I’ve used Geogebra in the past, but I often asked students to use their understanding of algebra to support their burgeoning knowledge of geometry and relate themes, such as similarity and rigid motions, to analytical actions positioned within the Cartesian coordinate system (National Governor’s Association, 2010). These investigations marked the first time I used Geogebra’s Geometric tools synthetically without having the grid as a background. They also expanded my ability to use transformations to explore congruence, similarity and symmetry as essential elements of Geometry (National Council of Teachers of Mathematics, 2018).
Geogebra is dynamic geometry software (DGS) “developed for and…used by students on computers and handheld calculators” (Hollebrands & Dove, 2011, p.33). In the book Focus in High School Mathematics: Using Technology to Support Reasoning and Sense Making, Hollebrands and Dove (2011) describe four types of geometrical tasks which can be explored using DGS: Constructions, Constraints, Equivalence and Generalization. It is also mentioned that most tasks are not “mutually exclusive” to the four types (p.51) and traverse one another, which was my experience with each Geogebra investigation.
In the earlier investigations, the objectives included the creation of isosceles triangles, rectangles, equilateral triangles and, in a method which may seem counterintuitive to the novice geometry learner, we used circles to do it.
What did we need to know about circles in order to create these shapes, and why does it matter?
Initially, I used the primitive objects (which are representations of certain shapes) to create shapes based on the properties and attributes I was most familiar with. I had a sense that I was missing out on the intention of the investigations and quickly realized that what I was doing was “drawing” where the creative objective of each investigation called for “constructing” (Hollebrands & Dove, 2011, p. 33). From an instructional standpoint, constructions are essential for understanding the attributes and relationships of geometric objects. A circle is defined as “a set of points in a plane that are equidistant from a given point” (WolframAlpha, 2019). The equidistant points on the circle allowed me to develop conjectures about shapes with congruent sides and angles.
Through a proper construction, I could also prove that regardless of a shape’s size or dimensions, the side lengths can be transformed by dragging a vertex and the properties remain the same. Hollebrands and Dove (2011) describe this as a shape retaining its integrity after a transformation has taken place.
In another investigation, I considered constraints when being asked to create a triangle with a fixed area. As a teacher, I’ve used and looked at examples where a triangle inside a square has a fixed area as the vertex slides along one side of the square. Using Geogebra to investigate this idea “help[s] students explore specific cases and identify unique relationships that occur under those constraints” (Hollebrands & Dove, 2011, p. 41). I discovered dragging one vertex along a curve or along a parallel line is one way to investigate constraints of a triangle’s area. I also engaged in tasks emphasizing equivalence and generalization, which required specific observations and conjectures to be made about the properties of transformations.
I am glad to have experienced Geogebra’s technology in new ways, because I see how well it encourages strategic competence – and promotes the appropriate use of DGS tools strategically – as well as adaptive reasoning in Geometry (National Council of Teachers of Mathematics, 2014; National Governor’s Association, 2010). As with other technology tools I’ve investigated this fall, teachers can create productive struggle by assigning an appropriately designed task where Geogebra is used as a tool for mathematical modeling and solving problems which may be difficult or impossible to solve without it.
References
Hollebrands, K. & Dove, A. (2011). Focus in High School Mathematics: Technology to Support Reasoning and Sense Making. T. Dick & K. Hollebrands (Ed.) Reston, VA: National Council of Teachers of Mathematics
National Council of Teachers of Mathematics. (2018). Catalyzing change in high school mathematics: Initiating critical conversations. Reston, VA.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA.
National Governor’s Association. (2010). Common Core State Standards for Mathematics. Washington, D.C. Retrieved from http://www.corestandards.org/Math/
Wolfram Alpha. (n.d.) Definition of a circle. Retrieved https://www.wolframalpha.com/input/?i=definition+of+a+circle
The Chronicles of Algebrea 