Using PC to Teach VD in PE
(part 2)

Bro. Patrick Carney (pcarney@dimacs.rutgers.edu)

General Information

The Problem
Part 1 of this pair dealt with the relatively symmetric and simplistic zone defenses in basketball. This led to fairly easy Voronoi Diagrams. This expands on Part I using football defenses which require the students to make many more decisions in creating their pictures and thus force them to see more of the interplay among the relative positions of the points (players).

Suggested Materials
As in the first part, this could be done with nothing more complicated than pencil and paper. In fact, it was done with the aid of the school's football coach (a highly recommended resource) and the PaintBrush program available on most MS-DOS machines. It is easily adaptable to other software packages which include drawing tools.

Prerequisites
Students should have done Part 1 of this pair and it should be reviewed before beginning. Thus they already have the concepts of "zone", Voronoi Diagram, perpendicular bisector, etc. If using a software package, they should have a working knowledge of using it for simple drawings.

Activity Description

Guided Exploration
Although the teacher could do the following, if possible it is suggested that the football coach be invited in. The excercise will continue from his or her point of view. The coach comes in as a guest lecturer and discusses the idea of a zone defense against passing as used by the particular school. He or she sketches a couple of them on the board (overhead) and shows where the responsibilities lie. If appropriate, this could be expanded (e.g., Coach Pete Kelley proceeded to explain how this was exploited in some famous Super Bowl touchdowns by putting "receivers in the seams". The students knew from the previous part that these seams were the edges of the Voronoi diagrams).

Some general formations are included with this packet, but there is much to be said for having the students work on the actual ones used by their school (or rival schools). They will learn just as much about Voronoi diagrams (since football has very little to do with it -- it is the hook to get them interested) but take much more of an interest and ownership in the process.

Generally 3 or 4 of the 11 football players have the responsibility of rushing the passer. That leaves 7 or 8 to cover these zones. Some of the formations are symmetrical and some are not. The defense depends on the formation of the offense. These formations could be assigned so that the more difficult ones go to those students who had the easiest time with basketball or each student be allowed to pick one of those discussed. At any rate, there should be a variety being done in the class.

Students then use Paint Brush and proceed to draw the Voronoi Diagrams. They find this more difficult than basketball. Many have to be constantly reminded that they MUST consider all relevant pairs of 2 players. When they are incorrect, it frequently helps to show them a part of one player's zone that is clearly closer to a teammate than to the person whose color makes that zone. They are then asked to reconsider their zones.

Concluding the Exploration
As students finish, ask them to share their results with other students who are finished. They should note their differences and be asked to explain why they think they are different. All students should be asked where the receivers should run to be able to best exploit the zone defense and WHY! (running along the seams not only might cause confusion with regard to who should cover the receiver, but also the receiver is as far as possible from the defender). They should also address the question of what the implications are for other applications of Voronoi Diagrams -- e.g., where is the weakest place in phone cells?

Sample Results from Exploration
Many students began with an overly simplistic view in which they did not consider the effect of all the players. They would be correct for two players but fail to include the influence of the third (or three and fail to consider a fourth, etc). Eventually when forced to answer why B was closer to parts of A's area than A was, they refined the process and as they did, noticed a much more "elegant" picture that whenever two perpendicular bisectors met at a point, so did at least one more. They noticed the differences symmetry made as well as even or odd numbers of linebackers (the pass defenders closest to the ball) and their effect on shapes.

A Mathematical Approach
The football examples made it clearer that the perpendicular bisectors of the sides of a triangle meet at a point. Initially this was not happening in the students' work. As they got better, they realized from experimentation that this should happen. They also saw how a little error in measurement or estimation could be magnified when a line is extended the length of a football field. They realized that this could not be "eye-balled" like the simple and symmetric basketball zones.

Further Problems for Review or Assessment
Ask students to consider why the zones are unequal in area. Where are the smaller areas? Why would that make sense?

Consider what effect adding or deleting points would have on the picture. What would happen if a team "blitzes" (i.e., decides to rush some of these defenders leaving their teammates to divide up the field differently)or has linemen drop back in pass coverage? Where would an offensive team want to attack? What effect would these changes have on those patterns? This was a particularly good assignment for those who finished earlier than the others.

Writing Assignment
Any of the problems in the previous section would lend themselves to good writing assignments. In addition, students could be asked to actually watch a football game on TV and see if they could spot where the zones are (teams might try to hide them) and how successful they were as well as how successful offenses were at exploiting the zones (note announcers refer to the edges as "seams").

Where 3 zones come together at a point, what are the advantages and disadvantages of trying to catch a pass thrown to that point? Why?

Followup Problems or Activities
Everyone was asked to apply this to some aspect other than what we studied or mentioned in class. In addition some of the writing assignments above would be followup problems.

One of the interesting plusses to come from the exercise is that the school's basketball coach was interested and asked if we would explain our findings to the team since he was having trouble keeping them in the zones.

Attachments

Teaching Notes

References
To date I have found no references to this particular application.

General references include:

Drysdale, Scot, Lecture notes from DREI96, Princeton University, 1996 (Published on the DREI Web Site).

Dickerson, Matthew and Drysdale, Scot Voronoi Diagrams and Proximity Problems, COMAP, Lexington MA, 1996.

Comments from Teachers who have Used this Unit
I cannot stress enough the authenticity that the football coach gave to the process. He is highly respected and one sensed an attitude of "gee, if this guy uses math, maybe there is something to this stuff" beneath the surface of a project that all took very seriously.