Level 4: Calculus
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.
(Top) Dynamic sketch on the Voyage 200 with initial dimensions scaled by a factor of 0.5
(Middle) Data (QR and total rope length) collected using the animate and tabulate features of CABRI
(Bottom) A plot of total rope length versus QR
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.
(Top) Calculations associated with quadratic regression on rope length versus QR
(Bottom) Graphical superposition of quadratic fit equation on data, with minimum calculated
A cubic model generates a coefficient of determination even closer to 1, namely R2
= 0.999367, as illustrated in Figure 16.
(Top) Calculations associated with cubic regression on rope length versus QR
(Right) Graphical superposition of cubic fit equation on data, with minimum calculated
Because the coefficient of determination is closer to 1 for the cubic model, one might expect that the cubic model will estimate minimum rope length more accurately. At this point, students may wonder if the cubic is the best model. Uncertainty regarding the accuracy of various models provides motivation for "exact" calculus-based procedures.
As Figure 17 suggests, our calculus students use CAS to define a rope length function, r(x), and then find critical points by setting the derivative equal to zero and solving for x .
Figure 17. CAS used to calculate critical points for the rope length function r(x)
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.
Figure 18. The second derivative test confirms that a minimum occurs at the
critical value, and the exact value of minimal rope length is calculated.
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.
Figure 19. "Two Towers" problem script and execution
of the script in the Voyage 200 environment
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.
Figure 20. With a power model stored in y1, list c3 is defined as y1(c1)-c2
(i.e., predicted - fitted value). Similarly, with a quartic model stored in y2,
list c4 is defined as y2(c1)-c2.
A cursory examination of residuals for these two models suggests the quartic polynomial produces a better fit than the power model.