- Christian Dorner
We have already worked our way through the first two steps, so it remains to implement the GeoGebra applet. In this model we quickly create two sliders in GeoGebra, one for the key interest rate p and the other for the exchange rate er. In the spreadsheet-view one calculates the yearly debt level for the four credits by using the recurrence relation (1). For Example in cell B1 one nds the formula "=B2*(1+p/100)-8 400", analogous calculations for credit 2 (pay attention to the rst two periods!) and credit 3 (use if- statements!). The calculations of credit 4, see table 1, lead to the same yearly debt levels as for credit 1.
We follow the idea of Heugl and create a list of points for every credit (see , p. 76). The result is a nice visualization of the yearly debt levels of each credit. Different point styles enable easy distinguishing between the four credits.
In school a stage of experiments might be of advantage, like manipulating slider bars and observing its outcomes. In case of less experienced pupils the teacher should prepare guiding questions, e.g. "What will happen, if the value of the key interest rate in- creases/decreases?" or "At which values is credit 1 better than credit 2?" These heuristic observations can be collected and shared in class. Afterwards there is an opportunity to justify some of these findings by using mathe- matical argumentations. In this deterministic model one gets for every value of the key interest rate p one best credit or for some values p two best credits, see below.
One gets such values by pairwise comparing the credits. For example one obtains 1.477 by using the equation (2) of credit 1 and 2 (slightly modified) and set Sn = 0. We express the number of years n explicitly. In case of credit 1 we get per CAS (for convenience we set q := 1 + p/100 ):
We equal the two terms of the equations (3) and (4) and solve the resulting equation for the variable q. Finally, we obtain q = 1.01477 and therefore p = 1.477. In a validation of this model one will recognize some of its wrong results. It is impossible that the yearly debt level of the foreign currency loan equals the yearly debt level of credit 1. Empirical investigations like reading newspaper reveal this mistake in the model above. Foreign currency loans have a bad press, actually in Austria they are forbidden for hypothecary credits.