NUMB3RS Activity: Location! Location! Location! Episode: "Bones of Contention"
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1 NUMRS ctivity Teacher Page Episode: "ones of ontention NUMRS ctivity: Location! Location! Location! Episode: "ones of ontention" Topic: Voronoi Diagrams Grade Level: 8 - Objectives: Understand the basic idea of a Voronoi diagram. Draw Voronoi diagrams in simple cases. Time: bout 0 minutes Materials: ruler (Optional: colored pencils, compass, geometry software such as abri Jr. on the TI-8 Plus) Introduction When you go into the takeout section of your favorite pizza store, you might see a map on the wall that highlights the part of the city where this store delivers, as well as the delivery zones for other nearby stores in this chain. simplified version of such a map could be modeled by a Voronoi diagram where a set of points (sites) in the plane are associated with service regions or areas of influence. Every point in a given (convex) region is closer to that site than to any other site. The boundaries between the regions are called Voronoi edges. In ones of ontention, harlie talks about Voronoi regions associated with fast food restaurants. The goal of this activity is to investigate and construct Voronoi diagrams with or sites. Discuss with Students Explore this idea with students using a simple drawing. Suppose you had two pizza stores which make deliveries to a certain area. How can you decide which part of the area each store will serve? For example, students may say that the homes closer to one store should get pizzas from that store. However, how do you determine the boundaries of each part of the area so that each home is served by the closer store? The key to determining the boundaries, or edges, in a Voronoi diagram is finding the perpendicular bisector of a line segment. To find this bisector, you need to know the midpoint and the slope of the segment. Review with students how to find the perpendicular bisector of a line segment using a, geoboard, a coordinate plane, or a geometry drawing program. For the abri Jr. figures to accompany this activity, visit education.ti.com/exchange and search for 75. The suggestions below can be used in class as review before completing the activities. education.ti.com/go/numrs Tom utts, UT Dallas
2 NUMRS ctivity Teacher Page Geoboard pproach Midpoint: The midpoint of is located at half of the horizontal distance from to and half of the vertical distance from to. From, the midpoint has a vertical distance and horizontal distance. Slope: The slope of is the vertical distance from to divided by the horizontal distance from to : =. Slope of Perpendicular isector: This slope is the opposite reciprocal of the slope of :. Episode: "ones of ontention oordinate pproach oordinates: (0,) and (,8) Midpoint:, = (, 5 ) Slope: 8 = = 0 Slope of Perpendicular isector: (, 8) M 5 M (, 5) (0, ) Student page answers: a. : slope =, midpoint = (, 5), slope of perpendicular bisector = ; : slope =, midpoint = (, ), slope of perpendicular bisector = ; : slope =, midpoint = (, ), slope of perpendicular bisector = b and c. See diagram below. Store ; the house at (5, 5) is in the same region (zone) as Store, so Store is closest to this house.. Either Store or Store ; because this line is the perpendicular bisector of, a point on the line will be the same distance from Store and Store. See diagram below 5a. The best location is the Voronoi vertex located near (.,.) 5b. See diagram below. This will happen only when the four points lie on a circle. 7. Sample answer: Driving times are assumed to be uniform and independent of traffic. c.. 5. D N Note: the dashed lines are not part of the Voronoi diagram. education.ti.com/go/numrs Tom utts, UT Dallas
3 NUMRS ctivity Student Page Name: Episode: "ones of ontention Date: NUMRS ctivity: Location! Location! Location! When you go into the takeout section of your favorite pizza store, you might see a map on the wall that highlights the part of the city where this store delivers, as well as the delivery zones for other nearby stores in this chain. simplified version of such a map could be modeled by a Voronoi diagram. In this activity, you will learn how to draw a Voronoi diagram for and stores First, we will investigate constructing a delivery zone map for stores [sites]. 5. Pizza World owns pizza stores,,, and. onstruct a delivery zone map for stores,, and. a) omplete the table below. Slope of line segment: Midpoint: Slope of Perpendicular isector: b) Draw the perpendicular of bisector of. Then do the same for and. c) In the final diagram, the three perpendicular bisectors will intersect in a common point in the interior of called a Voronoi vertex. Each Voronoi edge should stop at this vertex. education.ti.com/go/numrs Tom utts, UT Dallas
4 NUMRS ctivity Episode: "ones of ontention Student Page Use your Voronoi diagram from Question # to answer the following questions.. Which store should deliver to a house located at (5, 5)? Why?. Which store should deliver to a house on the boundary between and? Why?. The owner of Pizza World decided to open a fourth suburban store at D. Draw the Voronoi diagram or delivery zone map for these four stores. Hint: Think about how you can modify your map for three stores. D 5. The owner then changed his mind and decided to locate the fourth store downtown near the center of the city a. Where should the downtown store be located so it is in the center of the city but as far away from each of the other three stores as possible? Place this point in the diagram and estimate its coordinates. b. Draw the Voronoi diagram or delivery zone map for these four stores.. Suppose the Voronoi diagram or zone map for four stores X, Y, Z, and W has exactly four Voronoi edges. How are the locations of the sites X, Y, Z, and W related to each other? Draw an example of such a Voronoi diagram. 7. zone map or Voronoi diagram is a mathematical model in which several simplifying assumptions are made. One of these assumptions is that the population is uniformly distributed throughout the city at least the population who wish to buy pizza. What is another simplifying assumption made in using this model? education.ti.com/go/numrs Tom utts, UT Dallas
5 NUMRS ctivity Episode: "ones of ontention The goal of this activity is to give your students a short and simple snapshot into a very extensive math topic. TI and NTM encourage you and your students to learn more about this topic using the extensions provided below and through your own independent research. Extensions For the Teacher nother way to introduce the idea of a Voronoi diagram is through an activity involving the use of a zone defense in basketball. See the website for more details. lessons/pat For the Student Find several elementary schools, post offices, fire stations, etc. in your city. Draw a Voronoi diagram using their locations as sites on a city map. Then check to see how the Voronoi regions compare with the school attendance zones, postal delivery zones, fire station response zones, etc. Voronoi diagram with n sites cannot have more than n Voronoi edges. What is the minimum number of edges in a Voronoi diagram with n sites? How would the sites be positioned so that it would be possible for a minimum number to exist? dditional Resources There are applications of Voronoi diagrams in many areas including archaeology, astronomy, biology, and marketing. Examine "pplications from rcheology through Zoology at or Voronoi Diagrams and Proximity Problems, Matthew Dickerson and Scot Drysdale, OMP, 99. heck for information about how to purchase this module. Geometry and Its pplications, Walter Meyer, Harcourt cademic Press, 999 hapter has a section on Voronoi diagrams. In a Voronoi diagram, each Voronoi vertex is the circumcenter of a circle containing three sites. Students can explore this concept using these applets for constructing Voronoi diagrams and Delaunay Triangulations: and The abri Jr. pp for the TI-8 Plus and TI-8 Plus calculators can be downloaded for free from Related Topic The Delaunay Triangulation/Tessellation is the geometric dual of the Voronoi Diagram. It can be defined as a triangulation of the sites (constructing triangles with the sites as vertices) with the additional property that for each triangle of the triangulation, the circumcircle of that triangle does not contain any other site. See: education.ti.com/go/numrs Tom utts, UT Dallas
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