Level 2: Geometric/Dynamic Geometry
A handful of our students are able to solve the TTP without the use of technology -- e.g., by invoking the optical principle of incidence and reflection with straight-edge and compass. However, technology greatly broadens the group of students who have access to the ideas of optimization. Students with previous exposure to dynamic geometry software may perform the data collection in the previous section more precisely. Using software such as CABRI Geometry II (Texas Instruments, 1996) or Geometer's Sketchpad (Key Curriculum Press, 2002), our students examine the relationship of angle PRQ and angle TRS to total rope length without resorting to use of trigonometric functions or even protractors or rulers.
Use of dynamic geometry software has become commonplace in many middle and secondary school classrooms -- in fact, there are free versions of CABRI for TI-83+, TI-92+ and Voyage 200 calculators. With the software, instructors have the ability to create Java-based models of the Two Towers problem that students explore either in class or at home in a web browser. Similarly, students with access to the appropriate software can construct a CABRI model or Sketchpad model of the problem situation.
Unlike a "static" sketch drawn with pencil on a worksheet, a "dynamic" sketch may be modified by clicking and dragging. Measurements of various geometric objects (e.g. segments, angles) are updated automatically as the user moves objects along the screen. In Figure 5, we show a screenshot generated from a dynamic sketch of the Two Towers problem, which illustrates collection of various measures of angle PRQ, angle TRS, and total rope length in CABRI Geometry II. By animating point R within CABRI, our students capture numerous measures of angle PRQ, angle SRT, and total rope length quickly and easily. A CABRI-generated table of the results is shown in Figure 6.
Figure 5. Selecting the Tabulate menu item within CABRI to collect
data regarding measurements that appear within a sketch
After our students select values to tabulate, they drag on point R to animate their sketches. As measures of angle PRQ, angle TRS, and rope length (i.e. PR+RT) change, these values are collected in the table -- see Figure 6.
Figure 6. After the animation, the table of requested measures
is populated with values.
As Figure 6 suggests, the length of the rope appears to be minimal (16.97 units) when angles PRQ and TRS are congruent.
Alternatives to animation and tabulation exist. For instance, once the dynamic model is built, students may use ''trial and error'' to approximate a minimum. Paying careful attention to the value of Tot.Rope, students click and drag on point R until the smallest value of PR+RT is found.
Geometry students may take a more sophisticated approach by reflecting line segments RT and TS over segment QS. In this manner, the original exercise is recast as the problem of finding the shortest path between points P, R, and T'. Figure 7 illustrates an initial construction, and Figure 8 shows a construction in which PR+RT is minimized.
Figure 7. A new model constructed by reflecting RT and TS over QS
When a line segment is used to connect T' (i.e. the reflection of T over QS) with P, triangle PRT' is formed. By the triangle inequality, PR + RT' is greater than or equal to PT', with equality holding when points P, R, and T' are collinear.
Figure 8. The shortest path connecting points P, R, and T' occurs when the
points are collinear, in which case angles PRQ and TRS are congruent.
This reflection argument is equivalent to the argument that some (certainly not all) students see in physics to justify the equality of angles of incidence and reflection. However, even the students who do encounter this idea are likely to see it after having access to dynamic tools in a geometry course.