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Locus and Trace in Cabrigéomètre: relationships between geometric and functional aspects in a study of transformations Ana Paula Jahn, São Paulo (Brasil) Abstract: The present text describes and characterises

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Locus and Trace in Cabrigéomètre: relationships between geometric and functional aspects in a study of transformations Ana Paula Jahn, São Paulo (Brasil) Abstract: The present text describes and characterises the tools Locus and Trace of Cabri-géomètre II, in relations to a study of geometric transformation, more precisely, the passage from the notion of transformation of figures to the notion of applications 1 that map points on the plane onto the plane itself. In particular it discusses how the conception of image of a figure under a transformation can evolve through interaction in a milieu organised around Cabri-géomètre such that students move from views of figure-images as undecomposible entities to see them as sets of image-points. Moreover, the study allowed the identification that the notion of trajectory (in a dynamic interpretation) has an important role in this conceptually difficult passage and that dynamic geometry environments renovate this notion. Kurzreferat: Der Text beschreibt die Werkzeuge Ortskurve und Spur der Software Cabri-géomètre-II und deren Rolle beim Studium geometrischer Abbildungen. Genauer wird der Übergang von den Abbildungen einer Figur zu den Abbildungen aller Punkte der Ebene untersucht. Im Einzelnen studieren wir, wie sich der Begriff des Bildes einer Figur unter einer Abbildung entwickelt, wenn sich diese Entwicklung in einer Umgebung ( milieu ) vollzieht, welche durch die Nutzung von Cabri-géomètre gekennzeichnet ist. Die Lernenden gehen dabei von einer Sichtweise der Bildfigur als unzerlegbare Einheit über zu einer Sichtweise als Menge von Bildpunkten. Außerdem erlaubt die Untersuchung die Feststellung, daß der Begriff der Spur einer Bewegung (in einer dynamischen Deutung) eine wichtige Rolle in diesem begrifflich schwierigen Übergang spielt und daß Dynamische Geometrie-Software (DGS) dieser Vorstellung neues Leben einhaucht. ZDM-Classifikation: C30, C70, G50, N80, U70 0 Introduction and context The notion of geometric transformation occupies an important place in mathematics teaching in various countries and, in particular, in France, which is the context for this research. A number of Mathematics Education research studies have been devoted to the understanding of transformations (see, for example, Grenier, 1990; Healy, 2002; Küchemann, 1981). According to Grenier&Laborde (1988), transformation can be understood at different levels. Among others, 1 In this paper we do not distinguish between the terms transformation and application. It is interesting to observe however that this is not always the case in secondary school teaching, where transformation is reserved to designate bijections of R 2 (or R 3 ). these include: Level 1 relationships between two figures or two parts of the same figure. The concept at this level seems to be connected to the context figures and hence involves the transformation of figures. Level 2 applications that map points in the plane onto the plane itself. In the French lower secondary school, geometric transformations are introduced at level 1 (Collège, years), but from Lycée (15-18 years) they should be studied at the second level. This is a change from a transformation that operates globally on a figure (a movement) to an application that operates on points and over figures as parts of the plane composed by points. We analysed the students possibilities to establish relations between these two aspects which we name global (in synthetic geometry) and point-wise (in a functional interpretation). It is in this context that we highlight the intervention of the concept of locus and the use of an dynamic geometry environment. 1 Sets of points, trajectories and loci in the functional approach The notion of Locus is introduced naturally in the study of transformations, once they are defined while application that map points on the plane are not as easily understood. Indeed, studying the French textbooks, we perceive that this notion appears, generally, in the transformations context (case 1), showing that these transformations are very efficient tools to solve loci problems. On the other hand, the term locus may also be used in the application context (case 2) whose meaning is not the same. Let us highlight the differences. In a last case (2), locus is a set of points having a certain property: a characteristic condition determines whether a point belongs to the set. This is the case, for instance, of the classic point-wise characterisation of objects, such as perpendicular bisector, angle bisector, conic, and more particularly circle. In a first case (1), we refer to a set of points that are images of a set of points defined as the image of an object under an application or transformation. If we call f