is that a transformation group are used to depict and reflect the space form. By this method, the nature of the stability of different levels of space forms are studied, thus to study the properties of the graph.

7. Contragradient transformation • The concept of contragradient transformation comes from physical phenomenon of the rigid motion. A rigid object, after motion, only change the position and does not change the shape and size. It's the most basic variable is the distance between any two points, therefore, to contragradient transformation, can make the following mathematical definition.Definition 1 A plane w to its own transformation f, if for any two points A and B on plane, the distance D(A,B)always equal to the distance D(A’,B’)between their corresponding points A’ and B’, called f is the contragradient transformation in the plane .

8. In the national nine-year compulsory education mathematics curriculum standard, the field of "space and figure" includes "understanding of graphics" the "graphics and coordinates" “graphics and transformation” "graphics and proof" four parts. This embodies the thinking researching and exploring the nature of graphics in many ways .

9. Suppose A、A’B，B’C、C’are three pairs of corresponding not collinear points of a contragradient transformation, Then to any point P on the plane, the three pairs of corresponding points can determine the position of the corresponding point P’. • In fact if P, A,B on the same line, suppose A locate between A and B, then D(A,B)=D(A,P)+D(P,B)That is，P is on segment AB, and determined by AB. • If P is not on AB, suppose it is on the same side as C, PA=P’A’,PB=P’B’,P is the intersection of ⊙A’AP and ⊙B’BP. The location is determined. • We can see, contrgradient transformation is completely determined by a corresponding triangle.

10. Here are three kinds of basic contracgradient transformation: translation transform, rotation, reflection transformation. • They are three transformation we studied in middle school mathematics

11. 4.1 translation transform A set of translational transform constitute an Abelian group, it is the subgroup of transformation group.

12. D D′ D D B B’ C′ C C C C′ A’ A A A′ A A B B′ B B C’ C

13. × × B A × C √ D × × E F

14. 4.2 Rotation transformation Rotation transformation with the same center constitute a rotation transformation group, it is also a subgroup of transformation group contract.

16. O

18. 4.3 Reflection Transformation Product of two reflection transformation with different reflection axis is not necessarily a reflection transformation. So the set of all reflection transformation does not constitute a transformation group.

19. Axial symmetryandCentralsymmetry A graph overturn and overlap with another graph A graph rotate 1800 and overlap with another graph

20. 4.4 The relation among the three transformations • As mentioned above, the product of the two translational transformation is still a translation transformation, the product of two rotation with the same center is still the rotation transformation, the product of two reflection transformation with the same reflection axis is of identical transformation. So • what is the produce of reflection transformation with different reflection axis ? • what is the product of two rotation transformation with different center ?

22. Theorem 4 Any contragradient transformation can be expressed as the product of three reflection.

23. Part two: Analysis on High School Geometry Course Content

24. 性质 等角的余角相等 等角的补角相等 和 为1800 余角.补角 定义 性质 相等 点到直线 的距离 角平分线 角的计算 一“放”二“靠” 三“推”四“画” 垂 直 对邻 顶补 角角 叠合法 角的比较 画法 度量法 角的比较与运算 同位角相等 内错角相等 条件 相交线 定义.表示 判定 同旁内角互补 借助角研究平面内两条直线的位置关系 度.分.秒互化 角的度量 进位.计算 平行公理.推论 尺规作角 平行线 关系 同位角相等 直线 图形认识初步 相交线.平行线 直线公理 性质 内错角相等 表示与画法 直线.射线.线段 射线 同旁内角互补 寻找射线方法 表示与画法 多姿多彩的图形 线段 分类 计算与比较 命题 性质 平面图形 结构 立体图形 展开与折叠 三视图 知名称 辨认 展开图 点与直线位置关系 确定有标记 的相对图 图形认识初步相交线平行线

