A. Breda, D. Nogueira

University of Aveiro (PORTUGAL)
Our aim is to explore the potential advantages of integrating dynamic environments in the study of trigonometric functions, through a geometric and historical approach, bringing to light the way trigonometry was applied in ancient times.

Living in a world which becomes every day more and more technological, it is crucial that this be felt in classrooms. Looking at free softwares, GeoGebra is a good choice. It combines a spreadsheet and an algebraic and graphic view, all operating simultaneously which gives a tremendous potential for the creation of efficient learning environments. Unfortunately, some teachers still do not feel comfortable to conceive work tasks where dynamic geometry plays a central role, thus significant activities must be continuously created.

The introduction of trigonometry in a vast majority of textbooks involves the definition of the basic trigonometric ratio and the calculations of these ratios for special angles namely 30, 45, and 60 degrees, using some elementary geometry, assuming that their values for other angles will be given by the calculator. Students are thus led to believe that their calculators are “magic boxes’’. Furthermore, when the formulas for the sine and cosine of the half angle, sum, and difference are derived, students wonder why they are important and what the purpose of such formulas is.
What if we combine the past with the future? What if we bring to light Ptolemy´s amazing trigonometry using technology available to everyone in our days?

It seems to us, to be much more natural to adapt the Ptolemy original treatment of trigonometry and develop this subject inspired in the history of mathematics aggregating to this approach the potential of new technologies.

In order to do so, we purpose an activity for secondary school children, grade 11, with GeoGebra as a partner. We will follow the Ptolemy steps, realizing and understanding how the values of “sine” were born, in the Almagest, figuring out what was the purpose of “trigonometry” in those ancient times. In the proposed activity, the student will be first invited to build a chord table following the Ptolemaic ideas. Then, through several types of mathematics regression, the student will be invited to compare Ptolemy´s “sine” values to the real ones. Finally, the student will found out how Ptolemy got the value for earth’s eccentricity from his chord table.