Dynamic geometry software has been available on graphing calculators with symbolic manipulation (CAS) capabilities (e.g. TI's Voyage 200) since 1995. With the introduction of CABRI Junior, dynamic geometry software is now available for non-CAS calculators such as the TI-83+. Although the viewscreen of a handheld calculator lacks the larger screen and color capabilities of a desktop computer, investigating problems with calculator-based geometry software provides students with access to the calculator's more powerful, integrated data analysis utilities. Figure 14 illustrates a dynamic sketch and accompanying data analysis using a Voyage 200 calculator.
Once our students generate and collect data using dynamic geometry tools, they study it in detail with the calculator's extensive data analysis utilities. The table in Figure 14 contains 292 points that can be plotted immediately. Constructing a model that fits the plot of total rope length versus QR is not obvious -- it requires significant algebraic thought.
Students who are familiar with calculator-based non-linear regression and who want to avoid a "synthetic" algebraic approach (such as that proposed on the preceding page) may hypothesize that a quadratic or a cubic model will fit the data reasonably well. As Figure 15 illustrates, the relatively large value of the coefficient of determination (R2 = 0.989455) and graphical superposition suggest a good fit for a quadratic model.
Our students use the second derivative test in CAS to confirm that a minimum occurs at the critical value -- even though a graph of r(x) would make this evident. By evaluating r(x) at the critical value (in "exact calculation mode"), students calculate minimal rope length exactly, as shown in Figure 18. Note that this closed-form solution contains significantly less information about the problem than either the Level 2 or Level 3 approaches on the preceding two pages.
Up to 30 commands may be stored in the Voyage 200 home screen for review at a later time. Any work session may also be saved as a text file, called a script, that can be recalled, commented, and executed at any time. Students may use a "Two Towers" script as a template to solve similar problems. Hence, a script can be defined as a pseudo-program consisting of a set of commands to accomplish a task (Quesada, 2000). To successfully use a script, the user must provide necessary input and manually execute all template commands one after another.
Non-executable comments may be added to "document" the script. If well-written, the comments describe key goals and steps of the algorithm being performed, thus encouraging the user to review the main ideas of the script each time it is executed. Figure 19 shows a script we've used with our students to automate the calculus-based solution strategy shown in Figures 17 and 18. The use of scripts facilitates the inclusion of new topics and applications, while scaffolding cumbersome calculations that our students have yet to master (Kutzler, 1996). Scripts help our students focus on key algorithmic steps without arithmetic errors and with a minimum investment of time.
Students may use a power model (which yields no regression coefficient) or a quartic polynomial to fit the data. To compare the relative "goodness of fit" of the models, students calculate the residuals, i.e., the differences between predicted and fitted values for each model. Figure 20 shows the calculation of residuals for power and quartic models.
A cursory examination of residuals for these two models suggests the quartic polynomial produces a better fit than the power model.