the function M N = f(m), searching the locus of N is searching the set of all points f(m), that is searching the image of line L under f (translation RS 2 ); see Antibi&Barra, 1996: Transmath 1 ère S, p So, locus is defined in a functional form, theoretically we have to consider that a variable point P which belongs to a figure F considered as a set of points (a straight line, a circle,...) corresponds to a point P, image of P under an application f. Locus has a double meaning: it legitimates the change from a synthetic figure (a global point of view) to a figure as a set of points, and it allows to recompose the figure. When analysing textbooks we also notice that the notion of locus is often presented - on the high school 2 Si on appelle f la fonction: M N = f(m), chercher le lieu géométrique de N c est chercher l ensemble de tous les points f(m), c est donc chercher l image par f de la ligne (L). 78 level - in a dynamic way: Supposed a point M moves on a fixed line (L), another point N which is linked to M, will move too. The problem consists of finding out on which fixed line (L1) point N will move. This line (L1) is called 'geometrical locus' of N if point M moves on line (L) (translation RS 3 ; see Antibi&Barra, 1996: Transmath 1 ère S, p. 376). The locus (or the transformed figure) is then identified with the trajectory of a point N constructed as the image of a mobile point M in the figure (L). This dynamic point of view avoids the use of quantifiers and the language of set theory. Even though this didactic choice doesn t correspond to the notion of movement presented in Physics, inspired by a historic study, we adopted the dynamic interpretation mobile point on a curve as an intermediate phase with the aim that the students may grasp a point oriented conception of a transformation. The didactic necessities that determine the distinction between trajectory and locus are equally present in the creation of the tools Trace and Locus of Cabri II, which are described in the following section. 2 Locus and Trace in Cabri-géomètre The distinction between trajectory and locus as described above is reflected in the form of two distinct tools present in version II of Cabri-géomètre, Locus and Trace. A locus as produced by the Locus tool behaves in many ways like other Cabri objects: for example, it remains visible on screen, moving accordingly as the elements upon which it depends are manipulated (no longer disappearing as in the preceding version). A point can be constructed on it and moved in the same way as a point on other Cabri objects and it can be defined as a final object in a macro-construction. In addition, it is possible to obtain a locus of objects such as lines, rays, segments and circles and hence generate their envelopes (see Figure 1). However, a Cabri-locus differs from other Cabri objects in that it cannot be used in the construction of intersection points. reported in this article, the Locus tool of Cabri II produces a set of points, L, such that each element is defined in function of an element from the set E: L = {f(p), P E}, and will appear on the screen as a sketch (a representation) of the set L for a finite number of f(p) 4. To define such a locus it is necessary to select a point P' for which the locus is desired and then the point P on which P' depends (where a functional relationship exists between P and P'). The point P is a variable point that belongs to particular set of points of the plane (a line, a circle, a line segment...) and the point P' is related to P by a geometric construction. The points P' of the locus are calculated by the software and obtained directly and it is not necessary to drag the point P. The locus is immediately represented in its entirety which was not the case in Cabri I where the Locus tool is related to dragging (that is, has a dynamic aspect). In fact, in the first version of the software, using a point on object M (with one degree of freedom) and a point M' (with zero degrees of freedom) that depends on M, the locus of the point M' is produced as the trace of its successive positions when M is dragged. A (manual) locus can also be produced by selecting any point on the screen including free points (points with two degrees of freedom) and observing the trajectory generated when this point is moved. This second use of the Locus tool of Cabri I corresponds to the Trace tool in Cabri II. Trace allows the user to instruct certain objects on screen to leave a trace when they are moved, either manually using the mouse or through the use of the Animation tool. The trace does not exist as an object of Cabri, only as a set of pixels highlighted on the screen. Roughly speaking then, in Cabri II, Trace emphasises a dynamic interpretation of the representation of a trajectory of a point, while Locus is characterised in a functional manner by a one-to-one correspondence between two points P and P', representing, at least implicitly, the image of a set of points for a certain application. Under the conditions described above, not all