The Problem Situation
Display the Pizza Parlor Proximity overhead and read the problem with students.
To make sure they understand the problem and the solution
expected, ask students to determine which pizzeria is closest to
specific locations.
- Which pizzeria is closest to a house on the corner of D Street and 5th Street?
- Which pizzeria is closest to a house on G Street?
After you get a sense that students understand the problem, suggest
that in order to solve a complicated problem it often helps to start
with a simpler version of the problem and then work your way up to the
more complicated problem. Ask students how that could apply to this
situation. In this lesson, students will first solve the problem with
two pizzerias, and then three pizzerias, before trying to solve the
problem using five pizzerias.
Make sure students have all the materials they need, as listed
above. The Regions for Pizzerias activity sheets can be distributed in
the beginning of the activity or individually as students work through
each scenario.
Note: Below is an outline for each of the two, three,
four, and five pizzerias scenarios. If you find that students are
successful with the two and three pizzerias scenarios, you may choose
to skip the scenario of four pizzerias to minimize the repetition.
The Two Pizzerias Scenario
Hand out the Regions for Two Pizzerias
activity sheet. You may choose to point out a few locations on the map
and have students determine which pizzeria would deliver pizza to those
locations.
Dividing the Town into Regions
Help students to arrive at the fact that the perpendicular
bisector of the line segment connecting the two pizzeria locations
would best divide the town into two regions for pizza delivery. Here
are some questions you might ask:
- If we wanted to find the point halfway between the pizzerias, what point would that be?
[The midpoint.]
- If we wanted to find other points that are equidistant from the pizzerias, how could we do that?
[Using each pizzeria as a center, draw circles with equal
radii. The intersection of the circles will be a point equidistant from
both pizzerias. Draw the circles using a ruler, compass, or string.]
- What is common about all of these points?
[They lie on the perpendicular bisector of the segment from pizzeria A to pizzeria B.]
Constructing a Perpendicular Bisector
Have students draw a line segment to connect the two points
representing the pizzerias. Depending on the type of materials you have
for your students (Mira, patty paper, or straightedge and compass),
students will need to use a method to construct the perpendicular
bisector. Before constructing the perpendicular bisector on the
activity sheet, it might be practical to have students practice their
constructions on a separate sheet of paper.
When students construct the perpendicular bisector, make sure
they extend that line on both ends until it intersects with the borders
of Squaresville. The perpendicular bisector divides the town into two
regions, each serviced by one pizzeria.
Approximating the Area of Each Region
Suggest that students approximate the area covered by each
pizzeria by counting up the whole blocks. Then students should estimate
partial blocks to the nearest quarter, half, or three-quarter block.
When done, have them find the total number and round to the nearest
half block.
[The answers are below.]
Determining Number of Workers Needed at Each Pizzeria
If one region is significantly larger than the other, then the
pizzeria servicing the larger region should have more staff. How many
more staff? How do we determine how to assign a given number of staff
members to the pizzerias if the regions are not the same size?
Proportions!
Review solving proportions. Remind students if ^{a}/_{b} = ^{c}/_{d}, then a · d = b · c.
Using this method, set up the proportion | area of parlor region | = | number of workers for parlor |
total area of Squaresville | total number of workers |
Solve for number of workers for each pizzeria.
Answers
| Approximate Area | Number of Workers |
Pizzeria A | ≈ 30 blocks | ^{30}/_{64} · 40 ≈ 19 workers |
Pizzeria B | ≈ 34 blocks | ^{34}/_{64} · 40 ≈ 21 workers |
Placing Pizzerias So the Areas Are Equal
Answers will vary, but here are some possibilities:
The Three Pizzerias Scenario
Hand out the Regions for Three Pizzerias
activity sheet. Just as with the Two Pizzerias Scenario, you may point
out a few locations on the map and ask students to determine the
closest pizzeria.
Dividing the Town into Regions
Help students see that they will need to construct three perpendicular bisectors to divide the town into three regions.
Some questions you may ask:
- When we had two pizzerias, what did we construct to divide the town?
[The perpendicular bisector between A and B.]
- How do we divide the town between B and C? A and C?
[By constructing perpendicular bisectors between each pair of pizzerias.]
- Ideally, the three perpendicular bisectors should intersect at
a point. What is special about this point? What is the name of this
point? What is unique about this point?
[This point is the circumcenter. It's the point equidistant from all the pizzerias.]
- Do we need to use the whole perpendicular line for each of these regions, or do we use only a line segment?
