- Guided Exploration
- Ask the class if anyone plays on a basketball team. In any class, there seem to be many. Select two of these players, perferably one boy and one girl. Place the waste basket on a student's desk in the front of the room. Tell these two students that they must defend the basket against the teacher in a zone defense. (A zone defense is not explained yet). The students have their backs to the class facing the teacher. See diagram:
CLASS HERE FACING DOWN PAGE
basket
STUDENT 1 STUDENT 2
The teacher, possibly holding a basketball, moves in this area facing class.
Move to the extreme left and ask the class which student should cover him or her. The class will generally realize it is STUDENT 1. Then move to the extreme right and repeat the question. Obviously they will say Student 2. Then get closer to the middle and see when the answer changes. Move backward and forward right to left as the class calls out answers until there is some disagreement.
- Concluding the Exploration
- Once there is some disagreement (only a minute or two), ask how they decide when to switch defenders. They will usually say something about half way. My experience is that they will tell me to draw a "straight line" which I faithfully do but draw it on a diagonal. The usual reaction is "No, we said straight". Of course they mean that it should be vertical in the context of this diagram. So I usually draw exactly what they say, but in such a wild manner that it cannot be correct until we establish the concept of the perpendicular bisector of the segment joining the 2 players. This takes at least two or three times as long as moving around the basket did. We now draw a picture. It is merely the diagram above with the perpendicular bisector of the segment joining the 2 players drawn in. Students are then asked what the word "zone" means in the term "zone defense". Most students clearly see the areas of responsibility of each defending player.
Next add a third player who will be positioned as in the diagram below:
STUDENT 1 STUDENT 2
STUDENT 3
Or, just using the numbers so that they will look more like points:
Now ask how the floor would be divided. Eventually, they realize that there is a perpendicular bisector between each pair of points and they are drawn in. Each defender takes his or her own side of that line. But the students are then asked if ALL of the lines are needed as they were with 2 defenders. With a little bit of leading, they see that we can erase some of the lines so that we are left with an inverted Y since, for example, in the area directly behind #3, there is no need to know which side of the perpendicular bisector of the segment joining #1 and #2 the person with the ball is standing since #3 will guard the whole area.
Since basketball has 5-player teams, ask five students to arrange themselves in a 2-1-2 zone which looks like this:
The students now move to the computers and are asked to use Paint Brush to design the zones and color them differently for each player. This could just as easily be done on paper or with software that actually does compute accurate perpendicular bisectors. The thought here however was to do it as an athlete would -- i.e., with careful use of hand-eye coordination.
- Sample Results from Exploration
- When tested with an ordinary 8th grade class, most of them came up with the following pattern. An actual student's work appears below:
Where the diamond in the middle is actually much larger than this figure.
- A Mathematical Approach
- There are many lessons to be learned here in addition to the Voronoi Diagrams. For example, we not only discovered the concept of the perpendicular bisector, but also, that "for some strange reason" no matter what we did, the perpendicular bisectors of any 3 non-collinear points always seemed to meet at a point! This made the drawings easier. Of course it remained to be seen if this would hold up in part 2. But if it did, it made some aspects much simpler!
- Further Problems for Review or Assessment
- Ask students if they know what the classic attack on such a 2-1-2 defense is. In my class, I was surprised that the majority knew it was a 1-3-1. We then drew those players in and found that 4 of the 5 would be right on a seam in our diagram. So the question was posed to the class: Why would that be the best way to attack ? (See attached diagrams for what is meant by 2-1-2 and 1-3-1.)
Since not all students work at the same pace, it is suggested that when some finish their PaintBrush diagram, they can be challenged to do one for another defense -- say a 1-3-1. This keeps everyone on the task but gives those who are ahead more to think about without boring repetition.
Compare the 2-1-2 result with the 1-3-1 by a 45 degree rotation.
- Writing Assignment
- Students could be asked to write about the previous question concerning the best way to attack a certain zone in the light of their pictures. They could also be asked to write what they observed. Another possible topic would be to predict what will happen the next day when we do Part 2 -- football zone defenses.
Of course, at this point, they can be introduced to the terminology and asked to find out the more practical applications of what they were doing -- e.g., cell phones, radio stations, postal delivery zones, assignment to schools, etc.
- Followup Problems or Activities
- See Lesson Plan for Part 2 when we invite the football coach in and apply our knowledge to more complex and less symmetrical zones! Previews of coming attractions!
Teaching Notes
- References
- While there are many references for Voronoi Diagrams, I could find nothing on this particular application. This was taught to an 8th grade class and I did not find a lot of things on Voronoi Diagrams that would not scare them. I think it should be filtered through the teacher.
For general reference, the following are recommended:
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
- Although I wrote it, since I also taught it, I thought I should comment here. This was taught to 8th graders but it was obvious that it was easily adaptable to the other grades. The students were quite enthusiastic and succeeded very well with this part. They certainly appeared to enjoy the activities and were eager to continue.
Also I am well aware that this does not follow true to form with regard to sections suitable for student handouts since this is taught in a Socratic method and does not lend itself to handouts -- rather they create the diagrams on the screen.
