## Dividing a Town into Pizza Delivery Regions

- Lesson

Students will construct perpendicular bisectors, find circumcenters, calculate area, and use proportions to explore the following problem:

You are the owner of five pizzerias in the town of Squaresville. To ensure minimal delivery times, you devise a system in which customers call a central phone number and get transferred to the pizzeria that is closest to them. How should you divide the town into five regions so that every house receives delivery from the closest pizzeria? Also, how many people should staff each location based on coverage area?

### The Problem Situation

Display the Pizza Parlor Proximity overhead and read the problem with students.

Pizza Parlor Proximity Overhead

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.

Regions for Two Pizzerias Activity Sheet

*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.

Regions for Three Pizzerias Activity Sheet

**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.

Regions for Four Pizzerias Activity Sheet

**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.

- Have students ignore pizzeria D for the moment and reconstruct the boundary lines for the regions from the Regions for Three Pizzerias activity.
- Then, students should construct boundaries between A and C, A and D, and C and D.
Ask students if they have to construct all three boundary lines?

[No, we already have the boundary between A and C constructed.]

- Use the boundaries to determine the regions for pizzeria A, B, C, and D.

**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.

Regions for Five Pizzerias Activity Sheet

**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:

- Compass and straight edge
- Mira (optional alternative to compass and straight edge)
- 4 sheets of patty paper or transparency paper (optional alternative to compass and straight edge)
- Access to Geometer's Sketchpad or Cabri Jr. (optional alternative to compass and straight edge)
- Pizza Parlor Proximity Overhead
- Regions for Two Pizzerias Activity Sheet
- Regions for Three Pizzerias Activity Sheet
- Constructing a Circumcenter Activity Sheet (optional)
- Regions for Four Pizzerias Activity Sheet
- Regions for Five Pizzerias Activity Sheet
- Regions for Pizza Delivery Answer Keys

**Assessment Options**

- Ask students to find the regions for a map with four pizzerias or with six pizzerias.
- Ask students to design a map and then make posters of the delivery regions.
- Have students write in their own words why they think
*perpendicular bisectors*and*circumcenters*are helpful in finding the delivery regions for the pizzerias.

**Extensions**

- Instead of points A, B, C, D, and E being pizzerias, let them represent rain gauges holding different amounts of rain measured in inches. Tell students that each block on the grid represents one square mile, and ask them to estimate the total volume of rainfall in cubic inches that fell on Squaresville.
- Set up Squaresville on a coordinate plane with the bottom left corner being the origin with four locations identified. Rather than the points A, B, C, and D being pizzerias, let them represent obstacles in a square room (with a side length of 8 feet) around which a robot needs to maneuver. Assuming the robot should stay the same distance from the two nearest obstacles, ask students to find the equations of the lines that the robot can travel (and their domains).

**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?

### Learning Objectives

Students will:

- Construct the perpendicular bisector between two points.
- Construct the circumcenter of a triangle.
- Use circumcenters and perpendicular bisectors to construct a Voronoi diagram for a series of points.
- Approximate the area of regions by counting the number of unit squares in the region.
- Use ratios and proportions to solve problems.

### NCTM Standards and Expectations

- Draw reasonable conclusions about a situation being modeled.

- Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.

- Use geometric models to gain insights into, and answer questions in, other areas of mathematics.