25. 等 边 对 等 角 三 线 合 一 等 角 对 等 边 表 示 方 法 等 边 三 角 形 已知两边求第三边 要 素 定 义 弦图 毕达哥拉斯苏菲尔德 条件 应用 镶嵌 定义 证明 性质 判定 特例 概念 文字.符号图形 内容 外角和 定理 互逆命题 多边形 及其 内角和 内角和 勾股定理 文字.符号图形 等腰三角形 内容 逆定理 定义 全等 证明 直角三角形 内角和 知三边定形状 应用 有关的角 锐角三角函数 三角形 外角的性质 锐角三角函数 解直角三角形 特殊值的运算 有关线段 高.中线.角平分线 定义 定义 计算 正切 应用 正弦 余弦 三边关系 三边关系锐角关系边角关系 坡度 仰.俯角方位角 符号.几何意义. 特殊角的值 三角形

26. 基 本 图 形 对 称 轴 成轴对称的两图形全等 对称轴垂直平分对称点的连线 翻折后与 另一图形重合 作对称轴 作等腰三角形 作一点到两点距离相等 作一点到三点距离相等（外心） 用 坐标 表示 轴对 称 作：关于x轴、 y轴的对称点 对称点 定义 特征 要素 解决几何中的 极值问题 到两点距离相等的点 应用 利用轴对称制作图案 判定 点到两点 的距离相等 关于轴对称 性质 垂直平分线 基本图形 轴对称变换 旋转中心 一条直线 要 素 轴对称图形 旋转方向 对称轴 静 动 旋转角 静 翻折后与 两部分重合 对应点到旋转中心的距离相等 定义 轴对称 旋转前.后的图形全等 特 征 基本图形 对应点与旋转中心所连线段的夹角=旋转角 图形的旋转 方向 要素 距离 旋转1800后与 其自身重合 平移 旋转角=1800 中心对称图形 前.后图形全等 旋转 对称中心是对称点连线的中点 中心对称 特征 对应线段 平行且相等 关于中心对称 两图形全等 应用 动 旋转1800后与 另一图形重合 图案设计 利用平移制作图案 关于原点对称 （x,y）平移后（x±a,y±b） 平移过程 对应点坐标 的变化规律 对称点的坐标符号相反 用平移.轴对称和旋转的组合设计图案 用坐标表示 旋转 右加左减 上加下减 图形的全等变换

27. 两图形相似 对应顶点的连线交于一点对应边平行 对应点的坐标比为k或-k 到角两边距离相等的点 放大或缩小图形 外位似内位似 动 用坐标表示 位似变换 点到角两边 的距离相等 适用于 直角三角形 应用 位似 中心是原点 性质 判定 特征 对应角相等， 对应边成比例， 周长的比=相似比 面积的比=相似比的平方 应用 性质 HL 角平分线 AAS 关系 位似变换 两角对应 相等 适合 判定 所有 三角 形 全 等 性质 拓展、延伸 ASA 类比 两边成比例 且夹角相等 条件 相似三角形 全等三角形 SAS 判定 三边对应 成比例 SSS 性质 平行 相似多边形 A字型X字型 定义 对应边、角、周长 面积、中线、高线、 角平分线相等 表示方法 性质 比例线段 相似图形 完全重合 两个三角形 对应角相等， 对应边成比例， 周长的比=相似比 面积的比=相似比的平方 两个三角形 用符号≌连接 形状相同 全等 三角 形与 相似 三角 形

28. 正方形 对角线垂直 对角线相等 一组邻边相等 一个直角 ① ② ③ ④ 中 任 意 满 足 两 个 条 件 形状：取决于原四边形对角线的 相等或垂直 ① ② 矩形 中点 四边形 对角线垂直 一组邻边相等 一个直角 对角线相等 三角形中位线 性质 ③ 对边平行且相等 边 判定 平行四边形 对角相等邻角互补 角 判定 对角线 对角线 互相平分 四边形 圆 判定 性质 等腰 梯形 判定 直角 辅助线 平移对角线 延长两腰 平移两腰 利用腰中点 割补成--- 全等三角形、 平行四边形 作高线 外心：是三边垂直平 分线的交点. 到三顶点的距离相等锐—形内；直—斜边上；钝—形外 旋转 不变性 轴对称性 等 对 等 定 理 圆 周 角 定 理 垂 径 定 理 内心：是三角平分线的交点. 到三边的距离相等在三角形内 ④ ③ 圆 上 圆 外 圆 内 菱形 相交 切线的 性质.判定 点与圆 基本性质 相切 切线长 定理 直线与圆 有关位置 ① 相离 ④ ② 圆与圆 相 交 性质 外 离 内 切 外 切 内 含 弧等 性质 弦等 等分圆周 正多边形 圆心角等 有关计算： 中心.中心角. 半径.边心距 四 边 形 与 圆 正多边形 弧长.扇形 圆锥的 侧面积、全面积

