Attractive projects
This book gathers the ideas I presented on 12-16-2018 (my birthday) under the title of [i]Attractive projects[/i], in the seminar [i]Classroom experiences with GeoGebra[/i], held at CIEM (International Centre for Mathematical Meetings, Castro Urdiales, Spain). [b]Summary[/b] One of the challenges that a maths teacher have to face when he wants his students to learn how to make GeoGebra constructions, is to decide what kind of projects he can propose, as well as what are the keys to their development. What kind of "geogebraic" projects can be attractive to students? What prior information should we provide them so they can address them? I will present some ideas that can be classified into three groups: Design projects, Model projects and Robot projects. In all of them, the notion of "movement" will be esential, both from the mathematical point of view (vectors and parametric equations) and the use of GeoGebra (sliders, paths, automatic animations and GeoGebra scripts). In addition, I will be offer undocumented technical information (in Spanish) on parametrizations of the loci used by GeoGebra.
Table of Contents
- Information
- Information to students
- Automatic animations
- Same absolute speed
- Relative reference system
- Design Projects
- Static designs
- 3D dynamic designs
- 2D dynamic designs
- Vector graphics
- Model Projects
- Modeling the everyday
- Barber's pole
- Modeling physical reality
- Modeling nature
- Deconstructing a model
- Robot Projects
- Dynamic balances
- Robot between enemies
- Elliptical orbits
- Robot reciprocal of Pithagorean
- Robot Fermat point
- Geometric median of 4 points
- Geometric median of tetrahedron
- Weighted geometric median (Weber problem)
- Robot Steiner points
- Labyrinth
Information to students
[color=#999999][color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/dm9prd7h]Attractive projects.[/url][/color][/color][br][br][b]Contents [/b][br][br]Naturally, the information provided to the students will vary according to the project to be carried out. However, the following contents of GeoGebra are of special interest:[br][list][*]Points, vectors and parametric curves.[/*][*]Sliders and animations.[/*][*]Lists and sequences.[/*][*]Basic scripts.[/*][/list]Note: The students will receive technical assistance whenever they need it, but they will have to explain to the rest of the class and the teacher how they prepared the project, in all its phases. [br][br][b]Guideline procedures [/b][br][br]To expose some procedures used in GeoGebra, it is very useful to have a template already prepared (like the one shown here), in which the only thing to vary in each case is the list of texts with the guidelines. The objects used by the template itself are auxiliary, to hide them from the Algebraic View. [br][br]Note: To perform a line break in a text in the list, use \\n.
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
Static designs
[color=#999999][color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/dm9prd7h]Attractive projects.[/url][/color][/color][br][br][b]Project 2D[/b]: [i]recreate a collection of designs, such as flags, isotypes, logos, signs, emoticons, ideograms, pictograms, rosettes, etc.[/i]
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
Modeling the everyday
[color=#999999][color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/dm9prd7h]Attractive projects.[/url][/color][/color][b][br][br]2D project[/b]: [i]model everyday situations that involves a change (a before and after).[/i][br][br]Note: this type of project is difficult, since the students must search a specific objective and make all of the construction from scratch.[br][br]In this example, a young couple sits down to dinner at a restaurant. The square table where the subject sits, has a tablecloth with rectangular shape on the table and the couple will sit on opposite sides of the table. "Too far away!", say both. So, they change to contiguous sides of the table. When the waiter arrives, he repositions the tablecloth to adapt it to the new situation.[br][br]The project consists on modeling this situation with GeoGebra, as observed in the construction. To do this, just calculate the new position of one of the corners of the mantle (since the angle will be 45°). This requires various schemes and the systematic use of the Pythagorean theorem.
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
Dynamic balances
[color=#999999][color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/dm9prd7h]Attractive projects.[/url][/color][/color][br][br]There is a big difference between condition and calculation. For example, the condition for a real number to be a root of a function is that the numerical value of the function, for that number, must be zero. To calculate that root is another thing.[br][br]Usually, calculations require procedures whose learning is long and tedious. But the fact that we do not know how to perform those calculations is not an impediment to appropriate the mathematical idea that they approach. These ideas may seem much more attractive if we sacrifice some calculation ... or simply postpone it to a more advanced level.[br][br][b]2D project[/b]: [i]create dynamic systems that stabilize by themselves.[/i][br][br]Let's put two points inside a circle. Imagine that both the points and the edge of the circle are electrically charged, with the same charge. The two points repel each other, and are repelled by the circumference, with inversely proportional intensity to the square of the distance. Immediately , points will look for the balance, which will be reached when the two points are arranged symmetrically with respect to the center of the circle and a distance between them equal to one third of the diameter.
If we add more points, the equilibrium will be regular polygons, as you would expect. Here we see how 10 points are balanced forming the regular decagon (it could also be formed an enneagon with its center).
The result is not always as intuitive. In this case, apart from the five points in a square expected, four other possible distributions appear, symmetrical with each other.
You can even observe situations in which you can differentiate in the initial conditions the order of chaos. In this example, the points repel each other as before, but now they are also attracted, with double intensity, to the origin of the coordinates.
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]