loci can be obtained through the use of the Locus tool of Cabri II. The restrictions relate to the type of transformations of geometrical configurations that are possible using the drag-mode. For example, in Figure 2, since point M is a free point, it is only possible to sketch the required locus (a circle) using the Trace tool, whereas Figure 3 below Figure 1 Ellipse as envelope of a line In the case of the loci of points, the subject of the work 3 on suppose qu'un point M se déplace sur une ligne fixe (L). Alors, un autre point N, associé à M, bouge aussi, et le problème consiste à chercher quelle est la ligne fixe (L1) décrite par le point N. Cette ligne (L1) est appelée lieu geométrique du point N lorsque le point M décrit (L). Figure 2 Visualisation of the locus of M using Trace 4 n: number of points of the locus, with 5 n Figure 3 Perpendicular bisector of AB using Locus presents a case in which, although the Locus tool can be used, an auxiliary construction is needed in order that the desired locus (the perpendicular bisector of AB) may be generated. Schumann and Green (1997), mentioning Cabri I, they show some uses of the interactive generation of loci. In our work, we are interested in a specific kind of problem described by them as in investigations of the position and shape of the image of a transformed original shape (ibid., p. 80). From the characterisation of the tools Locus and Trace of Cabri II, we formulate the hypothesis that they offer new possibilities of interpretation of functional dependence, being able to follow the didactical introduction of the concept of function in Geometry. With this in mind, a study was designed to investigate how the Cabri-géomètre environment, and especially the distinctive and comparative use of the Locus and Trace tools, accommodates (or perhaps favours) an approach to the notion of geometric transformations that brings into evidence both their functional character and the importance of the preservation of properties. The remainder of the paper presents some of the situations from the didactic sequence designed during the study. The overall aim of the sequence was to problematise the construction of an image under a transformation and to relate this to the notion of locus. During six of the research sessions, the class was divided into two groups, while the seventh session was conducted with the whole class. During their interactions with the proposed activities, the students worked in pairs in the computer laboratory. Five pairs were selected for case-study. The analyses 6 presented in this paper mainly refer to two of the four situations of the sequence entitled Affinity and Oblique symmetry respectively and focus upon aspects related to the understanding and use of the Trace and Locus tools of the case-study pairs (for a complete description of the sequence, see Jahn 1998). 3.1 Generating a conic This situation was designed with the aim of characterising a locus as the image-set of another set by an application applied to points in a geometric setting. The problem proposed to the students was to construct, point by point, an ellipse as the image of a circle (C) whose centre lay on a line (d) under an orthogonal affinity with the line d as axis and a ratio of 1/2. This affinity was introduced as a simple geometric construction (see, figure 4a), whose steps, along with the associated Cabri-tools, were presented on a worksheet given to the students (a guided construction). Starting from this construction, the idea was that students would examine the correspondence between a point M on the circle C and its image-point M' (without speaking of transformations) and, as a consequence, would consider the set of points M' as point M describes the circle; that is, point M was to be treated as a variable point. Figure 4b reproduces part of the worksheet given to students. 3 An experimental study in the context of transformations The experimental sequence was composed of four situations to be worked upon during seven one hour sessions by a class of 33 students (aged years) from a public school in the south-east of France. It is important to observe that the students who participated already had had around six months experience with Cabri in their mathematics lessons as part of a project 5 aimed at the design of Cabri-integrated learning scenarios (of with the transformation study formed a part). This meant that the students were familiar with most of the Cabri tools and that the principle of robusticity of constructions in Cabri had been previously negotiated with, and accepted by, the majority of students. 