In the First International GeoGebra Congress, held last year (2023) in Córdoba, I presented some methods that make it easier for students to engage in investigations with minimal algebraic apparatus. In this way, middle school students can present projects with significant mathematical content without being held back by a lack of experience in using functions and algebraic transformations. For example, we can use the CAS view to define an arbitrary point X and the distance between it and other already defined objects, such as a fixed point A or a line r:

X:= (x, y)  XA(x,y):= Distance(X, A)  Xr(x,y):= Distance(X, r)

In this simple way, we can pose equations like the following:

XA / Xr = k

and observe the resulting conic, with eccentricity k.

Or, for example, we can confine several free points within a sphere and, simply by using vectors between them, make them automatically and mutually repel each other, observing what kind of configuration emerges depending on the number of confined points.

This automatic repulsion is made possible by a continuously animated slider (which I’ve called “anima”) that, each time it changes value, runs the script that “animates” the points to repel each other. In this presentation, a similar, continuously animated slider will undoubtedly take center stage. Since Mathematics, for better or worse, is present in countless cross-disciplinary contexts, one might think it's easy to choose some that allow for assessing mathematical competence at each level. However, most "real" situations are too "complex" (pardon the wordplay) to simplify without losing their essential content. Of all the scientific fields, Physics is undoubtedly the most "mathematized." This mathematization carries certain risks, such as overuse of formulas and the premature introduction of mathematical concepts (particularly trigonometric and complex functions, and those related to infinitesimal calculus, like derivatives and integrals). Here, I will present a method that allows for approaching some physical relationships inherent to motion, such as velocity and acceleration, based solely on their conceptual definition (in classical, non-relativistic mechanics). The original idea for this method was presented as variation tables by Richard Feynman in his famous book The Feynman Lectures on Physics (Volume I, 9-7, Planetary Motions). We will see that this method seems purpose-built for incorporation as a script into the "anima" slider. For this, we will use real-time synchronization with the user's computer (or other device) clock. This will allow us to simulate various situations fairly accurately, where a mass subjected to acceleration g moves its position M with velocity v, without resorting to higher mathematics.

• Note: The letters g and v appear in bold because they represent vectors, not scalars. (Formally, the letter M would also represent the position vector of point M, so an expression like M + dt v is a vector sum.)

The animations do not use formulas (no pre-set equations, no trigonometry, no differential calculus); they simply make the necessary variations in the vectors that guide the motion. These variations essentially reduce to the execution of two instructions:

SetValue(v, v + dt g) SetValue(M, M + dt v)

where dt is a very small time interval (the time it takes for the slider to change value, just a few hundredths of a second). In other words, every short time interval, v changes by an amount equal to "a little bit of g", and the position M of the mass shifts by "a little bit of v". It's important to note that these sums are not numerical but vectorial, meaning a certain amount is added in a specific direction. This method allows GeoGebra to be used as a kinematic lab accessible to middle school students, as it only requires the introduction of a slider with a few lines of script that, with slight variations, are essentially always the same. This lab can be used as a starting point for proposing physical-mathematical projects in this educational stage or later ones, all aimed at the acquisition of genuine mathematical competence. Author of the activity and GeoGebra constructions: Rafael Losada.