Geometric Transformations in Geometry Teaching: different appropriations in modern mathematics

Leme da Silva, Maria Célia; Jahn, Ana Paula

doi:10.1590/1980-4415v38a230289

person Maria Célia Leme da Silva

school Doutora em Educação (Currículo) na Pontifícia Universidade Católica de São Paulo (PUC/SP). Professora Associada da Universidade Federal de São Paulo (UNIFESP), Diadema e do Programa de Pós-graduação da Universidade Estadual Paulista (UNESP), Bauru e Rio Claro, SP, Brasil. E-mail: celia.leme@unifesp.br.

0000-0001-6029-0490

person Ana Paula Jahn

school Doutora em Didática da Matemática na Universidade Joseph Fourier (UJF-Grenoble, França). Professora Doutora do Instituto de Matemática e Estatística da Universidade de São Paulo (IME-USP), São Paulo, SP, Brasil e Pós-doutoranda do Programa de Pós-graduação em Educação Matemática da UNESP de Rio Claro, SP. E-mail: anajahn@ime.usp.br.

0000-0003-0515-7536

Title	Author	Publisher	Year
Matemática – Curso Moderno para os ginásios – 3º vol.	Osvaldo Sangiorgi	Cia. Editora Nacional São Paulo	1967
Matemática – Curso Moderno – 3 Ciclo Ginasial	Alcides Bóscolo, Benedito Castrucci	Editora FTD São Paulo	1970
Ensino Atualizado da Matemática – curso ginasial – vol. 3	Omar Catunda, Martha M. de Souza Dantas, Eliana Costa Nogueira, Norma Coelho de Araújo, Eunice da C. Guimarães e Neide C. de Pinho e Souza	Edart São Paulo	1971
Curso Moderno de Matemática para ensino de primeiro grau – 7	Anna Averbuch, Franca Cohen Gottlieb, Lucília Bechara Sanchez, Manhúcia Liberman e Jacy Monteiro (supervisão)	Cia. Editora Nacional São Paulo	1975

Geometric plane transformations - OS	Transformations of the plane – BC
Translation Group: oriented segment, algebraic measure, equipollent segment, translation in the plane, translation of plane figures, addition of translation, verification that translations in the plane, in relation to the addition operation, have a commutative group structure.	*Plane transformations*: identity and isometry.
*Rotation Group:* rotation at a point, in the figures, co-terminal arcs, addition of rotations and check the four properties to justify the structure of the commutative group.	*Translation:* defines translation at a point, highlights the properties of the transformation – isometry, that a straight line not parallel to the amplitude corresponds to a straight line parallel to it, a straight line parallel to the amplitude corresponds to the straight line itself, an angle corresponds to an angle congruent to it –, equal and inverse translations, the composition of translations and verifies the properties of the transformation with the composition.
*Axial Symmetry:* defines a symmetrical point, axis of symmetry, axial symmetry, presents the symmetry of a quadrilateral and a triangle.	*Central symmetry*: defines a point, highlights the properties of the transformation, and presents the composition of two central symmetries of center O and O’ obtaining the translation of amplitude OO’ and parallel to OO’.
*Central Symmetry:* defines a symmetrical point, central symmetry, and presents as an example the central symmetry of a triangle.	*Rotation:* Sets rotation at a point, highlights the transformation properties and checks the transformation properties with the composition.
Practical application of axial symmetry – shortest path problem.	*Axial Symmetry:* defines a point, highlights the properties of the transformation, and presents two possibilities for composing axial symmetries: one with parallel axes that result in a translation and another with perpendicular axes that result in central symmetry of the center at the intersection of the axes.
*Attention test:* set of ten exercises: 4 translations, 1 rotation, 1 translation followed by rotation, 1 axial symmetry, 1 central symmetry, and 2 applications of TG to shortest path problems.	*Isometria Direta e Inversa*: relaciona com a ideia de deslizamento e sobreposição.
	*Transformation Group:* revisits the examples of addition and multiplication in the numerical sets studied previously (Z, Q, and R) as commutative groups, and announces that some transformations have the same properties with the composition operation, exemplifying the case of the commutative group of translations in relation to composition and says that it is possible to verify the same for rotations.

Chapter II Straight Line	*Translation* on the real line; *Symmetry* on the real line; Set of translations and symmetries; Affine transformation or affinity on the line; Homothety on the line.
		Chapter III Affine Plane	*Translations in the plane:* definition, composition of translations (addition of vectors), translations form a commutative group; Dilations: definition (multiplication of vector by scalar), composition of dilations; Properties of translations and dilations to define vector space.
*Transformations in the affine plane:* definition of transformation in the plane (pointwise correspondence); definitions of translation; central symmetry; homothety; Theorems on the product of two symmetries in the plane and on the product of a symmetry by a translation; Elementary affine group: set of translations and pointwise symmetries. Congruent figures: which can be transformed into each other by a transformation of the elementary affine group; Homothety in the Triangle.
Chapter IV Euclidian Geometry	Axial symmetry (symmetry in relation to the straight line); Properties of axial symmetry (fixed points; half-plane in opposite half-plane; composition of symmetry with itself is Id; symmetrical figure of a straight line is another straight line); Definitions: two figures, symmetrical to each other, in relation to an axis r, are said to be congruent (congruence by symmetry); Symmetrical figure and axis of symmetry (a figure that transforms into itself by an axial symmetry); Perpendicular line from axial symmetry; Symmetrical property of orthogonality; Composition of symmetries (with coordinate system); Theorems: composed of 2 axial symmetries in relation to parallel axes and in relation to orthogonal axes; group of congruences (isometric or Euclidean): set of transformations that decompose into axial symmetries.
	*Rotation:* defined by the composition of two symmetries whose axes are neither parallel nor orthogonal; Transport of figures (by transporting segments and angles); cases of congruence of triangles (by transporting segments and angles).
	*Similar figures: definition: a figure F is similar to another figure F’ when F is congruent to a figure F’’ which is homothetic to F; similarity is obtained by the composition of a homothety with a congruence,* properties of similarity (homothety theorems). Cases of similarity of triangles and of any figures.

Author(s)	GT Approach (definition)	Articulation between GT and EG	Contents
OS	Translation and Rotation, sets equipped with the addition operation to justify the commutative group structure.	There was not an articulation; GT’s isolated in the Appendix of the book	Translations Rotations Axial Symmetry Central Symmetry
BC	Transformations in the plane as functions and study of their properties. Transformation Group, exemplifying with the commutative group of translations in relation to composition	There was not an articulation; GT’s isolated in the Appendix of the book	Transformations Translations Central Symmetry, Rotation Axial Symmetry Direct and Inverse Isometry
CD	Transformations in the affine plane as functions and study of their properties. Elementary affine group, congruence group (isometric or Euclidean)	Integrates GT with EG in chapters III and IV	Translations, Dilations Transformation in the affine plane Elementary affine group Homothety Axial symmetry Congruence group Rotation
GRUEMA	Axial Symmetry as a Function and Study of its Properties	Integrates GT with EG in the topics Symmetry and Congruence	Axial symmetry

[1] Source: elaborated by the authors based on: Sangiorgi (1967), Bóscolo and Castrucci (1970),

[TFN1001] Catunda et al. (1971) e Averbuch et al. (1975)

Geometric Transformations in Geometry Teaching: different appropriations in modern mathematics

Abstract