[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br] [br][b]Abstract[/b][br][br]Since its inception, GeoGebra has been specifically designed to display the dual representation, both graphical and algebraic, of mathematical objects. In this presentation, as the central focus, I will show some procedures that exploit the didactic possibilities of this duality. [br][br]These procedures, presented to students aged 15 or 16, are so simple, engaging and quick to create that they allow the students themselves to generate and use them from scratch... with great success! [br] [br]Despite their simplicity, we will see that they are so powerful that they enable us to delve into mathematical depths that are practically unapproachable in the high school classroom without the assistance of GeoGebra, ranging from algebraic structures (such as fields) to non-Euclidean metrics (like the [i]taxicab [/i]metric).[br][list][*][color=#808080][size=85]Note: All the GeoGebra constructions linked on this GeoGebra Book have been created by the presenter of this content. None of them, except for the [url=https://www.geogebra.org/m/sw2cat9w#material/nhnxt7am]Bubbles[/url] construction, make use of [i]JavaScript [/i]programming.[/size][/color][/*][/list]

[b]The Author[/b][br][br]In my 40 years of teaching as a High School Teacher, in the pursuit of fostering the interest of students, I have researched the relationship between Mathematics and other areas as diverse as Games [url=https://www.geogebra.org/m/u8gFwdZP][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], Perception [url=https://www.geogebra.org/m/DfxmG6Vz][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], and Music [url=https://www.geogebra.org/m/qg2gkkat][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]. The advent of Dynamic Geometry brought new and significant opportunities to engage students and promote the creation of their own constructions.[br][br]My relationship with GeoGebra dates back to 2005 when I first encountered this program created by [b]Markus Hohenwarter[/b] [[url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]7[/url]], although I had already worked with other dynamic geometry software. Two years later, in 2007, Professor [b]Tomás Recio[/b] [url=https://personales.unican.es/reciot/][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] invited me to the [i]International Center for Mathematical Meetings[/i] (CIEM [url=https://www.ciem.unican.es/][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], Cantabria) which brought together several Spanish high school teachers who were pioneers in the educational use of dynamic geometry. During that meeting, I defended the efficiency of GeoGebra GeoGebra [[url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]10[/url]] compared to other software like Cabri. One outcome of that gathering was the formation of the [b]G⁴D[/b][i] [/i][url=https://www.geogebra.org/u/g4d][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] group, consisting of J.M. Arranz [url=https://www.geogebra.org/u/arranz][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], J.A. Mora [url=https://www.geogebra.org/u/jamora][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], M. Sada [url=https://www.geogebra.org/u/manuel+sada][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] and the author of this text. [br][br]Two years later, from the Ministry of Education of Spain, Antonio Pérez [url=https://aperez4.blogspot.com/][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], who was then the director of the [i]Institute of Educational Technologies[/i] (ITE, now INTEF [url=https://intef.es/][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]), entrusted me with the task of conducting courses for training Primary and Secondary Education teachers in GeoGebra [[url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]11[/url], [url=#Ref.12]1[/url][url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]3[/url]]. Additionally, I was tasked with creating a set of complete activities (including topic introductions, constructions to explore, and questionnaires) for students, categorized by subjects and levels, which we named the [i]Gauss Project[/i] [[url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]14[/url], [url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]1[/url]]. Simultaneously, Tomás launched the first Spanish-language GeoGebra Institute, the [i]GeoGebra Institute of Cantabria[/i] [url=https://geogebra.es/][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], of which I have been a Trainer since its inception.[br][br] [br][b]Introduction[/b][br][br]The main objective of this conference is to demonstrate the close relationship between algebraic and geometric procedures using GeoGebra. A significant portion of the time will be dedicated to presenting activities that can be approached by secondary education students through constructions carried out by themselves. Beyond sporadic use for exploring specific content, this type of construction gains its full didactic power in mathematics education based on competence acquisition.[br][br]In the first part of this presentation I will detail very simple procedures that harness the strong interconnection between [b]geometry and algebra[/b] that gives its name to GeoGebra (hence the title of[b] [i]Principia[/i][/b]) and enable secondary school students (around 15 or 16 years old) engage in mathematical explorations that are "in principle" beyond their reach, leaving the heavy algebraic and geometric calculations to the powerful tools of GeoGebra, much like we currently rely on calculators and spreadsheets for tedious arithmetic calculations.[br][br]In particular, the ease with which we can create parallel lines and circles will aid us in constructing a dynamic [b][i]offset[/i] [/b]whose colorful trace allows us to visualize a wide variety of geometric loci. Simultaneously, the Computer Algebra System [b]CAS[/b], applied to Euclidean distances, will facilitate the creation of implicit curves that conform to these loci. We will also see how to occasionally convert these implicit curves into algebraic equations and inequalities. [br] [br]We will expand the use of CAS to angles and also address other non-Euclidean distances, such as the [i]taxicab [/i]distance.[br][br]We will conclude this first part with a reciprocal example, wherein resorting to geometry will aid us in visualizing and manipulating the concepts and properties inherent to the algebraic structure of a [i]field[/i].[br][br]In the second part, I will showcase some ideas for creating slightly more sophisticated yet equally captivating constructions, which can serve as models to be analyzed or modified by students.[br][br]Firstly, we will explore how GeoGebra's [b]lists [/b]facilitate the incorporation of a substantial amount of information. As an example, we will represent the coastlines of continents in a single list, creating a template of the Earth.[br][br]Next, we will employ [b]vectors[/b] to instantaneously modify (thanks to the GeoGebra [b]scripts[/b]) the position of points according to our interests. These vectors can be used to combine repulsive forces (like particles of the same charge), attractive forces (similar to those used by Newton to formulate his universal law of gravitation), or simply reactive forces (as in elastic collisions). [br][br]Furthermore, we can use vectors to create an "elastic geometry". In this context, points do not possess a fixed location, but rather their position at each moment is the result of the application of the aforementioned forces. For instance, in elastic geometry, a point is not "on" a circle, but is inevitably drawn towards it, it has "its limit" in it. [br][br]Lastly, as an application of this type of elastic geometry, we will examine examples of tensegrities constructions.

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]