[You only need the line segment that connects the circumcenter to the border of the town.]
Note: A circumcenter of a
triangle is a point that is equidistant from each vertex. It's possible
to draw a circle that goes through each vertex, whose center is the
circumcenter. You may choose to introduce or elaborate on this idea
with the following activity sheet:
| | | Constructing a Circumcenter Activity Sheet |
| |
Answers
| Approximate Area | Number of Workers |
Pizzeria A | ≈ 23 blocks | ^{23}/_{64} · 60 ≈ 22 workers |
Pizzeria B | ≈ 14 blocks | ^{14}/_{64} · 60 ≈ 13 workers |
Pizzeria C | ≈ 27 blocks | ^{27}/_{64} · 60 ≈ 25 workers |
Placing Pizzerias So the Areas Are Equal
Answers will vary, but here are some possibilities:
The Four Pizzerias Scenario
(Consider this section optional. If you feel that there is too
much repetition for your students, move on to the challenge of the five
pizzerias problem.)
Hand out the Regions for Four Pizzerias
activity sheet. Once again, you might want to point out a few locations
on the map and have students determine which pizzeria is closest.
Dividing the Town into Regions
Lead a guided discussion to encourage students to think about a
good strategy for dividing the town into four regions. Hopefully, they
will arrive at a strategy to reconstruct the regions for three
pizzerias (i.e., use the work they have already done) and then work
from there.
Answers
| Approximate Area | Number of Workers |
Pizzeria A | ≈ 15 blocks | ^{15}/_{66} · 80 ≈ 19 workers |
Pizzeria B | ≈ 14 blocks | ^{14}/_{64} · 80 ≈ 18 workers |
Pizzeria C | ≈ 17.5 blocks | ^{17.5}/_{64} · 80 ≈ 22 workers |
Pizzeria D | ≈ 17.5 blocks | ^{17.5}/_{64} · 80 ≈ 22 workers |
Placing Pizzerias So the Areas Are Equal
Answers will vary, but here are some possibilities:
The Five Pizzerias Scenario
Hand out the Regions for Five Pizzerias activity sheet. You may point out a few of locations on the map and ask students to determine the closest pizzeria.
Dividing the Town into Regions
Lead a guided discussion to get students thinking about a good
strategy for dividing the town into five regions. Hopefully, they will
arrive at a strategy to reconstruct the regions for three or four
pizzerias (i.e., use the work they have already done) and then work
from there.
- Tell students to ignore pizzerias D and E for now and reconstruct
the boundary lines for the regions as they did in the Regions for Three
Pizzerias or Regions for Four Pizzerias activities.
- Then students should construct boundaries between E and A, E and B, E and C, and E and D.
- Use
the boundaries constructed to determine the regions for pizzerias A, B,
C, D, and E. Students should find that they have a total of 8 separate
boundary lines and 4 circumcenters.
Answers
| Approximate Area | Number of Workers |
Pizzeria A | ≈ 13 blocks | ^{13}/_{64} · 100 ≈ 20 workers |
Pizzeria B | ≈ 11 blocks | ^{11}/_{64} · 100 ≈ 17 workers |
Pizzeria C | ≈ 15.5 blocks | ^{15.5}/_{64} · 100 ≈ 24 workers |
Pizzeria D | ≈ 16.5 blocks | ^{16.5}/_{64} · 100 ≈ 26 workers |
Pizzeria E | ≈ 8 blocks | ^{8}/_{64} · 100 ≈ 13 workers |
Placing Pizzerias So the Areas Are Equal
Answers will vary, but here are some possibilities:
Questions for Students
1. The diagrams we constructed are called Voronoi diagrams. Besides
pizza delivery regions, in what other situations would these diagrams
be useful?
[There are a variety of answers.]
2. Conduct an Internet search to find answers to the questions
below. In your answers, include the URL of the web site in which you
found the information.
- Who are Voronoi diagrams named after? When and where did he live?
- What is a Voronoi tessellation?
- Explain how a Delaunay triangulation relates to a Voronoi diagram.
Teacher Reflection
- Were students able to accurately construct the perpendicular
bisectors and circumcenters using your method of choice (Mira, patty
paper, or straight edge and compass)? Why or why not? Would another
method have worked better?
- Was students’ level of enthusiasm/involvement high or low? Explain why.
- How did students demonstrate understanding of the materials presented?
- Were concepts presented too abstractly? too concretely? How would you change them?
- Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer?
- Did you find it necessary to make adjustments while teaching
the lesson? If so, what adjustments, and were these adjustments
effective?