29. 1、Solid geometry knowledge structure • Straight line and plane • to study the relations of point, line, plane, including (1) properties and determinant----qualitative description (2) the parallel and vertical --- focus on the location of the relationship (3) the Angle and distance - quantitative characterization • straight line and plane : • the lines relationship - line and plane relationship – the planes relation - the concept and calculation of the angle and distance

30. Polyhedron and rotator • the concept of cylinder, cone, and circular truncated cone, the ball, nature, drawing • polyhedron and the rotator approach • (1) the traditional way: definition, nature, lateral area, drawing, the application • (2) using space vector depict position, Angle and distance, using calculus to calculate area and volume • Two approaches to study polyhedron and rotator • straight line and plane ---the two straight line and plane,---polyhedron and rotator--- the integrated use of those content; • Model in real life--- polyhedron and rotator---line and plane

31. 2、Analytic geometry knowledge structure Addition and scalar-multiplication Basic theorem of plane vector Plane vector Point and real number to establish a one-to-one correspondence The coordinates representation of plane vector The coordinates of two vectors’ sum and difference The coordinates of vector product and real numbers Plane vector coordinate operations the score point of line segments Translation and translation formula Dot product coordinate representation of dot product

32. 2、Analytic geometry knowledge structure point-slope form Slope Angle and slope Two points form Linear equation and the equation of a straight line the equations of lines Parameter form The parallel and vertical General form The position relationship between two straight lines distance from point to line Intersection point and the Angle The application of linear equation The representation of a planar region linear programming problem Parameter equation of the circle General equation of the circle standard equation of a circle Curve and equation find curve equation Standard equations and geometric properties of the conic Use of translational simplified equation

33. 3、The geometry knowledge chain Math 2 preliminary of solid geometry、 preliminary of Plane analytic geometry Math 4 plane vector elective1, conic curve and equations elective2 conic curve and equations space vector and cubic geometry elective 3: Sphere geometry, Symmetry and group, Euler formula and closed surface, Trisection of an angle and number field expansion elective 4：Geometry prove；Coordinate system and parameter equation

34. 4、Stages, Progressive Design of Solid Geometry • The first level of progressive - the understanding of the spatial structure of graphics (math 2) • With the help of the abundant physical model or computer software rendering space geometry, to know spatial structure characteristic of simple graphics. • Features: perceptual intuition geometric, belong to use observation, experiment, operation method. • Overall research space graphics surface area and volume (does not require memory formula)

35. With the help of the abundant physical model or computer software, rendering space geometry, get to know spatial structure characteristic of simple graphics. • Features: perceptual intuition geometric, use observation, experiment, operation method. • Overall research space graphics’ surface area and volume (does not require memory formula)

36. To know Geometry on the structural characteristics

37. To know Geometry on the structural characteristics

38. To know Geometry on the structural characteristics

39. The second level of progressive - points, lines, the position relationship (required) • on the carrier of cuboid (students Most familiar geometry ), get the direct-viewing understanding and the understanding to experience space location relation between dot, line, face, abstracts the definition of the positional relationship of line, plane • learn some axioms and theorems as reasoning based • by intuitive perception and operation, summed up the nature of the vertical and parallel decision theorem, prove the nature of line and plane parallel or perpendicular theorem • can prove the positional relationship of lines and planes • Characteristics: the requirement of reasoning and calculation is reduced; using diverse ways ,such as the intuitive, experiment, operation, analysis