5 Conception et évaluation de scénarios d'enseignement avec Cabri-géomètre, a project of the team EIAH of the Leibniz-IMAG laboratory, IUFM of Grenoble, with a grant from the Région Rhône-Alpes and the INRP ( ). Figure 4a Affinity of circle (C) Drag M and observe the movement of M'! Description of observations: You have just built a construction that allows the association of each point M on the circle C with a point M'. # Determine the set of points M' as M varies on the circle.! Explain how you obtained this set. Figure 4b Students worksheet Students at this level of schooling were not expected to recognise the properties of an ellipse a topic they had 6 The analysis was based on data collected from each section. It contained experimental protocols (transcription from audio), observers notes, written work sheets from pairs of students and Cabri files (figures and macros). 80 not yet studied instead the activity enabled a response in the form of a representation of the image-curve on the computer screen, produced by means of the Cabri tools. 7 In practice, as the set they had asked to identify did not represent a figure known to the students (such as a line or a circle, for example), they needed to build it point by point; that is, producing a visual representation of the complete set necessitated the reproduction of the initial construction on various positions in the circle C or, the use of the tools Trace and Locus. It was effectively the last strategy that the students employed. Although they at first paid more attention to the geometric properties of M' than to the nature of the image set described by this point, two pairs (B1 and B4) concluded that M' described a curve. It was by dragging point M that they modified their original conjecture that the image of the circle C would also be a circle, with both pairs going on to suggest the image consisted of two arcs before the students in B1 settled on the term oval while those in B4 identified the set as an ellipse. Because they wanted to visualise the trajectory of M', already experienced dynamically through the dragging of M, these pairs privileged the use of the Trace tool and did not use Locus as their first option. S2 B1 Lud: Determine the set of points M' as M varies [reading from the worksheet] Lau: Isn t it this... it s a curve, isn t it? Lud: We have to have a Trace . Does it still have Trace ? [to the observer] Obs: Yes, I think so... It looks like you have the complete. Lud: Let s see what happens. [activates Trace of the point M' and drags M] These students attributed to the Trace tool a function of representing the curve of the trajectory of a point, allowing them to better visualise or understand the object in question. On top of this, the output of the Trace tools seemed to completely satisfy the students, to the extent that the Locus tool had only a contractual role (generating the robust construction emphasised by the teacher) or was used in the second part of the task as a means of constructing an object on which various points M' could be placed more precisely the five points that were necessary in order to define a conic using the Conic tool of Cabri. Three pairs did make use of the Locus tool although only after first obtaining an image of the locus using Trace. In fact, for the students the functions of the two tools are very similar (almost equivalent), except that they saw that the output of the Locus tool could be recognised by the software at least in terms of its points. S2 B2 Hor: Where is Refresh drawing ? Lil: In Edit Hor: Really, Trace and Locus are kind of the same, aren t they? Lil: I don t know! [...] Hor: OK, I put that we used Locus and... Lil: Of M'! And Trace as well. Hor: Yes, but there we have the locus, the actual curve. As previously described, the use of the Locus tool assumes some understanding of a functional relationship between two points and its application in practice reflects this relationship. Some hesitation on the part of all five of the case-study pairs was observed during considerations of the arguments of this tool. The excerpt below, for example, illustrates how the students in B5 were confused about the respective roles of the two points they were selecting: S2 B5 [Bea had selected M then M', as they tried to apply Locus ] Aman: With Locus , it didn t redo it! Bea: How do we use it? Wait, can you help me do it? Aman: Get Locus . Bea: Of M or of M'? Aman: Of M and afterwards click M'. Or the other way round... I don t know anything! Bea: This and this [choosing M' then M]... Magnificent! Moreover, the situation Affinity allowed the students to experience the differences in the ways of using and in the products of the Cabri-tools Trace and Locus of Cabri. At the end of this activity, related to Cabridragging, a first difference was established: this led the pairs to attribute to Trace the function of providing a provisional sketch and to Locus the function of determining the geometric object introduced by the initial construction the curve described by M' that could be used in attempts to validate their results. 3.2 Transformations which transform: the case of Oblique symmetry The situation Oblique symmetry 8 was directly located in the context of geometric transformations. It consisted of an investigation of the images of various objects points, lines, polygons and circles by a symmetry in a given axis and parallel to a given direction. Using the

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