Jul 6, 2017 - 2D vector drawing software like Adobe's Illustrator, Google's 3D SketchUp object ... Page 10 ... A tutorial of the GGB-ART is available at ...
Z. Kovács • Private Pädagogische Hochschule der Diözese Linz, Österreich T. Recio • Universidad de Cantabria, España P. Richard • Université de Montréal, Canada M.P. Vélez • Universidad Antonio de Nebrija, España
Lyon, July 3-6 , 2017
To present the very recent implementation (2016) of tools and commands for the automatic deriving, discovery and proving of geometric statements over the free dynamic geometry software GeoGebra
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Automated reasoning tools for math learning Use artifacts to achive some didactic goals
vs. Computer based learning for math reasoning Reconsider didactic goals of math education acording new computer tools
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
What one notes is: • The role of problem solving is to illustrate properties viewed in the classroom → few open actvities in the geometric model for exploring, deriving or discovering. • Predominance of the calculation of measures (carpenter's rule or the navigator's sustains the geometric approach), nor it is surprising if we consider the etymology of geometry (Earth measurements).
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
• We use Geometry to model everyday or scientific situations → very quick moving away from traditionally geometric approaches to situations in analysis or in algebra.
• We use proofs in Geometry to train in mathematical proving, but less frequently. • In some cases it seems that geometry is being studied at the end of the school year (before the holidays) or that the administration wishes to do more $tatistics.
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
• Now you can play (explore) with shapes: animate, transform, drag, etc. • The computer becomes the laboratory, as for physicists. Before, we have mathematical machines, something that is still laborious (you need to have the talent of carpenter, engineer, etc.)
* 2D vector drawing software like Adobe's Illustrator, Google's 3D SketchUp object construction, animation of space-time scenes (Scracht)
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
• Mathematically, we can conjecture → mathematical properties take meaning with an experimental rationality: the conviction of what may be true • Discover in the manner of Alexis Claude Clairaut this great master of the century of lights:
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Reasoning by abduction when the initiative of questioning belongs to the learner * Deduction.
Induction.
Hypothesis.
Rule: All the beans from this bag are white. Case: These beans are from this bag. Result: These beans are white.
Case: These beans are [randomly selected] from this bag. Result: These beans are white. Rule: All the beans from this bag are white.
Rule: All the beans from this bag are white. Result: These beans [oddly] are white. Case: These beans are from this bag.
* Classic example from Wiki
⚠️ [randomly selected] ⚠️ [oddly]
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Input: a geometric construction and a couple (eventually a collection) of elements in the construction Output: some/all geometric relations verified by such elements within the given construction Relation[ , ]
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Input: a conjectured geometric statement Output: systematically find complementary, necessary, hypotheses for the conjectured geometric statement to become true LocusEquation[ , ]
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
AUTOMATED PROVING Prove tool Input: a geometric statement Output: a mathematically rigorous (e.g. not based upon a probabilistic proof) yes/no answer to the truth of the given statement Relation - More Prove[ ] ProveDetails[ ]
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
GeoGebra Automated Reasoning Tools (GGB-ART) are a collection of GeoGebra 5.0 tools and commands ready to automatically derive, discover and/or prove geometric statements in a dynamic geometric construction*
* A tutorial of the GGB-ART is available at https://github.com/kovzol/gg-art-doc
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
• Start by drawing a geometric construction in GeoGebra. • GeoGebra has many ways to promote investigating geometrical proprieties: • • •
Visually we can investigate by dragging the free objects by setting the trace of a constructed object or by using sliders. Use Relation tool to obtain common properties to some objects. Use Locus tool to draw the trace of an object for while other one moves on a path.
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
• Relation tool and command give information about the relation between two or more objects. • LocusEquation command calculates the equation of a locus and plots this as an implicit curve. • Prove and ProveDetails commands decide if a statement is true in general and, eventually, give some additional conditions for its truth (low level GG tools).
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Relation[ , ] Relation[ , , ] Relation[ , , , ] Example: Take three aligned points A, B and C and a free point O, and the midpoints D, E and F of OA, OB and OC. Ask for relations between points or between segments or between lines in the configuration
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
2.1. The Relation tool and command Are D,E,F aligned? We can draw line DE and visually check that F is in this line, but we can also ask: Relation[{D,E,F}]
Now it seems that lines through ABC and DEF are parallel: Ask GeoGebra again!!
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
2.1. The Relation tool and command • The Relation command allows the user to find out numerically (that is, for the drawing with assigned coordinates) whether • • • • • • • •
two lines are perpendicular, two lines are parallel, two (or more) objects (points, lengths, areas) are equal, a point lies on a line or conic, a line is tangent or a passing line to a conic, three points are collinear, three lines are concurrent (or parallel), four points are concyclic (or collinear).
• Some of these checks can also be performed symbolically, that is, the statement can be verified rigorously for the general case (with arbitrary coordinates). Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
2.2. The LocusEquation command Implicit locus of a free point, or a point moving in a path, into a construction such that some property holds. LocusEquation[ , ].
Example 2: Consider a triangle with vertex A, B, C and sides a, b, c. Take D the middpoint of a and E any point on side b.
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Question 1: Where must we place E such that segments AB are DE are parallel? LocusEquation[AreParallel[f, c], E]
Drag the free objects and conjecture that object d must be the midpoint of b.
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Question 2: . To confirm this conjecture create midpoint F of segment b and prove now that they are parallel. For that, make the objects E, f and d invisible by hiding them. Join D and F by segment g. Use the Relation tool to compare c and g Relation[c,g]
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Question 3: c and g do not have the same length, can we find some relation between this lengths? Maybe c = 1,5 *g, or maybe more? Try Relation[c, 1.5*g] or Relation[c, 2*g]
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Question 4: Let F be the midpoint of side c, and let us denote segment CF by g. Let G be the intersection point of f and g. Finally, let us denote segments CG and FG by h and i, respectively. Conjecture about h and i.
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
The answer is:
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
2.3. The Prove and ProveDetails command Prove[ ] ProveDetails[ ] The Prove command decides if a geometric statement is true. It has three possible outputs: • True: the statement is always true, or true under some nondegeneracy conditions • False: means that the statement is false in general • Undefined: means that GeoGebra cannot decide because of some reason. The ProveDetails command has similar behavior like the Prove but it may provide information about non-degeneracy conditions Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Use Prove and ProveDetails for proving conjecture “h and i are equal” in previous Question 4
Prove [h==i] returns true ProveDetails[h==i] returns {true,{“AreCollinear[A,B,C]”}}
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
“It is unusual that the area of a rectangle agrees with its perimeter” Classical Geometry reasoning: consider 𝑥, 𝑦 the basis and the height and arrive to a conic 𝑥𝑦 − 2(𝑥 + 𝑦) = 0 Reasoning with GeoGebra: Draw a rectangle with vertex (x,y), (0, y), (x,0), (0,0).
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
• See Appendices: 1. 2.
Discourse and argumentation Use of ART for tutorial support in solving problems of proof
Lyon, July 3-6 , 2017
GeoGebra automated reasoning tools Z. Kovács, T. Recio, P. Richard & M.P. Vélez