40. The characteristics of the required content • From the overall to local, from the concrete to the abstract, the content order changed • Through intuitive perception - operation confirmation - speculative argument - metric calculation methods, such as understanding and exploring space geometry - the study way change • The requirement of Logic reasoning and computing is reduced; further reasoning and proof requirements are on the elective courses. • Emphasis on the connection and application of graphics and real life; • Add a content of drawing the pictorial diagram and view; embody the mathematical thinking of "dimension reduction“.

41. The third level of progressive ----space vector (elective) • Elective series 2, for students in science and technology, economic • content: space vector and its calculation • application : use vector express the location of the vertical and parallel relations; some positional theorem of lines and planes; can answer questions about the Angle. • Requirements: experience the thinking method of vector, learn to use vector to represent the dot, line, place and its position. neatly choose vector method or comprehensive method, solve three-dimensional geometry problems by different ways.

42. The fourth level of progressive ----reasoning and proof (elective) • A series of four thematic content: • reasoning and deductive reasoning • analysis and synthesis • direct and in direct prove • mathematical induction

43. 5. The content of the plane analytic geometry Compulsory 2 preliminary analytic geometry Elective 1-1,2-1 Conic curve and equations Elective 4 coordinate system and the parameter equation\

44. 6、Positioning of plane analytic geometry Highlight the process with algebraic method to solve the problem of geometry; Emphasize the geometric meaning of algebraic relationship; fully embodies the idea of combining algebra and geometry.

45. The first level of analytic geometry • The compulsory content: • (1) line and linear equation; • (2) circle and equation; • (3) the space rectangular coordinate system; • Learning requirements: • exploration geometric elements to determine the linear and circular---expressed geometry elements in coordinate, establish various forms of equation; • Learning and understanding with the analytic geometry method to solve the problem of "the trilogy" • understand the space rectangular coordinate system, to depict the location of a point; Explore and derive formula of distance between two spatial points

46. The second level of analytic geometry - conic curve and equation • emphasis on the cause and effect of conic, more emphasis on its geometric background. • For students seeking development in terms of humanity, the requirement is to have a fully comprehensive grasp of the ellipse. To the rest of the conic, they only need to get a general understanding. • For students seeking development in science and engineering, they should have a comprehensive grasp with elliptic, parabolic. • With the instance of curve and the equation has learned, understand the curve and equation corresponding relation, further realize the number form combining ideas.

47. The third level of analytic geometry - coordinate system and the parameter equation • Understand the difference between the polar coordinate system and rectangular coordinate system and mutual change • simple graphics (line, circle) in polar coordinates equation • cylindrical coordinates and spherical coordinates space application in the real life • analysis geometrical properties of lines, circles, conic; write the parameter equation; • With the aid of teaching AIDS and computer software, to observe some special curve (flat cycloidal and involute) generation process, write the parameter equation. • Know something by the curve of the parameter equations of the role of in real life.

48. Part three: Case Study Recreation under instruction—Can the Pythagorean theorem be explored by students?

49. 1. background a2+b2=c2 Pythagorean theorem is a barometer of mathematics educational reform: in the '50 s and' 60 s last century, it required strict proof in mathematics curriculum ; and then promote the “measure and calculate ",later “do the math ", after "tell the result", and "learning by doing" until now the inquiry learning, etc. Mathematics teaching should cultivate students' three big capacity : mathematical calculation, mathematical reasoning and decision-making . The Pythagorean theorem teaching is a case in point. 张波—个案学习模块二理论

50. 2. original teaching behavior Euclid approach(equivalence area transformation deduction require very high difficult skills Special context is some kind of direct suggestion. It is nothing else then telling the fact Provide pythagorean triple: 32+42=52 62+82=102 “measure and calculate” Can not get a2+b2=c2 Set operation context Come down to floor tile: “cut and put together” Students can not cut